Thinking in Project Management Terms – Basic Methods and Calculations

By Bob Hugg


In Unit 1 Project Management (PM) was discussed as a management philosophy. Within that philosophy there are many methods and techniques; most are industry specific and are designed to provide performance metrics meaningful to that industry. This unit will discus the three chief methods in wide use with the most focus paid to the most commonly used technique, PERT/CPM. PERT and CPM, though separate techniques, are commonly used in tandem because together they provide a stronger tool and will be discussed accordingly in this chapter.

 

A central weakness of both PERT and CPM is the inability to deal with resource dependencies As discussed in chapter 1, resource dependencies are those that concern the availability of resources whether they are human, mechanical or fiscal (PERT/CPM considers only causal dependencies, the completion of a prior task). PERT/CPM also assumes that additional resources can be shifted to a project as required. Because, in the real world, all projects have finite resources to draw on the estimates and expectations are frequently skewed. Because of this weakness, a significant portion of the PM community believes that PERT/CPM creates unrealistic expectations, at best. As a result, management of projects using only PERT/CPM can be difficult and frustrating for worker, Project Managers and stakeholders alike.

 

A newly emerging (within the last 10 years) methodology is Critical Chain Project Management (CCPM), also referred to as Theory of Constraints. In essence, CCPM focuses on managing constraints, the relationship between tasks within a project and resources within project. By actively managing these “hotspots” it is believed that CCPM decreases project conflict and tension and provides a more balanced expectation. Though an interesting theory, CCPM is largely unproven and appears to be most applicable in projects concerning highly dynamic tasks that can be grouped in modules. Module structure groups tasks where the completion of a module delivers some degree of function that can be used regardless of the status of the remainder of the project. An example would be software development, where a subroutine that is common to many applications can be completed and useful without the entire project is completed. Because the relationship between modules is not as critical, the modules themselves can be re-planned and re-scheduled as necessary, adding a degree of efficiency and decreasing conflict within a project or between projects. CCPM also focuses on overall project progress instead of individual task progress. A perceived strength of CCPM is that it is based on an absence of multi-tasking; a single resource is only assigned to a single task/project. A relatively humanistic approach, CCPM calculations also account for the inconsistent nature of human performance (good days, bad days, sick time, training needed, etc). CCPM estimates are much broader (50% probability, 90% probability, etc) and deal exclusively with a single “normal” completion date of the project as a whole. As such, it is believed that by identifying and grouping tasks and limiting constraints the project becomes more manageable while providing incremental value. Critics of CCPM argue that its assumptions (absence of multitasking, tasks may be grouped into semi-independent yet value-filled groups) create unrealistic expectations. In any event, CCPM seems applicable only in those industries where incremental progress can deliver incremental value or function. Clearly, only completing one wing of an airplane, 2 walls of a house or 1/3 of a city-wide traffic risk assessment would provide little value, so CCPM has found little acceptance outside of very specific hi-tech business areas.

 

The second method in use is a variation of PERT called Earned Value Method, introduced by the Department of Defense in the mid 60s. In the business world this method is synonymous with ROI (Return On Investment).Simply put, it examines the relationship between the cost of doing something and the value received by doing it. Earned value does not concentrate on probability of completion at a specific time, nor does it deal with a specific time or range of times, though a by –product of the analysis is a constantly moving completion projection. It tracks tasks and the project as a whole in terms of money by analysis that answers 3 specific questions:

 

1)      How does the cost of work performed compare to the value of the work performed?

2)      What is the value (in dollars) of work performed so far?

3)      How does the amount of money spent so far on a project compare to what should have been spent?

Using answers to those questions, Earned Value Method generates a variety of productivity indices that can be used to forecast a project completion date. Because Earned Method focuses on work performance in terms of cost and value, it is used extensively throughout the Department of Defense in contracts administration and in industries where significant amounts of work are performed either under contract or through contractors. It is not commonly used in Social and Behavioral sciences or technical production (software development, healthcare, etc) because, in those disciplines, the tangible value of the process and result is much more difficult to identify. Earned Value Method employs many fundamentals of WBS and PERT and is commonly found as an analysis tool in most mainstream PM software packages, including MS Project.

