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Reformulations by Discretization for Piecewise Linear Integer Multicommodity Network Flow Problems

Author

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  • Bernard Gendron

    (Département d’informatique et de recherche opérationnelle, Université de Montréal, Montréal, Québec H3T 1J4, Canada; and Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT), Montréal, Québec H3C 3J7, Canada)

  • Luis Gouveia

    (Department of Statistics and Operations Research–CMAFCIO, Faculdade de Ciencias da Universidade de Lisboa, 1749-016 Lisboa, Portugal)

Abstract

We consider the piecewise linear multicommodity network flow problem with the addition of a constraint specifying that the total flow on each arc must be an integer. This problem has applications in transportation and logistics, where total flows might represent vehicles or containers filled with different products. We introduce formulations that exploit this integrality constraint by adapting to our problem a technique known as discretization that has been used to derive mixed-integer programming models for several combinatorial optimization problems. We enhance the discretized models either by adding valid inequalities derived from cut-set inequalities or by using flow disaggregation techniques. Since the size of the formulations derived from discretization and flow disaggregation rapidly increases with problem dimensions, we develop an efficient and effective Lagrangian relaxation method to compute lower and upper bounds. We perform computational results on a large set of randomly generated instances that allow us to compare the relative efficiency of the different modeling alternatives (flow disaggregation plus addition of cut-set inequalities with or without discretization), when used within the Lagrangian relaxation approach.

Suggested Citation

  • Bernard Gendron & Luis Gouveia, 2017. "Reformulations by Discretization for Piecewise Linear Integer Multicommodity Network Flow Problems," Transportation Science, INFORMS, vol. 51(2), pages 629-649, May.
    • Handle: RePEc:inm:ortrsc:v:51:y:2017:i:2:p:629-649
    DOI: 10.1287/trsc.2015.0634
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    References listed on IDEAS

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    1. Luís Gouveia & Pedro Moura, 2012. "Enhancing discretized formulations: the knapsack reformulation and the star reformulation," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(1), pages 52-74, April.
    2. 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.
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    4. ORTEGA , Francisco & WOLSEY, Laurence A., 2003. "A branch-and-cut algorithm for the single-commodity, uncapacitated, fixed-charge network flow problem," LIDAM Reprints CORE 1611, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    5. 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.
    6. Mervat Chouman & Teodor Gabriel Crainic & Bernard Gendron, 2017. "Commodity Representations and Cut-Set-Based Inequalities for Multicommodity Capacitated Fixed-Charge Network Design," Transportation Science, INFORMS, vol. 51(2), pages 650-667, May.
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    Cited by:

    1. Fortz, Bernard & Gouveia, Luís & Joyce-Moniz, Martim, 2017. "Models for the piecewise linear unsplittable multicommodity flow problems," European Journal of Operational Research, Elsevier, vol. 261(1), pages 30-42.
    2. Gendron, Bernard & Hanafi, Saïd & Todosijević, Raca, 2018. "Matheuristics based on iterative linear programming and slope scaling for multicommodity capacitated fixed charge network design," European Journal of Operational Research, Elsevier, vol. 268(1), pages 70-81.

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