The Project Management Question and Answer Book

The Monte Carlo process is a simulation technique that is used to statistically predict the duration of a project when there is uncertainty about the duration of the activities of the project. Simulation techniques are used because the number of simultaneous algebraic equations would become quite large even for small projects.

PERT analysis of project schedules allows for there to be uncertainty in the durations of the project activities. In the PERT analysis, however, the critical path is usually held as a constant set of activities. In reality the critical path may actually be different from one set of project activity durations to another.

In the Monte Carlo process a duration is selected from the possible durations for each activity in the schedule. The project schedule and the finish date for the project are calculated. Once the project completion date is calculated and recorded, a new set of dates is picked for the activities in the project, and the next project finish date is calculated and saved. This process is repeated enough times to become stable, usually about 1,000 times, and the statistics are calculated. The most important statistics are the probability of the project's finishing on certain dates.

Tell me more …

The Monte Carlo process is used in project schedules to help predict project completion dates when there is uncertainty in the durations of the project activities. The problem with using uncertainty in project durations is the effect these uncertain durations have on the predicted project completion date. Since the completion date of the project is determined by the length of time contained in the critical path, it is important that we know what the critical path is. If there is uncertainty in the durations of the project activities, it simply means that the critical path could be one sequence of activities for one set of durations and a different sequence of activities for another set of durations. It would be incorrect, for example, to predict the completion date of the project by using one set of durations when another set of durations might yield a different critical path and a different project completion date.

The prediction of the critical path for the project when there is uncertainty in the activities' durations can be quite difficult. It would take an impossibly large set of simultaneous equations to analyze the possible critical paths of a project, predict the probability of each occurring, and then calculate the duration of the project. For this reason a simulation method is used instead. This is the Monte Carlo process.

The first things we need to consider in the Monte Carlo process are the durations of the project activities. Since we are considering the use of the Monte Carlo process in the first place, we are recognizing that there is uncertainty in the durations of the project activities. If the durations were known, we would not need a statistical technique at all. But how do the durations vary?

We can think of this variation of durations as a probability distribution. In Figure 5-23 the probability distribution relates a given duration to a probability of its occurring. The even distribution says that the probability of one duration's occurring is the same as the probability of the occurrence of any other duration in the range of possible values. The triangular distribution says that the probability of any duration's occurring increases in a linear way from some minimum value until a value for the most likely duration is reached. At that point the probability decreases linearly until the highest value of duration is reached. The familiar normal distribution says that the relationship between the duration value and the probability is similar to a normal distribution. In any Monte Carlo simulator there will be many choices given for probability distributions.

Figure 5-23: PROBABILITY DISTRIBUTIONS

The distributions given here are only three of many that can be selected for the project activities. Since Monte Carlo simulations must be done on a computer, we will be using software designed to do this process. The software will make available a wide choice of distributions.

Once we have assigned a probability distribution to each of the activities in the project, the computer will do most of the work for us. It will select a duration for each activity in such a way that the combinations of different durations for the different activities is random but they also fit the probability distribution we have specified. Each time a set of durations has been selected, the critical path schedule and the project completion date will be calculated.

Figure 5-24 shows a random number table. In the old days before computers, engineers and mathematicians would use tables like this one to generate random numbers. Random numbers are actually quite difficult to calculate because a truly random number has exactly the same probability of occurring as any other random number. In other words random numbers are the ultimate even distribution. Any number at all has exactly the same likelihood of being the next number in a series of numbers. If we were using two-digit random numbers and started with 56, any number in the range of numbers from 00 to 99 would have exactly the same probability of occurring.

Using a random number table is quite simple. Suppose we want a series of two-digit random numbers. We could pick any row or column in the table as a starting number. In the table in Figure 5-24 let us start with the second column, third row, first two digits, 87. We could then go to the next column and the next column to get successive random numbers. These would be 92 and 60. We could continue with 49 and 50. We could have just as easily continued down the column we were in with 80, 81, 95, and 29. In fact, any scheme for picking numbers is fine.

The random numbers are used to select the durations for the schedule simulation according to the probability distributions that were selected for each of the activities. In Figure 5-25, each of the possible durations for each of the activities is listed with its activity. The probability of each possible duration is listed next to each duration. The probability for each duration is proportional to the number of possible random numbers out of 100. In other words, the duration and probability of activity A could be 3, 4, 5, or 6 with respective probability of .25, .60, .10, and .05.

Figure 5-25: DURATIONS AND PROBABILITIES

We assign random numbers 00–24 to a duration of 3; random numbers 25–84 to a duration of 4; random numbers 85–94 to a duration of 5 and random numbers 95–99 to a duration of 6. This assignment of random numbers according to their respective probability distributions is done for all of the activities in the project. Of course, if an activity has a certain duration, any random number selected will select the same duration. This is all done automatically by the computer software once we have assigned probability distributions and a range of possible durations to each activity.

Once all of this is done, the computer can begin the simulation. It begins by selecting a random number for the first activity. The random number then selects the duration to be used for the first simulation. A second random number is then selected for the next activity, and a duration is selected for that activity. Once all of the activities in the project have been assigned a duration, the schedule can be calculated normally, the critical path identified, and the project completion date determined.

After the simulation has been run many times, the data are summarized. One popular way of looking at the data is to prepare a histogram for each possible date of the project completion. The height of the histogram bar is proportional to the number of times a particular project completion date has occurred in the simulations. A cumulative probability is also plotted to make reading the data easier. From the cumulative line we can read the probability that the project will finish by a certain date or earlier. In our example (Figure 5-26), we have run the simulation 1,000 times and the predicted calendar weeks that the project could end are shown. There is a 71 percent probability that the project will be completed on February 9 or sooner.

Figure 5-26: MONTE CARLO OUTPUT: 1,000 SIMULATIONS

Two other figures that are frequently calculated in Monte Carlo analysis are the criticality index and the critical value. The criticality index is a number between 0 and 1.0 that is equivalent to the percentage of the number of times an activity was found to be on the critical path. The critical value is the criticality index multiplied by the variance of the duration of that activity.

Both of these values are designed to help us manage the project. It is important for us to know the criticality index since this gives us guidance as to which activities in the project we must manage closely. As in the critical path method, we should be more concerned about activities with high criticality indexes since these are the ones that are most likely to cause delays in project completion should they be delayed.

The critical value tells us a bit more. It indicates activities that are likely to be on the critical path and also have large variations in their possible durations.