On a methodology for discrete-continuous scheduling

Discrete continuous problems of scheduling nonpreemptable jobs on parallel machines are considered. The problems arise e.g. when jobs are assigned to multiple parallel processors driven by a common electric, hydraulic or pneumatic power source. Existing models have assumed job processing rates as a function of the number of jobs currently being processed, or equivalently the number of machines currently in operation. In this paper a more general model is proposed in which processing rates of a job assigned to a machine depend on the amount of a continuous, i.e. continuously divisible resource (e.g. power) allotted to this job at a time. Thus the problem consists of two interrelated sub-problems: (i) to sequence jobs on machines, and (ii) to allocate the continuous resource among jobs already sequenced. We provide a comprehensive analysis of the problem. This includes properties of optimal schedules, efficiently (in particular analytically) solvable cases, formulations of the possibly simplest mathematical programming problems for finding optimal schedules in the general case, heuristics and the worst-case analysis. Although our objective function in this paper is to minimize makespan of a set of independent jobs, the presented methodology can be applied to other criteria, precedence-related jobs, and many resource types (apart from, or instead of machines).