Project Cost Estimation Methods

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== Abstract ==
 
== Abstract ==
  
  
  
Prior to the start of a project, cost estimate is critical. It is an important phase in project management since it is used to calculate and manage the project budget. The cost of a project is estimated for the first time at the start of the project or sometimes even before it starts. Afterwards, the cost is re-estimated on a regular basis to account for new information, scope changes, and the project's timetable. The cost of a project has the potential to impact nearly every area of the project, making it one of the most critical jobs for a project manager. A poorly written budget will result in incorrect asset allocation, unrealistic expectations, and, in the worst-case situation, project failure. Simply said, a project's budget must be accurate for it to succeed. Cost estimation is one of the most helpful tools in a project manager's arsenal for creating an adequate budget. There are different methods of cost estimation available in the literature such as analogue estimation, parametric estimation, bottom up estimation, expert judgement, and three-point estimation. Each method has its own pros and cons for different projects. The application of each cost estimation method varies from project to project and the nature of the project. Here, in this article we will focus on the three-point estimation as it is the most suitable for all kind of projects <ref name="budget"/>. This is due to the fact that the three-point estimation method provides a more precise estimate, hereby reducing the chance of failure and the possibility of unrealistic or excessively high estimates.
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Prior to the start of a project, cost estimation is critical. It is an important phase in project management, since it is used to calculate and manage the project budget, which is often the most important parameter within the standard success criteria of cost, schedule, and performance targets. The cost of a project is usually estimated at the start of the project or sometimes even before. Afterwards, the cost is re-estimated on a regular basis to account for new information, scope changes, and the timetable of the project <ref name="two"/>. The cost of a project has the potential to impact nearly every area of the project, making it one of the most critical jobs for a project manager. A poorly written budget will result in incorrect asset allocation, unrealistic expectations, and, in the worst-case situation, project failure & customer disappointment. Simply put, the budget of a project is essential for its success. Cost estimation is one of the most helpful tools in a project manager's arsenal for creating an adequate budget. There are different methods of cost estimation available such as analogous estimation, parametric estimation, bottom up estimation, expert judgement, and three-point estimation. Each method has its own pros and cons for different projects. The application of each cost estimation method varies from project to project and the nature of the project. <ref name="ref1"/> Throughout the article, different kinds of cost estimating techniques will be examined, to determine which approach is most appropriate for different types of projects, as well as different stages of the project.<ref name="budget"/>.
This article will examine all cost estimating techniques for different kinds of projects to determine which approach is most appropriate for whatever type of project and to determine the optimal method for all types of projects.
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== What is Cost Estimation ==
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A cost estimation is a calculated estimate of the number of resources necessary to accomplish a project or components of a project. These kind of cost estimates are often stated in monetary terms. If the monetary values are not appropriate or irrelevant, alternative units, such as man-days, might be used instead. It comes in a variety of forms. The rough order of magnitude (ROM) and definite estimate are included according to the Project Management Body of Knowledge. Both categories vary in terms of precision, project stages in which they are used, and tools and procedures accessible <ref name="two"/>. Other tools regarding cost estimation includes:
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Note that this article mainly provides a discussion of when to use a selection of cost estimation methods. For further insight into the concrete application of these methods, it is recommended to do further research through the annotated bibliography and references.
  
* Analogous estimation
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== About Cost Estimation ==
* Parametric estimation
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* Bottom up estimation
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* Expert Judgement
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* Three-point estimation
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* Cost of quality
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== Types of Cost Estimates ==
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A cost estimation is a calculated estimate of the number of resources necessary to accomplish a project or components of a project, which is typically done by dividing the projects into more manageable parts. Cost estimates are used to allocate funding to the deliverables and work packages of the project. These kinds of cost estimates are usually stated in monetary terms, to enable comparisons across projects. They can however also be stated in alternative units, such as staff hours or staff days, if the monetary values are not appropriate or relevant. The estimate of project costs is one of the most critical components of project planning and management. The reason for this being that every project relies upon three main components: scope, budget and timeline. Budget is often times one of the most essential things in order to gain approval from upper management or the person funding the project. The budget is typically very difficult to estimate accurately in the beginning of a project, due to lack of information and data, and estimates typically increase in accuracy as the project progresses. Therefore, cost estimation is divided into three different overall categories, according to PMBOK®, which is a set of standard terminology and guidelines for project management <ref name="three"/>:
  
There are two categories of cost estimates, according to PMBOK® <ref name="three"/>
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* Rough order of magnitude (ROM)
* Rough order of magnitude (ROM)   
+
* Budget Estimate  
 
* Definitive estimate  
 
* Definitive estimate  
  
'''[[Rough Order of Magnitude vs Definitive Estimate]]'''
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All three categories vary in terms of precision, project stages in which they are used, and tools and procedures accessible.
  
The rough order of magnitude is a ballpark number established at the very early phases of a project, such as during its inception or even earlier – for example, for the cost-benefit analysis during the project selection process. According to the PMBOK, its accuracy range is -25 percent to +75 percent. The range was formerly stated as +/-50 percent in prior versions of the PMBOK®, such as the 4th edition. This suggests that a project's actual cost is usually within the range of the original ROM estimate. ROM estimations are rather inaccurate because of the wide range of probable results. As a result, throughout the project, they are usually replaced by more precise estimates, such as the definitive estimate.
 
The definitive estimate is the PMBOK's most precise sort of estimate <ref name="three"/>. Its precision varies from -5 to +10 percent. This high degree of precision is generally only possible when the project has been meticulously planned and all required information for a credible estimate of the work is available. As a result, definite estimates are often established later in the project, while a rough order of magnitude is more prevalent in the early phases. Progressive elaboration refers to the process of refining preliminary estimations during the life of a project. When a definite estimate is created, it generally takes the place less accurate ROM estimate <ref name="three"/>.
 
