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The knapsack problem with a minimum filling constraint

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  • Zhou Xu

Abstract

We study a knapsack problem with an additional minimum filling constraint, such that the total weight of selected items cannot be less than a given threshold. The problem has several applications in shipping, e‐commerce, and transportation service procurement. When the threshold equals the knapsack capacity, even finding a feasible solution to the problem is NP‐hard. Therefore, we consider the case when the ratio α of threshold to capacity is less than 1. For this case, we develop an approximation scheme that returns a feasible solution with a total profit not less than (1 ‐ ε) times the total profit of an optimal solution for any ε > 0, and with a running time polynomial in the number of items, 1/ε, and 1/(1‐α). © 2012 Wiley Periodicals, Inc. Naval Research Logistics, 2013

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  • Zhou Xu, 2013. "The knapsack problem with a minimum filling constraint," Naval Research Logistics (NRL), John Wiley & Sons, vol. 60(1), pages 56-63, February.
    • Handle: RePEc:wly:navres:v:60:y:2013:i:1:p:56-63
    DOI: 10.1002/nav.21520
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    1. Hans Kellerer & Ulrich Pferschy, 2004. "Improved Dynamic Programming in Connection with an FPTAS for the Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 8(1), pages 5-11, March.
    2. Harvey M. Salkin & Cornelis A. De Kluyver, 1975. "The knapsack problem: A survey," Naval Research Logistics Quarterly, John Wiley & Sons, vol. 22(1), pages 127-144, March.
    3. Chi To Ng & Mikhail Yakovlevich Kovalyov & Tai Chiu Edwin Cheng, 2008. "An FPTAS for a supply scheduling problem with non‐monotone cost functions," Naval Research Logistics (NRL), John Wiley & Sons, vol. 55(3), pages 194-199, April.
    4. Hans Kellerer & Ulrich Pferschy, 1999. "A New Fully Polynomial Time Approximation Scheme for the Knapsack Problem," Journal of Combinatorial Optimization, Springer, vol. 3(1), pages 59-71, July.
    5. Eugene L. Lawler, 1979. "Fast Approximation Algorithms for Knapsack Problems," Mathematics of Operations Research, INFORMS, vol. 4(4), pages 339-356, November.
    6. Andrea Bettinelli & Alberto Ceselli & Giovanni Righini, 2010. "A branch-and-price algorithm for the variable size bin packing problem with minimum filling constraint," Annals of Operations Research, Springer, vol. 179(1), pages 221-241, September.
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    Cited by:

    1. Feng Li & Zhou Xu & Zhi-Long Chen, 2020. "Production and Transportation Integration for Commit-to-Delivery Mode with General Shipping Costs," INFORMS Journal on Computing, INFORMS, vol. 32(4), pages 1012-1029, October.
    2. Muter, İbrahim & Sezer, Zeynep, 2018. "Algorithms for the one-dimensional two-stage cutting stock problem," European Journal of Operational Research, Elsevier, vol. 271(1), pages 20-32.
    3. Wang, Danni & Xiao, Fan & Zhou, Lei & Liang, Zhe, 2020. "Two-dimensional skiving and cutting stock problem with setup cost based on column-and-row generation," European Journal of Operational Research, Elsevier, vol. 286(2), pages 547-563.

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