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The fractional multidimensional knapsack problem: solution and uniqueness

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  • John Y. Zhu

    (University of Kansas)

Abstract

Every solution to the fractional multidimensional knapsack problem admits a cost–benefit interpretation: for each solution, it is possible to associate a price to each dimension such that an object is placed (not placed) in the knapsack if its cost under the associated price system is strictly less (greater) than its benefit. In particular, an object is fractionally placed in the knapsack only if its cost equals its benefit. The characterization is sharper than what is implied by strong duality. Practical conditions are then provided that guarantee the solution is unique. The results yield a simple criterion for when an incremental addition to the collection of objects placeable in the knapsack is worthwhile.

Suggested Citation

  • John Y. Zhu, 2022. "The fractional multidimensional knapsack problem: solution and uniqueness," Economic Theory Bulletin, Springer;Society for the Advancement of Economic Theory (SAET), vol. 10(1), pages 95-103, May.
    • Handle: RePEc:spr:etbull:v:10:y:2022:i:1:d:10.1007_s40505-022-00217-3
    DOI: 10.1007/s40505-022-00217-3
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    References listed on IDEAS

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    1. Sven de Vries & Rakesh V. Vohra, 2003. "Combinatorial Auctions: A Survey," INFORMS Journal on Computing, INFORMS, vol. 15(3), pages 284-309, August.
    2. Hai Nguyen & Thành Nguyen & Alexander Teytelboym, 2021. "Stability in Matching Markets with Complex Constraints," Management Science, INFORMS, vol. 67(12), pages 7438-7454, December.
    3. Mansini, Renata & Speranza, Maria Grazia, 1999. "Heuristic algorithms for the portfolio selection problem with minimum transaction lots," European Journal of Operational Research, Elsevier, vol. 114(2), pages 219-233, April.
    4. Jayaraman, Vaidyanathan & Pirkul, Hasan, 2001. "Planning and coordination of production and distribution facilities for multiple commodities," European Journal of Operational Research, Elsevier, vol. 133(2), pages 394-408, January.
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    More about this item

    Keywords

    Multidimensional knapsack; Linear programming; Optimization; Production; Resource allocation;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D24 - Microeconomics - - Production and Organizations - - - Production; Cost; Capital; Capital, Total Factor, and Multifactor Productivity; Capacity
    • H40 - Public Economics - - Publicly Provided Goods - - - General

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