Date: 01 Jun 2005

Scheduling unit jobs with compatible release dates on parallel machines with nonstationary speeds

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We consider the problem of nonpreemptively scheduling a set of jobs with identical processing requirements (unit jobs) on parallel machines with nonstationary speeds. In addition to the case of uniform machines, this allows for such predictable effects as operator learning and tool wear and tear, as well as such planned activities as machine upgrades, maintenance and the preassignment of other operations, all of which may affect the available processing speed of the machine at different points in time. We also allow release dates that satisfy a certain compatibility property. We show that the convex hull of feasible completion time vectors is a supermodular polyhedron. For nonidentical but compatible release dates, the supermodular function defining this polyhedron is the Dilworth truncation of a (non supermodular) function defined in a natural way from the release dates. This supermodularity result implies that the total weighted flow time can be minimized by a greedy algorithm. Supermodular polyhedra thus provide a general framework for several unit job, parallel machine scheduling problems and for their solution methods.

This work was done in part during mutual visits of the authors to each other's institution. This research was supported in part by a research grant of the Natural Sciences and Engineering Research Council (NSERC) of Canada to the first author, and by the graduate school “Algorithmische Diskrete Mathematik”. The latter is supported by the Deutsche Forschungsgemeinschaft, grant We 1265-1.