DTU ProjectLab - User contributions [en-gb]

MCDM-AHP method in decision making

2023-05-04T22:48:01Z

S174669: /* The analytic hierarchy process */

''Developed by Mansoor Samadi''

The MCDM-AHP (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is AHP which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the AHP-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=AHP for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the AHP method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The AHP metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale. <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The AHP method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

After the AHP method calculations are completed the comapany that provides the best services compared to your needs is shown as company #1. This method can be scaled to larger organisation such as NASA, seen in the "implementation of AHP method in organisations".

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

=Annotated bibliography=

As the AHP method is widely used method in the industry, there are multiple papers that are very intersting and informative that i was not able to incorpriate in this wiki but future AHP nerds may implement it:

This paper implents the AHP method for a building project where they use a software named 'Expert choise', it would be intersting to see what the commerical programs can do and how they solve some of the limitations of the AHP methods
*Kamal M.Al-Subhi A., "Application of the AHP in project management", 2001, ResearchGate.
*Erick L., Ewerton G., et. al."Applying machine learning to AHP multicriteria decision making method to assets prioritization in the context of industrial maintenance 4.0", 2019, ResearchGate

This paper goes deep into project delivery method and descripes the importance for the project owners to have a delivery method, they have developed a methodology that incorporates the AHP method:
* Mohammed K., "Selecting the appropriate project delivery method using AHP", 2002, ResearchGate

Even tho this source was used in the wiki, i higly reccormed that it is studied more because the source descripes in detatail and in a rational way the limitations that the AHP method have, it uses real life examples and some things to be aware of as project managers.
*Nolberto M., Eloy H., "Uses and Limitations of the AHP Method - A Non-Mathematical and Rational Analysis", 2021, Springer.

= References =
<references/>

MCDM-AHP method in decision making

2023-04-07T12:47:11Z

S174669: /* Calculate the global priority of each criteria and ranking */

''Developed by Mansoor Samadi''

The MCDM-AHP (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is AHP which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the AHP-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=AHP for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the AHP method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The AHP metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The AHP method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

After the AHP method calculations are completed the comapany that provides the best services compared to your needs is shown as company #1. This method can be scaled to larger organisation such as NASA, seen in the "implementation of AHP method in organisations".

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

=Annotated bibliography=

As the AHP method is widely used method in the industry, there are multiple papers that are very intersting and informative that i was not able to incorpriate in this wiki but future AHP nerds may implement it:

This paper implents the AHP method for a building project where they use a software named 'Expert choise', it would be intersting to see what the commerical programs can do and how they solve some of the limitations of the AHP methods
*Kamal M.Al-Subhi A., "Application of the AHP in project management", 2001, ResearchGate.
*Erick L., Ewerton G., et. al."Applying machine learning to AHP multicriteria decision making method to assets prioritization in the context of industrial maintenance 4.0", 2019, ResearchGate

This paper goes deep into project delivery method and descripes the importance for the project owners to have a delivery method, they have developed a methodology that incorporates the AHP method:
* Mohammed K., "Selecting the appropriate project delivery method using AHP", 2002, ResearchGate

Even tho this source was used in the wiki, i higly reccormed that it is studied more because the source descripes in detatail and in a rational way the limitations that the AHP method have, it uses real life examples and some things to be aware of as project managers.
*Nolberto M., Eloy H., "Uses and Limitations of the AHP Method - A Non-Mathematical and Rational Analysis", 2021, Springer.

= References =
<references/>

MCDM-AHP method in decision making

2023-04-07T12:46:49Z

S174669: /* Calculate the global priority of each criteria and ranking */

''Developed by Mansoor Samadi''

The MCDM-AHP (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is AHP which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the AHP-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=AHP for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the AHP method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The AHP metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The AHP method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

After the AHP method calculations are completed the comapany that provides the best services compared to your needs is shown as company #1. This method can be scaled to larger organisation such as NASA, seen in the "implementation of

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

=Annotated bibliography=

As the AHP method is widely used method in the industry, there are multiple papers that are very intersting and informative that i was not able to incorpriate in this wiki but future AHP nerds may implement it:

This paper implents the AHP method for a building project where they use a software named 'Expert choise', it would be intersting to see what the commerical programs can do and how they solve some of the limitations of the AHP methods
*Kamal M.Al-Subhi A., "Application of the AHP in project management", 2001, ResearchGate.
*Erick L., Ewerton G., et. al."Applying machine learning to AHP multicriteria decision making method to assets prioritization in the context of industrial maintenance 4.0", 2019, ResearchGate

This paper goes deep into project delivery method and descripes the importance for the project owners to have a delivery method, they have developed a methodology that incorporates the AHP method:
* Mohammed K., "Selecting the appropriate project delivery method using AHP", 2002, ResearchGate

Even tho this source was used in the wiki, i higly reccormed that it is studied more because the source descripes in detatail and in a rational way the limitations that the AHP method have, it uses real life examples and some things to be aware of as project managers.
*Nolberto M., Eloy H., "Uses and Limitations of the AHP Method - A Non-Mathematical and Rational Analysis", 2021, Springer.

