View source for The Critical Path Method (CPM) in Project Management



According to the <span class="plainlinks">[https://en.wikipedia.org/wiki/Project_Management_Institute '''Project Management Institute (PMI)''']</span>, project management is the application of knowledge, skills, tools and techniques to project activities in order to meet project requirements and objectives<ref name=PMI2008>Project Management Institute. (2008). <i> A Guide to the Project Management Body of Knowledge</i>. 4th Edition. USA. ISBN 9781933890517</ref>. The challenging task of managing projects can be supported by an <span class="plainlinks">[https://en.wikipedia.org/wiki/Operations_research operation research]</span> technique called the <span class="plainlinks">[https://en.wikipedia.org/wiki/Critical_path_method <b>Critical Path Method (CPM)</b>]</span>. The <b>CPM</b> is a mathematically-network based algorithm in which can be used for planning, scheduling and monitoring project progress.<ref name=Larson2014> Larson, E. W & Gray, C. F. (2014). <i>Project Management - The Managerial Process</i>. 6th edition. USA: NY. ISBN 9781259010705</ref><ref name=Jesper>Larsen, J. & Clausen, J., (2009). Course material in Networks and Integer Programming Supplementary at DTU - Notes to Networks and Integer Programming. Retrieved 04. September 2016 from campusnet.dtu.dk</ref>

The technique developed in late 1950s utilizes information from a <span class="plainlinks">[https://en.wikipedia.org/wiki/Work_breakdown_structure <b>work breakdown structure (WBS)</b>]</span> in a network representation to display interrelationships and dependencies between project activities that must be accomplished to complete a '''project'''. The technique is used for analyzing projects by determining the longest sequence of tasks through a project network, called the <b>critical path</b>.  This determines the shortest possible time to complete the project as well as which activities should be given a particular focus within the project.<ref name=Larson2014/><ref name=Newbold1998>Newbold, R.C. (1998). <i>Project Management in the Fast Lane – Applying the Theory of Constraint.</i> USA: FL. ISBN 9781498738064</ref>

Based on acquired information from the '''CPM''', the next question could be if it is possible to shorten the project in order to finish within certain deadline and what is the least expensive way to do it. In today’s competitive environment, there is an increasing pressure to get projects done earlier and quicker, either from the very beginning, prior to setting the project baseline, or in the midst of project execution in order to get the project back on schedule. Comparison of the benefits by reducing project time with the total cost associated with it can often be a huge challenge for project managers, thus, the decision to reduce the project duration should be based on an analysis of the '''trade-offs''' between '''time''' and '''cost'''.<ref name=Larson2014/><ref name=Jesper/><ref name = Komesh2014>Sahu, K. & Sahu, M. (2014). <i>Cost & Time and Also Minimum Project Duration Using Alternative Method </i> International Review of Applied Engineering Research.  5(4), pp. 403-412. Retrieved 10. September 2016. [http://www.ripublication.com/iraer-spl/iraerv4n5spl_03.pdf Available Online]</ref>

== Overview ==

=== Introduction ===


Organizations across the world and within different sectors have been using project management as a way to improve project results. The increased awareness and acceptance supports that the application of knowledge, skills, tools and techniques can have significant impact on project success.<ref name=PMI2008/><ref name=PMIVOPM> Project Management Institute. (2010). The Value of Project Management. Retrieved 05. September 2016. [http://www.pmi.org/-/media/pmi/documents/public/pdf/white-papers/value-of-project-management.pdf Available Online Version]</ref>

[[File:The_triad_constraints1.png|220px|thumb|right|<b>Figure 1:</B> The Project Management Iron Triangle <ref name=WIKITRIANGLE>Wikipedia. <span class="plainlinks">[https://en.wikipedia.org/wiki/Project_management_triangleProject <i>The Project management Triangle.</i>]</span> Retrieved 10.08.2016</ref> ]]

According to the <span class="plainlinks">[https://en.wikipedia.org/wiki/ISO_21500 '''ISO 21500''']</span> a project is defined as a<i> “unique set of processes consisting of coordinated and controlled activities with start and end dates, performed to achieve project objectives. Achievement of the project objectives requires the provision of deliverables conforming to specific requirements.” </i><ref name=ISO2012> International Organization for Standardization (2012) ISO 21500 – Guidance on Project Management. </ref> 

Each and every project is carried out under certain constraints. The three constraints of cost, time and scope represent together the <span class="plainlinks">[https://en.wikipedia.org/wiki/Project_management_triangle '''iron triangle''']</span> of project management, which can be seen in <i>Figure 1</i>. Each constraint forms the vertices with quality as the central theme and together indicates that projects must be delivered within agreed time and cost and furthermore to meet the agreed scope and customer quality requirements. The three constraints are aligned and changes in one will likely affect the others, or impact the quality of the project. This emphasizes the importance of project management and its challenging area in which is not at all easy to deal with.<ref name=WIKITRIANGLE/><ref name=MINDTOOLS>Mind Tools.<i> The Iron Triangle of Project Management.</i> Retrieved 10. September 2016. [https://www.mindtools.com/pages/article/newPPM_54.htm Available Online]</ref>

