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A multi-criteria approach to approximate solution of multiple-choice knapsack problem

Author

Listed:
  • Ewa M. Bednarczuk

    (Systems Research Institute, Polish Academy of Sciences
    Warsaw University of Technology)

  • Janusz Miroforidis

    (Systems Research Institute, Polish Academy of Sciences)

  • Przemysław Pyzel

    (Systems Research Institute, Polish Academy of Sciences)

Abstract

We propose a method for finding approximate solutions to multiple-choice knapsack problems. To this aim we transform the multiple-choice knapsack problem into a bi-objective optimization problem whose solution set contains solutions of the original multiple-choice knapsack problem. The method relies on solving a series of suitably defined linearly scalarized bi-objective problems. The novelty which makes the method attractive from the computational point of view is that we are able to solve explicitly those linearly scalarized bi-objective problems with the help of the closed-form formulae. The method is computationally analyzed on a set of large-scale problem instances (test problems) of two categories: uncorrelated and weakly correlated. Computational results show that after solving, in average 10 scalarized bi-objective problems, the optimal value of the original knapsack problem is approximated with the accuracy comparable to the accuracies obtained by the greedy algorithm and an exact algorithm. More importantly, the respective approximate solution to the original knapsack problem (for which the approximate optimal value is attained) can be found without resorting to the dynamic programming. In the test problems, the number of multiple-choice constraints ranges up to hundreds with hundreds variables in each constraint.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:coopap:v:70:y:2018:i:3:d:10.1007_s10589-018-9988-z
    DOI: 10.1007/s10589-018-9988-z
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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.
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    Cited by:

    1. Yu-Hsien Liao, 2021. "Axiomatic Results for Weighted Allocation Rules under Multiattribute Situations," Mathematics, MDPI, vol. 9(6), pages 1-14, March.
    2. Yu-Hsien Liao, 2024. "Evaluating Mechanism and Related Axiomatic Results under Multiple Considerations," Mathematics, MDPI, vol. 12(9), pages 1-14, May.
    3. Ewa M. Bednarczuk & Ignacy Kaliszewski & Janusz Miroforidis, 2024. "Approximate solutions to the multiple-choice knapsack problem by multiobjectivization and Chebyshev scalarization," Operations Research and Decisions, Wroclaw University of Science and Technology, Faculty of Management, vol. 34(4), pages 53-66.
    4. Yu-Hsien Liao, 2024. "Fuzzy Assessment Mechanisms under Multi-Objective Considerations," Mathematics, MDPI, vol. 12(19), pages 1-15, September.

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