  
== When are Costs Estimated ==
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'''[[Rough Order of Magnitude]]'''
  
Costs are calculated at various stages of the project. According to the PMBOK, the procedure is carried out "periodically during the duration of the project as required.“ <ref name="three"/> Cost estimation begins at the beginning phase, for example, when the project charter or business case is being prepared. A project manager must identify the resources necessary to finish these papers. Due to the lack of specificity in the information available at that stage, the project manager is likely to provide a rough order estimate according to the project tasks and its complexity rather than a precise estimate<ref name="three"/>.
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The ROM is a rough numerical estimate or approximation, established at the very early phases of a project when there is not much information available, such as during its inception or even earlier. In the case of the cost-benefit analysis, this would mean during the project selection process. This gives the decision makers the possibility to asses early on whether to proceed with the project or not. According to PMBOK, the accuracy range of ROM is -25 percent to +75 percent. The range was formerly stated as +/-50 percent in prior versions of the PMBOK, such as the 4th edition. As ROM estimations are rather inaccurate because of the wide range of probable results, they are usually replaced by more precise estimates, such as the definitive estimate. Analogous/ top-down estimation or expert judgment is most commonly used in this situation<ref name="three"/>.
When further information becomes available later in the project, the rough order estimate is replaced with a definite estimate. Following the project's inception phase, the budget will be reassessed using the approaches described in this article throughout the planning phase. Costs are often re-estimated in succeeding stages as relevant new info and details become available or as the scope of the project or timetable changes. One of the most typical reasons for re-estimating costs is when the project's controlling indicators indicate that the initial budget baseline cannot be fulfilled.
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== Importance of Cost Estimation ==
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[[File:BudgetImage.png|400px|thumb|right|Figure 1: Estimate Ranges ]]
  
The estimate of project costs is one of the most critical components of project planning and management. The reason for this is because every project must comply to at least three fundamental constraints: scope, budget and timeline. Due to the fact that cost estimates are by their very nature limited in scope, they are essential for project administration. It is common practice to include the initial tentative cost estimate in the project charter as well as in the project's business case when developing a project charter.
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'''[[Budget Estimate]]'''
Cost estimate is also required for calculating the project budget that is directly relevant to project sponsor approval (s). In reality, "determine budget" process employs a method known as "cost aggregation," which refers to the "estimate cost" phase's outputs directly. Cost estimates are used to allocate funding to the deliverables and work packages of the project, which may be politically charged within a project and among its stakeholders. As a result, budget planning and allocation need stakeholder participation, communication, and, in many circumstances, approval.
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The budget estimate is typically applied in the beginning phase of a project, when the final specifications of the project are still unclear, but there is a good understanding of the primary features and technical requirements. In this kind of situation, it can assist in finding a preliminary cost and budget strategy that is more accurate than that of the ROM, as it has a -10 percent to +25 percent accuracy. This approach also most commonly makes use of the top-down approach, as well as expert judgement<ref name="seven"/>.
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'''[[Definitive Estimate]]'''
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 +
The definitive estimate is the PMBOK's most precise type of estimate. Its precision varies from -5 to +10 percent. This high degree of precision is generally only possible when the project has been meticulously planned and all required information for a credible estimate of the work is available. As a result, definite estimates are often established later in the project. This is referred to as "Progressive elaboration", here addressing the process of refining preliminary estimations during the life of a project. This approach typically includes bottom-op-, parametric-, & three point estimation<ref name="three"/>.
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===Cost Estimation at Different Stages===
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As mentioned, costs are calculated at various stages of the project <ref name="ref1"/> and cost estimation procedures are typically "carried out periodically during the duration of the project" according to PMBOK <ref name="three"/>. At the early stage of a project, when the project charter or business case is being prepared, a project manager must identify the resources necessary to finish. Due to the lack of data available at that stage, the project manager is likely to provide a ROM according to the project tasks and its complexity, rather than a precise estimate <ref name="three"/>. Although this estimate is not very precise, it gives the stakeholders and project manager a good idea about whether or not to proceed with the project.
 +
As further information becomes available later in the project, the ROM is replaced with a more precise budget estimate, and lastly a definite estimate. Of course this may vary from project to project, and some may skip the budget estimate or start directly with the definite estimate.
 +
 
 +
Following the project's inception phase, the budget will be reassessed using the approaches described in this article throughout the different phases. Costs are often re-estimated in succeeding stages, as relevant new info and details become available or as the scope of the project or timetable changes. One of the most typical reasons for re-estimating costs is that the controlling indicators of the project indicate that the initial budget baseline cannot be fulfilled.
  
 
== Project Cost Estimation Techniques ==
 
== Project Cost Estimation Techniques ==
  
Project managers' cost estimations are influenced by a number of factors. Some companies, for example, require that all projects must be funded in line with strict laws, while others may depend on the project manager's expertise. Similarly, many firms may depend on estimates rather than more accurate forecasts in the early stages of project development. Five of the most prevalent cost estimating approaches include<ref name="two"/>:
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The choice of project managers cost estimations techniques is influenced by a number of factors. This includes the size and scope of the project, the availability of data from previous projects, and the stage which the project is in. Some companies may also require that all projects must be funded in line with strict principles, while others may depend on the project manager's expertise. Many firms tend to depend on estimates in the early stages of project development, rather than more accurate forecasts. All these factors directly influence whether a ROM, a budget estimate or a definite estimate is being made. Five of the most commonly applied techniques for completing such estimates are described in the following section <ref name="three"/>:
  
 
'''[[Analogous Estimating]]'''
 
'''[[Analogous Estimating]]'''
  
Analogous estimating is the process of applying previously observed cost figures and variables to new projects or segments of projects. The kind and structure of the referenced project activities must be similar to the present project in order to ensure accuracy. This method determines the predicted resource needs of a present project by analyzing the past data in terms of numbers and parameters. For the present project, the past values are used, and they may be changed to account for variations in project scope or its complexity. Analogous estimating falls under the category of gross value estimation. Comparable estimates are used when a project has access to previous data on similar types of work but lacks the specifics and resources to make more exact estimates <ref name="two"/>.
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Analogous estimating, also referred to as top-down estimating, is the process of applying previously observed cost figures and variables to new projects or segments of projects. The type and structure of the referenced project activities must be similar to the present project in order to ensure accuracy. This method determines the predicted resource needs of a present project by analyzing the past data in terms of numbers and parameters. For the present project, the past values are used, and they may be changed to account for variations in project scope or its complexity. Analogous estimating falls under the category of gross value estimation, as is is often used when completing estimates without having a lot of available information. Comparable estimates are used when a project has access to previous data on similar types of work, but lacks the specifics and resources for more exact estimates <ref name="three"/>.
  
