Contemplate ‘Complexity’ constraint to Mathematical Techniques in Project Estimations


Selection of suitable estimation techniques in a project is crucial. Estimations may occur at different times during the lifecycle of a project, typically most of which is completed during the initiation phase. Most commonly estimations are tied to costs and resources in project management knowledge areas. The emphasis of this publication is on all estimation techniques however the author would like to embrace a variable in those mathematical formulae inclining them to additional constraints, often missed during the estimation process. The author has focused on estimation technique known as ‘Three Point Estimation’, which used weightage approach and redefined the formulae for this technique. This technique is commonly used in conventional IT Project Management methods.

In project management engineering science, every project component is broken down into smaller modules. They may or may not be broken down many levels down as long as it is manageable, measurable, and it organizes the project teams’ scope of work to be delivered. The Project Management Body of Knowledge (PMBOK 6, 2016, PMI Institute USA) explains the work-breakdown structure as “A hierarchical decomposition of the total scope of work to be carried out by the project team to accomplish the project objectives and create the required deliverables”.

The WBS plays an important element for project planning and especially in cost & schedule planning of a project. Every WBS component is scheduled which collectively forms the schedule baseline of a project. Similarly, cost is assigned to all WBS components and aggregated to obtain a cost baseline. Both belong to different knowledge areas however they are very must interrelated to each other. PMI Inc. says cost baseline is the agreed version of the time-phased project budget (PMBOK, 6th Edition, 2017). The key word ‘time-phased project budget’ indicates the relationship between project costs and project schedule. As the cost estimates make up the cost baseline and are directly knotted to the schedule activities process, this empowers a time-phased view of the cost baseline. This explains how project estimation play a key role to obtain important components of project engineering science.

Figure 1: Work breakdown structure representation from Wikipedia

The figure above is from Wikipedia and exhibits an example of a Work Breakdown Structure.


Although project management engineering has multiple processes, procedures & areas, Cost Management & Schedule Management are key wisdom expanses especially in project estimations in project management excellence. PMI Inc. has intensely demarcated all Project Management artefacts architecturally in the PMBOK (Project Management Body of Knowledge) guide (PMBOK, 6th Edition, 2017). As per the PMBOK, project management administration has 49 different Processes across its 5 Project Phases and 10 Knowledge areas. In the above text we covered WBS which is a part of ‘Create WBS’ process (PMBOK, Chapter 5.4, 6th Edition, 2017) in Scope Management knowledge area. The WBS components are estimated in ‘Estimate Activity Duration’ process (PMBOK, Chapter 6.4, 6th Edition, 2017) in terms of time and schedule. Similarly, cost is aligned to each activity in the ‘Estimate Costs’ process (PMBOK, Chapter 7.2, 6th Edition, 2017).


PMI Inc. in the PMBOK guide has listed tools and techniques for cost and activity duration processes. Similar techniques are used in these processes. The figure below represents the tools and techniques defined in the All estimations gathered from both these processes are aggregated to device the project budget.

Figure 2: Tools & Techniques defined by the PMI in PMBOK

The researcher has identified few methods and has used mathematical formulae to introduce a variable which contemplates constraints in estimation process.


As indicated in the image above, similar techniques are used by various industries. Some techniques have defined formulae to deduce the end estimates and some may not have any defined formulae. However, the estimators always use mathematical techniques to come to a number.

Generally, in projects estimations are not done by people doing the work. The researcher identifies this as a significant problem in the setup, although it is so common in companies. Estimations are completed by architects, which is obviously more experienced or skilled. The team delivering the work may not have the same skill sets as that of an architect. Hence, it is important to restructure the formulae so that estimates are calculated based on skillset and experience of people doing the work.


The project management processes use common techniques as listed down in the figure above. Some of these tools and techniques have well defined formulae which is used in the estimation process. The author will lay emphasis on some of these techniques and showcase how these techniques need to introduce a variable that should account for complexity or any other constraint(s) when performing estimations. The variable could be one or many constraint(s) together.

2.3.1 Parametric Estimating

Compared to that of Analogous estimating, parametric estimating technique is relatively more accurate. In this technique the estimations are obtained with historical data. Fundamentally, an estimate is obtained by detecting the unit cost or duration, followed by the number of number of units for the entire activity. Let us consider using a mathematical formulae with an example.

For example, if it took a resource paint a wall in 8 hours in a 10 story apartment last week and this week 3 resources will paint similar apartments. The same can be represented as follows.

ePE = 8, than 3 times ePE = 3 times 8 = 24 hours

In the above example, ePE indicates the estimate. A key point in the example above, 8 hours of work is estimated from historical data from previous projects and from a particular resource. However, the forecasted work will be done with 2 other resources. Possibly the 2 other resources may do not have the same skillset or experience of the first resource. Hence, we need to add another variable that accounts for these buffers. Having said that, we could device the formulae as below.

ePE = 8+x so 3 times would be 3 * 8+x where x tends to a variable constraint

This method will always be subjective and will account for the skillset of the actual resources delivering the work. With this method you are building your estimates based on historical data which is primarily what Parametric estimates are about however we are also accounting for constraints like complexity or skillset of the actual resource which may or may not be the same as that from the historical data.

