The PERT (Program Evaluation and Review Technique) has been around for some time. ["Application of a technique for Research and Development Program Evaluation," Malcolm, Roseboom, and Fazar, *Operations Research*, Volume 7, Number 5, Sep-Oct, 1959].

There are some who pride themselves on knowing every intricate detail of the source and evolution of PERT and its partner CPM (Critical Path Method). I'm not one of those. But the problems of scheduling and creating a credible schedule are still with us, even after all the historical assessment of who, why, and how we got to where we are.

**The PERT Formula**

First let's look at the PERT formula

T = (a + 4b + c) / 6

- T = expected duration (there is no cost in the PERT model, except when resource loading the schedule)
- a = the optimistic duration of the task
- b = the most likely duration of the task, the Mode, the most recurring value
- c = the pessimistic duration of the task

This is the "classical" PERT formula. This is the one implemented by the scheduling tools. Plug in the three value to the Microsoft Project columns and it will tell you the value of T, the expected time. This formula is in the text book, taught by all the teachers and possibly even used. The formula has been reverse engineered thought. The Standard deviation for the probability distribution function for all the possible value of a, b, and c is:

σ = (c - a)/6

This is where the reverse engineering took place. And the source of the problem of using PERT in any credible manner.

There are two fundamental statistical problems with PERT.

- The estimate of a and c. These are the endpoints of the Beta distribution from which their values are drawn. The probability elicitation literature tells us that is it difficult for a person to accurately estimate the absolute endpoints of a stochastic quantity. A stichastic quantity is a random variable whose underlying probability distributuion changes its shape with time. Stchastic quantities easily represnt realworld processes. Traffic flow, packet flow, bank teller line length, things like that.
- The estimate of b, the most likely time is also a problem. The original b is the Modal value of all the possible values of the task duration. But since the estimators have large amounts of empirical knowledge in their heads, they tend to estimate the median instead of the mode. The median is the middle most value. The Mode is the most recurring value. The difference between using the median and the mode is a significant asymmetric that PERT and the Beta distributions are not designed to handle. Remember the Beta and the classic formula were reverse engineered from the field data. This is called "curve fitting" and is a powerful way to forecast the future - IF the underlying statistical processes stay the same. Since they don't - they are stochastic - this is a problem.

There are some other limitations to PERT:

- Activity times are not stochastically independent
- The critical path comprises fewer activities than reasonable application of the Central Limit Theorem requires to have the modes of the most likely be credible.
- The approximations of activity duration mean and variance can deviate significantly from reasonable and accurate approximations of the underlying activities.
- Since only the mean and variance of the Beta distribution on which PERT is built is being used, there is little explicit attention being paid to the shape of the distribution. Mean and Variance do not define the shape of the Beta distribution.

**Another Major Source of Bias**

- The near critical path activities become critical in every real project. PERT does not make any consideration for these.
- Many parallel activities of roughly the same duration with differing variances produce wide swings in the estimated total duration. This occurs on small projects as well as large.
- PERT makes a very specific estimate of the variance (remember the reverse engineering). The Coon-Donaldson estimator ("Note on William A. Donaldson's 'The Estimation of the Mean and Variance of a PERT Activity Time,'" H. Coon,
*Operations Research*, Volume 13, Number 3, May-June, 1965) creates serious errors that are not accounted for in the normal "teaching" literature.

**Actionable Recommendations**

- Exercise great care of the myopic focus on the critical path. Near critical paths are the "show stoppers" of PERT.
- The difficulties of mis-specifying of activity duration - Mode versus Median
- Move beyond PERT to Monte Carlo simulation.