= References =
<references/>

MCDM-AHP method in decision making

2023-04-07T12:43:29Z

S174669: /* Calculate the global priority of each criteria and ranking */

''Developed by Mansoor Samadi''

The MCDM-AHP (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is AHP which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the AHP-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=AHP for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the AHP method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The AHP metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The AHP method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

After the calculations are completed the #1 company to get your

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

=Annotated bibliography=

As the AHP method is widely used method in the industry, there are multiple papers that are very intersting and informative that i was not able to incorpriate in this wiki but future AHP nerds may implement it:

This paper implents the AHP method for a building project where they use a software named 'Expert choise', it would be intersting to see what the commerical programs can do and how they solve some of the limitations of the AHP methods
*Kamal M.Al-Subhi A., "Application of the AHP in project management", 2001, ResearchGate.
*Erick L., Ewerton G., et. al."Applying machine learning to AHP multicriteria decision making method to assets prioritization in the context of industrial maintenance 4.0", 2019, ResearchGate

This paper goes deep into project delivery method and descripes the importance for the project owners to have a delivery method, they have developed a methodology that incorporates the AHP method:
* Mohammed K., "Selecting the appropriate project delivery method using AHP", 2002, ResearchGate

Even tho this source was used in the wiki, i higly reccormed that it is studied more because the source descripes in detatail and in a rational way the limitations that the AHP method have, it uses real life examples and some things to be aware of as project managers.
*Nolberto M., Eloy H., "Uses and Limitations of the AHP Method - A Non-Mathematical and Rational Analysis", 2021, Springer.

= References =
<references/>

MCDM-AHP method in decision making

2023-04-07T12:42:40Z

S174669: /* Calculate the global priority of each criteria and ranking */

''Developed by Mansoor Samadi''

The MCDM-AHP (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is AHP which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the AHP-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=AHP for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the AHP method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The AHP metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The AHP method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

=Annotated bibliography=

As the AHP method is widely used method in the industry, there are multiple papers that are very intersting and informative that i was not able to incorpriate in this wiki but future AHP nerds may implement it:

This paper implents the AHP method for a building project where they use a software named 'Expert choise', it would be intersting to see what the commerical programs can do and how they solve some of the limitations of the AHP methods
*Kamal M.Al-Subhi A., "Application of the AHP in project management", 2001, ResearchGate.
*Erick L., Ewerton G., et. al."Applying machine learning to AHP multicriteria decision making method to assets prioritization in the context of industrial maintenance 4.0", 2019, ResearchGate

This paper goes deep into project delivery method and descripes the importance for the project owners to have a delivery method, they have developed a methodology that incorporates the AHP method:
* Mohammed K., "Selecting the appropriate project delivery method using AHP", 2002, ResearchGate

Even tho this source was used in the wiki, i higly reccormed that it is studied more because the source descripes in detatail and in a rational way the limitations that the AHP method have, it uses real life examples and some things to be aware of as project managers.
*Nolberto M., Eloy H., "Uses and Limitations of the AHP Method - A Non-Mathematical and Rational Analysis", 2021, Springer.

= References =
<references/>

MCDM-AHP method in decision making

2023-04-07T12:31:49Z

S174669: /* Annotated bibliography */

''Developed by Mansoor Samadi''

The MCDM-AHP (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is AHP which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the AHP-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=AHP for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the AHP method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The AHP metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The AHP method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

=Annotated bibliography=

As the AHP method is widely used method in the industry, there are multiple papers that are very intersting and informative that i was not able to incorpriate in this wiki but future AHP nerds may implement it:

This paper implents the AHP method for a building project where they use a software named 'Expert choise', it would be intersting to see what the commerical programs can do and how they solve some of the limitations of the AHP methods
*Kamal M.Al-Subhi A., "Application of the AHP in project management", 2001, ResearchGate.
*Erick L., Ewerton G., et. al."Applying machine learning to AHP multicriteria decision making method to assets prioritization in the context of industrial maintenance 4.0", 2019, ResearchGate

This paper goes deep into project delivery method and descripes the importance for the project owners to have a delivery method, they have developed a methodology that incorporates the AHP method:
* Mohammed K., "Selecting the appropriate project delivery method using AHP", 2002, ResearchGate

Even tho this source was used in the wiki, i higly reccormed that it is studied more because the source descripes in detatail and in a rational way the limitations that the AHP method have, it uses real life examples and some things to be aware of as project managers.
*Nolberto M., Eloy H., "Uses and Limitations of the AHP Method - A Non-Mathematical and Rational Analysis", 2021, Springer.

= References =
<references/>

MCDM-AHP method in decision making

2023-04-07T12:18:29Z

S174669: /* Annotated bibliography */

''Developed by Mansoor Samadi''

The MCDM-AHP (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is AHP which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the AHP-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=AHP for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the AHP method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The AHP metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The AHP method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

=Annotated bibliography=

As the AHP method is widely used method in the industry, there are multiple papers that are very intersting and informative that i was not able to incorpriate in this wiki but future AHP nerds may implement it:

This paper implents the AHP method for a building project where they use a software named 'Expert choise', it would be intersting to see what the commerical programs can do and how they solve some of the limitations of the AHP methods
*Kamal M.Al-Subhi A., "Application of the AHP in project management", 2001, ResearchGate.

= References =
<references/>

MCDM-AHP method in decision making

2023-04-07T12:05:45Z

S174669: /* Implementation of AHP method in organisations */

''Developed by Mansoor Samadi''

The MCDM-AHP (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is AHP which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the AHP-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=AHP for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the AHP method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The AHP metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The AHP method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

=Annotated bibliography=

= References =
<references/>

MCDM-AHP method in decision making

2023-04-07T12:05:12Z

S174669:

''Developed by Mansoor Samadi''

The MCDM-AHP (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is AHP which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the AHP-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=AHP for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the AHP method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The AHP metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The AHP method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Annotated bibliography=

= References =
<references/>

Articles Spring Term 2023

2023-04-07T12:03:19Z

S174669:

