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Column generation decomposition with the degenerate constraints in the subproblem

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  • Metrane, Abdelmoutalib
  • Soumis, François
  • Elhallaoui, Issmail

Abstract

In this paper, we propose a new Dantzig-Wolfe decomposition for degenerate linear programs with the non degenerate constraints in the master problem and the degenerate ones in the subproblem. We propose three algorithms. The first one, where some set of variables of the original problem are added to the master problem, corresponds to the Improved Primal Simplex algorithm (IPS) presented recently by Elhallaoui et al. [7]. In the second one, some extreme points of the subproblem are added as columns in the master problem. The third algorithm is a mixed implementation that adds some original variables and some extreme points of a subproblem to the master problem. Experimental results on some degenerate instances show that the proposed algorithms yield computational times that are reduced by an average factor ranging from 3.32 to 13.16 compared to the primal simplex of CPLEX.

Suggested Citation

  • Metrane, Abdelmoutalib & Soumis, François & Elhallaoui, Issmail, 2010. "Column generation decomposition with the degenerate constraints in the subproblem," European Journal of Operational Research, Elsevier, vol. 207(1), pages 37-44, November.
    • Handle: RePEc:eee:ejores:v:207:y:2010:i:1:p:37-44
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    References listed on IDEAS

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    1. P. C. Gilmore & R. E. Gomory, 1963. "A Linear Programming Approach to the Cutting Stock Problem---Part II," Operations Research, INFORMS, vol. 11(6), pages 863-888, December.
    2. George B. Dantzig & Philip Wolfe, 1960. "Decomposition Principle for Linear Programs," Operations Research, INFORMS, vol. 8(1), pages 101-111, February.
    3. Knut Haase & Guy Desaulniers & Jacques Desrosiers, 2001. "Simultaneous Vehicle and Crew Scheduling in Urban Mass Transit Systems," Transportation Science, INFORMS, vol. 35(3), pages 286-303, August.
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    Cited by:

    1. Jérémy Omer & Samuel Rosat & Vincent Raymond & François Soumis, 2015. "Improved Primal Simplex: A More General Theoretical Framework and an Extended Experimental Analysis," INFORMS Journal on Computing, INFORMS, vol. 27(4), pages 773-787, November.
    2. Bouarab, Hocine & El Hallaoui, Issmail & Metrane, Abdelmoutalib & Soumis, François, 2017. "Dynamic constraint and variable aggregation in column generation," European Journal of Operational Research, Elsevier, vol. 262(3), pages 835-850.
    3. Jean Bertrand Gauthier & Jacques Desrosiers & Marco E. Lübbecke, 2016. "Tools for primal degenerate linear programs: IPS, DCA, and PE," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 5(2), pages 161-204, June.
    4. Abdelouahab Zaghrouti & François Soumis & Issmail El Hallaoui, 2014. "Integral Simplex Using Decomposition for the Set Partitioning Problem," Operations Research, INFORMS, vol. 62(2), pages 435-449, April.

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