Abstract
We prove the NP-hardness of constructing a minmax regret solution for the two-machine flow shop problem under the interval uncertainty of the job processing times. The problem complexity status has been an open question for over the past 20 years. We establish the NP-hardness of this problem using a so-called alternative scheme for proving the NP-hardness of optimization problems. Also, we show that the problem is non-approximable in polynomial time.
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Cheng, T. C. E., Shafransky, Y., & Ng, C. T. (2016). An alternative approach for proving the NP-hardness of optimization problems. European Journal of Operational Research, 248, 52–58.
Garey, M. R., & Johnson, D. S. (1979). Computers and intractability: A guide to the theory of NP-completeness. San Francisco, CA: Freeman.
Johnson, S. M. (1954). Optimal two and three stage production schedules with set up times included. Naval Research Logistics Quarterly, 1, 61–68.
Kasperski, A., & Zieliński, P. (2014). Minmax (regret) scheduling problems. In Y. N. Sotskov, F. Werner (Eds.) Sequencing and scheduling with inaccurate data, Nova Science Publishers, pp. 159–210.
Kasperski, A. (2008). Discrete optimization with interval data. Minmax regret and fuzzy approach. Berlin, Heidelberg: Springer.
Kasperski, A., Kurpisz, A., & Zieliński, P. (2012). Approximating a two-machine flow shop scheduling under discrete scenario uncertainty. European Journal of Operational Research, 217, 36–43.
Kouvelis, P., Daniels, R. L., & Vairaktarakis, G. (2000). Robust scheduling of two-machine flow shop with uncertain processing times. IIE Transactions, 32, 421–432.
Kouvelis, P., & Yu, G. (1997). Robust discrete optimization and its applications. Dordrecht: Kluwer.
Lin, Y., & Deng, J. (1999). On the structure of all optimal solutions of the two-machine flow shop scheduling problem. OR Transactions, 3(2), 10–20.
Lin, Y., & Wang, X. (2007). Necessary and sufficient conditions of optimality for some classical scheduling problems. European Journal of Operational Research, 176, 809–818.
Acknowledgements
The authors are deeply grateful to an anonymous referee for the valuable comments and suggestions. The research of the first author has been partially supported by the BRFFR project \(\Phi \)15CO-043.
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Shafransky, Y., Shinkarevich, V. On the complexity of constructing a minmax regret solution for the two-machine flow shop problem under the interval uncertainty. J Sched 23, 745–749 (2020). https://doi.org/10.1007/s10951-020-00663-6
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DOI: https://doi.org/10.1007/s10951-020-00663-6