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Parallel Machine Scheduling by Column Generation

Author

Listed:
  • J. M. van den Akker

    (Department of Mathematical Models and Methods, National Aerospace Laboratory NLR, P.O. Box 90502, 1006 BM Amsterdam, The Netherlands)

  • J. A. Hoogeveen

    (Department of Mathematics and Computing Science, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands)

  • S. L. van de Velde

    (Rotterdam School of Management, Erasmus University, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands)

Abstract

Parallel machine scheduling problems concern the scheduling of n jobs on m machines to minimize some function of the job completion times. If preemption is not allowed, then most problems are not only (N-script)(P-script)-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, n covering 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 m identical 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.

Suggested Citation

  • J. M. van den Akker & J. A. Hoogeveen & S. L. van de Velde, 1999. "Parallel Machine Scheduling by Column Generation," Operations Research, INFORMS, vol. 47(6), pages 862-872, December.
    • Handle: RePEc:inm:oropre:v:47:y:1999:i:6:p:862-872
    DOI: 10.1287/opre.47.6.862
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    References listed on IDEAS

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