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The PERT method is used when there is uncertainty in the duration of the activities in a project.

Figure 5-8 shows what would be expected if you were to plot the probability distribution of the expected dates for completing a project. On the left side of the diagram, we have the optimistic completion date for the project. On the right side, we have the pessimistic completion date for the project. The optimistic and pessimistic dates for the project completion are the earliest and latest dates that are reasonable for the project's completion; they should not be dates that are impossibly early or impossibly late. Notice that the curve of the probability distribution is skewed to the right. This is because it is increasingly unusual for the project to be done earlier and earlier. As most of us have experienced, when things begin to go wrong in terms of project lateness, it seems that they get worse and worse or later and later. So, in this characteristic plot of projects we see more dates later than the most likely date and fewer dates eerlier than the most likely date. This causes the PERT weighted average date to shift to a position later Chan the mosC likely date. The PERT weighted average is not the most likely date for finishing the project. It is shifted somewhat because the probability distribution is not symmetrical.

Figure 5-9 shows the range of values that is plus or minus two standard deviations from the PERT weighted average value. The standard deviation is always a positive number and is the distance from the expected va°ef In the case of PERT, it is the distance measured from the PERT weighted average.

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Figure 5-9: SKEWED PROBABILITY DISTRIBUTION

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Figure 5-9: SKEWED PROBABILITY DISTRIBUTION

The standard deviation of the project completion is the sum of the standard deviations of the durations of the activities that make up the critical path. Since the durations of the activities on the critical path are the only ones that should go into the total that is the project duration, only the critical path items' standard deviations should be used to determine the standard deviation of the project completion.

To the scheduling example we had been working with earlier, we now add estimated values for optimistic, pessimistic, and most likely. From these we calculate the expected value and the standard deviation (see Figure 5-10).

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Figure 5-10: PERT EXAMPLE

When we add up the standard deviations of the activities to get the standard deviation of the project, we must first square each one of the standard deviations of each activity, add them up, and then take the square root of the total.

In the example shown in Figure 5-11, the expected value for each activity is calculated by taking four times the most likely value of the duration for each activity and adding this to the optimistic and pessimistic values. The total of the three values is then divided by six. To calculate the standard deviation for each activity, we simply subtract the optimistic duration from the pessimistic duration and divide by six Strictly speaking we should always take the absolute value of this, but since it is quite unusual for the °essimistic date to be earlier than the optimistic date, it is really not necessary.

 Activity Optimistic Pessimistic Most Likely Expected Value Std. Dev. (Std.Dev)2 1 1 2 1 1.17 0.17 0.029 2 2 3 2 2.17 0.17 0.029 3 4 7 5 5.17 0.50 0.250 4 10 17 12 12.50 1.17 1.369 5 1 1 1 1.00 0.00 0.000 6 2 4 3 3.00 0.33 0.109 7 2 3 2 2.17 0.17 0.029 8 1 2 1 1.17 0.17 0.029 Total 18 SD 1.206 95% Probability Range 20.41-15.59

Figure 5-11: EXAMPLE: PERT CALCULATIONS

To get the project total standard deviation, we add the square of the standard deviation of activities 1, 2, 4, 5, and 8 and then take the square root of the total. Since we are generally interested in a probability of 95 percent, we add and subtract two standard deviations to the expected value of the project duration to get the range of values for project duration that will give us a 95 percent probability of predicting the actual project completion date.

Actually, in most projects we are interested in knowing whether the project will finish late or not, and we are not as concerned about whether the project will finish early. In other words we are interested in the project's finishing earlier than the date that is the PERT weighted average plus two standard deviations, and we are not concerned about the date that is the PERT weighted average minus two standard deviations. This raises the probability of the prediction to about 97 percent instead of 95

percent.

One of the problems that many people avoid talking about in PERT analysis is the problem of what happens when the critical path changes during the actual project. When the project is actually done, the tasks will only have one duration each, the actual duration. Since the duration of the task can be any value in the possible range ofvalues forthattask, the project could have a combination of durations that would cause the critical path to be differentthan originally predicted bythe expected value durations for each task.

What we mean by this is that when the critical path is calculated in PERT analysis, there is only one critical path and that is determined by using only the expected value of the duration for each activity. Once these durations are found, the critical path is determined, and the project's expected duration is calculated by adding thee expected durations of the activities on the critical path. The durations of all the other activities in the project are not added because they are being done in parallel with the critical path activities. The expected value of the duration of the project is then adjusted by adding and subtracting two standard deviations. This results in a range of values for the project. This actual project duration has a 95 percent probability of falling inside of this range.

But what if the durations ol the actual project are such that a new critical path forms? To solve this problem we cannot practically solve the equations for all the possible values of all the durations of all the activities in the project. Instead wa nse computer simulation. This simulation is called Monte Carlo simulation and is discussed Ister in tiiis ^hespter.

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