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Approximating a two-machine flow shop scheduling under discrete scenario uncertainty

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  • Kasperski, Adam
  • Kurpisz, Adam
  • Zieliński, Paweł

Abstract

This paper deals with the two machine permutation flow shop problem with uncertain data, whose deterministic counterpart is known to be polynomially solvable. In this paper, it is assumed that job processing times are uncertain and they are specified as a discrete scenario set. For this uncertainty representation, the min–max and min–max regret criteria are adopted. The min–max regret version of the problem is known to be weakly NP-hard even for two processing time scenarios. In this paper, it is shown that the min–max and min–max regret versions of the problem are strongly NP-hard even for two scenarios. Furthermore, the min–max version admits a polynomial time approximation scheme if the number of scenarios is constant and it is approximable with performance ratio of 2 and not (4/3−ϵ)-approximable for any ϵ>0 unless P=NP if the number of scenarios is a part of the input. On the other hand, the min–max regret version is not at all approximable even for two scenarios.

Suggested Citation

  • Kasperski, Adam & Kurpisz, Adam & Zieliński, Paweł, 2012. "Approximating a two-machine flow shop scheduling under discrete scenario uncertainty," European Journal of Operational Research, Elsevier, vol. 217(1), pages 36-43.
  • Handle: RePEc:eee:ejores:v:217:y:2012:i:1:p:36-43
    DOI: 10.1016/j.ejor.2011.08.029
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    References listed on IDEAS

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    1. Jian Yang & Gang Yu, 2002. "On the Robust Single Machine Scheduling Problem," Journal of Combinatorial Optimization, Springer, vol. 6(1), pages 17-33, March.
    2. M. R. Garey & D. S. Johnson & Ravi Sethi, 1976. "The Complexity of Flowshop and Jobshop Scheduling," Mathematics of Operations Research, INFORMS, vol. 1(2), pages 117-129, May.
    3. Richard L. Daniels & Panagiotis Kouvelis, 1995. "Robust Scheduling to Hedge Against Processing Time Uncertainty in Single-Stage Production," Management Science, INFORMS, vol. 41(2), pages 363-376, February.
    4. Aissi, Hassene & Bazgan, Cristina & Vanderpooten, Daniel, 2009. "Min-max and min-max regret versions of combinatorial optimization problems: A survey," European Journal of Operational Research, Elsevier, vol. 197(2), pages 427-438, September.
    5. Roy, Bernard, 2010. "Robustness in operational research and decision aiding: A multi-faceted issue," European Journal of Operational Research, Elsevier, vol. 200(3), pages 629-638, February.
    6. Averbakh, Igor, 2006. "The minmax regret permutation flow-shop problem with two jobs," European Journal of Operational Research, Elsevier, vol. 169(3), pages 761-766, March.
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    Citations

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    Cited by:

    1. Choi, Byung-Cheon & Chung, Kwanghun, 2016. "Min–max regret version of a scheduling problem with outsourcing decisions under processing time uncertainty," European Journal of Operational Research, Elsevier, vol. 252(2), pages 367-375.
    2. Shabtay, Dvir & Gilenson, Miri, 2023. "A state-of-the-art survey on multi-scenario scheduling," European Journal of Operational Research, Elsevier, vol. 310(1), pages 3-23.
    3. Levorato, Mario & Figueiredo, Rosa & Frota, Yuri, 2022. "Exact solutions for the two-machine robust flow shop with budgeted uncertainty," European Journal of Operational Research, Elsevier, vol. 300(1), pages 46-57.
    4. Michał Ćwik & Jerzy Józefczyk, 2018. "Heuristic algorithms for the minmax regret flow-shop problem with interval processing times," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 26(1), pages 215-238, March.
    5. Goerigk, Marc & Khosravi, Mohammad, 2023. "Optimal scenario reduction for one- and two-stage robust optimization with discrete uncertainty in the objective," European Journal of Operational Research, Elsevier, vol. 310(2), pages 529-551.

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