Project management is accomplished through the appropriate application of five processes: <ref name=PMI2008/>

# Initiating
# Planning
# Executing
# Monitoring and Controlling
# Closing

Each and every organization can be involved in a project of any size, duration and complexity level at any given time. It is notable that project planning, scheduling and monitoring is a major part involved in project management. Therefore the '''critical path method (CPM)''' can be of good support within the challenging process of managing projects and helps those involved to address questions such as: <ref name=NETWORK>Chapter on Deterministic Decision Models. (n.d.). Course material in Network Optimization at DTU autumn 2015. <i>Project Scheduling: PERT/CPM.</i> [https://www.google.dk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwixqcub74nPAhUMFSwKHS1pAFIQFggeMAA&url=http%3A%2F%2Fhighered.mheducation.com%2Fsites%2Fdl%2Ffree%2F0070813809%2F700204%2FsampleChapter8.pdf&usg=AFQjCNGErOnRK7sLSe6Ii4g7lsOCaWXpuA&sig2=ss1geYqxTa5GJ2feFpsi7Q Available online]</ref>

* What is the (minimum) total time required to complete the whole project? 
* What are the (earliest and latest) start and completion times for individual activities in the project?
* Which activities are critical and must be completed on time in order to complete the whole project on time?
* How much delay is tolerated of non-critical activities without impacting the overall project schedule and project completion time?
* With cost information on each activity: What is the least expensive way to reduce project duration in order to meet a targeted completion time?

All these questions are extremely valid and important to have under control when managing projects.

=== Background ===

The '''critical path method (CPM)''' is a mathematically-network based project modeling technique developed in late 1950s in order to plan, schedule and control large, complex projects with many activities. '''CPM''' is one approach of <b>network techniques</b> that has been widely used and was developed by a DuPont engineer Morgan R. Walker and a Remington Rand computer expert, James E. Kelly, Jr., to manage plant maintenance and construction work. Furthermore, <span class="plainlinks">[https://en.wikipedia.org/wiki/Program_evaluation_and_review_technique <b>PERT (Program Evaluation and Review Technique)</b>]</span>, a critical path network technique was developed simultaneously and independently from the '''CPM''' by the U.S. Navy for managing the Fleet Ballistic Missile <span class="plainlinks">[https://en.wikipedia.org/wiki/UGM-27_Polaris <b>(Polaris)</b>]</span> submarine project. <span class="plainlinks">[https://en.wikipedia.org/wiki/Program_evaluation_and_review_technique <b>PERT</b>]</span> is commonly used in conjunction with '''CPM''' and the two optimization techniques are often referred to collectively.  <ref name=Jesper/><ref name=Anderson1986>Anderson, E. B., & Hales, R. S., (1986).<i> Critical Path Method Applied to Project Planning: Fire Economics Evaluation Systems (FEES)</i>. United States Department of Agriculture. USA. [http://www.fs.fed.us/psw/publications/documents/psw_gtr093/psw_gtr093.pdf Available Online]</ref><ref name=STELTH2009>Stelth, P. (2009). <i>Project’s Analysis through CPM (Critical Path Method). </i>Isles International University. Retrieved 06. September 2016. [http://www.iiuedu.eu/press/journals/sds/sds1_july_2008/05_SECC_01.pdf Available Online]</ref>

The '''CPM''' technique has been used for many different forms of projects including construction of a new building, bridge or a road, construction of IT-systems, building of ships and research and development of new products. As can be imagined, these projects consist of multiple activities that have to be completed in order to finish the project as a whole.
It is important to note that the methodology is normally not applied to real large scale and complex problems with many activities, since developing the project network and applying the technique can be extremely time consuming. Today, there exists project management software packages based on the methodology, for example <span class="plainlinks">[https://en.wikipedia.org/wiki/Microsoft_Project <b>Microsoft Project</b>]</span>, which are used to calculate all completion times based on a input. The programs are further able to create a visualization of the whole project which is used to ease overview and understanding, as well as supporting the monitoring of the project. On the other hand, the methodology can easily be applied manually to smaller problems in which will be the case in this article. Despite what already mentioned, a knowledge of the concept is important, thus the ones who manage projects have to deal with multiple challenges within planning, coordinating and monitoring all project activities in order to complete the project of interest successfully within agreed time, cost and scope as well as according to customer quality requirements.<ref name=Jesper/>

== Methodology ==


The '''Critical Path Method (CPM)''' can be divided into the following steps:

*<b>STEP 1:</B>  <i>Developing the Project Network</i>

*<b>STEP 2: </B>  <i>Constructing the Project Network</i>

*<b>STEP 3: </B>  <i>The Computation Process of forward pass, backward pass and slack</i>

*<b>STEP 4: </B>  <i>Critical Path Identification</i>

*<b>STEP 5: </B>  <i>Update of Project Progress</i>

The methodology will be explained step-by-step as well as with an example in <span class="plainlinks">[http://apppm.man.dtu.dk/index.php/The_Critical_Path_Method_(CPM)_in_Project_Management#CPM_Example <i>Chapter 3: CPM Example</i>.]</span>

