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2-Path Cuts for the Vehicle Routing Problem with Time Windows

Author

Listed:
  • Niklas Kohl

    (University of Copenhagen, Denmark)

  • Jacques Desrosiers

    (École des Hautes Études Commerciales, Montréal, Quebec, Canada and GERAD)

  • Oli B. G. Madsen

    (Technical University of Denmark, Copenhagen, Denmark)

  • Marius M. Solomon

    (Northeastern University, Boston, Massachusetts and GERAD)

  • François Soumis

    (École Polytechnique de Montréal, Montréal, Quebec, Canada and GERAD)

Abstract

This paper introduces a strong valid inequality, the 2-path cut, to produce better lower bounds for the vehicle routing problem with time windows. It also develops an effective separation algorithm to find such inequalities. We next incorporate them as needed in the master problem of a Dantzig–Wolfe decomposition approach. In this enhanced optimization algorithm, the coupling constraints require that each customer be serviced. The subproblem is a shortest path problem with time window and capacity constraints. We apply branch and bound to obtain integer solutions. We first branch on the number of vehicles if this is fractional, and then on the flow variables. The algorithm has been implemented and tested on problems of up to 100 customers from the Solomon datasets. It has succeeded in solving to optimality several previously unsolved problems and a new 150-customer problem. In addition, the algorithm proved faster than algorithms previously considered in the literature. These computational results indicate the effectiveness of the valid inequalities we have developed.

Suggested Citation

  • Niklas Kohl & Jacques Desrosiers & Oli B. G. Madsen & Marius M. Solomon & François Soumis, 1999. "2-Path Cuts for the Vehicle Routing Problem with Time Windows," Transportation Science, INFORMS, vol. 33(1), pages 101-116, February.
    • Handle: RePEc:inm:ortrsc:v:33:y:1999:i:1:p:101-116
    DOI: 10.1287/trsc.33.1.101
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    References listed on IDEAS

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    1. Martin Desrochers & Jacques Desrosiers & Marius Solomon, 1992. "A New Optimization Algorithm for the Vehicle Routing Problem with Time Windows," Operations Research, INFORMS, vol. 40(2), pages 342-354, April.
    2. Yvan Dumas & Jacques Desrosiers & Eric Gelinas & Marius M. Solomon, 1995. "An Optimal Algorithm for the Traveling Salesman Problem with Time Windows," Operations Research, INFORMS, vol. 43(2), pages 367-371, April.
    3. Niklas Kohl & Oli B. G. Madsen, 1997. "An Optimization Algorithm for the Vehicle Routing Problem with Time Windows Based on Lagrangian Relaxation," Operations Research, INFORMS, vol. 45(3), pages 395-406, June.
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