=Overview of 2023 Wiki Collections=

{| class="wikitable sortable"
|+Spring Term 2023 Wiki Collections
|Group 3
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|-

|-
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|-

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|Ingimundardóttir
|s222538
|New? / revised?
|[[Personality Types and Leadership]]
|-
|-
|Group TBD
|Sasha
|Mantel
|s222742
|New
|[[Project Stakeholder Management]]
|-
|-
|Group 23
|Julie
|Hustoft
|s194359
|New
|[[The Benefits of Self-Awareness]]
|-
|-
|Group TBD
|Mikkel
|Anderson
|s184230
|New? / revised?
|[[Continuous Integration/Continuous Delivery (CI/CD)]]
|-
|-
|Group 12
|Emma
|Bigum
|s160048
|New
|[[Lippitt-Knoster Complex Change Management]]
|-
|-
|Group 22
|Ingibjörg
|Einarsdóttir
|s223228
|New? / revised?
|[[Mediating conflicts and controversy]]
|-
|-
|Group 4
|Peter
|Aggerholm
|s170890
|New? / revised?
|[[When: The Scientific Secrets of Perfect Timing]]
|-
|-
|Group 36
|Hekla Mist
|Valgeirsdóttir
|s223229
|New? / revised?
|[[Management vs. leadership]]
|-
|-
|Group 20
|Theis Rosenkvist
|Sørensen
|s195768
|New? / revised?
|[[Managing threats and opportunities in risk treatment]]
|-
|-
|Group 9
|Julie Amanda
|Busch
|s190974
|New
|[[Utilizing Value Functions for Evaluating the Performance of Project Alternatives]]
|-
|-
|Group 6
|Martin
|Fehst
|s222575
|New? / revised?
|[[Leading multidisciplinary Project Teams]]
|-
|-
|Group 11
|Eric Vincent
|Vong
|s222594
|New
|[[PDCA cycle]]
|-
|-
|Group 23
|Efthymios
|Samaras
|s223292
|New? / revised?
|[[Benefit Cost Ratio (BCR)]]
|-
|-
|Group 1
|Cirkeline
|Bräuner
|s193974
|Revised
|[[The Role of Program Management in an Organisational Change]]
|-
|-
|Group 2
|Hrannar
|Þórarinsson
|s222536
|New
|[[Balanced scorecard: connecting the performance measures]]
|-
|-
|Group 15
|Mar
|Rivera Hedo
|s222520
|New? / revised?
|[[The Importance of Psychological Safety in Team Development]]
|-
|-
|Group 22
|Thordis
|Ragnarsdottir
|s222535
|New
|[[Resource Conflicts]]
|-
|-
|Group 5
|Oriol
|Solans Ormo
|s222527
|New
|[[Sustainability Issue]]
|-
|-
|Group 15
|Naïl
|Mulatier
|s222590
|New
|[[Addressing diversity and inclusion in a short-term project]]

|-
|-
|Group 22
|Valdís
|Sigurdardóttir
|s2223305
|New
|[[Challenges and opportunities regarding diversity within teams work]]
|-
|-
|Group 12
|Sophia
|Fjeldsøe
|s184141
|New
|[[Stakeholder Mapping]]

|-
|-
|Group 29
|Miriam
|Khader
|s194717
|New
|[[Leadership development through MBTI]]
|-
|-
|Group 5
|Maria
|Arrillaga Tarazona
|s222708
|New? / revised?
|[[Program Evaluation and Review Technique (PERT)]]

|-
|-
|Group 24
|Josefine
|Steinfurth
|s183648
|New
|[[Project based organisations]]

|-
|-
|Group 5
|Patricio Alejandro
|Fabro
|s220269
|New
|[[Metrics in Portfolio management]]

|-
|-
|Group 10
|Jurek Noah
|von Petersdorff-Campen
|s222570
|New
|[[Competency Mapping for Project Management]]

|-
|-
|Group 15
|Inke
|Lindberg
|s223191
|New
|[[Post-occupancy evaluation (POE)]]
|-
|-
|Group 16
|Ahmet
|Ibre
|s222442
|New
|[[Social loafing in multidisciplinary teams]]

