Discrete-continuous project scheduling with preemptable activities

In this paper, discrete-continuous project scheduling problems with preemptable activities are considered. In these problems, activities of a project simultaneously require discrete and continuous resources for their execution. The activities are preemptable, and the processing rate of each activity is a continuous, increasing function of the amount of a single continuous resource allotted to the activity at a time. The problem is to find a precedence- and discrete resource-feasible schedule and, simultaneously, continuous resource allocation that would minimize the project duration. Convex and concave processing rate functions are considered separately. We show that for convex functions the problem is simple, whereas for concave functions a special methodology has to be developed. We discuss the methodology for three cases of the problem: no discrete resource constraints, one discrete resource being a set of parallel, identical machines, and an arbitrary number of discrete resources. In each case we analyze separately independent and precedence-related activities. Some conclusions and directions for future research are given.