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Pricing the multiple-choice nested knapsack problem

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  • Drexl, Andreas
  • Jørnsten, Kurt

Abstract

The multiple-choice nested knapsack problem (MCKP) is a generalization of the ordinary knapsack problem, where the set of items is partitioned into classes. The binary choice of selecting an item is replaced by taking exactly one item out of each class of items. Due to the fact that the MCKP is an NP-hard integer program dual prices are not readily available. In this paper we propose a family of linear programming models the optimal solution of which is integral for many instances. The models are evaluated experimentally using a well-defined testbed consisting of 9,000 instances. Overall our methodology produces an integral solution for 75% of the instances considered. In particular, for two out of five distribution types studied at least one of the models produces "almost always" an integral solution. Hence, in most of the cases there exists a linear price function that supports the optimal allocation.

Suggested Citation

  • Drexl, Andreas & Jørnsten, Kurt, 2007. "Pricing the multiple-choice nested knapsack problem," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 626, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.
    • Handle: RePEc:zbw:cauman:626
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    References listed on IDEAS

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    1. 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.
    2. Marshall L. Fisher, 1981. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 27(1), pages 1-18, January.
    3. Prabhakant Sinha & Andris A. Zoltners, 1979. "The Multiple-Choice Knapsack Problem," Operations Research, INFORMS, vol. 27(3), pages 503-515, June.
    4. 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.
    5. Barcia, Paulo & Jornsten, Kurt, 1990. "Improved Lagrangean decomposition: An application to the generalized assignment problem," European Journal of Operational Research, Elsevier, vol. 46(1), pages 84-92, May.
    6. 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.
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    Cited by:

    1. Briskorn, Dirk & Knust, Sigrid, 2008. "On Circular 2-Factorizations of the Complete Tripartite Graph," Manuskripte aus den Instituten für Betriebswirtschaftslehre der Universität Kiel 636, Christian-Albrechts-Universität zu Kiel, Institut für Betriebswirtschaftslehre.

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