 

By far, the most common method used is PERT/CPM. The remainder of this unit will focus on introducing basic methods and calculations in use. As discussed in Unit 1, PERT is based on a beta distribution that is useful in real-world planning because it accounts for a degree of randomness (that all humans bring to the table). Based on its theoretical model, PERT delivers a task or project completion estimate based on pessimistic, optimistic and most likely estimates provided by the user. PERT also provides a probability of completion on any date selected by the user. PERT calculations are simple and straightforward, but tend to get lengthy when many tasks are used. Before the task calculations can be made, however, 2 steps must be taken in any project planning:

 

1)      Define the goal of the project and the tasks required to complete it

2)      Place tasks in a logical order and determine the critical path (it is helpful to diagram the tasks)

a.       The critical path is the longest time path through the network of tasks

 

When these steps are complete, generate a set of duration estimates for each task; each set should contain a pessimistic, most likely and optimistic estimate. To keep the estimates straight, it is useful to label pessimistic estimates as TP, optimistic estimates as TO and most likely estimates as TL (any labeling system can be used, but these are fairly intuitive).For each task, calculate the PERT derived expected duration (TE) based on a formula, (TP + 4 TL + TO) / 6 = TE

 

1)      Read this formula as the sum of pessimistic plus 4 times likely plus optimistic divided by 6 equals the expected duration

2)      Compete this calculation for all tasks; making sure to group tasks on the critical path separately

a.       The critical path is the longest time path through the network of tasks

b.      The sum of duration of tasks on the critical path will determine the project duration

 

A second set of calculations are necessary to determine information that will be useful later in the process. These calculations will yield the Standard Deviation (SD) and Variance (V) for each task duration. The SD is the average deviation form the estimated time; as a general rule, the higher the SD is the greater amount of uncertainty exists. The V reflects the spread of a value over a normal distribution. The SD and V will be useful in determining the probability of the project meeting a desired completion date. The formulae for calculating SD and V are:

 

1)      SD=(TP-T0)/6 {read as (pessimistic-optimistic)/6}

2)      V=SD2 (Standard Deviation squared)

3)      Compete this calculation for all tasks; making sure to group tasks on the critical path separately

c.       The critical path is the longest time path through the network of tasks

d.      The sum of duration of tasks on the critical path will determine the project duration

 

Since most projects involve several tasks, it is helpful to construct a table to stay organized. A table might look like:

 

CRITICAL PATH TASKS (Longest Duration)

TASK

TO

TL

TP

TE

 

 

 

 

 

SD

V

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TOTAL

 

 

 

 

 

 

 

 

 

 

 

OTHER PROJECT TASKS

TASK

TO

TL

TP

TE

 

 

 

 

 

SD

V

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

TOTAL

 

 

 

 

 

 

 

 

 

 

 

Table 1, Sample table of estimates

 

Consider a sample project, planting flowers and trees. This project could involve 8 tasks; when diagramed it would look like:

 

 

Figure 1, PERT Diagram for sample project

 

For this sample project a table would be helpful in getting organized, and would yield more usable information than the PERT diagram.. This sample project, with 8 tasks, complete with optimistic, pessimistic and likely estimates could then populate the table. When PERT expected durations and SD and V are added using the formulae, the table would look like:

 

CRITICAL PATH TASKS (Longest Duration)

TASK

TO

TL

TP

TE

 

 

 

 

 

SD

V

1

1

3

5

3

 

 

 

 

 

.67

.44

2

2

4

7

4.17

 

 

 

 

 

.83

.69

5

1

3

6

3.17

 

 

 

 

 

.83

.69

6

1

3

5

3

 

 

 

 

 

.67

.44

8

1

2

4

2.17

 

 

 

 

 

.5

.25

TOTAL

7

15

28

15.51

 

 

 

 

 

3.5

2.51

OTHER PROJECT TASKS

TASK

TO

TL

TP

TE

 

 

 

 

 

SD

V

3

.5

1

3

1.25

 

 

 

 

 

.42

.17

4

.5

1

3

1.25

 

 

 

 

 

.42

.17

7

.5

1

3

1.25

 

 

 

 

 

.42

.17

TOTAL

1.5

3

9

3.75

 

 

 

 

 

1.26

.51

Table 2, Sample Project populating a table of estimates

 

This table now provides a wealth of information. It contains a list of required tasks and separates those tasks in critical path and non-critical path tasks. It also lists the best and worst case estimates and the expected duration for each task and the project as a whole (the sum of the expected critical path tasks). This table also lists the Standard Deviations and Variances for each task and the project as a whole, a valuable (but not intuitive) indicator of the probability of project completion by a desired date. A manual method to calculate the probability of meeting a desired date is somewhat more complicated than the other formulae used so far, but would look like:

 

1)      Denote the sum of all expected durations on the critical path as S

2)      Denote the sum of all variances on the critical path as V

3)      Select a desired completion time, denote this as D

4)      COMPUTE: (D-S)/square root (V) = Z ( Read as the result of D minus S divided by the square root of V equals Z)