====Step 1: Developing the Project Network====
<!-- Comment <b>Step 1: Developing the Project Network</b>====-->

When developing the project network, the following input requirements are needed in order to construct a network model of the project of interest: <ref name=STANFORD>Santiago, J. & Magallon, D. (2009). <I>The Critical Path Method</I> - (CEE 320 – VDC Seminar). Stanford University.  Retrieved 06. September 2016. [http://web.stanford.edu/class/cee320/CEE320B/CPM.pdf Available online]</ref>

* A list of all activities required to complete  the project also known as the <span class="plainlinks">[https://en.wikipedia.org/wiki/Work_breakdown_structure '''work breakdown structure (WBS)''']</span>
* The duration of each individual activity for completion
* The dependencies between the activities

With this information, the <b>critical path method (CPM)</b> is able to determine the longest path of activities through the '''project network''', defined as the <b>critical path</b> and the corresponding activities are defined as <b>critical activities</b>. Based on the fact that the entire project cannot finish until all the activities are completed, the longest path or the '''critical path''' gives also the minimum required time to complete the entire project.

====Step 2: Constructing the Project Network====
<!-- Comment <b>Step 2: Constructing the Project Network</b>====-->
The second step involves drawing the project network by placing the activities that need to be completed in the right logical sequence based on their interdependencies. The '''CPM''' notation used for each and every activity in the project network can be seen in <i>Figure 2</i>. Therefore, for all activities in the network the following information is registered and/or computed:  [[File:Notation.png|220px|thumb|right|<b>Figure 2:</b> Activity Notation in a Project Network <ref name=Larson2014/>]]

* '''ID:''' Identification code of individual activity based on the WBS
* '''Description:''' Description of individual activity based on the WBS
* '''Duration (DUR):''' Duration time of individual activity
* '''Early Start (ES):''' The earliest starting time of an individual activity 
* '''Early Finish (EF):''' The earliest finishing time of an individual activity
* '''Late Start (LS):''' The latest starting time of an individual activity
* '''Late Finish (LF):''' The latest finishing time of an individual activity
* '''SL (Slack):''' The tolerance of delay of an individual activity without affecting  project completion date

====Step 3: The Computation Process====
<!-- Comment <b>Step 3: The Computation Process</b>====-->

The third step involves the two procedures of '''forward pass''' used to compute earliest times, '''ES''' and '''EF''', and the '''backward pass''' used to compute latest times, '''LS''' and '''LF''' as well as <b>slack</b> computation. <ref name=Larson2014/><ref name=NETWORK/>

<b>Forward Pass</b> begins at the initial activity and traces each path through the network step-by-step, to the end of the last project activity/activities. An important rule of the forward pass procedure is that all immediate predecessors must be completed before a succeeding activity can start. That is, early start (ES) of an activity equals the <b>largest</b> early finish (EF) of the immediate predecessors. 

For the starting activity/activities of the network, early start (ES) is set as '''ES = 0''' (or some other known value) and each path is traced forward to the finishing activity/activities by applying '''EF = ES + DUR''' to all activities in the project network. When finished, the longest path denotes the project completion time for the plan, that is, '''largest EF = Total Time.''' 

<b>Backward Pass</b> is initiated after the forward pass has been completed and starts with the last project activity/activities without successors and the late finish (LF) is set as equal to the maximum early finish (EF) of those activities. An important rule of the backward pass is that at the latest time a succeeding activity can start is only when all immediate predecessors are completed. That is, late finish (LF) for an preceding activity equals the <b>smallest</b> late start (LS) of the immediate successor activities.

With that in mind, each path is traced backward towards the initial starting activity/activities by applying '''LS = LF - DUR'''. When this has been applied to all the activities in the project network, the total '''slack''' of an activity can be determined by using '''SL = LS – ES''' or '''LF – EF'''. <ref name=NETWORK/>

====Step 4: Critical Path Identification====
<!-- Comment <b>Step 4: Critical Path Identification</b>====-->

The '''critical path''' is of great importance since all '''critical activities''' must be completed as scheduled in order to meet the scheduled project completion time, that is, if an activity on the '''critical path''' is delayed, the project is delayed by the same amount of time. This is due to the fact that the activities on the '''critical path''' have a '''zero slack'''.  '''Slack (SL)''' is defined as the amount of time an activity can be delayed without delaying the whole project. Therefore, project managers need pay close attention to the '''critical path activities''', which can represent around 10% of all project activities, in order to ensure that they are not delayed and therefore put the entire project in the risk of being late. <ref name=Larson2014/>

====Step 5: Update of project progress====
<!-- ====<b>Step 4: Update of project pogress</b>====-->

As the project progresses, the actual activity completion times will be known and the project network can be updated in order to include this information. It is important to note that a new critical path may emerge and changes may be made in the network if project requirements suddenly change. <ref name=Komesh2014\>SmartSheet. <i>The Ultimate Guide to the Critical Path Method.</i> Retrieved 18.08.2016 from [https://www.smartsheet.com/critical-path-method SmartSheets online page]</ref>

The emerge of new critical path is related to the '''sensitivity''' of a project network. A network is considered '''sensitive''' when it has one or more critical path(s) and/or non-critical activities with little slack and therefore it is more likely that the original critical path(s) will change once the project has started. On the other hand, a network is considered '''insensitive''' when it has one critical path and non-critical activities with very large slack, and therefore less likely that the critical path will change once the project has started. <ref name=Larson2014/>

== CPM Example ==

In order to gain understanding of the critical path method, the following example is provided.

