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Lagrangian Relaxation Methods for Solving the Minimum Fleet Size Multiple Traveling Salesman Problem with Time Windows

Author

Listed:
  • Jacques Desrosiers

    (École des Hautes Études Commerciales and GERAD, Montréal, Québec, Canada H3T 1V6)

  • Michel Sauvé

    (École des Hautes Études Commerciales and GERAD, Montréal, Québec, Canada H3T 1V6)

  • François Soumis

    (École Polytechnique and Centre de Recherche sur les Transports, Montréal, Québec, Canada H3C 3A7)

Abstract

We consider the problem of finding the minimum number of vehicles required to visit once a set of nodes subject to time window constraints, for a homogeneous fleet of vehicles located at a common depot. This problem can be formulated as a network flow problem with additional time constraints. The paper presents an optimal solution approach using the augmented Lagrangian method. Two Lagrangian relaxations are studied. In the first one, the time constraints are relaxed producing network subproblems which are easy to solve, but the bound obtained is weak. In the second relaxation, constraints requiring that each node be visited are relaxed producing shortest path subproblems with time window constraints and integrality conditions. The bound produced is always excellent. Numerical results for several actual school busing problems with up to 223 nodes are discussed. Comparisons with a set partitioning formulation solved by column generation are given.

Suggested Citation

  • Jacques Desrosiers & Michel Sauvé & François Soumis, 1988. "Lagrangian Relaxation Methods for Solving the Minimum Fleet Size Multiple Traveling Salesman Problem with Time Windows," Management Science, INFORMS, vol. 34(8), pages 1005-1022, August.
    • Handle: RePEc:inm:ormnsc:v:34:y:1988:i:8:p:1005-1022
    DOI: 10.1287/mnsc.34.8.1005
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    Cited by:

    1. Dondo, Rodolfo & Cerda, Jaime, 2007. "A cluster-based optimization approach for the multi-depot heterogeneous fleet vehicle routing problem with time windows," European Journal of Operational Research, Elsevier, vol. 176(3), pages 1478-1507, February.
    2. Monique Guignard, 2003. "Lagrangean relaxation," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 11(2), pages 151-200, December.

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