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A common approximation framework for early work, late work, and resource leveling problems

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  • Györgyi, Péter
  • Kis, Tamás

Abstract

We study the approximability of four scheduling problems on identical parallel machines. In the late work minimization problem, the jobs have arbitrary processing times and a common due date, and the objective is to minimize the late work, defined as the sum of the portion of the jobs done after the due date. A related problem is the maximization of the early work, defined as the sum of the portion of the jobs done before the due date. We describe a polynomial time approximation scheme for the early work maximization problem, and we extended it to the late work minimization problem after shifting the objective function by a positive value that depends on the problem data. We also prove an inapproximability result for the latter problem if the objective function is shifted by a constant which does not depend on the input. These results remain valid even if the number of the jobs assigned to the same machine is bounded. This leads to an extension of our approximation scheme to two variants of the resource leveling problem with unit time jobs, for which no approximation algorithm is known.

Suggested Citation

  • Györgyi, Péter & Kis, Tamás, 2020. "A common approximation framework for early work, late work, and resource leveling problems," European Journal of Operational Research, Elsevier, vol. 286(1), pages 129-137.
    • Handle: RePEc:eee:ejores:v:286:y:2020:i:1:p:129-137
    DOI: 10.1016/j.ejor.2020.03.032
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    References listed on IDEAS

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    1. Xin Chen & Malgorzata Sterna & Xin Han & Jacek Blazewicz, 2016. "Scheduling on parallel identical machines with late work criterion: Offline and online cases," Journal of Scheduling, Springer, vol. 19(6), pages 729-736, December.
    2. Chen, Xin & Liang, Yage & Sterna, Małgorzata & Wang, Wen & Błażewicz, Jacek, 2020. "Fully polynomial time approximation scheme to maximize early work on parallel machines with common due date," European Journal of Operational Research, Elsevier, vol. 284(1), pages 67-74.
    3. C. N. Potts & L. N. Van Wassenhove, 1992. "Single Machine Scheduling to Minimize Total Late Work," Operations Research, INFORMS, vol. 40(3), pages 586-595, June.
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    6. Neumann, K. & Zimmermann, J., 2000. "Procedures for resource leveling and net present value problems in project scheduling with general temporal and resource constraints," European Journal of Operational Research, Elsevier, vol. 127(2), pages 425-443, December.
    7. Malgorzata Sterna & Kateryna Czerniachowska, 2017. "Polynomial Time Approximation Scheme for Two Parallel Machines Scheduling with a Common Due Date to Maximize Early Work," Journal of Optimization Theory and Applications, Springer, vol. 174(3), pages 927-944, September.
    8. Drótos, Márton & Kis, Tamás, 2011. "Resource leveling in a machine environment," European Journal of Operational Research, Elsevier, vol. 212(1), pages 12-21, July.
    9. Rieck, Julia & Zimmermann, Jürgen & Gather, Thorsten, 2012. "Mixed-integer linear programming for resource leveling problems," European Journal of Operational Research, Elsevier, vol. 221(1), pages 27-37.
    10. Cédric Verbeeck & Vincent Peteghem & Mario Vanhoucke & Pieter Vansteenwegen & El-Houssaine Aghezzaf, 2017. "A metaheuristic solution approach for the time-constrained project scheduling problem," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 39(2), pages 353-371, March.
    11. M. Y. Kovalyov & C. N. Potts & L. N. van Wassenhove, 1994. "A Fully Polynomial Approximation Scheme for Scheduling a Single Machine to Minimize Total Weighted Late Work," Mathematics of Operations Research, INFORMS, vol. 19(1), pages 86-93, February.
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    Cited by:

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    2. Sterna, Małgorzata, 2021. "Late and early work scheduling: A survey," Omega, Elsevier, vol. 104(C).
    3. Xiaofei Liu & Yajie Li & Weidong Li & Jinhua Yang, 2023. "Combinatorial approximation algorithms for the maximum bounded connected bipartition problem," Journal of Combinatorial Optimization, Springer, vol. 45(1), pages 1-21, January.

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