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Exact algorithms for solving the constrained parallel-machine scheduling problems with divisible processing times and penalties

Author

Listed:
  • Jianping Li

    (Yunnan University)

  • Runtao Xie

    (Yunnan University)

  • Junran Lichen

    (Beijing University of Chemical Technology)

  • Guojun Hu

    (Yunnan University)

  • Pengxiang Pan

    (Yunnan University)

  • Ping Yang

    (Yunnan University)

Abstract

In this paper, we address the constrained parallel-machine scheduling problem with divisible processing times and penalties (the CPS-DTP problem), which is a further generalization of the parallel-machine scheduling problem with divisible processing times (the PS-DT problem). Concretely, given a set M of m identical machines and a set J of n independent jobs, each job has a processing time and a penalty, the processing times of these n jobs are divisible, and we implement these n jobs under the requirement that each job in J must be either continuously executed on one machine with its processing time, or rejected with its penalty that we must pay for. We may consider three versions of the CPS-DTP problem, respectively. (1) The constrained parallel-machine scheduling problem with divisible processing times and total penalties (the CPS-DTTP problem) is asked to find a subset A of J and a schedule T for jobs in A to satisfy the aforementioned requirement, the objective is to minimize the makespan of such a schedule T for jobs in A plus the summation of penalties paid for jobs not in A; (2) The constrained parallel-machine scheduling problem with divisible processing times and maximum penalty (the CPS-DTMP problem) is asked to find a subset A of J and a schedule T for jobs in A to satisfy the aforementioned requirement, the objective is to minimize the makespan of such a schedule T for jobs in A plus maximum penalty paid for jobs not in A; (3) The constrained parallel-machine scheduling problem with divisible processing times and bounded penalty (the CPS-DTBP problem) is asked to find a subset A of J and a schedule T for jobs in A to satisfy the aforementioned requirement and the summation of penalties paid for jobs not in A is no more than a fixed bound, the objective is to minimize the makespan of such a schedule T for jobs in A. As our main contributions, we design three exact algorithms to solve the CPS-DTTP problem, the CPS-DTMP problem and the CPS-DTBP problem, and these three algorithms run in time $$O((n\log n+nm)C)$$ O ( ( n log n + n m ) C ) , $$O(n^{2}\log n)$$ O ( n 2 log n ) and $$O((n\log n+nm)\log C)$$ O ( ( n log n + n m ) log C ) , respectively, where C is the optimal value of same instance for the PS-DT problem.

Suggested Citation

  • Jianping Li & Runtao Xie & Junran Lichen & Guojun Hu & Pengxiang Pan & Ping Yang, 2023. "Exact algorithms for solving the constrained parallel-machine scheduling problems with divisible processing times and penalties," Journal of Combinatorial Optimization, Springer, vol. 45(4), pages 1-19, May.
    • Handle: RePEc:spr:jcomop:v:45:y:2023:i:4:d:10.1007_s10878-023-01028-3
    DOI: 10.1007/s10878-023-01028-3
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    References listed on IDEAS

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    1. C N Potts & V A Strusevich, 2009. "Fifty years of scheduling: a survey of milestones," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 60(1), pages 41-68, May.
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