Probabilistic Activity Times

In the project network for building a house in the previous section, all activity time estimates were single values. By using only a single activity time estimate, we are, in effect, assuming that activity times are known with certainty (i.e., they are deterministic). For example, in Figure 17.3, the time estimate for activity 2-3 (laying the foundation) is 2 months. Since only this one value is given, we must assume that the activity time does not vary (or varies very little) from 2 months. It is rare that activity time estimates can be made with certainty. Project activities are likely to be unique. There is little historical evidence that can be used as a basis to predict activity times. Recall that one of the primary differences between CPM and PERT is that PERT uses probabilistic activity times.

Probabilistic Time Estimates

In the PERT-type approach to estimating activity times, three time estimates for each activity are determined, which enables us to estimate the mean and variance of a beta distribution of the activity times.

We assume that the activity times can be described by a beta distribution for several reasons. The beta distribution mean and variance can be approximated with three time estimates. Also, the beta distribution is continuous, but it has no predetermined shape (such as the bell shape of the normal curve). It will take on the shape indicated--that is, be skewed--by the time estimates given. This is beneficial, since typically we have no prior knowledge of the shapes of the distributions of activity times in a unique project network. Although other types of distributions have been shown to be no more or less accurate than the beta, it has become traditional to use the beta distribution to estimate probabilistic activity times.

Repair Project Management at Sasol

Sasol is a leading South African company that converts coal to oil and chemicals. On March 8, 1994, a fire broke out in a regeneration column (e.g., chimney/ smokestack) used to process hydrogen in the Benfield Unit at Sasol Three, one of Sasol's factories. The column is 70m (231 feet) high, and the fire caused it to buckle in the middle so that it tilted to one side. Without the column, a large section of the factory could not function, resulting in a substantial loss of income. It was imperative that the Benfield column be repaired as soon as possible, which required the damaged portion of the column shell to be cut out and replaced. A goal was immediately established to have the column back in service within 47 days.

Sustech, a subsidiary of Sasol's, was assigned the repair project. A project team was put together consisting of 27 members including 4 process engineers, 6 mechanical engineers, a pressure vessel specialist, a metallurgist, a welding engineer, a pipe stress engineer, a piping draftsman, a mechanical draftsman, a structural engineer and draftsman, 3 quality assurance inspectors, and commercial contract and procurement officers. Team members came not only from Sasol but also from the original column fabricators, Chicago Bridge and Iron Works, and various equipment and material suppliers.

The scope statement was brief--repair the Benfield column as soon as possible. First a work breakdown structure was developed. This was accomplished at open brainstorming meetings with all interested parties and team members present. Each topic identified was written on an adhesive note and attached to a huge white board. A project schedule was established using Microsoft Project for Windows. Special attention was focused on critical path activities. Team members responsible for critical path activities received voluntary help from all other members of the team. Quality control was strictly enforced. Not only would this result in a safe and durable new column; it would negate the need to rework poor quality work that might delay the project.

The repair project was completed in just 25 days--22 days ahead of schedule. The initial project budget was $85.28 million and the final cost was $63.74 million, a savings of 25 percent of the total estimated project cost. Keys to project success were a simple plan with good communication and leadership, and a motivated workforce. The Benfield column repair project was named the 1995 International Project of the Year by the Project Management Institute.

Source: I. Boggon, "The Benfield Column Repair Project," PM Network 10, no. 2 (February 1996): 25-30.

The three time estimates for each activity are the most likely time (m), the optimistic time (a), and the pessimistic time (b). The most likely time is a subjective estimate of the activity time that would most frequently occur if the activity were repeated many times. The optimistic time is the shortest possible time to complete the activity if everything went right. The pessimistic time is the longest possible time to complete the activity assuming everything went wrong. The person most familiar with an activity or the project manager makes these "subjective" estimates to the best of his or her knowledge and ability.

These three time estimates are used to estimate the mean and variance of a beta distribution, as follows:

These formulas provide a reasonable estimate of the mean and variance of the beta distribution, a distribution that is continuous and can take on various shapes, or exhibit skewness.

Figure 17.10 illustrates the general form of beta distributions for different relative values of a, m, and b.

A Project Network with Probabilistic Time Estimates

The Southern Textile Company has decided to install a new computerized order-processing system. In the past, orders were processed manually, which contributed to delays in delivery orders and resulted in lost sales. The new system will improve the quality of the service the company provides. The company wants to develop a project network for the installation of the new system.

