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Sensitivity of the Optimum to Perturbations of the Profit or Weight of an Item in the Binary Knapsack Problem

Author

Listed:
  • Mhand Hifi

    (Université de Picardie Jules Verne)

  • Hedi Mhalla

    (Université de Picardie Jules Verne)

  • Slim Sadfi

    (Université de Picardie Jules Verne)

Abstract

In the binary single constraint Knapsack Problem, denoted KP, we are given a knapsack of fixed capacity c and a set of n items. Each item j, j = 1,...,n, has an associated size or weight w j and a profit p j . The goal is to determine whether or not item j, j = 1,...,n, should be included in the knapsack. The objective is to maximize the total profit without exceeding the capacity c of the knapsack. In this paper, we study the sensitivity of the optimum of the KP to perturbations of either the profit or the weight of an item. We give approximate and exact interval limits for both cases (profit and weight) and propose several polynomial time algorithms able to reach these interval limits. The performance of the proposed algorithms are evaluated on a large number of problem instances.

Suggested Citation

  • Mhand Hifi & Hedi Mhalla & Slim Sadfi, 2005. "Sensitivity of the Optimum to Perturbations of the Profit or Weight of an Item in the Binary Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 10(3), pages 239-260, November.
    • Handle: RePEc:spr:jcomop:v:10:y:2005:i:3:d:10.1007_s10878-005-4105-5
    DOI: 10.1007/s10878-005-4105-5
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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. Pisinger, David, 1999. "An exact algorithm for large multiple knapsack problems," European Journal of Operational Research, Elsevier, vol. 114(3), pages 528-541, May.
    3. Valerio de Carvalho, J. M. & Guimaraes Rodrigues, A. J., 1995. "An LP-based approach to a two-stage cutting stock problem," European Journal of Operational Research, Elsevier, vol. 84(3), pages 580-589, August.
    4. George B. Dantzig, 1957. "Discrete-Variable Extremum Problems," Operations Research, INFORMS, vol. 5(2), pages 266-288, April.
    5. Martello, Silvano & Pisinger, David & Toth, Paolo, 2000. "New trends in exact algorithms for the 0-1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 123(2), pages 325-332, June.
    6. Egon Balas & Eitan Zemel, 1980. "An Algorithm for Large Zero-One Knapsack Problems," Operations Research, INFORMS, vol. 28(5), pages 1130-1154, October.
    7. Martello, Silvano & Toth, Paolo, 1977. "An upper bound for the zero-one knapsack problem and a branch and bound algorithm," European Journal of Operational Research, Elsevier, vol. 1(3), pages 169-175, May.
    8. David Pisinger, 1999. "Core Problems in Knapsack Algorithms," Operations Research, INFORMS, vol. 47(4), pages 570-575, August.
    9. Silvano Martello & David Pisinger & Paolo Toth, 1999. "Dynamic Programming and Strong Bounds for the 0-1 Knapsack Problem," Management Science, INFORMS, vol. 45(3), pages 414-424, March.
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    Cited by:

    1. Pisinger, David & Saidi, Alima, 2017. "Tolerance analysis for 0–1 knapsack problems," European Journal of Operational Research, Elsevier, vol. 258(3), pages 866-876.

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