|-
|-
|Group 16
|Marie-Therese
|Guidje
|s180863
|New? / revised?
|[[Sustainable Project management]]
|-
|-
|Group 12
|Peter
|Andresen
|s190517
|New
|[[Project performance with Value Stream management]]
|-
|-
|Group 3
|Annas
|Jadi
|s194831
|New/ Revised
|[[Key performance indicators for portfolio management]]
|-
|-
|Group X
|Fabio
|Colombo
|s212492
|New/ Revised
|[[Robust Decision Making: better decisions under uncertainty]]
|-
|-
|Group 20
|Rune
|Knudsen
|s193983
|New
|[[Pooled, Sequential & Reciprocal dependence]]
|-
|-
|Group 24
|Mia
|Hemmingsen
|s183639
|New/ Revised
|[[Seven stages of change]]
|-
|-
|Group 1
|Christina
|Kjær
|s183504
|New
|[[The 6C Model]]
|-
|-
|Group 21
|Jakob
|Berling
|s230256
|New
|[[Successful Teamwork]]
|-
|-
|Group 1
|Joakim
|Cayouette
|s193978
|Revised
|[[Forecasting and estimation techniques]]
|-
|-
|Group 30
|Amalie
|Pedersen
|s193967
|New
|[[Satisficing]]
|-
|-
|Group 30
|Filip Furbo
|Enevoldsen
|s193967
|New
|[[Optimism bias in teams]]
|-
|-
|Group 23
|Nicolai
|Frausing
|s175034
|New
|[[Benefit Realization Management]]
|-
|-
|Group 4
|Emma
|Egelund
|s171516
|Revised
|[[Network Planning in Project Management]]
|-
|-
|Group 21
|Henrik
|Svensson
|s230245
|New
|[[Participatory Design]]
|-
|-
|Group 11
|Luisa Fernanda
|Salazar Rivera
|s222401
|New
|[[Building Effective Work Breakdown Structures (WBS)]]
|-
|-
|Group 15
|Michele
|Ferrante
|s220282
|New
|[[Safety risk management in construction sites]]
|-
|-
|Group 4
|Helena
|Mladenovski
|s183367
| New
|[[Situational Leadership Theory (SLT)]]
|-
|-
|Group 20
|Rolf
|Wollesen
|s193972
| New
|[[Risk Burn Down Chart]]
|-
|-
|Group 6
|Gianluca
|Santinon
|s222630
|New
|[[The Rolling-Wave planning technique]]
|-
|-
|Group X
|Hanna
|Rakel Bjarnadóttir
|s223414
|New
|[[How to achieve sustainable management using ISO21500]]
|-
|-
|Group 5
|Marta
|Santos
|s226759
|New
|[[Risk Response Plan]]
|-
|-
|Group 16
|Hendrik
|Stegers
|s230258
|New
|[[Communication Theories in Project Management]]
|-
|-
|Group X
|Jiafei
|Xia
|s213084
|New
|[[Employee Loyalty Management]]
|-
|-
|Group 8
|Frederik
|Høgdal
|s173876
|New
|[[Levels of uncertainties]]
|-
|-
|Group 30
|Jianan
|Yang
|s230033
|New
|[[Communication Funnel Problems in Project Management]]
|-
|-
|Group 18
|Ellen Cathrine
|Luke
|s184404
|New & Revised
|[[Project War Rooms: Physical vs. Virtual]]
|-
|-
|Group 17
|Canovas Iglesias
|Carla
|s226573
|New
|[[Stage-Gate Project Management Model]]
|-
|-
|Group 2
|Ólafur Þorri
|Sigurjónsson
|s222540
|New
|[[Decision tree analysis]]
|-
|-
|Group 11
|Dario
|Fiorica
|s222631
|New
|[[Groups or teams?]]
|-
|-
|Group 10
|Benjamin
|Søndberg
|s184314
|New
|[[Top-down vs ground up estimations]]
|-
|-
|Group 24
|Kristian
|Brandstrup
|s183653
|New
|[[Adaptation and agile]]
|-
|-
|Group 14
|Mads
|Marum
|s222620
|New / Reviced
|[[High Performing Teams (HPT)]]
|-
|-
|Group 30
|Jeppe
|Paludan-Müller
|s185048
|New
|[[Feasibility risk assessments of transport projects using Monte Carlo-simulations]]
|-
|-
|Group 5
|Rejath
|Ramachandran
|s226673
|New? / revised?
|[[A modern re-think of Fayolism]]
|-
|-
|Group 29
|Onur
|Osman Mustafa
|s223710
|New
|[[Resource Leveling in Construction Projects]]
|-
|-
|Group 15
|Philip
|Wiehe Larsen
|s154259
|New
|[[Situational mapping]]
|-
|-
|Group 9
|Michael
|Vinther
|s163490
|New
|[[Transformational Leadership]]
|-
|-
|Group 21
|Alberto
|Pillon
|s222629
|New
|[[Driving Continuous Improvement with retrospective meetings]]
|-
|-
|Group 2
|Unnar Óli
|Arnarsson
|s222543
|New
|[[The Role of Due Diligence in Project Management]]
|-
|-
|Group 9
|Mateusz
|Szaryk
|s213423
|New
|[[Incentive contract]]
|-
|-
|Group 23
|Mariely
|Sousa
|s230555
|New
|[[Return on Investment (ROI)]]
|-
|-
|Group 31
|Vinay Kumar
|Meena
|s230019
|New
|[[The implication of ethics in project management]]
|-
|-
|Group 10
|Niels Peter
|Lindegaard
|s194473
|New
|[[Risk assessment using Failure mode and effects analysis]]
|-
|-
|Group 6
|Nils Hendrik
|Lange
|s223634
|New
|[[Eisenhower Decision Matrix in Project, Program and Portfolio Management]]
|-
|-
|Group 20
|Caroline Hellerung
|Holm
|s175026
|New
|[[Resistance to Change Mapping]]
|-
|-
|Group 24
|Radost
|Deneva
|s230780
|New
|[[Eisenhower decision matrix]]
|-
|-
|Group 4
|Marie
|Hoel
|s193960
|New
|[[The Hawthorne effect in management]]
|-
|-
|Group 4
|Carl Bygum
|Knudsen
|s193995
|New
|[[Assessment of stakeholders (analysis)]]
|-
|-
|Group 14
|Md Huzzatul Islam
|Shuvo
|s222392
|New
|[[Environmental sustainability]]
|-
|-
|Group 23
|Esther Kiara
|Pattipeilohy
|s230264
|New
|[[The Active Listening Technique]]
|-
|-

|-
|Group 6
|Lars Ole
|Holdensen
|s224931
|New
|[[Management vs Leadership qualities]]
|-

|-
|Group 11
|Ingeborg
|Rasmussen
|s22218
|New
|[[Conflict management using the Myers-Briggs Type Indicator (MBTI)]]
|-

|-
|Group 22
|Blomsterberg
|Gudrun
|s222534
|New
|[[Negotiations]]
|-

|-
|Group 1
|Thorgaard
|Sigrid
|s186044
|New
|[[Critical Chain]]
|-

|-
|Group 12
|Laurberg
|Emilie
|s193880
|New
|[[Choosing the right communication network for your project]]
|-

|-
|Group 2
|Bjarki Rúnar
|Sverrisson
|s222547
|New? / revised?
|[[The Two Systems of Thinking]]
|-

|-
|Group 25
|Anirudh Ram
|Sriram
|s220045
|New
|[[SMART goals in strategy planning]]
|-

|-
|Group 16
|Pavel
|Delimarschi
|s222606
|New
|[[Nominal group technique]]
|-
|Group 31
|Nik Rasmus
|Meier-Sawatzki
|s223232
|New
|[[Effort-Reward-Imbalance]]
|-
|-
|Group 31
|Antonia Leonie
|Kaul
|s222581
|New
|[[FMEA – Failure Mode and Effects Analysis]]
|-
|-