The network for the installation of the new order-processing system is shown in the accompanying figure. The network begins with three concurrent activities: The new computer equipment is installed (activity 1-2); the computerized order-processing system is developed (activity 1-3); and people are recruited to operate the system (activity 1-4). Once people are hired, they are trained for the job (activity 4-5), and other personnel in the company, such as marketing, accounting, and production personnel, are introduced to the new system (activity 4-8). Once the system is developed (activity 1-3) it is tested manually to make sure that it is logical (activity 3-5). Following activity 1-2, the new equipment is tested, any necessary modifications are made (activity 2-6), and the newly trained personnel begin training on the computerized system (activity 5-7). Also, event 5 begins the testing of the system on the computer to check for errors (activity 5-8). The final activities include a trial run and changeover to the system (activity 7-9), and final debugging of the computer system (activity 6-9).

The three time estimates, the mean, and the variance for all the activities in the network as shown in the figure are provided in the following table:


As an example of the computation of the individual activity mean times and variance, consider activity 1-2. The three time estimates (a = 6, m = 8, b = 10) are substituted in the formulas as follows:

The other values for the mean and variance are computed similarly.

Once the mean times have been computed for each activity, we can determine the critical path the same way we did in the deterministic time network, except that we use the expected activity times, t. Recall that in the home building project network, we identified the critical path as the one containing those activities with zero slack. This requires the determination of earliest and latest start and finish times for each activity, as shown in the following table and figures:

From the table, we can see that the critical path encompasses activities 1-3-5-7-9, since these activities have no available slack. We can also see that the expected project completion time (tp) is the same as the earliest or latest finish for activity 7-9, or tp = 25 weeks. To determine the project variance, we sum the variances for those activities on the critical path. Using the variances shown in the table for the critical path activities, the total project variance can be computed as follows:

CPM/PERT Network Analysis with POM for Windows

POM for Windows will provide the same scheduling analysis that we computed in Example 17.1. The POM for Windows solution screen for Example 17.1 is shown in Exhibit 17.1 with earliest start and latest finish times and slack for each activity. Notice that instead of activity variances the computer output provides activity standard deviations (i.e., s instead of s2).

Probabilistic Network Analysis

The CPM/PERT method assumes that the activity times are statistically independent, which allows us to sum the individual expected activity times and variances to get an expected project time and variance. It is further assumed that the network mean and variance are normally distributed. This assumption is based on the central limit theorem of probability, which for CPM/PERT analysis and our purposes states that if the number of activities is large enough and the activities are statistically independent, then the sum of the means of the activities along the critical path will approach the mean of a normal distribution. For the small examples in this chapter, it is questionable whether there are sufficient activities to guarantee that the mean project completion time and variance are normally distributed. Although it has become conventional in CPM/PERT analysis to employ probability analysis using the normal distribution regardless of the network size, the prudent user should bear this limitation in mind.

Probabilistic analysis of a CPM/PERT network is the determination of the probability that the project will be completed within a certain time period given the mean and variance of a normally distributed project completion time. This is illustrated in Figure 17.11. The value Z is computed using the following formula:


This value of Z is then used to find the corresponding probability in Table A.1 (Appendix A).

Probabilistic Analysis of the Project Network

The Southern Textile Company in Example 17.1 has told its customers that the new order-processing system will be operational in 30 weeks. What is the probability that the system will be ready by that time?


The probability that the project will be completed within 30 weeks is shown as the shaded area in the accompanying figure. To compute the Z value for a time of 30 weeks, we must first compute the standard deviation (s) from the variance (s2).

Next we substitute this value for the standard deviation along with the value for the mean, 25 weeks, and our proposed project completion time, 30 weeks, into the following formula:

A Z value of 1.91 corresponds to a probability of 0.4719 in Table A.1 in Appendix A. This means that there is a 0.9719 probability of completing the project in 30 weeks or less (adding the probability of the area to the left of m = 25, or .5000 to .4719).

Probabilistic Analysis of the Project Network

A customer of the Southern Textile Company has become frustrated with delayed orders and told the company that if the new ordering system is not working within 22 weeks, it will not do any more business with the textile company. What is the probability the order-processing system will be operational within 22 weeks?


The probability that the project will be completed within 22 weeks is shown as the shaded area in the accompanying figure.

The probability of the project's being completed within 22 weeks is computed as follows:

A Z value of -1.14 corresponds to a probability of 0.3729 in the normal table in Appendix A. Thus, there is only a 0.1271 (i.e., 0.5000 - 0.3729) probability that the system will be operational in 22 weeks.

17-12. Which method for determining activity time estimates, deterministic or probabilistic, do you perceive to be preferable? Explain.

*17-13. One of the most popular project management software is Microsoft Project. Look at the homepage for this product and print out sample screens that show its capabilities.

17-14. How are the mean activity times and activity variances computed in probabilistic CPM/PERT analysis?

17-15. How is total project variance determined in CPM/PERT analysis?

*These exercises require a direct link to a specific Web site. Click Internet Exercises for the list of internet links for these exercises.