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Necessary and sufficient optimality conditions for scheduling unit time jobs on identical parallel machines

Author

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  • Peter Brucker

    (Universität Osnabrück)

  • Natalia V. Shakhlevich

    (University of Leeds)

Abstract

In this paper we characterize optimal schedules for scheduling problems with parallel machines and unit processing times by providing necessary and sufficient conditions of optimality. We show that the optimality conditions for parallel machine scheduling are equivalent to detecting negative cycles in a specially defined graph. For a range of the objective functions, we give an insight into the underlying structure of the graph and specify the simplest types of cycles involved in the optimality conditions. Using our results we demonstrate that the optimality check can be performed by faster algorithms in comparison with existing approaches based on sufficient conditions.

Suggested Citation

  • Peter Brucker & Natalia V. Shakhlevich, 2016. "Necessary and sufficient optimality conditions for scheduling unit time jobs on identical parallel machines," Journal of Scheduling, Springer, vol. 19(6), pages 659-685, December.
    • Handle: RePEc:spr:jsched:v:19:y:2016:i:6:d:10.1007_s10951-016-0471-3
    DOI: 10.1007/s10951-016-0471-3
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    References listed on IDEAS

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    1. M.I. Dessouky & B.J. Lageweg & J.K. Lenstra & S.L. van de Velde, 1990. "Scheduling identical jobs on uniform parallel machines," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 44(3), pages 115-123, September.
    2. Lin, Yixun & Wang, Xiumei, 2007. "Necessary and sufficient conditions of optimality for some classical scheduling problems," European Journal of Operational Research, Elsevier, vol. 176(2), pages 809-818, January.
    3. Mitre Dourado & Rosiane Rodrigues & Jayme Szwarcfiter, 2009. "Scheduling unit time jobs with integer release dates to minimize the weighted number of tardy jobs," Annals of Operations Research, Springer, vol. 169(1), pages 81-91, July.
    4. Clemens Heuberger, 2004. "Inverse Combinatorial Optimization: A Survey on Problems, Methods, and Results," Journal of Combinatorial Optimization, Springer, vol. 8(3), pages 329-361, September.
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    Cited by:

    1. Zhijun Xu & Dehua Xu, 2018. "Single-machine scheduling with workload-dependent tool change durations and equal processing time jobs to minimize total completion time," Journal of Scheduling, Springer, vol. 21(4), pages 461-482, August.

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