'''[[Parametric Estimating]]'''
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'''[[Bottom Up Estimating]]'''
  
A statistical approach for determining the projected amount of funds or time required to complete a project, an activity, or a portion of a project is known as parametric estimation. An estimate is calculated using a statistical or ostensible relationship between a group of factors and a costing or time value. The size of the present project is then scaled in accordance with the observed association. For instance, in highway construction, the cost and timing for constructing a mile in a previous project might be used to estimate the resources and timetable for the current project. This, however, requires statistical proof of the association as well as a comparison of the two projects' features <ref name="two"/>.
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Bottom-up estimate is a method for calculating the cost of work units at the lowest possible level of detail. The cost estimates for all project components are then combined to arrive at a total project cost estimate. Generally, these estimations are often made after breaking down the project into smaller work packages and even individual tasks. Whereas there is no specific rule on who should do these estimates, it is typical to ask actively involved stakeholders to do so. As a result, the bottom-up estimating technique often produces substantially more accurate results than analogous estimations. Getting these estimates on the lowest level and integrating them, on the other hand, often demands significant resources and may become a political minefield, particularly for big or complicated projects <ref name="three"/>. This level of detail can however be achieved more easily by breaking the project down using a Work Breakdown Structure (WBS) tool[http://wiki.doing-projects.org/index.php/Work_Breakdown_Structure_(WBS)_in_Project_Management].
It is also considered a technique for estimating expenses at various degrees of granularity, the method by which it is implemented differs significantly. Some projects create complicated statistical models and do extensive regression analyses on a variety of variables. They may also create algorithms and provide a large amount of resources to the deployment and testing of such models. This is an approach that can be used for large projects, sometimes known as "large - scale projects," when even minor estimating mistakes can have a huge impact. On the other hand, smaller projects may employ parametric estimate by creating functions or simply using the 'rule of three' if there is proof or a plausible assumption that observable parameters and values correspond. It may also need expert judgement to determine if the predicted regressions are realistic and suitable to the task or the project.
+
  
'''[[Bottom Up Estimating]]'''
+
'''[[Parametric Estimating]]'''
 +
 
 +
Parametric estimating is a statistical approach for determining the cost, budget or duration required to complete a project, an activity, or a portion of a project. It uses a relationship between variables (a unit cost/duration and the number of units) to provide an accurate estimate. The size of the present project is then scaled in accordance with the observed association. For instance, in the construction of a highway, the cost and timing for constructing a mile in a previous project might be used to estimate the resources and timetable for the current project. This, however, requires statistical proof of the association as well as a comparison of the two projects' features <ref name="three"/>.
 +
Like analogous estimating, parametric estimating requires historical data, but it differs from analogous estimating in its algorithmic approach and its use of additional quantitative parameters. At the same time, it accounts for the difference between new and old data. Depending on the level of underlying data built into the model, a high accuracy can be achieved.
  
Bottom-up estimate is a method for calculating the cost of work units at the granular level. The cost estimates for all project components are then combined to arrive at a total project cost estimate. Generally, these estimations are often made after breaking down the project into smaller work packages and even individual tasks. Whereas there is no specific rule on who should do these estimates, it seems to be a good project management practice to ask those stakeholders who are operationally involved in appropriate activities and work packages to do so. As a result, this method of cost estimate often produces substantially more accurate results than top-down calculations. Getting these granular estimates and integrating them, on the other hand, often needs significant resources and may become a political minefield, particularly for big or complicated projects <ref name="two"/>.
 
  
 
'''[[Expert Judgement]]'''
 
'''[[Expert Judgement]]'''
  
Expert judgement may be used in both top-down and bottom-up estimation. Its precision is highly dependent on the quantity and expertise of the professionals participating, the understanding of the activities planned and phases, and the kind of project. If the primary stakeholder and team are familiar with the type of work that will be performed on the project, expert judgement may be used to provide an estimate. This involves a working knowledge of the project's topic and its environment, such as the organization and the industry <ref name="four"/>.
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The precision of this technique is highly dependent on the quantity and expertise of the professionals participating, the understanding of the activities planned and phases, and the type of project. If the primary stakeholder and team are familiar with the type of work that will be performed on the project, expert judgement may be used to provide an estimate. This involves a working knowledge of the project's topic and its environment, such as the organisation and the industry <ref name="four"/>.
Expert judgement may be seen in two ways:
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Estimation of the rough order magnitude at the outset of a project. As there are often not many team member at the beginning, and precise estimates are not available due to inexistence of data, estimates are often times performed by top-down estimation approach  
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Expert judgement can be done in two ways:
• Inquiring the people in charge of the various activities or work packages make an estimate of how much time and resources they think it will take to produce the deliverables outlined in the WBS. It is often times possible to get quite precise findings from this kind of expert judgement.
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* Estimation of the ROM at the outset of a project. As there are often not many team member at the beginning, and precise estimates are not available due to inexistence of data. Estimates are often times performed by a top-down estimation approach.
Expert judgement, in addition to being an estimating approach in and of itself, is also inherent in the other estimation procedures. Instances include determining if earlier work and the present project are comparable or determining whether revisions to parametric estimation are necessary.
+
 
 +
* Dividing the project into smaller tasks through the use of a WBS, and asking the people in charge of different tasks of the project to create estimates for how much time and how many resources they think it will take to produce the deliverables that are outlined. It is often times possible to get quite precise findings from this kind of expert judgement.
 +
 
 +
Expert judgement, in addition to being an estimating approach in and of itself, is also inherent in the other estimation procedures such as the top-down and bottom-up estimation, which are quite similar to the two approaches just described.
  
 
'''[[Three-point Estimation]]'''
 
'''[[Three-point Estimation]]'''
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[[File:TP.png|350px|thumb|right|Figure 2: Probability Density Curves of PERT & Triangular Distributions <ref name="eight"/>]]
  
The three-point estimate approach is a simple but efficient way of evaluating work time and cost. It is used in the process groups "Estimate Activity Duration" and "Estimate Costs." In the strategy, three different estimates are employed, which are usually gathered from experienced specialists:
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The three-point estimate approach is a simple but efficient way of evaluating work, time and cost. Three different estimates are employed here, which are usually gathered from experienced specialists:
* Optimistic estimate
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* Most likely estimate
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* Pessimistic estimate
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The optimistic estimate is the amount of work or time necessary to perform a job under ideal circumstances. It illustrates what is referred to as the "best-case scenario." Pessimism is based on the notion that the opposite is true — it shows the worst-case scenario. Both estimates are designed to be quite reasonable, despite the fact that they represent the most improbable scenarios<ref name="five"/>.
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The third point indicates the most likely situation; it is the work or time estimate that is deemed most fair. Without further consideration, one would be inclined to take the average of the pessimistic and optimistic estimates. That, however, may not be appropriate in many situations. In practice, calculating the most likely estimate accurately, like the other estimating elements, is typically beneficial.
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Three-point estimating produces durations or cost numbers in two ways one is a PERT distribution and the other one is a triangular distribution as shown in figure.
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[[File:PERT1.png|450px|thumb|right|PERT Probability Density Curve and Standard Deviations ]]
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* ''Most likely estimate''
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Realistic expectations about the duration of the different activities, with the resources most likely to be available, as well as potential interruptions etc.
  