|-
|Group 3
|Ioannis
|Marinopoulos
|s220152
|New
|[[Centralizing Project Communication and Decision Making with War Rooms]]
|-
|Group 31
|Alessia
|Pacani
|s230269
|New
|[[Hofstede's Cultural Dimensions Theory]]
|-
|-
|Group 3
|Stefano
|Vigato
|s230253
|New? / revised?
|[[The Agile Stage-Gate Model]]
|-
|Group 1
|Sophia Lykke
|Anderse
|s183592
|New
|[[Critical path vs. critical chain]]
|-
|Group 1
|Sebastian Petchkorn
|Følsgaard
|s193958
|New
|[[SWOT Analysis 101]]
|-
|Group 30
|Qian
|Xiao
|s230120
|New
|[[Robust decision making under risk management]]
|-
|Group 12
|Rhishihesan
|Kugathason
|s193873
|New
|[[Optimizing a company and it's PPM with a PMO system]]
|-

MCDM-APH method in decision making

2023-04-07T12:02:49Z

S174669: Replaced content with "delete please"

delete please

MCDM-AHP method in decision making

2023-04-07T12:02:19Z

S174669:

''Developed by Mansoor Samadi''

The MCDM-AHP (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is AHP which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the AHP-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=AHP for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the AHP method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The AHP metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The AHP method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

Editing MCDM-AHP method in decision making

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MCDM-AHP method in decision making

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S174669: Created page with "''Developed by Mansoor Samadi'' The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approa..."

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

Editing MCDM-AHP method in decision making

2023-04-07T11:59:52Z

MCDM-APH method in decision making

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S174669: /* Implementation of AHP method in organisations */

MCDM-APH method in decision making

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S174669: /* US China conflict */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====NASA=====
When National Aeronautics and Space Administration (NASA) had to decide which power source to be used for the lunar outpost a AHP analysis was conducted where the criterias were safety, performance, reliability and flexibility and many more. NASA have since then used it to many other projects such as the lunar landing propulsion system, their Computer aided systems engineering tool set (CEASETS), and many more.
f
<ref name="The AHP process">Ernest H. F., el al. "The Analytic Hierachy process - An Exposition" http://www.johnsaunders.com/papers/ahpexpo.pdf </ref>

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T11:45:46Z

S174669: /* Inplementation of AHP method in organisations */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Implementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====US China conflict=====
- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T11:45:30Z

S174669: /* Inplementation of AHP method in organisations */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Inplementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video without paying the owners. To see wether it was benefitial for the US to sanction China, an AHP analysis was made where benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China, in the references many more applications can be seen.
<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=====US China conflict=====
- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T11:42:45Z

S174669: /* Inplementation of AHP method in organisations */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Inplementation of AHP method in organisations=

=====US China conflict=====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T11:42:36Z

S174669: /* Inplementation of AHP method in organisations */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Inplementation of AHP method in organisations=

====US China conflict====
The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T11:36:09Z

S174669: /* Limitation of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity
It is fair to assume that all the DMs are knowledgeable in the subject and that they are unbias but each person have their own preferences and experence. When DMs confront an issue they will tackle the issue differentely dipending on their perefences. These preferences have no ground in mathematic formula or reasoning, the stakeholders are relying on the preferences of the DMs which can change depending on their mood and feeling. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T11:06:24Z

S174669: /* Limitation of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance. <ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T11:03:37Z

S174669: /* Limitation of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance.
<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis", Springer, https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T11:03:09Z

S174669: /* Limitation of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance.
<ref name="Uses and limitations of the AHP method">Hontoria, E., el al. "Uses and limitations of the AHP method, a Non-Mathematical and Rational Analysis"https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T11:01:46Z

S174669: /* Limitation of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias. Lets say the danish goverment had to decide the locations of the super hospitals, there are many criterias where some of them are political, economic, and so on, it becomes difficult to evaluate each criteria's reletive importance.
<ref name="Uses and limitations of the AHP method">Hontoria Eloy</ref>

- Time dependency in project and portfolio management

It takes a large sum of time for the decision makers to select alternatives. In the time where the DMs are selecting alternatives the project might have changed where a alternative is added or deleted, which then results in that the DMs have to make the entire process over again. In complex projects like govermental ones where the DM is excecuting a pairwise comparison to govermental policies, when the DM have finished the process, the goverment implements a new policy. The AHP method is great tool for small projects where there arent too many criterias and alternatives.

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T10:41:58Z

S174669: /* Limitation of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, the criterias in the example above have only 3 criterias which is manageable but imagine if there were 500 criterias.

https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T10:33:26Z

S174669: /* Limitation of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, as example the table X have only 3 criterias which is manageable but imagine if there were 1000 criterias.

https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T10:31:19Z

S174669: /* Limitation of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process, as example the table X have only 3 criterias which is manageable but imagine if there were 1000 criterias.
https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T10:28:51Z

S174669: /* Limitation of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially, the large number of criterias becomes less manageable as it becomes more dificult to evaluate each criteria's reletive importance which can result to inconsistency in the decision making process.
https://link.springer.com/chapter/10.1007/978-3-030-60392-2_5

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T10:17:08Z

S174669: /* Critique of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Limitation of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially,
-hard to apply to larger projects

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-04-07T10:13:24Z

S174669: /* Critique of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
As the projects becomes larger the complexity increases exponentially,
-hard to apply to larger projects

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T12:41:28Z

S174669: /* Inplementation of AHP method in organisations */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity

-hard to apply to larger projects

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China.
https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf

