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A Branch-and-Cut Algorithm for the Undirected Traveling Purchaser Problem

Author

Listed:
  • Gilbert Laporte

    (HEC Montréal, 3000 chemin de la Côte-Sainte-Catherine, Montréal, Quebec, Canada H3T 2A7)

  • Jorge Riera-Ledesma

    (DEIOC, Facultad de Matemáticas, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain)

  • Juan-José Salazar-González

    (DEIOC, Facultad de Matemáticas, Universidad de La Laguna, 38271 La Laguna, Tenerife, Spain)

Abstract

The purpose of this paper is to present a branch-and-cut algorithm for the undirected Traveling Purchaser Problem which consists of determining a minimum-cost route through a subset of markets, where the cost is the sum of travel and purchase costs. The problem is formulated as an integer linear program, and several families of valid inequalities are derived to strengthen the linear relaxation. The polyhedral structure of the formulation is analyzed and several classes of valid inequalities are proved to be facet defining. A branch-and-cut procedure is developed and tested over four classes of randomly generated instances. Results show that the proposed algorithm outperforms all previous approaches and can optimally solve instances containing up to 200 markets.

Suggested Citation

  • Gilbert Laporte & Jorge Riera-Ledesma & Juan-José Salazar-González, 2003. "A Branch-and-Cut Algorithm for the Undirected Traveling Purchaser Problem," Operations Research, INFORMS, vol. 51(6), pages 940-951, December.
    • Handle: RePEc:inm:oropre:v:51:y:2003:i:6:p:940-951
    DOI: 10.1287/opre.51.6.940.24921
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    References listed on IDEAS

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    1. Manfred W. Padberg & M. R. Rao, 1982. "Odd Minimum Cut-Sets and b -Matchings," Mathematics of Operations Research, INFORMS, vol. 7(1), pages 67-80, February.
    2. Singh, Kashi N. & van Oudheusden, Dirk L., 1997. "A branch and bound algorithm for the traveling purchaser problem," European Journal of Operational Research, Elsevier, vol. 97(3), pages 571-579, March.
    3. Golden, Bruce & Levy, Larry & Dahl, Roy, 1981. "Two generalizations of the traveling salesman problem," Omega, Elsevier, vol. 9(4), pages 439-441.
    4. Petra Bauer, 1997. "The Circuit Polytope: Facets," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 110-145, February.
    5. Harlan Crowder & Ellis L. Johnson & Manfred Padberg, 1983. "Solving Large-Scale Zero-One Linear Programming Problems," Operations Research, INFORMS, vol. 31(5), pages 803-834, October.
    6. Matteo Fischetti & Juan José Salazar González & Paolo Toth, 1997. "A Branch-and-Cut Algorithm for the Symmetric Generalized Traveling Salesman Problem," Operations Research, INFORMS, vol. 45(3), pages 378-394, June.
    7. Robert L. Karg & Gerald L. Thompson, 1964. "A Heuristic Approach to Solving Travelling Salesman Problems," Management Science, INFORMS, vol. 10(2), pages 225-248, January.
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    Cited by:

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