PERT: PROBABILISTIC PROJECT ANALYSIS

 
The PERT acronym stands for Program Evaluation and Review Technique. PERT was developed jointly by representatives of the U.S. Navy, Lockheed, and the consulting firm of Booz, Allen and Hamilton working on the Polaris missile system. Since many of the tasks required in the development of the Polaris had never been done before, the time required to complete the tasks was uncertain. Thus, PERT was designed for projects where the time required to perform each activity is essentially a random variable. PERT focuses on estimating the probability distribution for total project time based on the uncertainties in the durations of individual activities.

The PERT modeling technique involves probability distributions for each activity duration in the project, and hence is an example of a stochastic model. Instead of a single representative time, one takes three different times for each activity, namely
 

aj = optimistic time (shortest time)
 
mj = modal time (most likely time)
 
bj = pessimistic time (longest time)
 
In order to simulate the project duration statistic for a project, one must "fill in" a probability distribution between aj and bj which has a "shape" which matches the given parameters, or more accurately, which matches the PERT mean and variance formulas given below. It is customary to use members of the four parameter BETA distribution family for this purpose, since the beta distribution is defined on a finite interval and can be skewed right or skewed left as well as symmetric. This makes it both more realistic and more flexible than the family of normal distributions, which are symmetric and defined on an infinite interval. For this reason, simulations of project duration are therefore ordinarily done using the beta distribution for durations of individual activities.
 

BETA DISTRIBUTIONS
 

To get an idea of the variety of shapes available within the family of beta distributions, we have plotted three examples below of which the first is skewed right, the second is symmetric, and the third is skewed left. By variation of the mean and variance statistics with repect to the minimum and maximum parameters, one gets a wide variety of shapes which can be expected to fit most practical situations pretty well. In these examples the range has been normalized from zero to one, the mean for the symmetric case is at the midpoint 0.5, whereas for the skewed cases the mean has been placed at 0.33 and 0.67 respectively. The variance is the same in all cases, being determined by the PERT approximation formula as (1/6)2 or 1/36.
 

We will consider the beta distribution formulas in more detail below, but first we need to look at the PERT approximation formulas for mean and variance which will be used as a basis for selecting the appropriate beta distribution parameters.