'''[[PERT]]'''
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* ''Optimistic estimate''
 +
The amount of work or time necessary to perform a job under ideal circumstances in the most likely estimate. It illustrates what is referred to as the "best-case scenario".
  
 +
* ''Pessimistic estimate''
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The amount of work or time necessary to perform a job under the least ideal circumstances in the most likely estimate. It shows the worst-case scenario.
  
PERT (Program Evaluation and Review Technique) is an advanced project scheduling and management system. PERT is generally utilized as a complement to the Critical Path Method when it comes to activity scheduling. It may, however, be used for single-item and activity estimations. The PERT distribution is based on three-point estimate values. It may be used at all levels of planning, from individual actions to major projects. Finding the correct level of granularity for meaningful estimates, on the other hand, may need considerable critical and conceptual thought. Overweighting the ‘most probable' estimate is part of the PERT technique. It converts a three-point estimate into a bell-shaped curve, allowing probabilities of predicted value ranges to be calculated <ref name="five"/>.
 
  
'''[[Differences between Triangular and PERT Distribution]]'''
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Both the optimistic and pessimistic estimate are designed to be quite reasonable, despite the fact that they represent the most improbable scenarios<ref name="five"/>.
  
The PERT method converts the three-point estimate into a bell-shaped, fairly normally distributed curve, while the triangular distribution consists mostly of the three estimated points. As a consequence, it may be used to calculate the probability of projected time ranges. Figure 1 depicts the differences between the PERT distribution, the triangular distribution, and a presentation of the three-point estimate as if it were a normal distribution. The areas underneath the probability distribution curves indicate the cumulative probabilities of each range of estimates. Typically, these ranges are computed by multiplying the expected value +/- standard deviation by one, two, or three. Figure 2 illustrates this:
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The Three-point estimation, calculates the final durations or cost numbers in two ways:
This triangular distribution has the following formula:
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* '''The Triangular distribution'''
 +
Takes the average of the three different scenarios and is calculated by:
 
::::::::::(O + M + P) / 3
 
::::::::::(O + M + P) / 3
This PERT distribution's formula is as follows:
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 +
* '''The PERT distribution'''
 +
Uses a weighted average, hereby overweighting the ‘most probable' estimate. In this case, the most likely scenario is assigned a four times higher value then the other scenarios, hereby converting the three-point estimate into a bell-shaped curve. The PERT distribution is calculated by:
 
::::::::::(O + 4⋅M + P) / 6
 
::::::::::(O + 4⋅M + P) / 6
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 +
 
where:
 
where:
  
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P = Pessimistic estimate
 
P = Pessimistic estimate
  
== Comparison of Estimation Techniques ==
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The different distributions are illustrated in figure 2
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== Limitations ==
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Although the project cost estimation technique provide a good way of calculating the overall cost of the project, the different techniques have some limitations as well. Applications of each techniques varies from project to project and their complexities. In the case of analogous estimation, it is applicable at the very basic level of the project, and it does not provide the most precise estimates. It is more appropriate at initial planning phases of the project rather than in execution phases. On the other hand, Parametric estimation undoubtedly is a more accurate estimation technique, but it consumes more time and resources in the preparation of the model and gathering historical data from past projects to apply to current project for estimation.
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Bottom up estimation also has limitations in certain dimensions, as it does not provide the accurate estimate of the project as its estimate is based on the sum of the activities of the project. It ignores the additional effort required for integrating the activities, while executing the enterprise and complex projects. Furthermore, it can be very time consuming to break down the project into smaller tasks. Expert judgement also has some disadvantages, as experts may be prone to downplaying the fact that a task took longer than anticipated in the past. Besides the fact that it is time consuming, expert judgment can be costly as the company sometimes needs to hire experts as consultants. Lastly, in the three-point estimation, there are not any major limitations, but it can be very time consuming to provide an estimate for both the optimistic, pessimistic and most likely scenario.
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== Comparison of Cost Estimation Techniques ==
  
 
{| class="wikitable"  
 
{| class="wikitable"  
Line 104: Line 119:
 
! style="font-weight:normal;" | <br />
 
! style="font-weight:normal;" | <br />
 
! <br />Analogous Estimating
 
! <br />Analogous Estimating
! <br />Parametric Estimating
 
 
! <br />Bottom-up Estimating
 
! <br />Bottom-up Estimating
 +
! <br />Parametric Estimating
 
! <br />Expert Judgment<br />  <br />
 
! <br />Expert Judgment<br />  <br />
 
! <br />Three-Point Estimating
 
! <br />Three-Point Estimating
 
|-
 
|-
| style="font-weight:bold;" | <br />Input Data
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| style="font-weight:bold;" | <br />Input Data   <br /><ref name="six"/><br />
| <br />Data from the past or the market: <br />the values of previous  comparable<br /> initiatives
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| <br />Data from previous comparable<br /><br />projects
| <br />Data from the past or the market: <br />the  values of previous comparable <br />initiatives
+
| <br />Activities & their scope of labour<br /><br />
| <br />Activities & their scope of work
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| <br />Data from previous comparable<br />projects<br />
 
| <br />The professionals' knowledge <br />and experience
 
| <br />The professionals' knowledge <br />and experience
| <br />Techniques for making estimates
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| <br />Different techniques for estimating <br />costs
 
|-
 
|-
 
| style="font-weight:bold;" | <br />Method
 
| style="font-weight:bold;" | <br />Method
| <br />Historic values for comparable sorts <br />of work  are adopted and adjusted<br /> (top-down)
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| Using data of past similar<br />projects & adapting it<br />
| <br />Using historical data to estimate the<br />present project's cost per parameter unit.
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| Estimating costs at the lowest <br />possible detail, then summing up<br />the different components<br />
| <br />Work breakdown structure (WBS) <br />cost estimations at a granular level <br />and resource needs aggregation <br />for a project or a  portion of a project
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| Using data of past projects the<br />present project's cost per parameter unit.<br />
| <br />Experts provide estimates of the<br />resources required to perform the<br /> job within the scope of the project, <br />either  from the top-down or from <br />the bottom-up.
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| Experts provide estimates of the<br />resources required to perform the<br /> job within the scope of the project, <br />either  from the top-down or from <br />the bottom-up.
| <br />Three-point cost estimates such as <br />optimistic, pessimistic, and most<br />probable are produced using one of <br />the previously outlined approaches <br />and are then converted into a weighted<br /> average value based on the Pert/Beta <br />distribution or triangular.
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| An optimistic, pessimistic, and most<br />probable estimate is produced using one of <br />the four other approaches <br />and are then based on the Pert distribution or triangular <br />converted into a weighted average value .
 
