Abstract
In project or production management, an activity network is classically defined by a set of tasks (activities) and a set of precedence constraints expressing which tasks cannot start before others are completed. When there are no resource constraints, we can display the network as a directed acyclic graph. With such a network the goal is to find critical activities, and to determine optimal starting times of activities, so as to minimize the makespan. The first step is to determine the earliest ending time of the project. This problem was posed in the fifties, in the framework of project management, by Malcolm et al. [32] and the basic underlying graph-theoretic approach, called Project Evaluation and Review Technique, is now popularized under the acronym PERT. The determination of critical activities is carried out via the so-called critical path method (Kelley [29]). The usual assumption in scheduling is that the duration of each task is precisely known, so that solving the PERT problem is rather simple. However, in project management, the durations of tasks are seldom precisely known in advance, at the time when the plan of the project is designed. Detailed specifications of the methods and resources involved for the realization of activities are often not available when the tentative plan is made up. This difficulty has been noticed very early by the authors that introduced the PERT approach. They proposed to model the duration of tasks by probability distributions, and tried to estimate the mean value and standard deviation of earliest starting times of activities. Since then, there has been an extensive literature on probabilistic PERT (see Adlakha and Kulkarni [1] and Elmaghraby [18] for a bibliography and recent views). Even if the task duration times are independent random variables, it is admitted that the problem of finding the distribution of the ending time of a project is intractable, due to the dependencies induced by the topology of the network [25]. Another difficulty, not always pointed out, is the possible lack of statistical data validating the choice of activity duration distributions. Even if statistical data are available, they may be partially inadequate because each project takes place in a specific environment, and is not the exact replica of past projects.