Stochastic network project scheduling with non-consumable limited resources

This paper presents a newly developed resource constrained scheduling model for a PERT type project. Several non-consumable activity related resources, such as machines or manpower, are imbedded in the model. Each activity in a project requires resources of various types with fixed capacities. Each type of resource is in limited supply with a resource limit that is fixed at the same level throughout the project duration. For each activity, its duration is a random variable with given density function. The problem is to determine starting lime values S-ij for each activity (i,j) entering the project, i.e., the timing of feeding-in resources for that activity. Values S-ij. are not calculated beforehand and are random values conditional on our decisions. The model's objective is to minimize the expected project duration. Determination of values S-ij is carried out at decision points when at least one activity is ready to be operated and there are free available resources. If, at a certain point of time, more than one activity is ready to be operated but the available amount of resources is limited, a competition among the activities is carried out in order to choose those activities which can be supplied by the resources and which have to be operated first. We suggest carrying out the competition by solving a zero-one integer programming problem to maximize the total contribution of the accepted activities to the expected project duration. For each activity, its contribution is the product of the average duration of the activity and its probability of being on the critical path in the course of the project's realization. Those probability values are calculated via simulation. Solving a zero-one integer programming problem at each decision point results in the following policy: the project management takes all measures to first operate those activities that, being realized, have the greatest effect of decreasing the expected project duration. Only afterwards, does the management take care of other activities. A heuristic algorithm for resource constrained project scheduling is developed. A numerical example is presented.