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Knapsack problem with probability constraints

Author

Listed:
  • Alexei Gaivoronski
  • Abdel Lisser

  • Rafael Lopez
  • Hu Xu

Abstract

This paper is dedicated to a study of different extensions of the classical knapsack problem to the case when different elements of the problem formulation are subject to a degree of uncertainty described by random variables. This brings the knapsack problem into the realm of stochastic programming. Two different model formulations are proposed, based on the introduction of probability constraints. The first one is a static quadratic knapsack with a probability constraint on the capacity of the knapsack. The second one is a two-stage quadratic knapsack model, with recourse, where we introduce a probability constraint on the capacity of the knapsack in the second stage. As far as we know, this is the first time such a constraint has been used in a two-stage model. The solution techniques are based on the semidefinite relaxations. This allows for solving large instances, for which exact methods cannot be used. Numerical experiments on a set of randomly generated instances are discussed below. Copyright Springer Science+Business Media, LLC. 2011

Suggested Citation

  • Alexei Gaivoronski & Abdel Lisser & Rafael Lopez & Hu Xu, 2011. "Knapsack problem with probability constraints," Journal of Global Optimization, Springer, vol. 49(3), pages 397-413, March.
    • Handle: RePEc:spr:jglopt:v:49:y:2011:i:3:p:397-413
    DOI: 10.1007/s10898-010-9566-0
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    References listed on IDEAS

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    1. Rendl, F. & Sotirov, R., 2007. "Bounds for the quadratic assignment problem using the bundle method," Other publications TiSEM b6d298bc-77c9-4a6d-a043-5, Tilburg University, School of Economics and Management.
    2. Anton J. Kleywegt & Jason D. Papastavrou, 1998. "The Dynamic and Stochastic Knapsack Problem," Operations Research, INFORMS, vol. 46(1), pages 17-35, February.
    3. C. Helmberg & F. Rendl & R. Weismantel, 2000. "A Semidefinite Programming Approach to the Quadratic Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 4(2), pages 197-215, June.
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    1. Dogacan Yilmaz & İ. Esra Büyüktahtakın, 2025. "A non-anticipative learning-optimization framework for solving multi-stage stochastic programs," Annals of Operations Research, Springer, vol. 355(3), pages 2859-2899, December.
    2. Zahra Beheshti & Siti Shamsuddin & Siti Yuhaniz, 2013. "Binary Accelerated Particle Swarm Algorithm (BAPSA) for discrete optimization problems," Journal of Global Optimization, Springer, vol. 57(2), pages 549-573, October.
    3. Agnès Gorge & Abdel Lisser & Riadh Zorgati, 2012. "Stochastic nuclear outages semidefinite relaxations," Computational Management Science, Springer, vol. 9(3), pages 363-379, August.
    4. Timonina-Farkas, Anna & Katsifou, Argyro & Seifert, Ralf W., 2020. "Product assortment and space allocation strategies to attract loyal and non-loyal customers," European Journal of Operational Research, Elsevier, vol. 285(3), pages 1058-1076.

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