- Xerox
http://www.johnsaunders.com/papers/ahpexpo.pdf

- NASA
http://www.johnsaunders.com/papers/ahpexpo.pdf

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T12:37:49Z

S174669: /* Inplementation of AHP method in organisations */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity

-hard to apply to larger projects

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

- The American DOD uses it very often to allocate their resources most efficiently to thieir activities. https://www.researchgate.net/publication/264823559_Military_applications_of_the_analytic_hierarchy_process

- The process was used when back in the 1995 when US and china was in conflict regardin intellectural property rights, the conflict was regarding chineses civilians copying music, video and CDs. To see wether it was benefitial for the US to sanction China a AHP analysis was made wheer benefit, cost and risk was taken into account and from the analysis it would be much more benefitial to not sanction China. Some time after a

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T11:44:29Z

S174669: /* Critique of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity

-hard to apply to larger projects

Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity

-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias

Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive

-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T11:44:07Z

S174669: /* Critique of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
-hard to apply to larger projects
Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity
-At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias
Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive
-The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T11:43:53Z

S174669: /* Critique of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
-hard to apply to larger projects
Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity
- At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias
Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive
- The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T11:43:41Z

S174669: /* Critique of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

- complexity
- hard to apply to larger projects
Soares, A. L., Cardoso, L. A., Coelho, P. G., & de Almeida, A. T. (2015). Strategic decision making in a large-scale hydropower project using AHP: The Belo Monte case. Energy Policy, 87, 482-494.

- Subjectivity
- At the end the AHP method relys heavily on the biasis of the decision-makers, the decision makers needs to be trained so they are knowledable and unbias
Ramanathan, M. M., & Ramanathan, V. (2003). AHP-based methodology to rank sustainable development indicators. Journal of Environmental Management, 69(4), 343-357.

- Sensetive
- The AHP is very sensetive to input data, small changes such as the weight or preferences of the decision criteria and alternatives can have huge changes of the decision outcome.

a study by M. X. Li et al. discusses the sensitivity analysis of AHP for sustainable supplier selection in supply chain management. (Reference: Li, M. X., Xie, G. H., & Wang, L. (2017). An AHP-based approach for sustainable supplier selection in supply chain management. Journal of Cleaner Production, 147, 336-347.)

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T11:21:45Z

S174669: /* Project selection */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and it could also remove some of the biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and introduce more bias in the project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T11:12:51Z

S174669: /* Critique of the AHP method */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and also removing the personal biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and hinder the unbias project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

While AHP is a have been implemented in many cases and it is an effective method for decision-making in project management, it is not without limitations. Some of the limitations of the AHP method are listed below:

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T10:40:02Z

S174669: /* Example for AHP */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and also removing the personal biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and hinder the unbias project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T10:39:54Z

S174669: /* Example for AHP */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and also removing the personal biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and hinder the unbias project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
There are many ways to preform the AHP analysis by making it more complex or less depending on the needs of the PM, here is an example:
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T10:39:00Z

S174669: /* Picking suppliers */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and also removing the personal biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and hinder the unbias project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
'''NOTE TO ME:------------------------------------------------------------ NEED INTRO'''
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T10:38:54Z

S174669: /* APH for Project management */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and also removing the personal biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and hinder the unbias project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

===Risk Management===
There are a multitude of risks that an orginisation have to account for and from there they can preform some risk management strategies and mitigate said risk based on the likelihood. For example a project manager in a construction company is about to build a commercial building, there are many ways that the project could be delayed such as weather, cost overruns, delays and safety hazards. The PM can weight each of the criterias based on the impact on the project and the likelihood of it happening. When the risks are in order the PM can make risk management strategies to mitigate the risks.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
'''NOTE TO ME:------------------------------------------------------------ NEED INTRO'''
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T10:25:25Z

S174669: /* Picking suppliers */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and also removing the personal biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and hinder the unbias project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>. There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
'''NOTE TO ME:------------------------------------------------------------ NEED INTRO'''
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T10:17:31Z

S174669: /* APH for Project management */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and also removing the personal biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and hinder the unbias project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>.

There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
'''NOTE TO ME:------------------------------------------------------------ NEED INTRO'''
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T10:16:56Z

S174669: /* Project selection */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and also removing the personal biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and hinder the unbias project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26 https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Risk assessment===
===Resource allocation===
===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>.

There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
'''NOTE TO ME:------------------------------------------------------------ NEED INTRO'''
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T10:16:21Z

S174669: /* Project selection */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and also removing the personal biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and hinder the unbias project selesction.<ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Risk assessment===
===Resource allocation===
===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>.

There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
'''NOTE TO ME:------------------------------------------------------------ NEED INTRO'''
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T10:16:02Z

S174669: /* Project selection */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and also removing the personal biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and hinder the unbias project selesction.</ref> <ref name="Evaluating projects"> Palcic, I.; Lalic, B. Analytical Hierarchy Process as a tool for selecting and evaluating projects. Int. J. Simul. Model. (IJSIMM) 2009,
8, 16–26https://www.researchgate.net/publication/238195934_Analytical_Hierarchy_Process_as_a_tool_for_selecting_and_evaluating_projects</ref>.

===Risk assessment===
===Resource allocation===
===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>.

There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
'''NOTE TO ME:------------------------------------------------------------ NEED INTRO'''
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>

MCDM-APH method in decision making

2023-03-30T10:12:39Z

S174669: /* Project selection */

''Developed by Mansoor Samadi''

The MCDM-APH (Multi-Criteria Decision Making - Analytic Hierarchy Process) is a tool that makes complex decisions. MCDM is a systematic approach that evaluate conflicting criterias in decision making, one of these approaches is APH which uses a mathematic method to derive a relative importance of criteria in a decision problem.