===Constructing the CPM Project Network===
<i>Table 1</i> provides the required information of <b>Step 1</B>, that is, 9 activities required to complete a project of interest as well as their dependencies and duration times. Based on those information, the projects needs can be translated into a network diagram in which the CPM can be utilized. The diagram in <b>Step 2</b> graphically shows the project flow and precedence relationships among all project activities and gives the project manager a visual representation of the entire project. 
{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+ Table 1: Project Information for the CPM 
|-
! Activity
! Duration [Weeks]
! Predecessor
! Description
|-
| <center>A</center>
| <center>1</center>
| <center>-</center>
| Constructing a base
|-
| <center>B</center>
| <center>5</center>
| <center>A</center>
| Build and construct wall structure
|-
| <center>C</center>
| <center>4</center>
| <center>A</center>
| Create bearings for the roof 
|-
| <center>D</center>
| <center>10</center>
| <center>B, C</center>
| Raise the roof
|-
| <center>E</center>
| <center>5</center>
| <center>C</center>
| Isolate and close the walls 
|-
| <center>F</center>
| <center>7</center>
| <center>D, E</center>
| Insert windows and doors
|-
| <center>G</center>
| <center>5</center>
| <center>B, D</center>
| Prepare floors and lay parquet
|-
| <center>H</center>
| <center>4</center>
| <center>E</center>
| Prepare the outside area  
|-
| <center>I</center>
| <center>6</center>
| <center>G, F, H</center>
| Paint walls and interior
|}

A network model of the project of interest represented in <i>Table 1</i> can be seen in <i>Figure 3</i> where all project activities have been placed in the right logical sequence based on their interdependencies and using the activity notation presented in <i>Figure 2</i>. This is the initial starting point of the critical path method before computation process, <b>Step 3</B>, takes place. 


[[File:ProjectNetwork2.png|800px|thumb|center|<b>Figure 3:</b> Network Representsation of the Project Example presented in Table 1]]

===The CPM Computation Process===
Following the <b>CPM</b> methodology presented earlier, the first step of the computation process is to perform the <b>forward pass</b>. Activity A has no predecessors and therefore we assume that the activity can start as soon as the project can start, at time 0. The earliest finish (EF) date for activity A can now be computed as <i>0 + 1 = 1</i>. Activities B and C have activity A as their immediate predecessor with early start (ES) as 1 and early finish (EF) as <i>1 + 5 = 6</i> and <i>1 + 4 = 5</i>. Activity D is a <b>merge activity</b> since it is preceded by both B and C. The early start (ES) of a merge activity depends on the largest early finish (EF) of all activities that merge to it, since the activity cannot start until both the predeceasing activities have finished. Therefore, early start of activity D is <i>max(6<sub>(EF-B)</sub>,5<sub>(EF-C)</sub>) = 6</i>. By continuing the forward pass, the early start (ES) and early finish (EF) for all activities can be computed as shown in <i>Figure 4</i>. When the forward pass has been completed,  it can be seen that the early finish (EF) for activity I, or the project in total is <b>29 weeks</b>. 

The second step in the CPM methodology is the <b>backward pass</b> starting with the last project activity, which in this case is I.  The late finish (LF) of the finishing node I is set equal to its early finish (EF) of 29 weeks and late start (LS) can be computed as <i>29 - 6 = 23.</i> The finishing node I is an immediate successor of activities G, F and H, hence the latest finishing (LF) time for all the three activities is set to 23 and their late starts (LS) are computed as <i>23 - 5 = 18</i>, <i>23 - 7 = 16</i> and <i>23 - 4 = 19</i>. Activity E is a <b>burst activity</b> since it is a immediate predecessor for both activities F and H. Hence, the late finish (LF) for activity E is controlled by the smallest late starts (LS) of the two succeeding activities since activity E cannot finish later than the latest start of an immediate successor. Therefore, late finish (LF) of activity E is <i>min(16<sub>(LS-F)</sub>,19<sub>(LS-H)</sub>) = 16</i>. If the backward pass is continued until the starting node, the late start (LS) and late finish (LF) for all activities can be computed as shown in <i>Figure 4</i>.  


