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New convergent heuristics for 0-1 mixed integer programming

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  • Wilbaut, Christophe
  • Hanafi, Said

Abstract

Several hybrid methods have recently been proposed for solving 0-1 mixed integer programming problems. Some of these methods are based on the complete exploration of small neighborhoods. In this paper, we present several convergent algorithms that solve a series of small sub-problems generated by exploiting information obtained from a series of relaxations. These algorithms generate a sequence of upper bounds and a sequence of lower bounds around the optimal value. First, the principle of a linear programming-based algorithm is summarized, and several enhancements of this algorithm are presented. Next, new hybrid heuristics that use linear programming and/or mixed integer programming relaxations are proposed. The mixed integer programming (MIP) relaxation diversifies the search process and introduces new constraints in the problem. This MIP relaxation also helps to reduce the gap between the final upper bound and lower bound. Our algorithms improved 14 best-known solutions from a set of 108 available and correlated instances of the 0-1 multidimensional Knapsack problem. Other encouraging results obtained for 0-1 MIP problems are also presented.

Suggested Citation

  • 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.
    • Handle: RePEc:eee:ejores:v:195:y:2009:i:1:p:62-74
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    References listed on IDEAS

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    1. Hanafi, Said & Glover, Fred, 2007. "Exploiting nested inequalities and surrogate constraints," European Journal of Operational Research, Elsevier, vol. 179(1), pages 50-63, May.
    2. Soyster, A. L. & Lev, B. & Slivka, W., 1978. "Zero-one programming with many variables and few constraints," European Journal of Operational Research, Elsevier, vol. 2(3), pages 195-201, May.
    3. 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.
    4. María Osorio & Fred Glover & Peter Hammer, 2002. "Cutting and Surrogate Constraint Analysis for Improved Multidimensional Knapsack Solutions," Annals of Operations Research, Springer, vol. 117(1), pages 71-93, November.
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    3. Setzer, Thomas & Blanc, Sebastian M., 2020. "Empirical orthogonal constraint generation for Multidimensional 0/1 Knapsack Problems," European Journal of Operational Research, Elsevier, vol. 282(1), pages 58-70.
    4. Renata Mansini & M. Grazia Speranza, 2012. "CORAL: An Exact Algorithm for the Multidimensional Knapsack Problem," INFORMS Journal on Computing, INFORMS, vol. 24(3), pages 399-415, August.
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    6. Chen, Yuning & Hao, Jin-Kao, 2014. "A “reduce and solve” approach for the multiple-choice multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 239(2), pages 313-322.
    7. Al-Shihabi, Sameh, 2021. "A Novel Core-Based Optimization Framework for Binary Integer Programs- the Multidemand Multidimesional Knapsack Problem as a Test Problem," Operations Research Perspectives, Elsevier, vol. 8(C).
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    9. Saïd Hanafi & Christophe Wilbaut, 2011. "Improved convergent heuristics for the 0-1 multidimensional knapsack problem," Annals of Operations Research, Springer, vol. 183(1), pages 125-142, March.
    10. 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.
    11. Yoon, Yourim & Kim, Yong-Hyuk & Moon, Byung-Ro, 2012. "A theoretical and empirical investigation on the Lagrangian capacities of the 0-1 multidimensional knapsack problem," European Journal of Operational Research, Elsevier, vol. 218(2), pages 366-376.
    12. Rong, Aiying & Figueira, José Rui & Lahdelma, Risto, 2015. "A two phase approach for the bi-objective non-convex combined heat and power production planning problem," European Journal of Operational Research, Elsevier, vol. 245(1), pages 296-308.

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