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A theoretical and empirical investigation on the Lagrangian capacities of the 0-1 multidimensional knapsack problem

  • Yoon, Yourim
  • Kim, Yong-Hyuk
  • Moon, Byung-Ro
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    We present a novel Lagrangian method to find good feasible solutions in theoretical and empirical aspects. After investigating the concept of Lagrangian capacity, which is the value of the capacity constraint that Lagrangian relaxation can find an optimal solution, we formally reintroduce Lagrangian capacity suitable to the 0-1 multidimensional knapsack problem and present its new geometric equivalent condition including a new associated property. Based on the property, we propose a new Lagrangian heuristic that finds high-quality feasible solutions of the 0-1 multidimensional knapsack problem. We verify the advantage of the proposed heuristic by experiments. We make comparisons with existing Lagrangian approaches on benchmark data and show that the proposed method performs well on large-scale data.

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    Article provided by Elsevier in its journal European Journal of Operational Research.

    Volume (Year): 218 (2012)
    Issue (Month): 2 ()
    Pages: 366-376

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    Handle: RePEc:eee:ejores:v:218:y:2012:i:2:p:366-376
    Contact details of provider: Web page: http://www.elsevier.com/locate/eor

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    1. Magazine, M. J. & Oguz, Osman, 1984. "A heuristic algorithm for the multidimensional zero-one knapsack problem," European Journal of Operational Research, Elsevier, vol. 16(3), pages 319-326, June.
    2. Frieze, A. M. & Clarke, M. R. B., 1984. "Approximation algorithms for the m-dimensional 0-1 knapsack problem: Worst-case and probabilistic analyses," European Journal of Operational Research, Elsevier, vol. 15(1), pages 100-109, January.
    3. Balev, Stefan & Yanev, Nicola & Freville, Arnaud & Andonov, Rumen, 2008. "A dynamic programming based reduction procedure for the multidimensional 0-1 knapsack problem," European Journal of Operational Research, Elsevier, vol. 186(1), pages 63-76, April.
    4. Marshall L. Fisher, 1981. "The Lagrangian Relaxation Method for Solving Integer Programming Problems," Management Science, INFORMS, vol. 27(1), pages 1-18, January.
    5. Wilbaut, Christophe & Salhi, Saïd & Hanafi, Saïd, 2009. "An iterative variable-based fixation heuristic for the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 199(2), pages 339-348, December.
    6. Kaparis, Konstantinos & Letchford, Adam N., 2008. "Local and global lifted cover inequalities for the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 186(1), pages 91-103, April.
    7. Bagchi, Ansuman & Bhattacharyya, Nalinaksha & Chakravarti, Nilotpal, 1996. "LP relaxation of the two dimensional knapsack problem with box and GUB constraints," European Journal of Operational Research, Elsevier, vol. 89(3), pages 609-617, March.
    8. Boyer, V. & Elkihel, M. & El Baz, D., 2009. "Heuristics for the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 199(3), pages 658-664, December.
    9. Vasquez, Michel & Vimont, Yannick, 2005. "Improved results on the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 165(1), pages 70-81, August.
    10. Wilbaut, Christophe & Hanafi, Said, 2009. "New convergent heuristics for 0-1 mixed integer programming," European Journal of Operational Research, Elsevier, vol. 195(1), pages 62-74, May.
    11. Freville, Arnaud & Plateau, Gerard, 1993. "An exact search for the solution of the surrogate dual of the 0-1 bidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 68(3), pages 413-421, August.
    12. Caprara, Alberto & Kellerer, Hans & Pferschy, Ulrich & Pisinger, David, 2000. "Approximation algorithms for knapsack problems with cardinality constraints," European Journal of Operational Research, Elsevier, vol. 123(2), pages 333-345, June.
    13. Hanafi, Said & Freville, Arnaud, 1998. "An efficient tabu search approach for the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 106(2-3), pages 659-675, April.
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