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Lawler’s minmax cost problem under uncertainty

Author

Listed:
  • Nadia Brauner

    (Université Grenoble Alpes, G-SCOP
    CNRS, G-SCOP)

  • Gerd Finke

    (Université Grenoble Alpes, G-SCOP
    CNRS, G-SCOP)

  • Yakov Shafransky

    (NAS of Belarus)

Abstract

The well-known $$O(n^2)$$ O ( n 2 ) minmax cost algorithm of Lawler (Manag Sci 19(5):544–546, 1973) was developed to minimize the maximum cost of jobs processed by a single machine under precedence constraints. We first develop a fast updating algorithm to obtain optimal solutions for two neighboring instances. This method will be used throughout this article. Then we consider job cost functions that depend on the completion time and on one or more additional numerical parameters. The parameters are uncertain and take values from given intervals. Under the uncertainty, we apply the minmax regret criterion for choosing a solution. We generalize results by Brauner et al. (J Sched, 2015) for decomposable cost functions with deterministic processing times and a single uncertain parameter to general cost functions. We describe different conditions, under which minmax regret solutions can be obtained with the time complexity $$O(n^3)$$ O ( n 3 ) or $$O(n^2)$$ O ( n 2 ) . Then the updating algorithm is applied to the lateness model by Kasperski (Oper Res Lett 33:431–436, 2005) where both the processing time and the due date of each job are uncertain. The original $$O(n^4)$$ O ( n 4 ) running time is improved to the time complexity $$O(n^3)$$ O ( n 3 ) . Finally, we extend the cost functions with a single uncertain parameter to those with a vector of additional uncertain parameters, improve one of the complexity results by Volgenant and Duin (Comput Oper Res 37:909–915, 2010) and solve some new problems.

Suggested Citation

  • Nadia Brauner & Gerd Finke & Yakov Shafransky, 2017. "Lawler’s minmax cost problem under uncertainty," Journal of Combinatorial Optimization, Springer, vol. 34(1), pages 31-46, July.
  • Handle: RePEc:spr:jcomop:v:34:y:2017:i:1:d:10.1007_s10878-016-0051-7
    DOI: 10.1007/s10878-016-0051-7
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    References listed on IDEAS

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    1. E. L. Lawler, 1973. "Optimal Sequencing of a Single Machine Subject to Precedence Constraints," Management Science, INFORMS, vol. 19(5), pages 544-546, January.
    2. ,, 2000. "Problems And Solutions," Econometric Theory, Cambridge University Press, vol. 16(2), pages 287-299, April.
    3. J. Michael Moore, 1968. "An n Job, One Machine Sequencing Algorithm for Minimizing the Number of Late Jobs," Management Science, INFORMS, vol. 15(1), pages 102-109, September.
    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.
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    Cited by:

    1. Fridman, Ilia & Pesch, Erwin & Shafransky, Yakov, 2020. "Minimizing maximum cost for a single machine under uncertainty of processing times," European Journal of Operational Research, Elsevier, vol. 286(2), pages 444-457.
    2. Rubing Chen & Jinjiang Yuan, 2020. "Single-machine scheduling of proportional-linearly deteriorating jobs with positional due indices," 4OR, Springer, vol. 18(2), pages 177-196, June.
    3. Xue Li & Zhengwen He & Nengmin Wang & Mario Vanhoucke, 2022. "Multimode time-cost-robustness trade-off project scheduling problem under uncertainty," Journal of Combinatorial Optimization, Springer, vol. 43(5), pages 1173-1202, July.

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