The method was developed in the 1970s by Thomas Saaty <ref name="Saaty">Saaty, T. L. How to make a decision: The analytic hierarchy process. European Journal of Operational Research, 48(1), 9-26, 1990. https://www.sciencedirect.com/science/article/pii/037722179090057I </ref>, AHP is a mathematical method that is used to derive the relative importance of criteria in a decision problem. The idea behind the method is that a decision problem can be represented as a hierarchical structure, with the most important criteria at the top followed by the lesser important criteria at the bottom.

Decision-making is at the core of project management and yet it can be a daunting task, especially when the projects are complex and multi-facted projects. The pressure of making decisions in a timely and informed manner can be compounded when the criteria for success is not defined clearly, At time like this it not unusual that project managers may feel overwhelmed and unable to determine the most important factors to consider. This is where the APH-decision-making method provides a systematic approach to this challenge. This approach is addressing the complexity of decision making by Segmenting the process into manageable chunks. By following these steps the project manager can get a obtain a holistic view of the project's key performance indicators and success factors.

=APH for Project management=

A project manager in the current day is faced with a dynamic environment, where making the right choices based on fitting objective is a crucial factor for weather a organization is profitable or not. Simply completing a project and hope for the best outcome is no longer enough, as the projects are more complex with more risks and dynamic challenges this means that there is a higher demand for a strategic approach.

In order to accomplish this, organizations are using more resources in portfolio management which helps them structurize the projects so they are in alignment with the organizations strategic goals. One of the main part of portfolio management is cost-benefit, where projects with high benefits is given higher priority. The analysis takes into account the benefits of completing a project as well as the cost, as projects become more complex the cost-benefit analysis becomes as complex and hectic.

Here are three exampels on where the APH method is applied for project, portfolio and program management.

===Project selection===

In project sellection the AHP method can be used in various of ways such as financial return, strategic alignment, resource avaliblility, risk. As the projects become more complicated so does the project selection process, companies uses a lot of resources and money on projects that at the end does not fulfill the customers needs and demands. The APH metod can solve this dellemma by improving the process of developing project proposals, and also removing the personal biasis that managers have. The weakness in project selection using AHP is that if some criterias are ignored, it will give some bad results, The AHP cannot fortune tell so it is exepcted to give all the criterias at the beginning of the selection process and as all other processes a powerfull member can promote their project and hinder the unbias project selesction.

===Risk assessment===
===Resource allocation===
===Picking suppliers===
Across industries the AHP method have been used in selecting suppliers such as the automotive industry, construction and green supply chain management<ref name="Selecting supplier1">Wang, Y., Dong, Y., & Hao, J. (2018). AHP-based supplier selection for an automotive manufacturer in China. Journal of Manufacturing Systems, 46, 63-70.</ref> <ref name="Selecting supplier2">Chan, A. P., Chan, D. W., & Ho, K. S. (2016). Application of AHP in selecting suppliers for a construction project in China. Journal of Construction Engineering and Management, 142(6), 04016012.</ref> <ref name="Selecting supplier3">Chai, J., Liu, J., & Lu, Y. (2017). An AHP-based evaluation method for supplier</ref>.

There have been multiple papers that demonstrate the effectiveness of the AHP method in supplier selection.
With the AHP method it becomes easier to minimize supplier-related risks such as financial stability, quality control, and delivery reliability. The method can also be used to rank the supplier performance over time and adjust changes.

= The analytic hierarchy process =
Assuming that we have multiple criterias and alternatives, the weight of the criterias is first weighted using pairwise comparison, using saatys scale:

{| class="wikitable" style="vertical-align:middle;"
|- style="font-weight:bold; text-align:center;"
! colspan="3" | The foundation scale of absolute numbers
|-
| Intensity of importance
| Definition
| Explanation
|-
| 1
| Equal importance
| Two activities contribute equally to the objective
|-
| 2
| Weak or slightly
|
|-
| 3
| Moderate importance
| Experience and judgment slightly favor one activity over another
|-
| 4
| Moderate plus
|
|-
| 5
| Strong importance
| Experience and judgment strongly favor one activity over another
|-
| 6
| Strong plus
|
|-
| 7
| Very strong
| An activity is favored very strongly over another, its dominance demonstrated in practice
|-
| 8
| Very, very strong
|
|-
| 9
| Extreme importance
| The evidence favoring one activity over another is of the highest possible order of affirmation
|-
| colspan="3" | '''Table 1:''' ''Saatys scale <ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>''
|}

The APH method makes use of the hiracy approach, where the objective is at the top and criteria is below it and the alternatives are at the bottom.
[[File:The three level hierarchy.png|500px|center]]

== Example for AHP==
'''NOTE TO ME:------------------------------------------------------------ NEED INTRO'''
#Identify the objectives
#Assign weights to the criteria by performing pairwise comparisons
#Evaluate the performance of each alternative with respect to each criterion
#Preform Consistancy test
#Calculate the global priority of each criteria and ranking

===Idenrify the objective===
A company is in the process of implementing a PMO system and is considering 3 providors. The company have come up with three specific criteria, which are considered highly important when determining which product they should buy. These criteria are cost, technical support, and flexibility.

===Assign weights to the criteria by performing pairwise comparisons===

When the objective is identified and relevent emperical information is gathered, the next step is to determin the relative importance among the criteria that is under consideration. This process involves giving a specific weight to each criteria based on its percieved importance. This step establish a hierachy, it ensures that the most significant criteria is given appropriate attention. This is done by giving each criteria a relative importance in the Saatys scale 1-9 as shown in table 1, which a person decides. It should be noted that the sum of each row is computed, this is used for later calculations.

