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Parallel Machine Scheduling by Column Generation

Parallel Machine Scheduling by Column Generation Parallel machine scheduling problems concern the scheduling of njobs on mmachines to minimize some function of the job completion times. If preemption is not allowed, then most problems are not only 𝒩𝒫-hard, but also very hard from a practical point of view. In this paper, we show that strong and fast linear programming lower bounds can be computed for an important class of machine scheduling problems with additive objective functions. Characteristic of these problems is that on each machine the order of the jobs in the relevant part of the schedule is obtained through some priority rule. To that end, we formulate these parallel machine scheduling problems as a set covering problem with an exponential number of binary variables, ncovering constraints, and a single side constraint. We show that the linear programming relaxation can be solved efficiently by column generation because the pricing problem is solvable in pseudo-polynomial time. We display this approach on the problem of minimizing total weighted completion time on midentical machines. Our computational results show that the lower bound is singularly strong and that the outcome of the linear program is often integral. Moreover, they show that our branch-and-bound algorithm that uses the linear programming lower bound outperforms the previously best algorithm. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Operations Research INFORMS

Parallel Machine Scheduling by Column Generation

11 pages

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Publisher
INFORMS
Copyright
Copyright © INFORMS
Subject
Research Article
ISSN
0030-364X
eISSN
1526-5463
DOI
10.1287/opre.47.6.862
Publisher site
See Article on Publisher Site

Abstract

Parallel machine scheduling problems concern the scheduling of njobs on mmachines to minimize some function of the job completion times. If preemption is not allowed, then most problems are not only 𝒩𝒫-hard, but also very hard from a practical point of view. In this paper, we show that strong and fast linear programming lower bounds can be computed for an important class of machine scheduling problems with additive objective functions. Characteristic of these problems is that on each machine the order of the jobs in the relevant part of the schedule is obtained through some priority rule. To that end, we formulate these parallel machine scheduling problems as a set covering problem with an exponential number of binary variables, ncovering constraints, and a single side constraint. We show that the linear programming relaxation can be solved efficiently by column generation because the pricing problem is solvable in pseudo-polynomial time. We display this approach on the problem of minimizing total weighted completion time on midentical machines. Our computational results show that the lower bound is singularly strong and that the outcome of the linear program is often integral. Moreover, they show that our branch-and-bound algorithm that uses the linear programming lower bound outperforms the previously best algorithm.

Journal

Operations ResearchINFORMS

Published: Dec 1, 1999

Keywords: Keywords : production/scheduling ; sequencing ; deterministic ; multiple machine ; column generation

References