[[File:ProjectNetworkSolution1.png|800px|thumb|center|<b>Figure 4:</b> Solution of the Project Example presented in Table 1 based on CPM - <b>Critical Path</b>: A - B - D - F - I ]]

===Slack Computation and Critical Path Identification===
After the completion of forward and backward passes, it is possible to determine which activities can be delayed by calculating the slack for individual activities in the project network. The slack for activity A is 0 since there is no difference between ES/EF and LS/LF.  On the other hand, the slack for activity C is <i>2 - 1 = 6 - 5 = 1.</i> The slack for individual activities can be computed as shown in <i>Figure 4</i>.  After the slack computation, the critical path can easily be identified as the longest path through the network with critical activities consisting of <b>zero slack</b>. The <b>critical path</b> in this example is: <b><i>A – B – D – F – I. </i></B>. That is, if an activity on the path is delayed, the project is delayed by the same amount of time.

== Consideration of Time-Cost trade-offs ==

[[File:TC-tradeoff.png|260px|thumb|right|<b>Figure 5:</B> Project Time-Cost Trade-Off Curve. <ref name=Tradeoffpicture>Hegazy, T. (1999). <i>Optimization of construction time – Cost trade-off analysis using genetic algorithms.</i> Figure of Time-Cost Trade-Offs Retrieved 09. September 2016. [https://www.google.dk/search?q=project+time+cost+relationship&client=safari&rls=en&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiH5dKvk5LPAhUDXCwKHdZhBoUQ_AUICCgB&biw=1080&bih=634#imgrc=288dO6iwGcV2gM%3A Available online]</ref> ]]

The consideration of '''time-cost trade-offs''' for a project is often considered an essential part of a complete '''Critical Path Method (CPM)''' analysis <ref name=CPMTRADEOFF>Panagiotakopoulos, D. <i>A CPM Time-Cost Computational Algorithm for Arbitrary Activity Cost Functions.</i> Department of Civil Engineering, McGill University, Montreal.  Retrieved 20.08.2016.</ref>. Based on acquired information from the <b>CPM</b>, the next questions could be if it is possible to shorten the project in order to finish within certain deadline and what is the least expensive way to do it. <b>Project crashing</B> is a procedure to identify the cost of reducing project duration so that comparisons can be made with the benefits of getting the project completed earlier. It requires information of the '''total project cost''', which includes both '''direct''' and '''indirect costs'''. Shortening the project duration will normally increase the '''direct costs''', which can be assigned directly to project activities and commonly represents labor, materials and etc. Meanwhile, a reduction in project duration indicates a reduction in '''indirect costs''', necessary costs for doing work, which continues for the life of the project and is not related to individual project activities. Therefore, the '''time-cost trade-off analysis''' can help to identify the <b>optimum cost-time point</b> between increasing cost of individual activities and decreasing overall project costs. The outcome of such analysis results in a time-cost trade-off curve as shown in <i>Figure 5</i> which furthermore shows that the optimum project duration can be determined as the project duration that results in the least total project cost. <ref name=TRADEOFF>Elbeltagi, E. <i>Construction Management - Chapter on Project Time-Cost Trade-Offs.</i>  Retrieved 15. September 2016. [http://osp.mans.edu.eg/elbeltagi/CM%20CH8%20Time-Cost.pdf Available online]</ref><ref name=CPMtutor>CPM Tutor. (2010). <i>Time-Cost Trade-Off</i>. Retrieved 15. September 2016 [http://www.cpmtutor.com/c08/timecosttradeoff.html Available online]</ref>


[[File:AcTrelationship1.png|350px|thumb|right|<b>Figure 6:</B> Activity Cost & Time Relationship.<ref name=NETWORK/> ]]

Simplified represensation of the relationship between activity duration and its direct costs can be seen in <i>Figure 6</i>. <b>Normal time</B> for an activity represents estimated activity duration under normal conditions retrieved from the <B>CPM</B> and <b>normal cost</B> refers to the corresponding cost of that activity, which implies minimum direct cost. <b>Crashing</b> refers to speeding up the duration of an activity and the <b>crash time</B> is the shortest possible time an activity can be completed in. The direct cost for completing an activity in its crash time is called <b>crash cost</b> and <b>crash point</b> represents the maximum time an activity can be compressed. The cost-time relationship is assumed to be linear as shown in <i>Figure 6</i> and for each activity a <b>crash cost per period</b> can be derived as: 


<center><math> \text{crash cost period}  =  \tfrac{\text{Crash Cost - Normal Cost}}{\text{Normal Time - Crash Time}}</math></center>


That is, the <b>slope</b> gives information about the <b>cost per unit of time</b> for shortening an individual activity and therefore allows comparison of which <b>critical activities</b> to shorten in order to minimize the total direct cost. <ref name=Larson2014/><ref name=NETWORK/><ref name=TRADEOFF/>

After having utilized the '''CPM''', the procedure for '''project crashing''' involves the following steps: <ref name=NETWORK/><ref name=TRADEOFF/>

# Compute the '''crash cost per period''' for all activities in the project network
# Find '''critical path(s)''' and '''critical activities''' in the project network
# In the case of only '''one''' '''critical path''', identify a critical activity with the '''smallest crash cost per period''' that can still be crashed. '''Otherwise''', identify critical activity/activities from '''each critical path''' with the s'''mallest crash cost per period''', which can still be crashed.
# Reduce the duration of the critical activity/activities identified in step 3 by one time period.
# Reconstruct or adjust the project network based on changes made in step 4. Calculate the <b>total costs</b> = <b>directs</b> + <b>indirect costs</b>. '''Stop''' the procedure if the desired completion deadline is reached, otherwise return to step 3. Continue until the '''crash point''' has been reached, that is, no further shortening is possible.

