Scheduling jobs of equal length: Complexity, facets and computational results

The following problem was originally motivated by a question arising in the automated assembly of printed circuit boards. Given are n jobs, which have to be performed on a single machine within a fixed timespan [0, T], subdivided into T unit-length subperiods. The processing time (or length) of each job equals p, p is an element of N. The processing cost of each job is an arbitrary function of its start-time. The problem is to schedule all jobs so as to minimize the sum of the processing costs. This problem is proved to be NP-hard, already for p=2 and 0-1 processing costs. On the other hand, when T=np+c, with c constant, the problem can be solved in polynomial time. A partial polyhedral description of the set of feasible solutions is presented. In particular, two classes of facet-defining inequalities are described, for which the separation problem is polynomially solvable. Also, we exhibit a class of objective functions for which the inequalities in the LP-relaxation guarantee integral solutions. Finally, we present a simple cutting plane algorithm and report on its performance on randomly generated problem instances.