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Reduced costs propagation in an efficient implicit enumeration for the 01 multidimensional knapsack problem

Author

Listed:
  • Yannick Vimont

    (Parc Scientifique Georges Besse)

  • Sylvain Boussier

    (Parc Scientifique Georges Besse)

  • Michel Vasquez

    (Parc Scientifique Georges Besse)

Abstract

In a previous work we proposed a variable fixing heuristics for the 0-1 Multidimensional knapsack problem (01MDK). This approach uses fractional optima calculated in hyperplanes which contain the binary optimum. This algorithm obtained best lower bounds on the OR-Library benchmarks. Although it is very attractive in terms of results, this method does not prove the optimality of the solutions found and may fix variables to a non-optimal value. In this paper, we propose an implicit enumeration based on a reduced costs analysis which tends to fix non-basic variables to their exact values. The combination of two specific constraint propagations based on reduced costs and an efficient enumeration framework enable us to fix variables on the one hand and to prune significantly the search tree on the other hand. Experimentally, our work provides two main contributions: (1) we obtain several new optimal solutions on hard instances of the OR-Library and (2) we reduce the bounds of the number of items at the optimum on several harder instances.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:jcomop:v:15:y:2008:i:2:d:10.1007_s10878-007-9074-4
    DOI: 10.1007/s10878-007-9074-4
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    References listed on IDEAS

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    1. Vasquez, Michel & Vimont, Yannick, 2005. "Improved results on the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 165(1), pages 70-81, August.
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    Cited by:

    1. 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.
    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. 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.
    4. Chen, Yuning & Hao, Jin-Kao, 2014. "A “reduce and solve” approach for the multiple-choice multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 239(2), pages 313-322.
    5. Gao, Chao & Lu, Guanzhou & Yao, Xin & Li, Jinlong, 2017. "An iterative pseudo-gap enumeration approach for the Multidimensional Multiple-choice Knapsack Problem," European Journal of Operational Research, Elsevier, vol. 260(1), pages 1-11.
    6. José García & Paola Moraga & Matias Valenzuela & Hernan Pinto, 2020. "A db-Scan Hybrid Algorithm: An Application to the Multidimensional Knapsack Problem," Mathematics, MDPI, vol. 8(4), pages 1-22, April.
    7. Setzer, Thomas & Blanc, Sebastian M., 2020. "Empirical orthogonal constraint generation for Multidimensional 0/1 Knapsack Problems," European Journal of Operational Research, Elsevier, vol. 282(1), pages 58-70.
    8. Yoon, Yourim & Kim, Yong-Hyuk & Moon, Byung-Ro, 2012. "A theoretical and empirical investigation on the Lagrangian capacities of the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 218(2), pages 366-376.

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