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A Branch-and-Price Algorithm for the Multiple Knapsack Problem

Author

Listed:
  • Olivier Lalonde

    (Centre Interuniversitaire de Recherche sur les Réseaux d’Entreprise, la Logistique et le Transport (CIRRELT), Université de Montréal, Montreal, Quebec H3T 1J4, Canada)

  • Jean-François Côté

    (Département d’Informatique et de Recherche Opérationnelle, Université de Montréal, Montreal, Quebec H3T 1J4, Canada)

  • Bernard Gendron

    (Université Laval, Quebec, Quebec G1V 0A6, Canada)

Abstract

The multiple knapsack problem is a well-studied combinatorial optimization problem with several practical and theoretical applications. It consists of packing some subset of n items into m knapsacks such that the total profit of the chosen items is maximum. A new formulation of the problem is presented, where a Lagrangian relaxation is derived, and we prove that it dominates the commonly used relaxations for this problem. We also present a Dantzig-Wolfe decomposition of the new formulation that we solve to optimality using a branch-and-price algorithm, where its main advantage comes from the fact that it is possible to control whether an item is included in some knapsack or not. An improved algorithm for solving the resulting packing subproblems is also introduced. Computational experiments then show that the new approach achieves state-of-the-art results.

Suggested Citation

  • Olivier Lalonde & Jean-François Côté & Bernard Gendron, 2022. "A Branch-and-Price Algorithm for the Multiple Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 34(6), pages 3134-3150, November.
    • Handle: RePEc:inm:orijoc:v:34:y:2022:i:6:p:3134-3150
    DOI: 10.1287/ijoc.2022.1223
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    References listed on IDEAS

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    1. Martello, Silvano & Toth, Paolo, 1980. "Solution of the zero-one multiple knapsack problem," European Journal of Operational Research, Elsevier, vol. 4(4), pages 276-283, April.
    2. Jean-François Côté & Manuel Iori, 2018. "The Meet-in-the-Middle Principle for Cutting and Packing Problems," INFORMS Journal on Computing, INFORMS, vol. 30(4), pages 646-661, November.
    3. Ming S. Hung & John C. Fisk, 1978. "An algorithm for 0‐1 multiple‐knapsack problems," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 25(3), pages 571-579, September.
    4. Alex Fukunaga, 2011. "A branch-and-bound algorithm for hard multiple knapsack problems," Annals of Operations Research, Springer, vol. 184(1), pages 97-119, April.
    5. Dell’Amico, Mauro & Delorme, Maxence & Iori, Manuel & Martello, Silvano, 2019. "Mathematical models and decomposition methods for the multiple knapsack problem," European Journal of Operational Research, Elsevier, vol. 274(3), pages 886-899.
    6. Belov, G. & Scheithauer, G., 2002. "A cutting plane algorithm for the one-dimensional cutting stock problem with multiple stock lengths," European Journal of Operational Research, Elsevier, vol. 141(2), pages 274-294, September.
    7. Giorgio Ingargiola & James F. Korsh, 1975. "An Algorithm for the Solution of 0-1 Loading Problems," Operations Research, INFORMS, vol. 23(6), pages 1110-1119, December.
    8. John C. Fisk & Ming S. Hung, 1979. "A heuristic routine for solving large loading problems," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 26(4), pages 643-650, December.
    9. Silvano Martello & David Pisinger & Paolo Toth, 1999. "Dynamic Programming and Strong Bounds for the 0-1 Knapsack Problem," Management Science, INFORMS, vol. 45(3), pages 414-424, March.
    10. Maxence Delorme & Manuel Iori, 2020. "Enhanced Pseudo-polynomial Formulations for Bin Packing and Cutting Stock Problems," INFORMS Journal on Computing, INFORMS, vol. 32(1), pages 101-119, January.
    11. Pisinger, David, 1999. "An exact algorithm for large multiple knapsack problems," European Journal of Operational Research, Elsevier, vol. 114(3), pages 528-541, May.
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