5)      Enter a standard normal table to find a probability that corresponds with Z or go online to:

a.       http://math.uc.edu/statistics/statbook/tables.html) to enter a z number - the application will retrieve the probability from the very lengthy table

 

For our sample project, figure a probability based on a desired time, 15 days: ((D-S)/sqrt {V} =Z)

a.       (15-15.51)/square root(2.51) = (15-15.51)/1.59=-.321 (Z) (Rounded)

b.      A corresponding probability is 37.7% (Rounded)

c.       In other word, there is a 37.7% probability that the project will be completed within 15 days of the start date

 

It is also helpful to determine the earliest and latest dates a task can start to highlight areas that may be improved upon. For each task, determine the latest allowable time for moving to the next task. Think of these tasks as flexible tasks that can be started earlier or later in the process with no effect on the project duration. The difference between latest time and expected time is called slack time; Tasks with zero slack time are on the critical path. For our sample project, these dates would look like:

 

CRITICAL PATH TASKS (Longest Duration)

TASK

TO

TL

TP

TE

ES

EF

LS

LF

Slack

SD

V

1

1

3

5

3

0

3

0

3

0

.67

.44

2

2

4

7

4.17

3

7.17

3

7.17

0

.83

.69

5

1

3

6

3.17

7

10.17

7

10.17

0

.83

.69

6

1

3

5

3

10

13

10

13

0

.67

.44

8

1

2

4

2.17

13

15.17

13

15.17

0

.5

.25

TOTAL

7

15

28

15.51

 

 

 

 

 

3.5

2.51

OTHER PROJECT TASKS

TASK

TO

TL

TP

TE

ES

EF

LS

LF

Slack

SD

V

3

.5

1

3

1.25

0

1.25

3

4.25

3

.42

.17

4

.5

1

3

1.25

0

1.25

3

4.25

3

.42

.17

7

.5

1

3

1.25

1.25

2.50

4.25

5.50

3

.42

.17

TOTAL

1.5

3

9

3.75

 

 

 

 

 

1.26

.51

ES=Earliest Start EF= Earliest Finish LS=Latest Start LF=Latest Finish

Table 3, Sample Project populating a table of estimates with start dates

 

The table is now complete and is a treasure trove of project information, but has proven labor intensive due to the number of manual calculations (imagine a project with dozens or hundreds of tasks!). The same results can be obtained in much less time with much less effort using MS Excel.

Open a new workbook in Excel and structure a spreadsheet to resemble the table for the sample project. This spreadsheet can become a template for future project calculations and, in effect becomes a PERT calculator. It may look like:

 

Figure 2, PERT Analysis Calculator layout

 

Notice how the spreadsheet resembles the table used so far, with a notable addition. This calculator will include a function to calculate the probability of completing a project on a desired date. Use the dame formulae discussed so far to write equations for each cell address. To start, begin writing equations for the cell addresses for calculating the PERT expected duration.

 

Figure 3, PERT Analysis Calculator layout – PERT Expected duration equations

 

1)      For each task cell: (Optimistic + 4x Typical + Pessimistic)/6

2)      Adjust cell address for each task

Next, write equations to calculate the Variances for each task:

 

Figure 4, PERT Analysis Calculator layout – PERT Variance equations

 

1)      For each task cell: ((Pessimistic-Optimistic)/6)2

2)      Adjust cell address for each task

 

Next, write equations to calculate the Standard Deviations for each task:

 

Figure 5, PERT Analysis Calculator layout – PERT Standard Deviation equations

 

1)      For each task cell: sqrt (V) ( the square root of V for that task)

2)      Adjust cell address for each task

 

Next, write an equation to sum the PERT Expected Date for the project:

 

Figure 6, PERT Analysis Calculator layout – Summing Pert Expected Dates

 

Next, write an equation to sum the Variances for the project:

 

Figure 7, PERT Analysis Calculator layout – Summing Variances

 

Finally, write the equation to calculate the Probability of Completion for a desired project date:

 

Figure 8, PERT Analysis Calculator layout – Probability of Completion

 

Excel uses a formula designed to compute the probability of placement of a combination of elements in a normal distribution that is very accurate for use in real-world situations. The equation is NORMDIST(x, mean, standard_dev, cumulative) in which:

 

1)      X   is the value for which you want the distribution (desired date)

2)      Mean  is the arithmetic mean of the distribution (summed PERT expected durations)

3)      Standard_dev   is the standard deviation of the distribution (square root of the summed variances)

