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Solution Methods for the Multiple-Choice Knapsack Problem and Their Applications

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  • Tibor Szkaliczki

    (HUN-REN Institute for Computer Science and Control, 1111 Budapest, Hungary)

Abstract

The Knapsack Problem belongs to the best-studied classical problems in combinatorial optimization. The Multiple-choice Knapsack Problem (MCKP) represents a generalization of the problem, with various application fields such as industry, transportation, telecommunication, national defense, bioinformatics, finance, and life. We found a lack of survey papers on MCKP. This paper overviews MCKP and presents its variants, solution methods, and applications. Traditional operational research methods solving the knapsack problem, such as dynamic programming, greedy heuristics, and branch-and-bound algorithms, can be adapted to MCKP. Only a few algorithms appear to have solved the problem in recent years. We found various related problems during the literature study and explored the broad spectrum of application areas. We intend to inspire research into MCKP algorithms and motivate experts from different domains to apply MCKP.

Suggested Citation

  • Tibor Szkaliczki, 2025. "Solution Methods for the Multiple-Choice Knapsack Problem and Their Applications," Mathematics, MDPI, vol. 13(7), pages 1-35, March.
    • Handle: RePEc:gam:jmathe:v:13:y:2025:i:7:p:1097-:d:1621731
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    References listed on IDEAS

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    1. P. C. Gilmore & R. E. Gomory, 1966. "The Theory and Computation of Knapsack Functions," Operations Research, INFORMS, vol. 14(6), pages 1045-1074, December.
    2. Dudzinski, Krzysztof & Walukiewicz, Stanislaw, 1987. "Exact methods for the knapsack problem and its generalizations," European Journal of Operational Research, Elsevier, vol. 28(1), pages 3-21, January.
    3. Xue Bai, 2012. "A Mathematical Framework for Data Quality Management in Enterprise Systems," INFORMS Journal on Computing, INFORMS, vol. 24(4), pages 648-664, November.
    4. Agostinho Agra & Cristina Requejo & Eulália Santos, 2016. "Implicit cover inequalities," Journal of Combinatorial Optimization, Springer, vol. 31(3), pages 1111-1129, April.
    5. Yogesh K. Agarwal, 2002. "Design of Capacitated Multicommodity Networks with Multiple Facilities," Operations Research, INFORMS, vol. 50(2), pages 333-344, April.
    6. Prabhakant Sinha & Andris A. Zoltners, 1979. "The Multiple-Choice Knapsack Problem," Operations Research, INFORMS, vol. 27(3), pages 503-515, June.
    7. Hugh Everett, 1963. "Generalized Lagrange Multiplier Method for Solving Problems of Optimum Allocation of Resources," Operations Research, INFORMS, vol. 11(3), pages 399-417, June.
    8. Pisinger, David, 1995. "A minimal algorithm for the multiple-choice knapsack problem," European Journal of Operational Research, Elsevier, vol. 83(2), pages 394-410, June.
    9. 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.
    10. Alberto Ceselli & Giovanni Righini, 2006. "A Branch-and-Price Algorithm for the Multilevel Generalized Assignment Problem," Operations Research, INFORMS, vol. 54(6), pages 1172-1184, December.
    11. Yongchang Su & Xinran Li, 2024. "Treatment effect quantiles in stratified randomized experiments and matched observational studies," Biometrika, Biometrika Trust, vol. 111(1), pages 235-254.
    12. Martello, Silvano & Toth, Paolo, 1995. "A note on exact algorithms for the bottleneck generalized assignment problem," European Journal of Operational Research, Elsevier, vol. 83(3), pages 711-712, June.
    13. Bretthauer, Kurt M. & Shetty, Bala, 2002. "The nonlinear knapsack problem - algorithms and applications," European Journal of Operational Research, Elsevier, vol. 138(3), pages 459-472, May.
    14. Nauss, Robert M., 1978. "The 0-1 knapsack problem with multiple choice constraints," European Journal of Operational Research, Elsevier, vol. 2(2), pages 125-131, March.
    15. Tsesmetzis, Dimitrios & Roussaki, Ioanna & Sykas, Efstathios, 2008. "QoS-aware service evaluation and selection," European Journal of Operational Research, Elsevier, vol. 191(3), pages 1101-1112, December.
    16. Martello, Silvano & Toth, Paolo, 1995. "The bottleneck generalized assignment problem," European Journal of Operational Research, Elsevier, vol. 83(3), pages 621-638, June.
    17. Madjid Tavana & Kaveh Khalili-Damghani & Amir-Reza Abtahi, 2013. "A fuzzy multidimensional multiple-choice knapsack model for project portfolio selection using an evolutionary algorithm," Annals of Operations Research, Springer, vol. 206(1), pages 449-483, July.
    18. Pisinger, David, 2001. "Budgeting with bounded multiple-choice constraints," European Journal of Operational Research, Elsevier, vol. 129(3), pages 471-480, March.
    19. Stéphane Dauzère-Pérès & Michael Hassoun & Alejandro Sendon, 2016. "Allocating metrology capacity to multiple heterogeneous machines," International Journal of Production Research, Taylor & Francis Journals, vol. 54(20), pages 6082-6091, October.
    20. N. Samphaiboon & Y. Yamada, 2000. "Heuristic and Exact Algorithms for the Precedence-Constrained Knapsack Problem," Journal of Optimization Theory and Applications, Springer, vol. 105(3), pages 659-676, June.
    21. Kameshwaran, S. & Narahari, Y., 2009. "Nonconvex piecewise linear knapsack problems," European Journal of Operational Research, Elsevier, vol. 192(1), pages 56-68, January.
    22. George Kozanidis, 2009. "Solving the linear multiple choice knapsack problem with two objectives: profit and equity," Computational Optimization and Applications, Springer, vol. 43(2), pages 261-294, June.
    23. 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.
    24. Zhu, Xiaoyan & Wilhelm, Wilbert E., 2007. "Three-stage approaches for optimizing some variations of the resource constrained shortest-path sub-problem in a column generation context," European Journal of Operational Research, Elsevier, vol. 183(2), pages 564-577, December.
    25. András Prékopa & Merve Unuvar, 2015. "Single Commodity Stochastic Network Design Under Probabilistic Constraint with Discrete Random Variables," Operations Research, INFORMS, vol. 63(6), pages 1512-1527, December.
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