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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. 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.
    4. Golden, Bruce & Levy, Larry & Dahl, Roy, 1981. "Two generalizations of the traveling salesman problem," Omega, Elsevier, vol. 9(4), pages 439-441.
    5. Petra Bauer, 1997. "The Circuit Polytope: Facets," Mathematics of Operations Research, INFORMS, vol. 22(1), pages 110-145, February.
    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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    2. E. Angelelli & R. Mansini & M. Vindigni, 2016. "The Stochastic and Dynamic Traveling Purchaser Problem," Transportation Science, INFORMS, vol. 50(2), pages 642-658, May.
    3. Monkman, Susan K. & Morrice, Douglas J. & Bard, Jonathan F., 2008. "A production scheduling heuristic for an electronics manufacturer with sequence-dependent setup costs," European Journal of Operational Research, Elsevier, vol. 187(3), pages 1100-1114, June.
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    8. Mingyu Xiao & Jianan Zhang & Weibo Lin, 2022. "Parameterized algorithms and complexity for the traveling purchaser problem and its variants," Journal of Combinatorial Optimization, Springer, vol. 44(4), pages 2269-2285, November.
    9. Luigi Moccia & Jean‐François Cordeau & Manlio Gaudioso & Gilbert Laporte, 2006. "A branch‐and‐cut algorithm for the quay crane scheduling problem in a container terminal," Naval Research Logistics (NRL), John Wiley & Sons, vol. 53(1), pages 45-59, February.
    10. Manerba, Daniele & Mansini, Renata, 2012. "An exact algorithm for the Capacitated Total Quantity Discount Problem," European Journal of Operational Research, Elsevier, vol. 222(2), pages 287-300.
    11. Miranda, Pablo A. & Blazquez, Carola A. & Obreque, Carlos & Maturana-Ross, Javier & Gutierrez-Jarpa, Gabriel, 2018. "The bi-objective insular traveling salesman problem with maritime and ground transportation costs," European Journal of Operational Research, Elsevier, vol. 271(3), pages 1014-1036.
    12. Matteo Fischetti & Ivana Ljubić & Markus Sinnl, 2017. "Redesigning Benders Decomposition for Large-Scale Facility Location," Management Science, INFORMS, vol. 63(7), pages 2146-2162, July.
    13. Bianchessi, N. & Mansini, R. & Speranza, M.G., 2014. "The distance constrained multiple vehicle traveling purchaser problem," European Journal of Operational Research, Elsevier, vol. 235(1), pages 73-87.
    14. Jaehn, Florian & Meissner, Finn, 2022. "The rebound effect in transportation," Omega, Elsevier, vol. 108(C).
    15. Riera-Ledesma, Jorge & Salazar-Gonzalez, Juan Jose, 2005. "A heuristic approach for the Travelling Purchaser Problem," European Journal of Operational Research, Elsevier, vol. 162(1), pages 142-152, April.
    16. Manerba, Daniele & Mansini, Renata & Riera-Ledesma, Jorge, 2017. "The Traveling Purchaser Problem and its variants," European Journal of Operational Research, Elsevier, vol. 259(1), pages 1-18.
    17. Jorge Riera-Ledesma & Juan-José Salazar-González, 2006. "Solving the asymmetric traveling purchaser problem," Annals of Operations Research, Springer, vol. 144(1), pages 83-97, April.
    18. Susan Heath & Jonathan Bard & Douglas Morrice, 2013. "A GRASP for simultaneously assigning and sequencing product families on flexible assembly lines," Annals of Operations Research, Springer, vol. 203(1), pages 295-323, March.
    19. E. Angelelli & R. Mansini & M. Vindigni, 2009. "Exploring greedy criteria for the dynamic traveling purchaser problem," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 17(2), pages 141-158, June.
    20. Mansini, Renata & Savelsbergh, Martin W.P. & Tocchella, Barbara, 2012. "The supplier selection problem with quantity discounts and truckload shipping," Omega, Elsevier, vol. 40(4), pages 445-455.

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