2.3.2 Three Point Estimating

Three-point estimating is an easy and a very popular technique used. As per PMBOK (6th Edition, PMI Inc) it is used in both cost & schedule estimation techniques as indicated in figure 2 above. This method revolves around 3 expanses as listed below.

· Most Optimistic Estimate (mOE)

· Most Pessimistic Estimate (mPE) and,

· Most Likely Estimate (mLE)

As its name suggest, the optimistic estimates is the amount of time required to perform an activity with no issues and everything going well without impediments. For simplicity sake we could say the best scenario. The pessimistic estimate on the contrary the worst scenario. This range accounts for all the possible impediments playing a hindrance and delaying the work. Both these estimate ranges represent to extreme ranges of the outcomes.

The third estimate range is he most likely estimate which basically represents the most likely case. Often it is misunderstood that most likely estimate range is between the previous 2 ranges. It’s possible the most likely estimate is closer to either of the 2. In case the performing tasks is likely to experience some impediments it could be closer to the pessimistic estimate, but not the worst case.

Three-point estimating have 2 expansions in their formulae called the Triangular Distribution and Beta Distribution. In Triangular distribution technique is used when the historical data is insufficient. In Beta Distribution technique a weightage average is aligned with the most likely estimate. The idea is to increase accuracy on the estimate and is generally considered where stronger historical data is available. Below are how they are represented.

Triangular Distribution Technique 3PE = (mOE + mPE + mLE) / 3

Beta Distribution Technique 3PE = (mOE + mPE + 4 * mLE) / 6

Adding variable representing an unknown constraint is a key here. For both the methods, there are three different estimates. So each of these estimates will be obtained with a resource with a certain skillset or experience level, which may or may not be the same to that of who is doing the work. Hence it is vital we add a variable to not only the result or the most likely case, but to all the 3 estimates. Having said that, the formulae are re-articulated as follows.

Triangular Distribution Technique 3PE = (mOE x + mPE x + mLE x) / 3

Beta Distribution Technique 3PE = (mOE x + mPE x + 4 * mLE x) / 6

The question is how we account for the variable x. Most of these estimations come from previous projects or experiences. Historical data is always used for such practices. Organizations need to define all the possible scenarios of a resource based on their skillset or experience. For e.g., value of variable x could range between unit value 2 to 10 for a resource type with experience level between 5 to 20 years of experience. Similarly, complexity could also be added, or any other, or combination of multiple constraints. This is subjective and can vary case by case, organization by organization, however the important element here is to have a provision in place.


The research study is done on 3 popular estimation techniques widely used in the IT industry. Although there are multiple estimation techniques that could be possibly used, this study is limited to technique used in conventional IT organizations. Further research study could be implemented on the conventional techniques used.

Also, PMI lists many techniques which are not formally defined using formulae however the estimations carried out in practice do need to account for additional constraints or need to have the provision to be considered during estimation processes.

Project estimations can be carried out at different times within the lifecycle of the project. Any changes to the baselines also need to be added to the historical data so that estimations are completed more accurately.


Estimations are never accurate. The idea of performing estimations is to have a rough idea on the effort and cost required to complete a task. Estimations guide you obtain a holistic picture what you need to invest in order to complete the scope of task. Hence, estimations are vital in project management science.

The prime focus is to ensure the estimations relay as much accuracy as possible. With past practices and new lessons learned estimation artefacts need to be revisit and if needed be changed or updated. The idea of doing this exercise is to reduce the cone of uncertainty in estimations. Cone of uncertainty is the level of uncertainty when you forecast any work, especially when you are performing estimation based on the current trend. The same is explained below with an example.

Consider a burn up chart for a project which is due to be completed in 10 days, or being forecasted for 10 days. The value on the y axis indicate story points or any unit representation of the work being completed and the value on the x axis indicate the time / days to complete the task. On day 5, the project team makes an estimate to forecast how we will the remainder scope. There is an optimistic, pessimistic and most likely trend that is recorded. The cone formed between the optimistic & pessimistic trend is the cone of uncertainty.

Figure 3: Burnup Chart Representation

The idea behind adding a variable in these formulae estimations techniques is to narrow down the cone of uncertainty. Estimations can never be accurate however an efficient approach to obtain a project estimation would be, to be as close as possible to the realistic effort, or basically narrow down the uncertainty in estimation however much as possible. Hence, all project estimation needs to account the live and current scenarios for the project, and add a variable that represents the uncertainties of these current scenarios.

With Agile methodologies becoming so popular, more the techniques are being implemented and these techniques do cover constraints on practical scenarios. In Agile project management techniques, estimations are done by the team delivering the task. These practices may add up to historical data, however as per agile principles all estimations need to be done by the team. Hence although historical data could be used to gather an idea, estimations are redone. For example, in Scrum, estimations are done again by the entire scrum team in a Sprint Planning event. These estimations are carried out keeping the live scenarios in mind. It may be due to skillset, experience or any other internal or external factors. This implies all additional situational constraints are taken into consideration before estimations are finalized and baselined for a sprint.


Practice Standards for WORK BREAKDOWN STRUCTURES — [Published by Project Management Institute, Inc., Second Edition, 2006]

Estimation Tools and Techniques — [SimpliLearn — By Manish Sharma, 2017]

PMBOK (Project Management Body of Knowledge) — [PMI, Inc, 6th Edition, 2017]

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