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An exact algorithm for the fixed-charge multiple knapsack problem

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  • Yamada, Takeo
  • Takeoka, Takahiro

Abstract

We formulate the fixed-charge multiple knapsack problem (FCMKP) as an extension of the multiple knapsack problem (MKP). The Lagrangian relaxation problem is easily solved, and together with a greedy heuristic we obtain a pair of upper and lower bounds quickly. We make use of these bounds in the pegging test to reduce the problem size. We also present a branch-and-bound (B&B) algorithm to solve FCMKP to optimality. This algorithm exploits the Lagrangian upper bound as well as the pegging result for pruning, and at each terminal subproblem solve MKP exactly by invoking MULKNAP code developed by Pisinger [Pisinger, D., 1999. An exact algorithm for large multiple knapsack problems. European Journal of Operational Research 114, 528-541]. As a result, we are able to solve almost all test problems with up to 32,000 items and 50 knapsacks within a few seconds on an ordinary computing environment, although the algorithm remains some weakness for small instances with relatively many knapsacks.

Suggested Citation

  • Yamada, Takeo & Takeoka, Takahiro, 2009. "An exact algorithm for the fixed-charge multiple knapsack problem," European Journal of Operational Research, Elsevier, vol. 192(2), pages 700-705, January.
    • Handle: RePEc:eee:ejores:v:192:y:2009:i:2:p:700-705
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    References listed on IDEAS

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    1. Pisinger, David, 1995. "An expanding-core algorithm for the exact 0-1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 87(1), pages 175-187, November.
    2. Marshall L. Fisher, 1981. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 27(1), pages 1-18, January.
    3. Pisinger, David, 1999. "An exact algorithm for large multiple knapsack problems," European Journal of Operational Research, Elsevier, vol. 114(3), pages 528-541, May.
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    Cited by:

    1. Diaz, Juan Esteban & Handl, Julia & Xu, Dong-Ling, 2018. "Integrating meta-heuristics, simulation and exact techniques for production planning of a failure-prone manufacturing system," European Journal of Operational Research, Elsevier, vol. 266(3), pages 976-989.
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    3. Kataoka, Seiji & Yamada, Takeo, 2014. "Upper and lower bounding procedures for the multiple knapsack assignment problem," European Journal of Operational Research, Elsevier, vol. 237(2), pages 440-447.
    4. Martello, Silvano & Monaci, Michele, 2020. "Algorithmic approaches to the multiple knapsack assignment problem," Omega, Elsevier, vol. 90(C).

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