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A “reduce and solve” approach for the multiple-choice multidimensional knapsack problem

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  • Chen, Yuning
  • Hao, Jin-Kao

Abstract

The multiple-choice multidimensional knapsack problem (MMKP) is a well-known NP-hard combinatorial optimization problem with a number of important applications. In this paper, we present a “reduce and solve” heuristic approach which combines problem reduction techniques with an Integer Linear Programming (ILP) solver (CPLEX). The key ingredient of the proposed approach is a set of group fixing and variable fixing rules. These fixing rules rely mainly on information from the linear relaxation of the given problem and aim to generate reduced critical subproblem to be solved by the ILP solver. Additional strategies are used to explore the space of the reduced problems. Extensive experimental studies over two sets of 37 MMKP benchmark instances in the literature show that our approach competes favorably with the most recent state-of-the-art algorithms. In particular, for the set of 27 conventional benchmarks, the proposed approach finds an improved best lower bound for 11 instances and as a by-product improves all the previous best upper bounds. For the 10 additional instances with irregular structures, the method improves 7 best known results.

Suggested Citation

  • 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.
  • Handle: RePEc:eee:ejores:v:239:y:2014:i:2:p:313-322
    DOI: 10.1016/j.ejor.2014.05.025
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    References listed on IDEAS

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    1. Abdelkader Sbihi, 2007. "A best first search exact algorithm for the Multiple-choice Multidimensional Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 13(4), pages 337-351, May.
    2. 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.
    3. Pisinger, David, 2001. "Budgeting with bounded multiple-choice constraints," European Journal of Operational Research, Elsevier, vol. 129(3), pages 471-480, March.
    4. Jakob Puchinger & Günther R. Raidl & Ulrich Pferschy, 2010. "The Multidimensional Knapsack Problem: Structure and Algorithms," INFORMS Journal on Computing, INFORMS, vol. 22(2), pages 250-265, May.
    5. Wilbaut, Christophe & Hanafi, Said, 2009. "New convergent heuristics for 0-1 mixed integer programming," European Journal of Operational Research, Elsevier, vol. 195(1), pages 62-74, May.
    6. Nawal Cherfi & Mhand Hifi, 2009. "Hybrid algorithms for the Multiple-choice Multi-dimensional Knapsack Problem," International Journal of Operational Research, Inderscience Enterprises Ltd, vol. 5(1), pages 89-109.
    7. 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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    2. Ewa M. Bednarczuk & Janusz Miroforidis & Przemysław Pyzel, 2018. "A multi-criteria approach to approximate solution of multiple-choice knapsack problem," Computational Optimization and Applications, Springer, vol. 70(3), pages 889-910, July.
    3. Diallo, Claver & Venkatadri, Uday & Khatab, Abdelhakim & Liu, Zhuojun, 2018. "Optimal selective maintenance decisions for large serial k-out-of-n: G systems under imperfect maintenance," Reliability Engineering and System Safety, Elsevier, vol. 175(C), pages 234-245.
    4. 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.
    5. Renata Mansini & Roberto Zanotti, 2020. "A Core-Based Exact Algorithm for the Multidimensional Multiple Choice Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 32(4), pages 1061-1079, October.
    6. Caserta, Marco & Voß, Stefan, 2019. "The robust multiple-choice multidimensional knapsack problem," Omega, Elsevier, vol. 86(C), pages 16-27.
    7. Lamanna, Leonardo & Mansini, Renata & Zanotti, Roberto, 2022. "A two-phase kernel search variant for the multidimensional multiple-choice knapsack problem," European Journal of Operational Research, Elsevier, vol. 297(1), pages 53-65.

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