Approximating the Criticality Indices of the Activities in PERT Networks

Abstract

A stochastic PERT network is a directed acyclic network in which the arc lengths are independent random variables with known distributions. A fundamental problem in PERT networks is to identify the activities which are critical to the achievement of the project objectives. In an activity network if the duration of each activity is not a random variable, then it is easy to identify the criticality of each activity represented by its float time. However, when the duration of any activity is a random variable, it is not easy to identify the criticality of each activity. In this case the criticality of an activity is known as the “criticality index,” which is defined as the sum of the criticality indices of the paths containing it. The criticality index of a path is the probability that the duration of the path is greater than or equal to the duration of every other path in the network. Clearly, the criticality index of an activity can be obtained by determining the criticality indices of the paths, which requires identifying all the paths, determining their criticality indices, then identifying the paths containing the activity. In this paper we develop a theory which leads to a procedure to approximate the criticality indices of all the activities without going through the above three steps. The procedure has been applied to large size PERT networks generated at random, and the results are found to be very close to those obtained by extensive Monte Carlo sampling.