The Program Evaluation and Review Technique (PERT): Incorporating activity time variability in a project schedule

The Program Evaluation and Review Technique (PERT) is a project scheduling technique to analyze and represent the tasks involved in completing a given project. It incorporates activity duration variability and relies on similar concepts as the critical path method

Table 1: An example project for PERT calculations

Task Predecessors A M BA None 2 5 8B A 1 2 9C A .25 .5 3.75D B 1 1 7E B & C 1 2 9F D & E 1 3 11

In this article, the technique will be explained by means of the example project data of table 1. More precisely, the following elements, necessary to analyze a PERT project network, will be discussed:

The calculation of the activity duration estimates based on three input measures The calculation of the activity average and variance The calculation of the average critical path The use of the central limit theorem The presence of the normal distribution

Activity duration estimatesActivity durations are based on estimates made by human beings and are therefore error-prone. In PERT, the technique requires three duration estimates for each individual activity, as follows:

Optimistic time estimate: This is the shortest possible time in which the activity can be completed, and assumes that everything has to go perfect

Realistic time estimate: This is the most likely time in which the activity can be completed under normal circumstances Pessimistic time estimate: This is the longest possible time the activity might require, and assumes a worst-case scenario

Figure 1: A PERT (solid line) and triangular (dotted line) distribution

Activity average and varianceBased on the three estimates, a weighted average and variance is calculated for each activity duration as a measure of the average duration and the corresponding variability, respectively. The weighted average is equal to (a + 4m + b) / 6 to express that the likeliness that the real activity duration lies close to the realistic estimate (m) is larger than the likeliness that it lies closer to the two extreme values a or b.

The standard deviation is equal to (b - a) / 6 and is based on the principles of a three-sigma interval that states that 99.73% (i.e. almost all) of the observations lie in that interval when the variable is normally distributed. Although it is assumed that the activity duration is beta distributed, the general principle is that the standard deviation assumes that almost all observations (i.e. real durations) will lie between the extreme values a and b.

The average durations and their standard deviations are given in the second line above each node of figure 2.

The average critical pathThe expected duration D of a critical path is equal to the sum of the expected durations of the critical activities, i.e. E(D) = 5 + 3 + 3 + 4 = 15

The expected variance V of a critical path is equal to the sum of the variances of the critical activities, i.e. E(V) = 12 + (8/6)2 + (8/6)2 + (10/6)2 = 7.33. Note that only the activities on the (average) critical path are taken into account for the calculation of the variance.

Figure 2: The project network and the (average) critical path

The central limit theoremThe main idea of the central limit theorem (CLT) is that the average of a sample of observations drawn from a population with any distribution shape is approximately distributed as a normal distribution if certain conditions are met. More precisely, the central limit theorem states that given a distribution with a mean μ and variance σ², the sampling distribution of the mean approaches a normal distribution with a mean (μ) and a variance σ²/n as n, the sample size, increases. The amazing and counter-intuitive thing about the central limit theorem is that no matter what the shape of the original distribution is, the sampling distribution of the mean approaches a normal distribution.

In PERT, the project network of figure 1 represents the population, while the sample that is drawn from that population is equal to the average critical path (highlighted in red).

Consequently, the project duration follows a normal distribution with the following parameters:

Average = E(D) = expected project duration (based on critical path)

Variance = E(V) = expected variance (of the critical path)

The normal distributionUsing the characteristics of the normal distribution (N(15,7.33)) of the total project duration, basic statistical calculations can be applied to give answers to questions such as:

What is the probability that the project will be finished before ...? What is the expected project deadline? What is a reasonable project duration such that it can be met with a probability of ... %?

As an example, the P(project duration ≤ critical path length E(D)) = 50% (symmetrical normal distribution) which clearly shows that the deterministic critical path underestimates the likely project duration. Another example shows that the P(project duration ≤ 17) = P(z ≤ (17 - 15) / 2.71)) = P(z ≤ 0.74) = 77%

with E(D) = 15 and E(V) = 7.33 using the transformations from the normal distribution N(15,7.33) to a standardized normal distribution N(0,1). This value can be found using standardized normal tables or by the “=NORMDIST(17,15,SQRT(7.33),1)” function in Excel.

The Critical Path Method (CPM): Incorporating activity time/cost trade-offs in a project schedule

The Critical Path Method (CPM) is a project scheduling technique to analyze and represent the tasks involved in completing a given project. It incorporates a trade-off between an activity’s duration and cost and relies on concepts similar to the program evaluation and review technique

In this article, three CPM related topics will be discussed:

Time/cost trade-offs in activities Scheduling objectives of CPM The Project Scheduling Game (PSG)

Time/cost trade-offsThe critical path method assumes that the duration of an activity depends on the amount of resources assigned to it, and incorporates a trade-off between its duration and the cost of the assigned resources. More precisely, the more resources have been assigned to the activity, leading to an overall increase in the cost, the lower the expected duration of the activity.

Figure 1: Time/cost trade-offs for a project activity

This time/cost trade-off is given in figure 1. The CPM assumes four pieces of information for each project activity, as follows:

Normal duration: The maximum duration of the activity. Crash duration: The minimum duration of the activity. Normal cost: The cost associated with the normal duration. Crash cost: The cost associated with the crash duration.

The normal duration is equal to the minimal expected duration when the activity is performed with the lowest possible amount of resources, leading to the lowest total activity cost (normal cost). The crash duration is the shortest activity duration, when the maximum amount of resources has been assigned to the activity, leading to the highest activity cost (crash cost). Each dot in figure 1 represents a so-called activity mode to represent the list of time/cost combinations for this activity. The aim of the critical path method is to schedule the project under a pre-defined scheduling objective, i.e. the choice of an activity mode for each activity to optimize a scheduling objective. A decrease in an activity‘s duration and the corresponding increase in the activity cost is known as activity crashing.

Scheduling objectivesScheduling a project requires a scheduling objective in order to optimize the project schedule according to the wishes and needs of the project manager. The CPM has a dual view on the project schedule, i.e. a time or cost point-of-view, resulting in two possible scheduling objectives. Figure 2 illustrates the time/cost profile on the project level as a result of selected time/cost combinations (i.e. modes) for each project activity. The two scheduling objectives are given along the following lines:

Deadline objective: The objective is to determine the activity durations and to schedule the activities in order to minimize the project duration, given a predefined total budget restriction.

Budget objective: The objective is to determine the activity durations and to schedule the activities in order to minimize the project costs, given a predefined project deadline restriction.

Figure 2: The scheduling objective of the CPM: time or cost