Monte Carlo Simulation

When a schedule with activities that have uncertainty associated with their durations is encountered, the PERT method can be used to help predict the probability and range of values that will encompass the actual duration of the project. While the PERT technique uses the normal and beta distributions to determine this probability and range of values, there is a serious flaw in the results. The assumption made in the PERT analysis is that the critical path of the project remains the same under any of the possible conditions. This is, of course, a dangerous assumption. In any given set of possibilities, it is quite possible that the critical path may shift from one set of activities to another, thus changing the predicted completion date of the project.

In order to predict the project completion date when there is a possibility that the critical path will be different for a given set of project conditions, the Monte Carlo simulation must be used. The Monte Carlo simulation is not a deterministic method like many of the tools that we normally use. By that I mean that there is no exact solution that will come from the Monte Carlo analysis. What we will get instead is a probability distribution of the possible days for the completion of the project.

Monte Carlo simulations have been around for some time. It is only recently that personal computers and third party software for project management have become inexpensive enough for many project managers to afford.

The Simulation

In our project schedule, the predecessors and successors form a critical path. As I explained earlier, the critical path is the list of activities in the project schedule that cannot be delayed without affecting the completion date of the project. These are the activities that have zero float. Float is the number of days an activity can be delayed without affecting the completion date of the project.

When we have uncertainty in the duration times for the activities in the schedule, it means that there is at least a possibility that the activity will take more time or less time than our most likely estimate. If we used PERT to make these calculations, we already have calculated the mean value and the standard deviation for the project and all of the activities that have uncertainty.

The Monte Carlo simulator randomly selects values that are the possible durations for each of the activities having possible different durations. The selection of a duration for each activity is made, and the calculation of the project completion date is made for that specific set of data. The critical path is calculated, as well as the overall duration and completion date for the project.

The simulator usually allows for the selection of several probability distributions. This can be done for one activity, a group of activities, or the entire project. Depending on the software package being used, a selection of probability distributions is offered, such as: uniform, binomial, triangular, Poisson, beta, normal, and others. The Monte Carlo simulation works in a step-by-step way:

1. A range of values is determined for the duration of each activity in the schedule that has uncertainty in its duration.

2. A probability distribution is selected for each activity or group of activities.

3. If necessary, the mean and standard deviation are calculated for each activity.

4. The network relationships between the activities are entered.

5. The computer simulation is begun.

6. A duration time is selected for each activity in the schedule, whether it is on the critical path or not.

7. The critical path, duration of the project, float, and other schedule data are calculated.

8. This process is repeated many times until the repetitions reach a certain predefined number of cycles or until the results reach a certain accuracy.

9. Output reports are generated.

Output from the Monte Carlo Simulation

The most common output from a Monte Carlo simulation is a chart showing the probability of each possible completion date. This is usually shown as a frequency histogram. Generally, a cumulative plot is made as well. In this way you may see graphically the probability of each of the possible dates. This clearly shows the most likely dates for project completion. Because of the shifting of the critical path, it is quite possible for early dates and late dates to be the most likely, with unlikely dates in between them.

A cumulative curve is also generated showing the cumulative probability of completing the activity before a given date. The criticality index can also be calculated. This is the percentage of the time that a particular activity is on the critical path. In other words, if a simulation were run 1,000 times and a particular activity was on the critical path 212 times, its criticality index would be 21.2 percent.