# Probabilistic Network Methods

The network methods can be termed “deterministic” because estimated activity times are assumed to be fixed constants. No recognition is given to the fact activity times could be uncertain.

Probabilistic network methods assume the more realistic situation in which uncertain activity times are represented by probability distributions. With such a basic model of the network of activities, it is possible to develop additional data important to managerial decision. Such data help in assessing planning decisions that might revolve around such questions as the following: What is the probability that the completion of activity A will be later than January 10? What is probability that the activity will become critical and effect the project completion date? What is the probability of meeting a given target completion date for the project? The nature of the planning decisions based on such questions might involve the allocation or reallocation of personnel or other resources to the various activities in order to derive a more satisfactory plan. Thus, a “crash” schedule involving extra resources might be justified to ensure the on-time completion of certain activities. The extra resources needed are drawn from non-critical activities or from activities for which the probability of critically is small.

The discussion that follows is equally applicable to the node methods of network diagramming. We require the expected completion time and the variance in the completion time for each activity. Since the activity time is uncertain, a probability distribution for the activity time is requires. This probability distribution is assessed by the engineers, the project manager, or the appropriate consultants. Once the probability distribution for the activity time is specified, the expected activity time and the variance in activity time can be computed. Historically, however, the probability distribution of activity times is based on three time estimates made for each activity. This widely used procedure that we describe now is simple, but it requires some additional restrictive assumptions.

Optimistic Time:

Optimistic time, ‘a’ is the shortest possible time to complete the activity if all goes well. It is based on the assumptions that there is no more than one chance in a hundred of completing the activity in less than the optimistic time.

Pessimistic Time:

Pessimistic time, ‘b’ is the longest time or an activity under adverse conditions. It is based on the assumption that there is no more than one chance in a hundred of completing the activity in a time greater than b.

Most Likely Time:

Most likely time, m, is the single most likely modal value of the activity time distribution. The three time estimates are shown in relation to an activity completion time distribution in Figure.

Expected Activity Time:

By assuming that activity time follows a beta distribution, the expected time of an activity, te is computed as

te = a + 4m + b / 6

Variance of Activity:

The variance, σ2, of an activity time is computed using the formula

σ 2 = (b – a / 6) 2

Probability theory provides the basis for applying probabilistic network concepts. First, the sum of a reasonably large number (n>30) of identically distributed random variables is itself a random variable, but it is normally distributed even if the individual random variables are not. Second, the variance of the sum of statistically independent random variables is the sum of the variances of the original random variables.

Translated for network scheduling concepts, we have the following useful statements for large projects with independent activity times.

1) The mean completion time is the mean of a normal distribution, which is the simple sum of the t, values along a critical path.
2) The variance of the mean project completion time is the simple sum of the variances of the Individual activities along a critical path.

We can now use the method described earlier to determine the critical path by substituting te for the activity time. The mean and the variance of the project completion time are obtained by summing te and σ2, respectively for the activities on the critical path.

We can then use the normal tables to determine the probabilities for the occurrence of a given project completion time estimates. For example, the probability that a project will be completed in less than the mean time is only .50. The probability that a project will be completed in less than the mean time plus one standard deviation is about .84 in less than the mean time plus two standard deviations; .98; and so on.