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Matheuristic fixed set search applied to the multidimensional knapsack problem and the knapsack problem with forfeit sets

Author

Listed:
  • Raka Jovanovic

    (Hamad bin Khalifa University)

  • Stefan Voß

    (University of Hamburg
    Pontificia Universidad Católica de Valparaíso)

Abstract

In this paper, we present a solution method for the multidimensional knapsack problem (MKP) and the knapsack problem with forfeit sets (KPFS) using a population-based matheuristic approach. Specifically, the learning mechanism of the fixed set search (FSS) metaheuristic is combined with the use of integer programming for solving subproblems. This is achieved by introducing a new ground set of elements that can be used for both the MKP and the KPFS that aim to maximize the information provided by the fixed set. The method for creating fixed sets is also adjusted to enhance the diversity of generated solutions. Compared to state-of-the-art methods for the MKP and the KPFS, the proposed approach offers an implementation that can be easily extended to other variants of the knapsack problem. Computational experiments indicate that the matheuristic FSS is highly competitive to best-performing methods from the literature. The proposed approach is robust in the sense of having a good performance for a wide range of parameter values of the method.

Suggested Citation

  • Raka Jovanovic & Stefan Voß, 2024. "Matheuristic fixed set search applied to the multidimensional knapsack problem and the knapsack problem with forfeit sets," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 46(4), pages 1329-1365, December.
    • Handle: RePEc:spr:orspec:v:46:y:2024:i:4:d:10.1007_s00291-024-00746-2
    DOI: 10.1007/s00291-024-00746-2
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    References listed on IDEAS

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    1. Yannick Vimont & Sylvain Boussier & Michel Vasquez, 2008. "Reduced costs propagation in an efficient implicit enumeration for the 01 multidimensional knapsack problem," Journal of Combinatorial Optimization, Springer, vol. 15(2), pages 165-178, February.
    2. Lai, Xiangjing & Hao, Jin-Kao & Yue, Dong, 2019. "Two-stage solution-based tabu search for the multidemand multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 274(1), pages 35-48.
    3. Chuda Basnet, 2018. "Heuristics for the multiple knapsack problem with conflicts," International Journal of Operational Research, Inderscience Enterprises Ltd, vol. 32(4), pages 514-525.
    4. P. C. Gilmore & R. E. Gomory, 1966. "The Theory and Computation of Knapsack Functions," Operations Research, INFORMS, vol. 14(6), pages 1045-1074, December.
    5. Coniglio, Stefano & Furini, Fabio & San Segundo, Pablo, 2021. "A new combinatorial branch-and-bound algorithm for the Knapsack Problem with Conflicts," European Journal of Operational Research, Elsevier, vol. 289(2), pages 435-455.
    6. Renata Mansini & M. Grazia Speranza, 2012. "CORAL: An Exact Algorithm for the Multidimensional Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 24(3), pages 399-415, August.
    7. Rosing, K. E. & ReVelle, C. S., 1997. "Heuristic concentration: Two stage solution construction," European Journal of Operational Research, Elsevier, vol. 97(1), pages 75-86, February.
    8. Woodruff, David L., 1998. "Proposals for chunking and tabu search," European Journal of Operational Research, Elsevier, vol. 106(2-3), pages 585-598, April.
    9. Marco Antonio Boschetti & Vittorio Maniezzo, 2022. "Matheuristics: using mathematics for heuristic design," 4OR, Springer, vol. 20(2), pages 173-208, June.
    10. 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.
    11. Raka Jovanovic & Antonio P. Sanfilippo & Stefan Voß, 2022. "Fixed set search applied to the multi-objective minimum weighted vertex cover problem," Journal of Heuristics, Springer, vol. 28(4), pages 481-508, August.
    12. Jiaxin Li & Yan Lan & Feng Chen & Xin Han & Jacek Blazewicz, 2021. "A Fast Algorithm for Knapsack Problem with Conflict Graph," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 38(06), pages 1-34, December.
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    Cited by:

    1. Donatella Granata & Andrea Raiconi, 2026. "Bi-objective knapsack problem with conflicts," Annals of Operations Research, Springer, vol. 357(2), pages 979-1001, February.
    2. Raka Jovanovic & Stefan Voß, 2026. "Extending GRASP towards an improved Fixed Set Search for solving the maximum disjoint dominating sets problem," Journal of Heuristics, Springer, vol. 32(2), pages 1-36, June.

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