Calculating the z values

The z value is calculated for each node that has a target date. It is equivalent to the number of standard deviations between the node's expected and target dates. It is calculated using the formula s where te is the expected date and 7 the target date.

Exercise 7.7 The z value for event 4 is (10 - 9.00)/0.53 = 1.8867.

Calculate the z values for the other events with target dates in the network shown in Figure 7.4.

Converting z values to probabilities

A z value may be converted to the probability of not meeting the target date by using the graph in Figure 7.5.

This graph is the equivalent of tables of z values, also known as standard normal deviates, which may be found in most statistics textbooks

Figure 7.5 The probability of obtaining a value within z standard deviations of o z value

Figure 7.5 The probability of obtaining a value within z standard deviations of the mean for a normal distribution.

Exercise 7.8 The z value for the project completion (event 6) is 1.23. Using Figure 7.5 we can see that this equates to a probability of approximately 11 %, that is, there is an 11% risk of not meeting the target date of the end of week 15.

Find the probabilities of not achieving events 4 or 5 by their target dates of the end of week 10.

What is the likelihood of completing the project by week 14?

The advantages of PERT

We have seen that by requesting multivalued activity duration estimates and calculating expected dates, PERT focuses attention on the uncertainty of forecasting. We can use the technique to calculate the standard deviation for each task and use this to rank them according to their degree of risk. Using this ranking, we can see, for example, that activity F is the one over which we have greatest uncertainty, whereas activity C should, in principle, give us relatively little cause for concern.

If we use the expected times and standard deviations for forward passes through the network we can, for any event or activity completion, estimate the probability of meeting any set target. In particular, by setting target dates along the critical path, we can focus on those activities posing the greatest risk to the project's schedule.

Monte Carlo simulation

As an alternative to the PERT technique, and to provide a greater degree of flexibility in specifying likely activity durations, we can use Monte Carlo simulation techniques to evaluate the risks of not achieving deadlines. The basis of this technique is the calculation of event times for a project network a large number of times, each time selecting activity times randomly from a set of estimates for each activity. The results are then be tabulated, summarized or displayed as a graph such as that shown in Figure 7.6.

Monte Carlo simulation was also discussed in Section 3.7 in the context of project evaluation.

Activity start date Activity end date

Week number

Figure 7.6 Risk profile for an activity generated using Monte Carlo simulation.

Activity duration estimates can be specified in a variety of forms, depending upon the information available. If for example, we have historic data available about the durations of similar activities, we might be able to specify durations as a probability distribution. With less information available we should, at least, be able to provide three time estimates as used by PERT.

There are a number of packages available for carrying out Monte Carlo simulation. Some will exchange data with project scheduling applications and some interface to standard spreadsheet software. The majority of these packages will apply Monte Carlo risk analysis to cost and resource as well as duration estimates.

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