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Improving set partitioning problem solutions by zooming around an improving direction

Author

Listed:
  • Abdelouahab Zaghrouti

    (Polytechnique Montréal and GERAD)

  • Issmail El Hallaoui

    (Polytechnique Montréal and GERAD)

  • François Soumis

    (Polytechnique Montréal and GERAD)

Abstract

In this paper, we introduce a general framework for vector space decompositions that decompose the set partitioning problem into a reduced problem, defined in the vector subspace generated by the columns corresponding to nonzero variables in the current integer solution, and a complementary problem, defined in the complementary vector subspace. We show that the integral simplex using decomposition algorithm (ISUD) developed in Zaghrouti et al. (Oper Res 62:435–449, 2014. https://doi.org/10.1287/opre.2013.1247) uses a particular decomposition, in which integrality is handled mainly in the complementary problem, to find a sequence of integer solutions with decreasing objective values leading to an optimal solution. We introduce a new algorithm using a new dynamic decomposition where integrality is handled only in the reduced problem, and the complementary problem is only used to provide descent directions, needed to update the decomposition. The new algorithm improves, at each iteration, the current integer solution by solving a reduced problem very small compared the original problem, that we define by zooming around the descent direction (provided by the complementary problem). This zooming algorithm is superior than ISUD on set partitioning instances from the transportation industry. It rapidly reaches optimal or near-optimal solutions for all instances including those considered very difficult for both ISUD and CPLEX.

Suggested Citation

  • Abdelouahab Zaghrouti & Issmail El Hallaoui & François Soumis, 2020. "Improving set partitioning problem solutions by zooming around an improving direction," Annals of Operations Research, Springer, vol. 284(2), pages 645-671, January.
    • Handle: RePEc:spr:annopr:v:284:y:2020:i:2:d:10.1007_s10479-018-2868-1
    DOI: 10.1007/s10479-018-2868-1
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    1. Andrew J. Schaefer & Ellis L. Johnson & Anton J. Kleywegt & George L. Nemhauser, 2005. "Airline Crew Scheduling Under Uncertainty," Transportation Science, INFORMS, vol. 39(3), pages 340-348, August.
    2. Balaji Gopalakrishnan & Ellis. Johnson, 2005. "Airline Crew Scheduling: State-of-the-Art," Annals of Operations Research, Springer, vol. 140(1), pages 305-337, November.
    3. 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.
    4. Cordeau, Jean-François & Desaulniers, Guy & Lingaya, Norbert & Soumis, François & Desrosiers, Jacques, 2001. "Simultaneous locomotive and car assignment at VIA Rail Canada," Transportation Research Part B: Methodological, Elsevier, vol. 35(8), pages 767-787, September.
    5. Michel Gamache & François Soumis & Gérald Marquis & Jacques Desrosiers, 1999. "A Column Generation Approach for Large-Scale Aircrew Rostering Problems," Operations Research, INFORMS, vol. 47(2), pages 247-263, April.
    6. Agha Iqbal Ali, 1998. "Reformulation of the set partitioning problem as apure network with special order set constraints," Annals of Operations Research, Springer, vol. 81(0), pages 233-250, June.
    7. 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.
    8. Egon Balas & Manfred W. Padberg, 1972. "On the Set-Covering Problem," Operations Research, INFORMS, vol. 20(6), pages 1152-1161, December.
    9. 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.
    10. 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.
    11. M. L. Balinski & R. E. Quandt, 1964. "On an Integer Program for a Delivery Problem," Operations Research, INFORMS, vol. 12(2), pages 300-304, April.
    12. Martin Desrochers & Jacques Desrosiers & Marius Solomon, 1992. "A New Optimization Algorithm for the Vehicle Routing Problem with Time Windows," Operations Research, INFORMS, vol. 40(2), pages 342-354, April.
    13. Cynthia Barnhart & Ellis L. Johnson & George L. Nemhauser & Martin W. P. Savelsbergh & Pamela H. Vance, 1998. "Branch-and-Price: Column Generation for Solving Huge Integer Programs," Operations Research, INFORMS, vol. 46(3), pages 316-329, June.
    14. Adam N. Letchford & Andrea Lodi, 2002. "Primal cutting plane algorithms revisited," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 56(1), pages 67-81, August.
    15. Martin Desrochers & François Soumis, 1989. "A Column Generation Approach to the Urban Transit Crew Scheduling Problem," Transportation Science, INFORMS, vol. 23(1), pages 1-13, February.
    16. Marco E. Lübbecke & Jacques Desrosiers, 2005. "Selected Topics in Column Generation," Operations Research, INFORMS, vol. 53(6), pages 1007-1023, December.
    17. 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.
    18. Karla L. Hoffman & Manfred Padberg, 1993. "Solving Airline Crew Scheduling Problems by Branch-and-Cut," Management Science, INFORMS, vol. 39(6), pages 657-682, June.
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