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No-idle, no-wait: when shop scheduling meets dominoes, Eulerian paths and Hamiltonian paths

Author

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  • J.-C. Billaut

    (Université de Tours)

  • F. Della Croce

    (Politecnico di Torino
    CNR, IEIIT)

  • F. Salassa

    (Politecnico di Torino)

  • V. T’kindt

    (Université de Tours)

Abstract

In shop scheduling, several applications require that some components perform consecutively. We refer to “no-idle schedules” if machines are required to operate with no inserted idle time and to “no-wait schedules” if tasks cannot wait between the end of an operation and the start of the following one. We consider here no-idle/no-wait shop scheduling problems with makespan as the performance measure and determine related complexity results. We first analyse the two-machine no-idle/no-wait flow shop problem and show that it is equivalent to a special version of the game of dominoes which is polynomially solvable by tackling an Eulerian path problem on a directed graph. We present for this problem an O(n) exact algorithm. As a by-product, we show that the Hamiltonian path problem on a digraph G(V, A) with a special structure (where any two vertices i and j either have all successors in common or have no common successors) reduces to the two-machine no-idle/no-wait flow shop problem. Correspondingly, we provide a new polynomially solvable special case of the Hamiltonian path problem. Then, we show that also the m-machine no-idle/no-wait flow shop problem is polynomially solvable and provide an $$O(mn \log n)$$ O ( m n log n ) exact algorithm. Finally, we prove that the decision versions of the two-machine job shop problem and the two-machine open shop problem are NP-complete in the strong sense.

Suggested Citation

  • J.-C. Billaut & F. Della Croce & F. Salassa & V. T’kindt, 2019. "No-idle, no-wait: when shop scheduling meets dominoes, Eulerian paths and Hamiltonian paths," Journal of Scheduling, Springer, vol. 22(1), pages 59-68, February.
    • Handle: RePEc:spr:jsched:v:22:y:2019:i:1:d:10.1007_s10951-018-0562-4
    DOI: 10.1007/s10951-018-0562-4
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    References listed on IDEAS

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    1. Kalczynski, Pawel Jan & Kamburowski, Jerzy, 2007. "On no-wait and no-idle flow shops with makespan criterion," European Journal of Operational Research, Elsevier, vol. 178(3), pages 677-685, May.
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    Cited by:

    1. Ahmadian, Mohammad Mahdi & Khatami, Mostafa & Salehipour, Amir & Cheng, T.C.E., 2021. "Four decades of research on the open-shop scheduling problem to minimize the makespan," European Journal of Operational Research, Elsevier, vol. 295(2), pages 399-426.
    2. Federico Della Croce & Andrea Grosso & Fabio Salassa, 2021. "Minimizing total completion time in the two-machine no-idle no-wait flow shop problem," Journal of Heuristics, Springer, vol. 27(1), pages 159-173, April.

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