|-
 
|-
 
| style="font-weight:bold;" | <br />Output Type
 
| style="font-weight:bold;" | <br />Output Type
| <br />Estimated costs for each activity<br /> of the project
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| <br />Total cost of the project, as well<br />as a cost of each activity<br />
| <br />Estimated costs for each activity or <br />the project
+
| <br />Total cost of the project, as well<br />as a cost of each activity
| <br />Estimated costs for each activity <br />or the project
+
| <br />Total cost of the project, as well<br />as a cost of each activity<br />
| <br />Several
+
| <br />Varies from each project, but <br />often times several outputs
 
| <br />Improved estimates of costs, <br />aswell as the standard deviation<br /> of those costs
 
| <br />Improved estimates of costs, <br />aswell as the standard deviation<br /> of those costs
 +
|-
 +
| style="font-weight:bold;" | Limitations
 +
| Not very precise estimations<br />Mostly applicable in the beginning<br />of a project
 +
| Does not address for the <br />ressources required for<br />combining each activity
 +
| Requires historical data which is both<br />time consuming to gather and may <br />not always be available
 +
| Expensive & time consuming
 +
| Time consuming to provide an estimate<br />for three scenarios
 
|}
 
|}
  
== Limitations ==
+
== Conclusion ==  
 
+
Where the project cost estimation techniques provide a good way to calculate the overall cost of the project,  these techniques have some limitations as well. Applications of each techniques varies from project to project and their complexities. In case of analogous estimation, it is applicable at the very basic level of the project and it does not provide the precision estimates. It is appropriate at initial planning phases of the project rather than in execution phases. On the other hand, Parametric estimation undoubtedly is an accurate estimation technique, but it consumes more time and resources in preparing the model and gather historical data from the past projects to apply on current project for estimation. It is the riskiest estimation method as inaccuracy of model will affect the project on which it has been applied.
+
Bottom up estimation also has limitations in certain dimensions, as it does not provide the accurate estimate of the project as its estimate based on the sum of the activities of the project. It ignores the additional effort required of integrating the activities, while executing the enterprise and complex projects. On the contrary, expert judgement also has some disadvantages as it is also very expensive and time consuming, as the company needs to hire an expert from the outside in this method. Lastly, in three-point estimation, there is not any major limitation, but it a lot of time to provide the estimate for both optimistic, pessimistic and most likely for each task.
+
Conclusion
+
The methodologies of cost estimation recommended by the PMBOK have been reviewed in this article. It is worth noting that the estimates' degree of information and complexity tends to rise as the project progresses. Rough orders of magnitude (ROM) estimates is the only kind of estimation available at the start phase. Analogous, bottom-up, and parametric estimating methodologies, which may only be accessible later in a project's life cycle, are frequently required for definitive estimates. The methods that produce the most precise cost estimates are generally parametric and bottom-up estimates. They're typically employed when the budget has to be re-evaluated and a fresh estimate included at the end. Estimates that are more exact, such as parametric estimates based on historical statistical correlations of comparable projects, are not accessible in many projects. In such cases, the three-point estimation method is an effective way to analyse and balance subject matter expert estimates or top down estimation technique. The PERT distribution is undoubtedly the most precise method for condensing these worst-case, best-case, and most likely possibilities into a single value. The standard deviation is used to account for both the inherent errors and the possible dispersion of estimates. This might explain why, despite the fact that PERT has been available for decades and could have been used by our grandparents, it is still a popular tool for project estimating and scheduling.
+
 
+
  
 +
Different techniques for cost estimation, as well as when to apply them, have been reviewed in this article. ROM estimates are typically the only kind of estimation available at the start phase of a project, resulting in the analogous estimation and expert judgment most commonly being applied here. As a project progresses, the degree of information and data tend to rise and become more accurate, and the ROM can be replaced by budget estimates, followed by definite estimates. As Bottom-up, parametric and three-point estimation methods generally produce the most precise cost estimates, they are frequently required for definitive estimates. They're typically employed when the budget has to be re-evaluated and a fresh estimate included at the end. Estimates that are more exact, such as parametric estimates based on historical statistical correlations of comparable projects, are not accessible in many projects. In such cases, the three-point estimation method is an effective way to analyse and balance subject matter expert estimates or top down estimation techniques. The PERT distribution is undoubtedly the most precise method for condensing the worst-case, best-case, and most likely possibilities into a single value. This might explain why, despite the fact that PERT has been available for decades and could have been used by our grandparents, it is still a popular tool for project estimating and scheduling. However it is worth noting that all methods have their limitations, and producing an exact cost estimate can be very difficult and sometimes close to impossible.
  
 
== Annotated bibliography ==
 
== Annotated bibliography ==
Line 148: Line 165:
 
'''R. T. Hughes, “Expert judgement as an estimating method,” Information and Software Technology, vol. 38, no. 2, pp. 67–75, Jan. 1996.''' This article gives the reader an important information about expert judgment-based estimates of projects specifically in information technology and software domain
 
'''R. T. Hughes, “Expert judgement as an estimating method,” Information and Software Technology, vol. 38, no. 2, pp. 67–75, Jan. 1996.''' This article gives the reader an important information about expert judgment-based estimates of projects specifically in information technology and software domain
  
'''C. E. Clark, “Letter to the Editor—The PERT Model for the Distribution of an Activity Time,” Operations Research, vol. 10, no. 3, pp. 405–406, Jun. 1962.''' This article guide the reader that how PERT distribution estimate the activity duration and cost accurately and how it is efficient method as compared to other estimation methods.
+
'''C. E. Clark, “Letter to the Editor—The PERT Model for the Distribution of an Activity Time,” Operations Research, vol. 10, no. 3, pp. 405–406, Jun. 1962.''' This article guides the reader how to use the PERT distribution, estimate the activity duration and cost accurately and how it is efficient method as compared to other estimation methods.
  
