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Improved integral simplex using decomposition for the set partitioning problem

Author

Listed:
  • Abdelouahab Zaghrouti

    (Polytechnique Montreal and GERAD)

  • Issmail El Hallaoui

    (Polytechnique Montreal and GERAD)

  • François Soumis

    (Polytechnique Montreal and GERAD)

Abstract

Integral simplex using decomposition (ISUD) is a method that efficiently solves set partitioning problems. It is an iterative method that starts from a known integer solution and moves through a sequence of integer solutions, decreasing the cost at each iteration. At each iteration, the method decomposes the original problem into a reduced problem (RP) and a complementary problem (CP). Given an integer solution to RP (that is also solution to the original problem), CP finds a descent direction having the minimum ratio between its cost and the number of its positive variables. We loop on until an optimal or near-optimal solution to the original problem is reached. In this paper, we introduce a modified model for CP. The new model finds a descent direction that minimizes the ratio between the cost of the direction and an overestimation of the number of variables taking one in the next solution. The new CP presents higher chances of finding improved integer solutions without branching. We present results for the same large instances (with up to 570,000 columns) as the ones previously used to test ISUD. For all the instances, optimality is always reached with a speedup factor of at least five.

Suggested Citation

  • Abdelouahab Zaghrouti & Issmail El Hallaoui & François Soumis, 2018. "Improved integral simplex using decomposition for the set partitioning problem," EURO Journal on Computational Optimization, Springer;EURO - The Association of European Operational Research Societies, vol. 6(2), pages 185-206, June.
    • Handle: RePEc:spr:eurjco:v:6:y:2018:i:2:d:10.1007_s13675-018-0098-6
    DOI: 10.1007/s13675-018-0098-6
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    References listed on IDEAS

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    1. Rönnberg, Elina & Larsson, Torbjörn, 2009. "Column generation in the integral simplex method," European Journal of Operational Research, Elsevier, vol. 192(1), pages 333-342, January.
    2. Issmail Elhallaoui & Abdelmoutalib Metrane & Guy Desaulniers & François Soumis, 2011. "An Improved Primal Simplex Algorithm for Degenerate Linear Programs," INFORMS Journal on Computing, INFORMS, vol. 23(4), pages 569-577, November.
    3. Egon Balas & Manfred W. Padberg, 1972. "On the Set-Covering Problem," Operations Research, INFORMS, vol. 20(6), pages 1152-1161, December.
    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.
    5. Egon Balas & Manfred Padberg, 1975. "On the Set-Covering Problem: II. An Algorithm for Set Partitioning," Operations Research, INFORMS, vol. 23(1), pages 74-90, February.
    6. Utz-Uwe Haus & Matthias Köppe & Robert Weismantel, 2001. "The integral basis method for integer programming," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 53(3), pages 353-361, July.
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    Cited by:

    1. Adil Tahir & Guy Desaulniers & Issmail El Hallaoui, 2022. "Integral Column Generation for Set Partitioning Problems with Side Constraints," INFORMS Journal on Computing, INFORMS, vol. 34(4), pages 2313-2331, July.

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