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Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows

Author

Listed:
  • Mads Jepsen

    (Department of Computer Science (DIKU), University of Copenhagen, DK-2100 Copenhagen Ø, Denmark)

  • Bjørn Petersen

    (Department of Computer Science (DIKU), University of Copenhagen, DK-2100 Copenhagen Ø, Denmark)

  • Simon Spoorendonk

    (Department of Computer Science (DIKU), University of Copenhagen, DK-2100 Copenhagen Ø, Denmark)

  • David Pisinger

    (Department of Computer Science (DIKU), University of Copenhagen, DK-2100 Copenhagen Ø, Denmark)

Abstract

This paper presents a branch-and-cut-and-price algorithm for the vehicle-routing problem with time windows. The standard Dantzig-Wolfe decomposition of the arc flow formulation leads to a set-partitioning problem as the master problem and an elementary shortest-path problem with resource constraints as the pricing problem. We introduce the subset-row inequalities, which are Chvatal-Gomory rank-1 cuts based on a subset of the constraints in the master problem. Applying a subset-row inequality in the master problem increases the complexity of the label-setting algorithm used to solve the pricing problem because an additional resource is added for each inequality. We propose a modified dominance criterion that makes it possible to dominate more labels by exploiting the step-like structure of the objective function of the pricing problem. Computational experiments have been performed on the Solomon benchmarks where we were able to close several instances. The results show that applying subset-row inequalities in the master problem significantly improves the lower bound and, in many cases, makes it possible to prove optimality in the root node.

Suggested Citation

  • Mads Jepsen & Bjørn Petersen & Simon Spoorendonk & David Pisinger, 2008. "Subset-Row Inequalities Applied to the Vehicle-Routing Problem with Time Windows," Operations Research, INFORMS, vol. 56(2), pages 497-511, April.
    • Handle: RePEc:inm:oropre:v:56:y:2008:i:2:p:497-511
    DOI: 10.1287/opre.1070.0449
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    References listed on IDEAS

    as
    1. Brian Kallehauge & Jesper Larsen & Oli B.G. Madsen & Marius M. Solomon, 2005. "Vehicle Routing Problem with Time Windows," Springer Books, in: Guy Desaulniers & Jacques Desrosiers & Marius M. Solomon (ed.), Column Generation, chapter 0, pages 67-98, Springer.
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    3. Stefan Irnich & Daniel Villeneuve, 2006. "The Shortest-Path Problem with Resource Constraints and k -Cycle Elimination for k (ge) 3," INFORMS Journal on Computing, INFORMS, vol. 18(3), pages 391-406, August.
    4. Stefan Irnich & Guy Desaulniers, 2005. "Shortest Path Problems with Resource Constraints," Springer Books, in: Guy Desaulniers & Jacques Desrosiers & Marius M. Solomon (ed.), Column Generation, chapter 0, pages 33-65, Springer.
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    10. Marius M. Solomon, 1987. "Algorithms for the Vehicle Routing and Scheduling Problems with Time Window Constraints," Operations Research, INFORMS, vol. 35(2), pages 254-265, April.
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