 
=References=
 
=References=
Line 157: Line 174:
  
 
<ref name="budget">H. Kwon and C. W. Kang, “Improving Project Budget Estimation Accuracy and Precision by Analyzing Reserves for Both Identified and Unidentified Risks,” Project Management Journal, vol. 50, no. 1, pp. 86–100, Feb. 2019</ref>  
 
<ref name="budget">H. Kwon and C. W. Kang, “Improving Project Budget Estimation Accuracy and Precision by Analyzing Reserves for Both Identified and Unidentified Risks,” Project Management Journal, vol. 50, no. 1, pp. 86–100, Feb. 2019</ref>  
 +
 +
<ref name="ref1"> Project Management: "Managing Successful Projects with PRINCE2" 6th Edition (2017) pp. 109–111 </ref>
 +
 +
 +
  
 
<ref name="two">D. A. N. Gregory K. Mislick, Cost Estimation: Methods and Tools</ref>  
 
<ref name="two">D. A. N. Gregory K. Mislick, Cost Estimation: Methods and Tools</ref>  
  
<ref name="three">I. Project Management Institute, A Guide to the PROJECT MANAGEMENTBODY OF KNOWLEDGE. </ref>
+
<ref name="three"> Project Management: A guide to the Project Management Body of Knowledge (PMBOK guide). </ref>
  
 
<ref name="four">R. T. Hughes, “Expert judgement as an estimating method,” Information and Software Technology, vol. 38, no. 2, pp. 67–75, Jan. 1996</ref>  
 
<ref name="four">R. T. Hughes, “Expert judgement as an estimating method,” Information and Software Technology, vol. 38, no. 2, pp. 67–75, Jan. 1996</ref>  
Line 166: Line 188:
 
<ref name="five">C. E. Clark, “Letter to the Editor—The PERT Model for the Distribution of an Activity Time,” Operations Research, vol. 10, no. 3, pp. 405–406, Jun. 1962</ref>  
 
<ref name="five">C. E. Clark, “Letter to the Editor—The PERT Model for the Distribution of an Activity Time,” Operations Research, vol. 10, no. 3, pp. 405–406, Jun. 1962</ref>  
  
 +
<ref name="six"> Project Management - Three-Point Estimating and PERT Distribution (Cost & Time Estimation),” Available online: https://project-management.info/estimating-project-cost/#2-rough-order-of-magnitude-vs-definitive-estimate</ref>
  
  
 +
<ref name="seven"> Altexsoft - Rough Order of Magnitude: Making Initial Project Estimates with High Uncertainty,” Available online: https://www.altexsoft.com/blog/rough-order-of-magnitude/</ref>
  
 
+
<ref name="eight"> Researchgate - Triangular distribution vs. a PERT distribution,” Available online: https://www.researchgate.net/figure/Illustration-of-the-Triangular-distribution-vs-a-PERT-distribution-Vose-2006_fig6_224061457</ref>
  
  
 
</references>
 
</references>

Latest revision as of 22:20, 22 March 2022

Contents

[edit] Abstract

Prior to the start of a project, cost estimation is critical. It is an important phase in project management, since it is used to calculate and manage the project budget, which is often the most important parameter within the standard success criteria of cost, schedule, and performance targets. The cost of a project is usually estimated at the start of the project or sometimes even before. Afterwards, the cost is re-estimated on a regular basis to account for new information, scope changes, and the timetable of the project [1]. The cost of a project has the potential to impact nearly every area of the project, making it one of the most critical jobs for a project manager. A poorly written budget will result in incorrect asset allocation, unrealistic expectations, and, in the worst-case situation, project failure & customer disappointment. Simply put, the budget of a project is essential for its success. Cost estimation is one of the most helpful tools in a project manager's arsenal for creating an adequate budget. There are different methods of cost estimation available such as analogous estimation, parametric estimation, bottom up estimation, expert judgement, and three-point estimation. Each method has its own pros and cons for different projects. The application of each cost estimation method varies from project to project and the nature of the project. [2] Throughout the article, different kinds of cost estimating techniques will be examined, to determine which approach is most appropriate for different types of projects, as well as different stages of the project.[3].

Note that this article mainly provides a discussion of when to use a selection of cost estimation methods. For further insight into the concrete application of these methods, it is recommended to do further research through the annotated bibliography and references.

[edit] About Cost Estimation

A cost estimation is a calculated estimate of the number of resources necessary to accomplish a project or components of a project, which is typically done by dividing the projects into more manageable parts. Cost estimates are used to allocate funding to the deliverables and work packages of the project. These kinds of cost estimates are usually stated in monetary terms, to enable comparisons across projects. They can however also be stated in alternative units, such as staff hours or staff days, if the monetary values are not appropriate or relevant. The estimate of project costs is one of the most critical components of project planning and management. The reason for this being that every project relies upon three main components: scope, budget and timeline. Budget is often times one of the most essential things in order to gain approval from upper management or the person funding the project. The budget is typically very difficult to estimate accurately in the beginning of a project, due to lack of information and data, and estimates typically increase in accuracy as the project progresses. Therefore, cost estimation is divided into three different overall categories, according to PMBOK®, which is a set of standard terminology and guidelines for project management [4]:

  • Rough order of magnitude (ROM)
  • Budget Estimate
  • Definitive estimate

All three categories vary in terms of precision, project stages in which they are used, and tools and procedures accessible.


Rough Order of Magnitude

The ROM is a rough numerical estimate or approximation, established at the very early phases of a project when there is not much information available, such as during its inception or even earlier. In the case of the cost-benefit analysis, this would mean during the project selection process. This gives the decision makers the possibility to asses early on whether to proceed with the project or not. According to PMBOK, the accuracy range of ROM is -25 percent to +75 percent. The range was formerly stated as +/-50 percent in prior versions of the PMBOK, such as the 4th edition. As ROM estimations are rather inaccurate because of the wide range of probable results, they are usually replaced by more precise estimates, such as the definitive estimate. Analogous/ top-down estimation or expert judgment is most commonly used in this situation[4].

Figure 1: Estimate Ranges

Budget Estimate

The budget estimate is typically applied in the beginning phase of a project, when the final specifications of the project are still unclear, but there is a good understanding of the primary features and technical requirements. In this kind of situation, it can assist in finding a preliminary cost and budget strategy that is more accurate than that of the ROM, as it has a -10 percent to +25 percent accuracy. This approach also most commonly makes use of the top-down approach, as well as expert judgement[5].