An example of project crashing where the procedure is utilized and explained step-by-step is available <span class="plainlinks">[https://www.google.dk/url?sa=t&rct=j&q=&esrc=s&source=web&cd=1&ved=0ahUKEwixqcub74nPAhUMFSwKHS1pAFIQFggeMAA&url=http%3A%2F%2Fhighered.mheducation.com%2Fsites%2Fdl%2Ffree%2F0070813809%2F700204%2FsampleChapter8.pdf&usg=AFQjCNGErOnRK7sLSe6Ii4g7lsOCaWXpuA&sig2=ss1geYqxTa5GJ2feFpsi7Q <b>here</b>]</span> in <b>Chapter 8.7: Project Crashing</b>. The procedure can be used as a continuation of the example that was provided above in <span class="plainlinks">[http://apppm.man.dtu.dk/index.php/The_Critical_Path_Method_(CPM)_in_Project_Management#CPM_Example <i>Chapter 3: CPM Example</i>]</span>, with relatively small change. The only change required is that the duration time of activity A is set to 0 since it is considered as a dummy activity. For additional knowledge and understanding, good examples are provided <span class="plainlinks">[http://osp.mans.edu.eg/elbeltagi/CM%20CH8%20Time-Cost.pdf <b>here</b>]</span> in <b>Chapter 8.4: Shortening Project Duration</b>.


=== Project Crashing Example ===

In order to gain understanding of the project crashing procedure, the following example is provided.

{| class="wikitable" style="margin-left: auto; margin-right: auto; border: none;"
|+ Table 1: Project Information for the project crashing procedure.<ref name=Larson2014/>
! rowspan="1" | 
! colspan="2" | Normal 
! colspan="2" | Crash 
!
! 
|-
! Activity ID
! Time [Week]
! Cost [$]
! Time [Week]
! Cost [$]
! Max Crash Time [Week]
! Crash Cost per week [$]
|-
! <center>A</center>
| <center>3</center>
| <center>50</center>
| <center>2</center>
| <center>70</center>
| <center>1</center>
| <center>20</center>
|-
! <center>B</center>
| <center>6</center>
| <center>80</center>
| <center>4</center>
| <center>160</center>
| <center>2</center>
| <center>40</center>
|-
! <center>C</center>
| <center>10</center>
| <center>60</center>
| <center>9</center>
| <center>90</center>
| <center>1</center>
| <center>30</center>
|-
! D
| <center>11</center>
| <center>50</center>
| <center>7</center>
| <center>150</center>
| <center>4</center>
| <center>25</center>
|-
! E
| <center>8</center>
| <center>100</center>
| <center>6</center>
| <center>160</center>
| <center>2</center>
| <center>30</center>
|-
! F
| <center>5</center>
| <center>40</center>
| <center>4</center>
| <center>70</center>
| <center>1</center>
| <center>30</center>
|-
! G
| <center>6</center>
| <center>70</center>
| <center>6</center>
| <center>70</center>
| <center>0</center>
| <center>0</center>
|}

[[File:Iteration0.png|350px|thumb|right|<b>Figure 5:</B> Project Time-Cost Trade-Off Curve. <ref name=Tradeoffpicture>Hegazy, T. (1999). <i>Optimization of construction time – Cost trade-off analysis using genetic algorithms.</i> Figure of Time-Cost Trade-Offs Retrieved 09. September 2016. [https://www.google.dk/search?q=project+time+cost+relationship&client=safari&rls=en&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiH5dKvk5LPAhUDXCwKHdZhBoUQ_AUICCgB&biw=1080&bih=634#imgrc=288dO6iwGcV2gM%3A Available online]</ref> ]]
<b>Iteration 0</b>

Critical Path: 

Mimin

Total Cost: 

<b>Iteration 1</b>
[[File:Iteration0.png|350px|thumb|right|<b>Figure 5:</B> Project Time-Cost Trade-Off Curve. <ref name=Tradeoffpicture>Hegazy, T. (1999). <i>Optimization of construction time – Cost trade-off analysis using genetic algorithms.</i> Figure of Time-Cost Trade-Offs Retrieved 09. September 2016. [https://www.google.dk/search?q=project+time+cost+relationship&client=safari&rls=en&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiH5dKvk5LPAhUDXCwKHdZhBoUQ_AUICCgB&biw=1080&bih=634#imgrc=288dO6iwGcV2gM%3A Available online]</ref> ]]
[[File:Iteration0.png|350px|thumb|left|<b>Figure 5:</B> Project Time-Cost Trade-Off Curve. <ref name=Tradeoffpicture>Hegazy, T. (1999). <i>Optimization of construction time – Cost trade-off analysis using genetic algorithms.</i> Figure of Time-Cost Trade-Offs Retrieved 09. September 2016. [https://www.google.dk/search?q=project+time+cost+relationship&client=safari&rls=en&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiH5dKvk5LPAhUDXCwKHdZhBoUQ_AUICCgB&biw=1080&bih=634#imgrc=288dO6iwGcV2gM%3A Available online]</ref> ]]
<b>Iteration 0</b>
<b>Iteration 0</b>

