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Solving the Single-Sink, Fixed-Charge, Multiple-Choice Transportation Problem by Dynamic Programming

Author

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  • Tue R. L. Christensen

    (Department of Economics and Business, Aarhus University, 8210 Aarhus V, Denmark)

  • Kim Allan Andersen

    (Department of Economics and Business, Aarhus University, 8210 Aarhus V, Denmark)

  • Andreas Klose

    (Department of Mathematics, Aarhus University, 8000 Aarhus C, Denmark)

Abstract

This paper considers a minimum-cost network flow problem in a bipartite graph with a single sink. The transportation costs exhibit a staircase cost structure because such types of transportation cost functions are often found in practice. We present a dynamic programming algorithm for solving this so-called single-sink, fixed-charge, multiple-choice transportation problem exactly. The method exploits heuristics and lower bounds to peg binary variables, improve bounds on flow variables, and reduce the state-space variable. In this way, the dynamic programming method is able to solve large instances with up to 10,000 nodes and 10 different transportation modes in a few seconds, much less time than required by a widely used mixed-integer programming solver and other methods proposed in the literature for this problem.

Suggested Citation

  • Tue R. L. Christensen & Kim Allan Andersen & Andreas Klose, 2013. "Solving the Single-Sink, Fixed-Charge, Multiple-Choice Transportation Problem by Dynamic Programming," Transportation Science, INFORMS, vol. 47(3), pages 428-438, August.
    • Handle: RePEc:inm:ortrsc:v:47:y:2013:i:3:p:428-438
    DOI: 10.1287/trsc.1120.0431
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    References listed on IDEAS

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    1. Isabel Correia & M. Captivo, 2003. "A Lagrangean Heuristic for a Modular Capacitated Location Problem," Annals of Operations Research, Springer, vol. 122(1), pages 141-161, September.
    2. Prabhakant Sinha & Andris A. Zoltners, 1979. "The Multiple-Choice Knapsack Problem," Operations Research, INFORMS, vol. 27(3), pages 503-515, June.
    3. Keely L. Croxton & Bernard Gendron & Thomas L. Magnanti, 2007. "Variable Disaggregation in Network Flow Problems with Piecewise Linear Costs," Operations Research, INFORMS, vol. 55(1), pages 146-157, February.
    4. Pisinger, David, 1995. "A minimal algorithm for the multiple-choice knapsack problem," European Journal of Operational Research, Elsevier, vol. 83(2), pages 394-410, June.
    5. Kameshwaran, S. & Narahari, Y., 2009. "Nonconvex piecewise linear knapsack problems," European Journal of Operational Research, Elsevier, vol. 192(1), pages 56-68, January.
    6. Keely L. Croxton & Bernard Gendron & Thomas L. Magnanti, 2003. "A Comparison of Mixed-Integer Programming Models for Nonconvex Piecewise Linear Cost Minimization Problems," Management Science, INFORMS, vol. 49(9), pages 1268-1273, September.
    7. Y. T. Herer & M. J. Rosenblatt & I. Hefter, 1996. "Fast Algorithms for Single-Sink Fixed Charge Transportation Problems with Applications to Manufacturing and Transportation," Transportation Science, INFORMS, vol. 30(4), pages 276-290, November.
    8. Correia, Isabel & Gouveia, Luís & Saldanha-da-Gama, Francisco, 2010. "Discretized formulations for capacitated location problems with modular distribution costs," European Journal of Operational Research, Elsevier, vol. 204(2), pages 237-244, July.
    9. Eitan Zemel, 1980. "The Linear Multiple Choice Knapsack Problem," Operations Research, INFORMS, vol. 28(6), pages 1412-1423, December.
    10. Keely L. Croxton & Bernard Gendron & Thomas L. Magnanti, 2003. "Models and Methods for Merge-in-Transit Operations," Transportation Science, INFORMS, vol. 37(1), pages 1-22, February.
    11. Sophie D. Lapierre & Angel B. Ruiz & Patrick Soriano, 2004. "Designing Distribution Networks: Formulations and Solution Heuristic," Transportation Science, INFORMS, vol. 38(2), pages 174-187, May.
    12. Qi, Xiangtong, 2007. "Order splitting with multiple capacitated suppliers," European Journal of Operational Research, Elsevier, vol. 178(2), pages 421-432, April.
    13. Holmberg, Kaj, 1994. "Solving the staircase cost facility location problem with decomposition and piecewise linearization," European Journal of Operational Research, Elsevier, vol. 75(1), pages 41-61, May.
    14. Bahram Alidaee & Gary A. Kochenberger, 2005. "A Note on a Simple Dynamic Programming Approach to the Single-Sink, Fixed-Charge Transportation Problem," Transportation Science, INFORMS, vol. 39(1), pages 140-143, February.
    15. Holmberg, Kaj & Ling, Jonas, 1997. "A Lagrangean heuristic for the facility location problem with staircase costs," European Journal of Operational Research, Elsevier, vol. 97(1), pages 63-74, February.
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    Cited by:

    1. Amir Ardestani-Jaafari & Erick Delage, 2018. "The Value of Flexibility in Robust Location–Transportation Problems," Transportation Science, INFORMS, vol. 52(1), pages 189-209, January.
    2. Bertazzi, Luca & Maggioni, Francesca, 2018. "A stochastic multi-stage fixed charge transportation problem: Worst-case analysis of the rolling horizon approach," European Journal of Operational Research, Elsevier, vol. 267(2), pages 555-569.

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