Definitive Estimate

The definitive estimate is the PMBOK's most precise type of estimate. Its precision varies from -5 to +10 percent. This high degree of precision is generally only possible when the project has been meticulously planned and all required information for a credible estimate of the work is available. As a result, definite estimates are often established later in the project. This is referred to as "Progressive elaboration", here addressing the process of refining preliminary estimations during the life of a project. This approach typically includes bottom-op-, parametric-, & three point estimation[4].

[edit] Cost Estimation at Different Stages

As mentioned, costs are calculated at various stages of the project [2] and cost estimation procedures are typically "carried out periodically during the duration of the project" according to PMBOK [4]. At the early stage of a project, when the project charter or business case is being prepared, a project manager must identify the resources necessary to finish. Due to the lack of data available at that stage, the project manager is likely to provide a ROM according to the project tasks and its complexity, rather than a precise estimate [4]. Although this estimate is not very precise, it gives the stakeholders and project manager a good idea about whether or not to proceed with the project. As further information becomes available later in the project, the ROM is replaced with a more precise budget estimate, and lastly a definite estimate. Of course this may vary from project to project, and some may skip the budget estimate or start directly with the definite estimate.

Following the project's inception phase, the budget will be reassessed using the approaches described in this article throughout the different phases. Costs are often re-estimated in succeeding stages, as relevant new info and details become available or as the scope of the project or timetable changes. One of the most typical reasons for re-estimating costs is that the controlling indicators of the project indicate that the initial budget baseline cannot be fulfilled.

[edit] Project Cost Estimation Techniques

The choice of project managers cost estimations techniques is influenced by a number of factors. This includes the size and scope of the project, the availability of data from previous projects, and the stage which the project is in. Some companies may also require that all projects must be funded in line with strict principles, while others may depend on the project manager's expertise. Many firms tend to depend on estimates in the early stages of project development, rather than more accurate forecasts. All these factors directly influence whether a ROM, a budget estimate or a definite estimate is being made. Five of the most commonly applied techniques for completing such estimates are described in the following section [4]:

Analogous Estimating

Analogous estimating, also referred to as top-down estimating, is the process of applying previously observed cost figures and variables to new projects or segments of projects. The type and structure of the referenced project activities must be similar to the present project in order to ensure accuracy. This method determines the predicted resource needs of a present project by analyzing the past data in terms of numbers and parameters. For the present project, the past values are used, and they may be changed to account for variations in project scope or its complexity. Analogous estimating falls under the category of gross value estimation, as is is often used when completing estimates without having a lot of available information. Comparable estimates are used when a project has access to previous data on similar types of work, but lacks the specifics and resources for more exact estimates [4].

Bottom Up Estimating

Bottom-up estimate is a method for calculating the cost of work units at the lowest possible level of detail. The cost estimates for all project components are then combined to arrive at a total project cost estimate. Generally, these estimations are often made after breaking down the project into smaller work packages and even individual tasks. Whereas there is no specific rule on who should do these estimates, it is typical to ask actively involved stakeholders to do so. As a result, the bottom-up estimating technique often produces substantially more accurate results than analogous estimations. Getting these estimates on the lowest level and integrating them, on the other hand, often demands significant resources and may become a political minefield, particularly for big or complicated projects [4]. This level of detail can however be achieved more easily by breaking the project down using a Work Breakdown Structure (WBS) tool[1].

Parametric Estimating

Parametric estimating is a statistical approach for determining the cost, budget or duration required to complete a project, an activity, or a portion of a project. It uses a relationship between variables (a unit cost/duration and the number of units) to provide an accurate estimate. The size of the present project is then scaled in accordance with the observed association. For instance, in the construction of a highway, the cost and timing for constructing a mile in a previous project might be used to estimate the resources and timetable for the current project. This, however, requires statistical proof of the association as well as a comparison of the two projects' features [4]. Like analogous estimating, parametric estimating requires historical data, but it differs from analogous estimating in its algorithmic approach and its use of additional quantitative parameters. At the same time, it accounts for the difference between new and old data. Depending on the level of underlying data built into the model, a high accuracy can be achieved.


Expert Judgement

The precision of this technique is highly dependent on the quantity and expertise of the professionals participating, the understanding of the activities planned and phases, and the type of project. If the primary stakeholder and team are familiar with the type of work that will be performed on the project, expert judgement may be used to provide an estimate. This involves a working knowledge of the project's topic and its environment, such as the organisation and the industry [6].

Expert judgement can be done in two ways:

  • Estimation of the ROM at the outset of a project. As there are often not many team member at the beginning, and precise estimates are not available due to inexistence of data. Estimates are often times performed by a top-down estimation approach.
  • Dividing the project into smaller tasks through the use of a WBS, and asking the people in charge of different tasks of the project to create estimates for how much time and how many resources they think it will take to produce the deliverables that are outlined. It is often times possible to get quite precise findings from this kind of expert judgement.

Expert judgement, in addition to being an estimating approach in and of itself, is also inherent in the other estimation procedures such as the top-down and bottom-up estimation, which are quite similar to the two approaches just described.

Three-point Estimation

Figure 2: Probability Density Curves of PERT & Triangular Distributions [7]

The three-point estimate approach is a simple but efficient way of evaluating work, time and cost. Three different estimates are employed here, which are usually gathered from experienced specialists:

  • Most likely estimate

Realistic expectations about the duration of the different activities, with the resources most likely to be available, as well as potential interruptions etc.

  • Optimistic estimate

The amount of work or time necessary to perform a job under ideal circumstances in the most likely estimate. It illustrates what is referred to as the "best-case scenario".

  • Pessimistic estimate

The amount of work or time necessary to perform a job under the least ideal circumstances in the most likely estimate. It shows the worst-case scenario.


Both the optimistic and pessimistic estimate are designed to be quite reasonable, despite the fact that they represent the most improbable scenarios[8].

The Three-point estimation, calculates the final durations or cost numbers in two ways:

  • The Triangular distribution

Takes the average of the three different scenarios and is calculated by:

(O + M + P) / 3
  • The PERT distribution

Uses a weighted average, hereby overweighting the ‘most probable' estimate. In this case, the most likely scenario is assigned a four times higher value then the other scenarios, hereby converting the three-point estimate into a bell-shaped curve. The PERT distribution is calculated by:

(O + 4⋅M + P) / 6


where:

E = Expected amount of time or cost

O = Optimistic estimate

M = Most likely estimate

P = Pessimistic estimate

The different distributions are illustrated in figure 2

[edit] Limitations

Although the project cost estimation technique provide a good way of calculating the overall cost of the project, the different techniques have some limitations as well. Applications of each techniques varies from project to project and their complexities. In the case of analogous estimation, it is applicable at the very basic level of the project, and it does not provide the most precise estimates. It is more appropriate at initial planning phases of the project rather than in execution phases. On the other hand, Parametric estimation undoubtedly is a more accurate estimation technique, but it consumes more time and resources in the preparation of the model and gathering historical data from past projects to apply to current project for estimation.