<b>Iteration 2</b>


<b>Iteration 3</b>
[[File:Iteration0.png|350px|thumb|right|<b>Figure 5:</B> Project Time-Cost Trade-Off Curve. <ref name=Tradeoffpicture>Hegazy, T. (1999). <i>Optimization of construction time – Cost trade-off analysis using genetic algorithms.</i> Figure of Time-Cost Trade-Offs Retrieved 09. September 2016. [https://www.google.dk/search?q=project+time+cost+relationship&client=safari&rls=en&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiH5dKvk5LPAhUDXCwKHdZhBoUQ_AUICCgB&biw=1080&bih=634#imgrc=288dO6iwGcV2gM%3A Available online]</ref> ]]
<b>Iteration 0</b>

<b>Iteration 4</b>
[[File:Iteration0.png|350px|thumb|right|<b>Figure 5:</B> Project Time-Cost Trade-Off Curve. <ref name=Tradeoffpicture>Hegazy, T. (1999). <i>Optimization of construction time – Cost trade-off analysis using genetic algorithms.</i> Figure of Time-Cost Trade-Offs Retrieved 09. September 2016. [https://www.google.dk/search?q=project+time+cost+relationship&client=safari&rls=en&source=lnms&tbm=isch&sa=X&ved=0ahUKEwiH5dKvk5LPAhUDXCwKHdZhBoUQ_AUICCgB&biw=1080&bih=634#imgrc=288dO6iwGcV2gM%3A Available online]</ref> ]]
<b>Iteration 0</b>

== Advantages and Benefits of the CPM ==

The <b>critical path method (CPM)</b> has been widely used in planning, scheduling and monitoring of project progress by a variety of organizations and industries with great success. The advantages of the '''CPM''' are multiple which can be used as a framework  for project information and insight used by project managers and members concerning project time, cost and performance. Following is a list of advantages and benefit reasons of why <B>CPM</B> is utilized in organizations today: <ref name=Larson2014/><ref name= Anderson1986/><ref name=STELTH2009/><ref name=TheConstructor>The Constructor – Civil Engineering Home. <i>Advantages of Critical Path Method (CPM)</i>. Retrieved 16. September 2016 [http://theconstructor.org/construction/const-management/scheduling/advantages-of-critical-path-method-cpm/6873/ Available Online] </ref>

* The method encourages all project members to identify and graphically represent all various project activities that need to be accomplished together with their interrelationship and dependencies in a logical manner. This is beneficial in the planning stage since it requires that the project is thought through in sufficient details in the long range. This minimizes the chances of overlooking necessary project activities and goals.

*The network representation is a graphic visualization of the entire project flow, giving complete overview in which is understood by all members in the project. Furthermore, smaller project network diagrams can easily be modified or changed when unexpected events occur as the project progresses. 
 
* The method provides an estimate of the minimum project duration together with scheduling of individual project activities needed to complete the project of interest in an efficient manner. Furthermore, the method also identifies the slack or the tolerance of delay without affecting the imposed completion time for individual activities.

* The method provides a basis for documentation standard and enhances the communication of project plans, schedules and time-cost performances.  

* The method enables the identification of the critical path(s) and corresponding critical activities in which special attention should be on in the project due to risk of delay. This is extremely beneficial in the monitoring stage when tracking the project progress. That means regular updates on project status as well as the critical path(s) with the concept of network sensitivity in mind, since the critical paths do not necessary remain static for the life of the project. 

* The benefit of time-cost trade off optimization is possible with cost [normal and crash] information on each activity in the project. That is, the '''CPM''' can help to identify the steps to be taken in order to accelerate a project completion or identify the shortest possible time or least possible cost that is needed as well as the optimum point of time and cost.

* The method provides the basis for scheduling labor and equipment as well as budgeting the cash flow of the project.

== Disadvantages and Limitations of the CPM ==

Despite the multiple advantages and benefits of the <b>CPM</b> mentioned earlier, the method possesses of several limitations or disadvantages which can be listed as following: <ref name=Larson2014/><ref name=STELTH2009/><ref name=MindToolsPERT> Mind Tools. <i> Critical Path Analysis and PERT Charts.</i> Retrieved 15. September 2016 [https://www.mindtools.com/critpath.html Available online]</ref><ref name=BrightHub>Bright Hub Project Managmenet. (2013). <i>Advantages and Disadvantages of Critical Path Method</i>. Retrieved 17. September 2016. [http://www.brighthubpm.com/project-planning/60960-advantages-and-disadvantages-of-critical-path-method/ Available online]</ref>

* The '''CPM''' and the analysis process can become extremely complicated  as the scope and complexity of the project increases and without a software it might be hard to manage. For large and complex projects, there can be thousands of activities and dependency relationships and the risk of making a mistake in the network computation becoming very high.

