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Mathematical models and decomposition methods for the multiple knapsack problem

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  • Dell’Amico, Mauro
  • Delorme, Maxence
  • Iori, Manuel
  • Martello, Silvano

Abstract

We consider the multiple knapsack problem, that calls for the optimal assignment of a set of items, each having a profit and a weight, to a set of knapsacks, each having a maximum capacity. The problem has relevant managerial implications and is known to be very difficult to solve in practice for instances of realistic size. We review the main results from the literature, including a classical mathematical model and a number of improvement techniques. We then present two new pseudo-polynomial formulations, together with specifically tailored decomposition algorithms to tackle the practical difficulty of the problem. Extensive computational experiments show the effectiveness of the proposed approaches.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ejores:v:274:y:2019:i:3:p:886-899
    DOI: 10.1016/j.ejor.2018.10.043
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    1. WOLSEY, Laurence A., 1977. "Valid inequalities, covering problems and discrete dynamic programs," LIDAM Reprints CORE 302, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
    2. Simon, Jay & Apte, Aruna & Regnier, Eva, 2017. "An application of the multiple knapsack problem: The self-sufficient marine," European Journal of Operational Research, Elsevier, vol. 256(3), pages 868-876.
    3. Wascher, Gerhard & Hau[ss]ner, Heike & Schumann, Holger, 2007. "An improved typology of cutting and packing problems," European Journal of Operational Research, Elsevier, vol. 183(3), pages 1109-1130, December.
    4. 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.
    5. 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.
    6. 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.
    7. P. C. Gilmore & R. E. Gomory, 1961. "A Linear Programming Approach to the Cutting-Stock Problem," Operations Research, INFORMS, vol. 9(6), pages 849-859, December.
    8. Guntram Scheithauer, 2018. "One-Dimensional Cutting Stock," International Series in Operations Research & Management Science, in: Introduction to Cutting and Packing Optimization, chapter 0, pages 73-122, Springer.
    9. Egon Balas & Eitan Zemel, 1980. "An Algorithm for Large Zero-One Knapsack Problems," Operations Research, INFORMS, vol. 28(5), pages 1130-1154, October.
    10. Martinovic, J. & Scheithauer, G. & Valério de Carvalho, J.M., 2018. "A comparative study of the arcflow model and the one-cut model for one-dimensional cutting stock problems," European Journal of Operational Research, Elsevier, vol. 266(2), pages 458-471.
    11. Samuel Eilon & Nicos Christofides, 1971. "The Loading Problem," Management Science, INFORMS, vol. 17(5), pages 259-268, January.
    12. Harald Dyckhoff, 1981. "A New Linear Programming Approach to the Cutting Stock Problem," Operations Research, INFORMS, vol. 29(6), pages 1092-1104, December.
    13. Gianni Codato & Matteo Fischetti, 2006. "Combinatorial Benders' Cuts for Mixed-Integer Linear Programming," Operations Research, INFORMS, vol. 54(4), pages 756-766, August.
    14. Kataoka, Seiji & Yamada, Takeo, 2014. "Upper and lower bounding procedures for the multiple knapsack assignment problem," European Journal of Operational Research, Elsevier, vol. 237(2), pages 440-447.
    15. Dyckhoff, Harald, 1990. "A typology of cutting and packing problems," European Journal of Operational Research, Elsevier, vol. 44(2), pages 145-159, January.
    16. Pisinger, David, 1999. "An exact algorithm for large multiple knapsack problems," European Journal of Operational Research, Elsevier, vol. 114(3), pages 528-541, May.
    17. 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.
    18. Mohamed Esseghir Lalami & Moussa Elkihel & Didier El Baz & Vincent Boyer, 2012. "A procedure-based heuristic for 0-1 Multiple Knapsack Problems," International Journal of Mathematics in Operational Research, Inderscience Enterprises Ltd, vol. 4(3), pages 214-224.
    19. Jean-François Côté & Mauro Dell'Amico & Manuel Iori, 2014. "Combinatorial Benders' Cuts for the Strip Packing Problem," Operations Research, INFORMS, vol. 62(3), pages 643-661, June.
    20. Labbe, Martine & Laporte, Gilbert & Martello, Silvano, 2003. "Upper bounds and algorithms for the maximum cardinality bin packing problem," European Journal of Operational Research, Elsevier, vol. 149(3), pages 490-498, September.
    21. Clautiaux, François & Hanafi, Saïd & Macedo, Rita & Voge, Marie-Émilie & Alves, Cláudio, 2017. "Iterative aggregation and disaggregation algorithm for pseudo-polynomial network flow models with side constraints," European Journal of Operational Research, Elsevier, vol. 258(2), pages 467-477.
    22. Delorme, Maxence & Iori, Manuel & Martello, Silvano, 2016. "Bin packing and cutting stock problems: Mathematical models and exact algorithms," European Journal of Operational Research, Elsevier, vol. 255(1), pages 1-20.
    23. 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.
    24. 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.
    25. David Pisinger & Mikkel Sigurd, 2007. "Using Decomposition Techniques and Constraint Programming for Solving the Two-Dimensional Bin-Packing Problem," INFORMS Journal on Computing, INFORMS, vol. 19(1), pages 36-51, February.
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    5. 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.
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    9. Mancini, Simona & Ciavotta, Michele & Meloni, Carlo, 2021. "The Multiple Multidimensional Knapsack with Family-Split Penalties," European Journal of Operational Research, Elsevier, vol. 289(3), pages 987-998.

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