Bottom up estimation also has limitations in certain dimensions, as it does not provide the accurate estimate of the project as its estimate is based on the sum of the activities of the project. It ignores the additional effort required for integrating the activities, while executing the enterprise and complex projects. Furthermore, it can be very time consuming to break down the project into smaller tasks. Expert judgement also has some disadvantages, as experts may be prone to downplaying the fact that a task took longer than anticipated in the past. Besides the fact that it is time consuming, expert judgment can be costly as the company sometimes needs to hire experts as consultants. Lastly, in the three-point estimation, there are not any major limitations, but it can be very time consuming to provide an estimate for both the optimistic, pessimistic and most likely scenario.

[edit] Comparison of Cost Estimation Techniques



Analogous Estimating

Bottom-up Estimating

Parametric Estimating

Expert Judgment


Three-Point Estimating

Input Data
[9]

Data from previous comparable

projects

Activities & their scope of labour


Data from previous comparable
projects

The professionals' knowledge
and experience

Different techniques for estimating
costs

Method
Using data of past similar
projects & adapting it
Estimating costs at the lowest
possible detail, then summing up
the different components
Using data of past projects the
present project's cost per parameter unit.
Experts provide estimates of the
resources required to perform the
job within the scope of the project,
either from the top-down or from
the bottom-up.
An optimistic, pessimistic, and most
probable estimate is produced using one of
the four other approaches
and are then based on the Pert distribution or triangular
converted into a weighted average value .

Output Type

Total cost of the project, as well
as a cost of each activity

Total cost of the project, as well
as a cost of each activity

Total cost of the project, as well
as a cost of each activity

Varies from each project, but
often times several outputs

Improved estimates of costs,
aswell as the standard deviation
of those costs
Limitations Not very precise estimations
Mostly applicable in the beginning
of a project
Does not address for the
ressources required for
combining each activity
Requires historical data which is both
time consuming to gather and may
not always be available
Expensive & time consuming Time consuming to provide an estimate
for three scenarios

[edit] Conclusion

Different techniques for cost estimation, as well as when to apply them, have been reviewed in this article. ROM estimates are typically the only kind of estimation available at the start phase of a project, resulting in the analogous estimation and expert judgment most commonly being applied here. As a project progresses, the degree of information and data tend to rise and become more accurate, and the ROM can be replaced by budget estimates, followed by definite estimates. As Bottom-up, parametric and three-point estimation methods generally produce the most precise cost estimates, they are frequently required for definitive estimates. They're typically employed when the budget has to be re-evaluated and a fresh estimate included at the end. Estimates that are more exact, such as parametric estimates based on historical statistical correlations of comparable projects, are not accessible in many projects. In such cases, the three-point estimation method is an effective way to analyse and balance subject matter expert estimates or top down estimation techniques. The PERT distribution is undoubtedly the most precise method for condensing the worst-case, best-case, and most likely possibilities into a single value. This might explain why, despite the fact that PERT has been available for decades and could have been used by our grandparents, it is still a popular tool for project estimating and scheduling. However it is worth noting that all methods have their limitations, and producing an exact cost estimate can be very difficult and sometimes close to impossible.

[edit] Annotated bibliography

H. Kwon and C. W. Kang, “Improving Project Budget Estimation Accuracy and Precision by Analyzing Reserves for Both Identified and Unidentified Risks,” Project Management Journal, vol. 50, no. 1, pp. 86–100. This article provides the reader important information about how to estimate the budget accurately, when is the best time to calculate the budget and which cost estimation method is the optimal method for all types of project.

D. A. N. Gregory K. Mislick, Cost Estimation: Methods and Tools. This E-book provides the overview of all cost estimation methods and tools to estimate the cost for all types of project. It has detailed description of estimated tools which can estimate cost of enterprise projects with maximum precision.

R. T. Hughes, “Expert judgement as an estimating method,” Information and Software Technology, vol. 38, no. 2, pp. 67–75, Jan. 1996. This article gives the reader an important information about expert judgment-based estimates of projects specifically in information technology and software domain

C. E. Clark, “Letter to the Editor—The PERT Model for the Distribution of an Activity Time,” Operations Research, vol. 10, no. 3, pp. 405–406, Jun. 1962. This article guides the reader how to use the PERT distribution, estimate the activity duration and cost accurately and how it is efficient method as compared to other estimation methods.

[edit] References

  1. D. A. N. Gregory K. Mislick, Cost Estimation: Methods and Tools
  2. 2.0 2.1 Project Management: "Managing Successful Projects with PRINCE2" 6th Edition (2017) pp. 109–111
  3. H. Kwon and C. W. Kang, “Improving Project Budget Estimation Accuracy and Precision by Analyzing Reserves for Both Identified and Unidentified Risks,” Project Management Journal, vol. 50, no. 1, pp. 86–100, Feb. 2019
  4. 4.0 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 Project Management: A guide to the Project Management Body of Knowledge (PMBOK guide).
  5. Altexsoft - Rough Order of Magnitude: Making Initial Project Estimates with High Uncertainty,” Available online: https://www.altexsoft.com/blog/rough-order-of-magnitude/
  6. R. T. Hughes, “Expert judgement as an estimating method,” Information and Software Technology, vol. 38, no. 2, pp. 67–75, Jan. 1996
  7. Researchgate - Triangular distribution vs. a PERT distribution,” Available online: https://www.researchgate.net/figure/Illustration-of-the-Triangular-distribution-vs-a-PERT-distribution-Vose-2006_fig6_224061457
  8. C. E. Clark, “Letter to the Editor—The PERT Model for the Distribution of an Activity Time,” Operations Research, vol. 10, no. 3, pp. 405–406, Jun. 1962
  9. Project Management - Three-Point Estimating and PERT Distribution (Cost & Time Estimation),” Available online: https://project-management.info/estimating-project-cost/#2-rough-order-of-magnitude-vs-definitive-estimate
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