PERT: Modeling Activity Duration Uncertainty

In the last handout, we developed a network model for a project as shown below.

Analysis showed that the critical path was B-D-F with a length of 46. We recognized this as the critical path because it was the longest path through the network. In practice, activity durations are not known in advance, but instead are estimated. What is the impact on our management of the project?

Two major impacts: (1) We are actually uncertain about the length of the critical path, and (2) we do not really know which path is critical (which path is the longest)!

Uncertain activity times can be handled in two basic ways: (1) we can assume that the critical path will be the path we have identified, and make probabilistic arguments about the length of the known critical path, or (2) we can use simulation to generate the probability distribution associated with the length of the project.

In either case, it is necessary to gather information about the probability distribution of each activity. This can be accomplished in an easy way. We will assume that each activity time can be described by a triangular distribution as shown below.

where

a = the shortest time needed to complete the activity (most optimistic),

b = the longest time needed to complete the activity (most pessimistic),

m = the most common time needed to complete the activity.

Then the mean and variance of t can be estimated by

The computations for our project are shown in the table below.

Activity / a / b / m / Expected Value / Variance
A / 12 / 27 / 21 / 20 / 9.50
B / 25 / 35 / 30 / 30 / 4.17
C / 2 / 7 / 6 / 5 / 1.17
D / 7 / 12 / 11 / 10 / 1.17
E / 2 / 4 / 3 / 3 / 0.17
F / 2 / 8 / 8 / 6 / 2.00

If the critical (longest) path of our project is B-D-F, and if T is the duration of the project, then since we have:

and

This shows explicitly that the project duration is not fixed at 46 days. Instead, T has natural variation induced by the variability in the activity times. This does not tell us what the distribution of T looks like. We only know two pieces of information about that distribution, namely its mean and standard deviation (or variance).

It is interesting feature of nature that the form of the distribution of T is hinted at by the fact that T is a sum. Roughly speaking, when a quantity is derived from a sum and there is not a dominant summand in the sum, then the Central Limit Theorem can apply and we can deduce that T will have an approximate normal distribution! This is a small sum involving only three terms; however, the distribution of each activity time has a well-behaved distribution. In particular, the triangular distributions are unimodal. What does this mean?

In the worksheet PERT Simulation 1 of the file Dist of T assuming critical path.xlsx, we have the results of 1000 simulations of the duration of this project assuming that the critical path is B-D-F. We will consider how to replicate this simulation in a moment, but for our purposes the real issue is the shape of the distribution of T.

Using the Distribution of T

Knowing that T has an approximate normal distribution provides us with a lot of information. In particular, a normal distribution is entirely distinguished by its mean and standard deviation. But we know that

Thus we have specified the entire distribution of T. In the EXCEL worksheet Computing the dist of T, we have used the EXCEL function

norm.dist(x,mean,stdev,cumulative)

to compute the probability of completing the project by a specified day, starting from day 40 through day 54. This gives the decision maker much more information when committing to complete the project by a given day.

Alternatively, we have used the EXCEL function

norm.inv(prob, mean, stdev)

to compute the day of completion given the probability.

Next we remove our assumption about which path is critical by exploring how to simulate the project.