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|- style="text-align:left;"
!
! Cost
! Support
! Flexibility
|-
| style="text-align:left;" | Cost
| 1,00
| 2,00
| 3,00
|-
| style="text-align:left;" | Support
| 0,50
| 1,00
| 2,00
|-
| style="text-align:left;" | Flexibility
| 0,33
| 0,50
| 1,00
|-
| style="text-align:left;" | Sum
| 1,83
| 3,50
| 6,00
|}

===Evaluate the performance of each alternative with respect to each criterion===

The alternatives are assesed by making a pariwise comparision, just like the previous step. For this step the alternatives are compared against eachother with respect to each criterion the, where they are given a relative importance just like earlier, which a person decides. they are then compiled into tabels, shown below:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to cost
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 7,00
| 2,00
|-
| style="text-align:left;" | Company 2
| 0,14
| 1,00
| 0,25
|-
| style="text-align:left;" | Company 3
| 0,50
| 4,00
| 1,00
|- style="text-align:left;"
| sum
| 1,64
| 12,00
| 3,25
|}

{| class="wikitable" style="vertical-align:middle;"
|- style="text-align:center; vertical-align:bottom;"
! colspan="4" | Alternatives with respect to support
|- style="vertical-align:bottom;"
|
| Company 1
| Company 2
| Company 3
|-
| style="vertical-align:bottom;" | Company 1
| 1,00
| 5,00
| 3,00
|-
| style="vertical-align:bottom;" | Company 2
| 0,20
| 1,00
| 0,50
|-
| style="vertical-align:bottom;" | Company 3
| 0,33
| 2,00
| 1,00
|-
| style="vertical-align:bottom;" | Sum
| 1,53
| 8,00
| 4,50
|}

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="4" | Alternatives with respect to flexibility
|- style="text-align:left;"
|
| Company 1
| Company 2
| Company 3
|-
| style="text-align:left;" | Company 1
| 1,00
| 0,11
| 0,13
|-
| style="text-align:left;" | Company 2
| 9,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Company 3
| 8,00
| 1,00
| 1,00
|-
| style="text-align:left;" | Sum
| 18,00
| 2,11
| 2,13
|}

In order to give a relative weight to each criterium, the matrices are normalised. This is done by dividing each value with the sum of the column value. Afterwards the priority is calculated, this is done by taking the sum and dividing with the number of elements, 3 in this case:

{| class="wikitable" style="vertical-align:bottom;"
|- style="text-align:center;"
! colspan="5" | Criteria Normalised
|-
|
| Cost
| Support
| Flexibility
| Priority
|-
| Cost
| 0,55
| 0,57
| 0,50
| 0,54
|-
| Support
| 0,27
| 0,29
| 0,33
| 0,30
|-
| Flexibility
| 0,18
| 0,14
| 0,17
| 0,16
|}

The values calculated under priority have a meaning in AHP, the values represent the relative importance of the criteria to the overall goal. In this case the company have decided that flexibility have a importance of 16% (this is the priority calculated in the above table).

===Preform consistancy test===

When preforming the consistancy the following is computed:
# The largest eigenvalue, λ_max, of the matrix is calculated
#The Consistency index (CI) and the consistency ratio (CR) is calculated

Math software, like maple, was used to find the largest eigenvalue of the matrix. The consistency index is calculated with the following formula:

<math>CI =\frac{\lambda_{max} - n}{n-1}</math>

where n is the number of elements. RI is is called the random consistency index, to finx the consistency ratio the CI is divided by the RI. The consistency ratio can be calculated with the following expression:

<math>CR =\frac{CI}{RI}</math>

The values for RI is presented below:

{| class="wikitable" style="text-align:center;"
|- style="text-align:left;"
! colspan="6" | Consistency of random matrices
|-
| style="text-align:left;" | matrix order
| 1
| 2
| 3
| 4
| style="text-align:left;" | ...
|-
| style="text-align:left;" | RI
| 0.00
| 0.00
| 0.52
| 0.89
| style="text-align:left;" | ...
|}

The consistancy test is preformed for the criterum matrix,cost matrix, support matrix and flexibility matrix:

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="5" | Consistancy test
|- style="text-align:left;"
|
| Criterium
| Cost
| Support
| Flexibility
|-
| style="text-align:left;" | Lambda_max
| 3,009
| 3,002
| 3,004
| 3,002
|-
| style="text-align:left;" | CI
| 0,004
| 0,001
| 0,002
| 0,001
|-
| style="text-align:left;" | CR
| 0,008
| 0,002
| 0,003
| 0,001
|}

===Calculate the global priority of each criteria and ranking===

{| class="wikitable" style="text-align:center; vertical-align:bottom;"
|-
! colspan="6" | The AHP ranking
|- style="text-align:left;"
|
| Cost
| Support
| Flexibility
| Global Priority
| Rank
|-
| style="text-align:left;" | Company 1
| 0,602
| 0,648
| 0,056
| 0,526
| 1
|-
| style="text-align:left;" | Company 2
| 0,082
| 0,122
| 0,481
| 0,160
| 3
|-
| style="text-align:left;" | Company 3
| 0,315
| 0,230
| 0,463
| 0,314
| 2
|}

https://www.pmi.org/learning/library/analytic-hierarchy-process-prioritize-projects-6608#

=Critique of the AHP method=

'''NOTE TO ME: look here'''<ref name="Critique of AHP">Asadabadi et al., Are MCDM methods useful? A critical review of Analytic Hierarchy Process (AHP) and AnalyticNetwork Process (ANP), Cogent Engineering(2019), 6: 1623153 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

=Inplementation of AHP method in organisations=

'''NOTE TO ME Look at the conclusion'''<ref name="The AHP process">Saaty, Int. J. Services Sciences, Vol. 1, No. 1, 2008 https://www.rafikulislam.com/uploads/resourses/197245512559a37aadea6d.pdf </ref>

= References =
<references/>