4)      Cumulative is a logical value that determines the form of the function. If cumulative is TRUE, NORMDIST returns the cumulative distribution function (probability of completion on the date entered)

Some things to consider when setting up this equation:

 

1)      Be sure to adjust formulae as necessary when adding additional tasks

a.       If a error message shows up check cell addresses in the formulae first – formulae must reflect intent

2)      This set of formulae mirrors the manual calculations but takes less time for the user

3)      Because PERT is a probabilistic approach, these formulae can deliver a 100% probability – but no plan is perfect – these are always estimates

4)      Never feel there is a 100% probability of a project completing on the estimated date

 

Armed with a substantial tool to compute PERT expectations and probabilities, all that remains is to complete a few simple CPM calculations. As discussed in Unit 1, and earlier in this unit, CPM deals with a single expected date and anticipates that a project may be crashed. Crashing a project is reducing the project to its shortest duration by adding resources. It is important to note that the effort required to complete a task or project remains the same, only the duration may be shortened. By its nature, crashing a project is disruptive (it pulls resources from other tasks) and increases the project cost. Careful consideration must be made whether the effects of crashing a project are worth completing a project earlier. The answer will vary from project to project and situation to situation. Again, using a table to get organized, a very simple example might look like:

 

The basic steps in completing a CPM analysis are:

1)      Develop time and cost data ("normal" and "crashed") for all tasks

2)      Develop cost-per-week for crashing (crashed costs divided by time saved)

3)      Develop project network (PERT)

4)      Crash the activity on the critical path with the lowest cost-for-crashing

5)      Recalculate the project network (the critical path might change!)

6)      Repeat steps 4 & 5 until all the paths have been crashed.

7)      Ease up on all non-critical paths, just to the point that all paths are critical

Activity

Begin

End

Time

(Crashed)

Time (Normal)

Cost (Crashed)

Cost (Normal)

Time

Saved

Cost Increase

Cost / Week

Foundation

1

2

1

2

4000

3000

1

1000

1000

Frame

2

3

1

4

8000

4000

3

4000

1333

cost-per-week for crashing = crashed costs divided by time saved

Table 4, Sample Project populating a table of CPM estimates of time and costs

 

In this example the time saved (4 weeks) is substantial but the cost increase is also substantial ($5,000). At an aggregate increase of $2,333 per week for crashing this project, this would be a course of action not lightly taken. Sometimes crashing a project is unavoidable but a serious consideration of the tangible and intangible costs must be undertaken.

 

When used together, PERT and CPM can provide:

 

1)      A range of time estimates (PERT)

2)      Likely time estimates (PERT and CPM)

3)      Cost estimates (CPM)

4)      Time and costs if crashed (CPM)

5)      Probabilities of completion on time for a range of times (PERT)

6)      A clear path of tasks that are critical to the project (PERT and CPM)

7)      A central focus for solid communications on project issues (PERT and CPM)

 

All plans are estimates and should be viewed as such. When used together, PERT and CPM provide a valuable tool for organizing and tracking projects as well as providing a usable “what if” forum. Care must be taken in collecting estimates used in planning – any plan is only as good as the most unrealistic estimate. By using simple but complementary formulae, managers at all levels can get, and keep, a good handle on a project or its tasks.

 

A step-by-step tutorial for using both Microsoft Project and setting up an Excel spreadsheet for estimating the times on a PERT analysis, refer to the powerpoint tutorials

 

Resources Used in This Unit

 

Bonini, Charles, et al, Quantitative Analysis for Management, Columbus: McGraw Hill, 1997.

 

Goldratt, Eli, Dr., The Goal: A Process of Ongoing Improvement, Great Barrington: New River Press, 1996.

 

Mednick, Barry, PERT-CPM on Excel, Fullerton: Cal State, 2000.

 

MS Project, by Microsoft Corporation.

 

MS Excel, by Microsoft Corporation.

 

PM Body of Knowledge (PMBOK), Philadelphia: PMI, 2000.

 

Project Management Institute (PMI) Resource Center

<Project Management Institute Website>.

 

ProjeX, by WAA, Inc .

Systema, Sid, Probabilistic Solutions to Project Scheduling, Ferris State, 1999.

 

US National Performance Survey, The Standish Group, 1998.

 

Verma, Vijay K., Managing the Project Team: The Human Aspects of Project Management, Philadelphia: PMI, 1997.

 

Wiest, Jerome D., and Levy, Ferdinand K., A Management Guide to PERT/CPM, New Delhi: Prentice-Hall of India Private Limited, 1974.

 


609

 

1996 A.J.Filipovitch
Revised 11 March 2005