Stochastic pert networks: OP diagrams, critical paths and the project completion time

Abstract

Fisher and Goldstein [5–7] have recently derived closed form expressions for the expectation, variance and distribution of the project completion time of stochastic PERT networks when the durations of the tasks in the network can be written as mutually independent, probability mixtures of general-gamma [8] random variables (a more general result is described in [9]). In this article, it is shown that their methods can also be used to obtain the probability that a particular path in the PERT network is critical [6]. One possible criticism of their approach is that computation of both the probability that a particular path is critical and the summary characteristics of the reaction time distribution require the translation of a PERT network into an OP (Order-of-ftocessing) diagram. Currently, this translation can only be done manually, a very laborious process for even moderately complex networks. In this article, an algorithm is developed which can be used to generate the OP diagram for a given PERT network. The algorithm has been implemented in FORTRAN. Once the OP diagram is constructed, computation of both the probability that a particular path is critical and the expectation, variance and distribution of the project completion time requires only simple matrix operations when the durations of the tasks in the network are mutually independent probability mixtures of general-gamma random variables. More complex calculations are required when other distributions of the task durations are involved.