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Very Large-Scale Linear Programming: A Case Study in Combining Interior Point and Simplex Methods

Author

Listed:
  • Robert E. Bixby

    (Rice University, Houston, Texas)

  • John W. Gregory

    (Cray Research, Inc., Eagan, Minnesota)

  • Irvin J. Lustig

    (Princeton University, Princeton, New Jersey)

  • Roy E. Marsten

    (Georgia Institute of Technology, Atlanta, Georgia)

  • David F. Shanno

    (Rutgers University, New Brunswick, New Jersey)

Abstract

Experience with solving a 12.753.313 variable linear program is described. This problem is the linear programming relaxation of a set partitioning problem arising from an airline crew scheduling application. A scheme is described that requires successive solutions of small subproblems, yielding a procedure that has little growth in solution time in terms of the number of variables. Experience using the simplex method as implemented in CPLEX , an interior point method as implemented in OBI , and a hybrid interior point/simplex approach is reported. The resulting procedure illustrates the power of an interior point/simplex combination for solving very large-scale linear programs.

Suggested Citation

  • Robert E. Bixby & John W. Gregory & Irvin J. Lustig & Roy E. Marsten & David F. Shanno, 1992. "Very Large-Scale Linear Programming: A Case Study in Combining Interior Point and Simplex Methods," Operations Research, INFORMS, vol. 40(5), pages 885-897, October.
    • Handle: RePEc:inm:oropre:v:40:y:1992:i:5:p:885-897
    DOI: 10.1287/opre.40.5.885
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    Citations

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    Cited by:

    1. Huisman, D. & Jans, R.F. & Peeters, M. & Wagelmans, A.P.M., 2003. "Combining Column Generation and Lagrangian Relaxation," ERIM Report Series Research in Management ERS-2003-092-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    2. Aghayi, Nazila & Maleki, Bentolhoda, 2016. "Efficiency measurement of DMUs with undesirable outputs under uncertainty based on the directional distance function: Application on bank industry," Energy, Elsevier, vol. 112(C), pages 376-387.
    3. Irvin Lustig & Patricia Randall & Robert Randall, 2021. "Formulation Matters: Reciprocating Integer Programming for Birchbox Product Assortment," Interfaces, INFORMS, vol. 51(5), pages 347-360, September.
    4. Hong‐Chih Huang, 2010. "Optimal Multiperiod Asset Allocation: Matching Assets to Liabilities in a Discrete Model," Journal of Risk & Insurance, The American Risk and Insurance Association, vol. 77(2), pages 451-472, June.
    5. Vahid Zeighami & François Soumis, 2019. "Combining Benders’ Decomposition and Column Generation for Integrated Crew Pairing and Personalized Crew Assignment Problems," Transportation Science, INFORMS, vol. 53(5), pages 1479-1499, September.
    6. Freling, R. & Lentink, R.M. & Wagelmans, A.P.M., 2001. "A decision support system for crew planning in passenger transportation using a flexible branch-and-price algorithm," ERIM Report Series Research in Management ERS-2001-57-LIS, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
    7. Neufeld, Janis S. & Scheffler, Martin & Tamke, Felix & Hoffmann, Kirsten & Buscher, Udo, 2021. "An efficient column generation approach for practical railway crew scheduling with attendance rates," European Journal of Operational Research, Elsevier, vol. 293(3), pages 1113-1130.
    8. Messina, E. & Mitra, G., 1997. "Modelling and analysis of multistage stochastic programming problems: A software environment," European Journal of Operational Research, Elsevier, vol. 101(2), pages 343-359, September.
    9. Marco E. Lübbecke & Jacques Desrosiers, 2005. "Selected Topics in Column Generation," Operations Research, INFORMS, vol. 53(6), pages 1007-1023, December.
    10. Saito, G. & Corley, H.W. & Rosenberger, Jay M. & Sung, Tai-Kuan & Noroziroshan, Alireza, 2015. "Constraint Optimal Selection Techniques (COSTs) for nonnegative linear programming problems," Applied Mathematics and Computation, Elsevier, vol. 251(C), pages 586-598.
    11. Alexandra Makri & Diego Klabjan, 2004. "A New Pricing Scheme for Airline Crew Scheduling," INFORMS Journal on Computing, INFORMS, vol. 16(1), pages 56-67, February.
    12. Walteros, Jose L. & Vogiatzis, Chrysafis & Pasiliao, Eduardo L. & Pardalos, Panos M., 2014. "Integer programming models for the multidimensional assignment problem with star costs," European Journal of Operational Research, Elsevier, vol. 235(3), pages 553-568.
    13. Hernández-Leandro, Noberto A. & Boyer, Vincent & Salazar-Aguilar, M. Angélica & Rousseau, Louis-Martin, 2019. "A matheuristic based on Lagrangian relaxation for the multi-activity shift scheduling problem," European Journal of Operational Research, Elsevier, vol. 272(3), pages 859-867.
    14. Atoosa Kasirzadeh & Mohammed Saddoune & François Soumis, 2017. "Airline crew scheduling: models, algorithms, and data sets," EURO Journal on Transportation and Logistics, Springer;EURO - The Association of European Operational Research Societies, vol. 6(2), pages 111-137, June.
    15. Kavinesh J. Singh & Andy B. Philpott & R. Kevin Wood, 2009. "Dantzig-Wolfe Decomposition for Solving Multistage Stochastic Capacity-Planning Problems," Operations Research, INFORMS, vol. 57(5), pages 1271-1286, October.
    16. Alain, Guinet & Angel, Ruiz, 2016. "Modeling the logistics response to a bioterrorist anthrax attackAuthor-Name: Wanying, Chen," European Journal of Operational Research, Elsevier, vol. 254(2), pages 458-471.
    17. Robert E. Bixby & Alexander Martin, 2000. "Parallelizing the Dual Simplex Method," INFORMS Journal on Computing, INFORMS, vol. 12(1), pages 45-56, February.
    18. M. Xiong & J. Wang & P. Wang, 2002. "Differential-Algebraic Approach to Linear Programming," Journal of Optimization Theory and Applications, Springer, vol. 114(2), pages 443-470, August.

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    Keywords

    programming; linear: large-scale systems;

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