*The '''CPM''' and network diagrams are highly dependent on information technology and computer software in which can be of high initial and usage cost for organizations.

*The method does not include labor, equipment scheduling and resource allocation, only provides the basis for it.  

*The method requires time in order to develop the project network and relies heavily on project managers and members involved in that stage. Poorly defined project scope and activities can result in ineffective use of '''CPM''', which becomes difficult to manage. 

*The method operates based on the assumption that the duration time of individual activities in the project are known with certainty, which may not always be the case. This is where the network technique variation of '''CPM''', <span class="plainlinks">[https://en.wikipedia.org/wiki/Program_evaluation_and_review_technique <b>PERT</b>]</span> can be useful which takes more skeptical view of the time needed to complete an individual activity. 

*The method cannot effectively handle changes in the project plan during the project execution. Changes require that the project network is redrawn and the computation process repeated according to new information midway in the project.

*Critical path(s) is/are not always clear and can sometimes be hard to identify and follow when monitoring the project progress. Constant review of the network diagram is necessary to identify the shifting and movement of the critical path(s) over time.

==Annotated Bibliography==

<b>Project Management Institute. (2008). A Guide to the Project Management Body of Knowledge.</b><ref name=PMI2008/> 

This is a recognized formal standard for the project management profession and describes established norms, methods, processes and practices for project, program and portfolio management. 

<b>Larson, E. W & Gray, C. F. (2014). Project Management - The Managerial Process.</b> <ref name=Larson2014/>

This book provides a holistic, socio-technical view of project management and focuses on the integration of project management into the organization as a whole. The focus is on the essential tools and processes used to manage projects as well as the human dimension and how they interact to determine the effectiveness and outcomes of projects. <i>Chapter 6</i> focuses on the development of project plan and the use of project networks. In addition, the chapter builds on and extends the network technique by introducing different lag relationships between project activities. <i>Chapter 9</i> focuses on project duration reduction and consideration on time-cost trade-offs. For a detailed example of the project crashing procedure, the reader is recommended to look up pages 316-318.  

<b>Chapter on Deterministic Decision Models. (n.d.). Course material in Network Optimization at DTU autumn 2015. Project Scheduling: PERT/CPM.</b><ref name=NETWORK/>

This chapter based on Deterministic Decision Models is extremely good reading material for better understanding and knowledge of the two modeling techniques, PERT/CPM, as well as their connection/differences. It introduces the basics concepts and describes in a useful and comprehensive way the methodologies behind. Detailed example is provided on project crashing in which can be used as a continuation on the example provided above (<span class="plainlinks">[http://apppm.man.dtu.dk/index.php/The_Critical_Path_Method_(CPM)_in_Project_Management#CPM_Example <i>Chapter 3: CPM Example</i>]</span>) with relatively small changes. Furthermore, based on that example, the use of linear programming in project acceleration and project scheduling is introduced and explained with an example.  

<b>Anderson, E. B., & Hales, R. S., (1986). Critical Path Method Applied to Project Planning: Fire Economics Evaluation Systems (FEES).</b><ref name= Anderson1986/>

The article provides a good introduction of the critical path method as a technique for planning, scheduling and monitoring projects together with its advantages in all the three phases. It provides the basic knowledge of the methodology and the different steps that have to be taken. Additionally to what has been described above, the construction of time chart is embedded in the '''CPM''' method as one step. Furthermore, the article provides a short introduction on resource as well as time-cost analysis. 

<b>Stelth, P. (2009). Project’s Analysis through CPM (Critical Path Method). Isles International University.</b><ref name= STELTH2009/>

The focus of the study is to understand and evaluate critical paths and critical chains in a project. The study identifies the salient features of the '''critical path (CPM)''' and critical chain method (CCM) together with potential problematic areas. Furthermore, it provides the reader with evaluation of the similarities and differences of the two methods,  the ability to use them in conjunction as well as the impact the two methods can have on project management. The study covers various points under each of those areas and provides the reader with good insight and knowledge by good and detailed discussion. For more detailed and good summation on the advantages and disadvantages of the <B>CPM</B>, the reader is recommended to look up pages 21-25.

<b>Elbeltagi, E. Construction Management - Chapter on Project Time-Cost Trade-Offs.</b><ref name=TRADEOFF/>

This chapter on project time-cost trade-off gives good understanding of the relationship between those two important constraints in a project.  It provides detailed discussion of the basic concepts of project & activity time-cost relationships as well as the project crashing procedure. The reader is highly recommended to look at the many different examples provided based on the project crashing procedure to exercise and gain further understanding.

== References ==
<references />