Using PERT/CPM to determine variability in project completion time

Using PERT/CPM to determine variability in project completion time

The Program Evaluation and Review Technique (PERT) and the Critical Path Method (CPM) are widely recognized as valuable tools in the field of project management. PERT was developed in the 1950s to deal with the unprecedented management needs of the Polaris missile project. CPM, on the other hand, was developed primarily for industrial projects involving well-known activity times. The best features of PERT and CPM have been combined into a single technique, known as PERT/CPM, that enables project managers to schedule projects with a good deal of efficiency. The reliability of the technique depends heavily, however, on the accuracy of the estimated activity times.

The PERT/CPM technique

Creating a project network diagram

The first step is to prepare a list of the activities that make up the project and identify the predecessor activities of each project activity. A predecessor activity is one that must be completed before the activity of which it is a predecessor can be started. When all the estimates have been compiled, a project network diagram representing the flow of project activities is created. The following is a typical project network diagram.

Since several activity paths can be identified from start to finish in a typical project, there will be a path that takes the longest time to complete. This path is known as the "critical path". Clearly the completion time of the project is dependent on the time taken to complete the critical path. In the example above the activities that make up the critical path - the critical activities comprising activities A, D, G, and K - are colored in green.

Estimating activity times

Activity time estimation can involve several techniques

The completion time for each activity is estimated. This entails assessing the number of work periods and the period of calendar time likely to be required to complete the activity. Activity time estimation can involve several techniques. One such technique is expert judgment. It is carried out by an individual or group with relevant specialized knowledge or training, and is generally guided by historical information relating to previous projects. A useful variant of expert judgment is analogous estimating (also known as top-down estimating). This technique is most commonly used in projects for which little detailed information is currently available. It bases activity time estimates on the duration of previous similar "analogous" activities. Another estimation technique is simulation. This involves the use of a computerized approximation algorithm, such as Monte Carlo analysis, that estimates project activity times using different sets of initial assumptions.

PERT/CPM with uncertain activity times

Optimistic time, modal time, and pessimistic time

The PERT/CPM technique takes uncertainties into account

Many projects, such as research and development projects, involve uncertain activity dates. The PERT/CPM technique takes these uncertainties into account to determine the probability of completing the project by a certain date. It does this by using three estimates for each activity - the optimistic time (a), the modal - "most probable" - time (m), and the pessimistic time (b). The optimistic time is the minimum activity time if the activity progresses under optimum conditions. The most probable time is an estimate of the duration of the activity under normal conditions. The pessimistic time is the longest activity time envisaged if significant problems are encountered.

Suppose that you are the project manager of a certain project for which you have identified activities A, D, G, and K as the critical activities. You have established estimates of the optimistic time, the modal time, and the pessimistic time for each of these activities as follows:

 

Critical path activity	Optimistic (days)	Modal (days)	Pessimistic (days)
A	5	8	17
D	5	8	23
G	3	6	9
K	5	8	11

 

Let's see how you deal use PERT/CPM to deal with this scenario.

Finding the expected time and the variance

The following expression is used to calculate a weighted average known as the expected time (t) for each activity:

So you calculate the expected time for each critical activity to produce the following results:

 

Critical path activity	Expected time (days)
A	9
D	10
G	6
K	8

 

You can use the expected times to draw up a project network diagram and create a project schedule. By identifying the critical path you can determine the expected time of completion of the entire project - this is the sum of the expected times of the critical activities. In this example the expected time of the project is 33 days.

Variance of critical activities and total variance

The variance (s 2) - the dispersion of activity time values - of each of the critical activity times is given by the following expression:

As the expression clearly indicates, your estimates of the optimistic time and the pessimistic time have a very significant bearing on the variance of any given activity. In this example the variances of the project's critical activities are as follows:
 

Critical path activity	Variance s 2
A	4
D	9
G	1
K	1

 

Variances in critical activity times automatically affect the time of completion of the entire project, either upwards or downwards. The assumption is that the variance of the project completion time equals the sum of the variances of the critical activities. That is:

This assumption is correct if all the activity times are independent of each other, but is decreasingly useful as an approximation as activity times become more mutually dependent. In this example, the total variance of the project is calculated to be 15 days.

The standard deviation (s) of the project completion time is simply the square root of the project completion time variance. So the standard deviation in this example is 3.87.

Probability of meeting a project completion date

Now you can investigate the probability of the project being completed on a particular date. For example, suppose penalties will begin to accrue if it takes you more than 40 days to complete the project. Your critical path indicates a project completion time of 33 days. So you need to determine the probability of the project taking 7 days longer than the time indicated by the critical path. You do this by calculating the z value - the normal probability distribution - of the scenario in which the project takes 40 days. The following expression yields the z value:

Here T is the project completion time you are investigating - 40 days - and T_C is the project completion time indicated by the critical path - 33 days. So the expression becomes

Using a table of areas for the standard normal distribution, this z value yields an area for the standard normal distribution of 0.9649. So your initial estimates predict a 96% probability of your project being completed on or before the 40-day milestone when penalties begin to take effect.

Let's investigate the probability of the project being completed in 30 days. This yields a z-value expression of

The table of areas for the standard normal distribution yields an area of 0.2206. So there is a 22% probability of the project being completed 3 days earlier than the estimated critical path time.

Now let's look at the probability of the project being completed within the range 29-37 days - 4 days before and after the estimated critical path completion date. Our technique produces a probability of 15% that the project will be completed in 29 days or sooner and a probability of 85% that it will be completed in 37 days or sooner. So there is a 70% probability that the project will be completed in the range 29-37 days.