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Approximation algorithms for the workload partition problem and applications to scheduling with variable processing times

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  • Oron, Daniel
  • Shabtay, Dvir
  • Steiner, George

Abstract

In the Workload Partition Problem(WPP) we are given a set of n jobs to be scheduled on a set of m identical parallel machines. Each job has its own workload and the scheduling cost on each machine is a convex function of the total workload of the jobs assigned to it. The objective is to minimize the total cost on the set of m machines. Shabtay and Kaspi (2006) showed that the WPP is equivalent to a scheduling problem on m identical machines with controllable processing times and with the scheduling criterion of minimizing the makespan. They also proved that the WPP is NP-hard when m=2. However, they left as an open question whether the problem is ordinary or strongly NP-hard. Moreover, they provided no practical tools to solve the problem. We bridge those gaps in the literature by showing that the WWP problem is strongly NP-hard when m is part of the input. Furthermore, we present two different approximation algorithms for solving the WWP problem. The first one is a fully polynomial time approximation scheme (FPTAS) for a fixed number of machines, while the second is a modification of the well-known longest processing time (LPT) heuristic. We show that our modified LPT heuristic guarantees a solution with a constant approximation ratio, whose value depends on the instance parameters.

Suggested Citation

  • Oron, Daniel & Shabtay, Dvir & Steiner, George, 2017. "Approximation algorithms for the workload partition problem and applications to scheduling with variable processing times," European Journal of Operational Research, Elsevier, vol. 256(2), pages 384-391.
    • Handle: RePEc:eee:ejores:v:256:y:2017:i:2:p:384-391
    DOI: 10.1016/j.ejor.2016.06.062
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    References listed on IDEAS

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    1. Shabtay, Dvir & Kaspi, Moshe, 2006. "Parallel machine scheduling with a convex resource consumption function," European Journal of Operational Research, Elsevier, vol. 173(1), pages 92-107, August.
    2. Clyde L. Monma & Alexander Schrijver & Michael J. Todd & Victor K. Wei, 1990. "Convex Resource Allocation Problems on Directed Acyclic Graphs: Duality, Complexity, Special Cases, and Extensions," Mathematics of Operations Research, INFORMS, vol. 15(4), pages 736-748, November.
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    Cited by:

    1. Biber Nurit & Mor Baruch & Schlissel Yitzhak & Shapira Dana, 2023. "Lot scheduling involving completion time problems on identical parallel machines," Operational Research, Springer, vol. 23(1), pages 1-29, March.

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