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Methods for the Solution of the Multidimensional 0/1 Knapsack Problem

Author

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  • H. Martin Weingartner

    (The University of Rochester, Rochester, New York)

  • David N. Ness

    (Massachusetts Institute of Technology, Cambridge, Massachusetts)

Abstract

In the knapsack problem, given the desirability of each of a number of items, one seeks to find that subset which satisfies a constraint on total weight. The multidimensional variant imposes constraints on additional variables of the items; the 0/1 specification means that an item is either taken or not, i.e., multiples of the same item are not considered, except possibly indirectly. Traditionally the one-dimensional knapsack problem is solved by means of dynamic programming. The multidimensional problem is usually reduced to a one-dimensional one by use of Lagrangian Multipliers that, however, do not generally yield the exact solution to the problem posed. Here some new algorithms are developed that are applied within a dynamic programming framework. Additionally, the methods make integral use of an interactive computer system in which the heuristics of the problem solver are applied and changed as the character of the solution process evolves. The problem arises in the context of capital budgeting, but has obvious applications in a variety of other areas. The methods have been employed for solving numerical problems with as many as 105 items, the parameters having been obtained from industrial applications.

Suggested Citation

  • H. Martin Weingartner & David N. Ness, 1967. "Methods for the Solution of the Multidimensional 0/1 Knapsack Problem," Operations Research, INFORMS, vol. 15(1), pages 83-103, February.
    • Handle: RePEc:inm:oropre:v:15:y:1967:i:1:p:83-103
    DOI: 10.1287/opre.15.1.83
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    Cited by:

    1. Yanhong Feng & Hongmei Wang & Zhaoquan Cai & Mingliang Li & Xi Li, 2023. "Hybrid Learning Moth Search Algorithm for Solving Multidimensional Knapsack Problems," Mathematics, MDPI, vol. 11(8), pages 1-28, April.
    2. Silvano Martello & Paolo Toth, 2003. "An Exact Algorithm for the Two-Constraint 0--1 Knapsack Problem," Operations Research, INFORMS, vol. 51(5), pages 826-835, October.
    3. Raquel Sanchis & Raúl Poler, 2019. "Enterprise Resilience Assessment—A Quantitative Approach," Sustainability, MDPI, vol. 11(16), pages 1-13, August.
    4. Nils Boysen & Simon Emde & Malte Fliedner, 2016. "The basic train makeup problem in shunting yards," OR Spectrum: Quantitative Approaches in Management, Springer;Gesellschaft für Operations Research e.V., vol. 38(1), pages 207-233, January.
    5. 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.
    6. Yalçın Akçay & Haijun Li & Susan Xu, 2007. "Greedy algorithm for the general multidimensional knapsack problem," Annals of Operations Research, Springer, vol. 150(1), pages 17-29, March.
    7. 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.
    8. Matteo Fischetti & Ivana Ljubić & Michele Monaci & Markus Sinnl, 2019. "Interdiction Games and Monotonicity, with Application to Knapsack Problems," INFORMS Journal on Computing, INFORMS, vol. 31(2), pages 390-410, April.

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