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Lagrangian Solution Techniques and Bounds for Loosely Coupled Mixed-Integer Stochastic Programs

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  • Samer Takriti

    (Mathematical Sciences Department, IBM Thomas J. Watson Research Center, Yorktown Heights, New York 10598)

  • John R. Birge

    (Department of Industrial and Operations Engineering, University of Michigan, Ann Arbor, Michigan 48109-2117)

Abstract

Many production problems involve facility setups that lead to integer variables, production decisions that are continuous, and demands that are likely to be random. While these problems can be quite difficult to solve, we propose a model and an efficient solution technique for this basic class of stochastic mixed-integer programs. We use a set of scenarios to reflect uncertainty. The resulting mathematical model is solved using Lagrangian relaxation. We show that the duality gap of our relaxation is bounded above by a constant that depends on the cost function and the number of branching points in the scenario tree. We apply our technique to the problem of generating electric power. Numerical results indicate significant savings when the stochastic model is used instead of a deterministic one.

Suggested Citation

  • Samer Takriti & John R. Birge, 2000. "Lagrangian Solution Techniques and Bounds for Loosely Coupled Mixed-Integer Stochastic Programs," Operations Research, INFORMS, vol. 48(1), pages 91-98, February.
    • Handle: RePEc:inm:oropre:v:48:y:2000:i:1:p:91-98
    DOI: 10.1287/opre.48.1.91.12450
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    References listed on IDEAS

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    Citations

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    Cited by:

    1. Escudero Bueno, Laureano F. & Garín Martín, María Araceli & Merino Maestre, María & Pérez Sainz de Rozas, Gloria, 2005. "A two-stage stochastic integer programming approach," BILTOKI 1134-8984, Universidad del País Vasco - Departamento de Economía Aplicada III (Econometría y Estadística).
    2. Serhat Gul & Brian T. Denton & John W. Fowler, 2015. "A Progressive Hedging Approach for Surgery Planning Under Uncertainty," INFORMS Journal on Computing, INFORMS, vol. 27(4), pages 755-772, November.
    3. M A Lejeune, 2008. "Preprocessing techniques and column generation algorithms for stochastically efficient demand," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 59(9), pages 1239-1252, September.
    4. Escudero Bueno, Laureano F. & Garín Martín, María Araceli & Pérez Sainz de Rozas, Gloria & Unzueta Inchaurbe, Aitziber, 2010. "Lagrangean decomposition for large-scale two-stage stochastic mixed 0-1 problems," BILTOKI 1134-8984, Universidad del País Vasco - Departamento de Economía Aplicada III (Econometría y Estadística).
    5. Borodin, Valeria & Bourtembourg, Jean & Hnaien, Faicel & Labadie, Nacima, 2015. "A multi-step rolled forward chance-constrained model and a proactive dynamic approach for the wheat crop quality control problem," European Journal of Operational Research, Elsevier, vol. 246(2), pages 631-640.
    6. Fangzhou Yan & Huaxin Qiu & Dongya Han, 2023. "Lagrangian Heuristic for Multi-Depot Technician Planning of Product Distribution and Installation with a Lunch Break," Mathematics, MDPI, vol. 11(3), pages 1-22, January.
    7. L. Escudero & A. Garín & M. Merino & G. Pérez, 2007. "A two-stage stochastic integer programming approach as a mixture of Branch-and-Fix Coordination and Benders Decomposition schemes," Annals of Operations Research, Springer, vol. 152(1), pages 395-420, July.
    8. Antonio Frangioni, 2005. "About Lagrangian Methods in Integer Optimization," Annals of Operations Research, Springer, vol. 139(1), pages 163-193, October.
    9. Jiang, Ruiwei & Zhang, Muhong & Li, Guang & Guan, Yongpei, 2014. "Two-stage network constrained robust unit commitment problem," European Journal of Operational Research, Elsevier, vol. 234(3), pages 751-762.
    10. L. Escudero & M. Garín & G. Pérez & A. Unzueta, 2012. "Lagrangian Decomposition for large-scale two-stage stochastic mixed 0-1 problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 20(2), pages 347-374, July.
    11. Beltran-Royo, C., 2017. "Two-stage stochastic mixed-integer linear programming: The conditional scenario approach," Omega, Elsevier, vol. 70(C), pages 31-42.
    12. Eguía Ribero, María Isabel & Garín Martín, María Araceli & Unzueta Inchaurbe, Aitziber, 2018. "Generating cluster submodels from two-stage stochastic mixed integer optimization models," BILTOKI 31248, Universidad del País Vasco - Departamento de Economía Aplicada III (Econometría y Estadística).
    13. Steeger, Gregory & Rebennack, Steffen, 2017. "Dynamic convexification within nested Benders decomposition using Lagrangian relaxation: An application to the strategic bidding problem," European Journal of Operational Research, Elsevier, vol. 257(2), pages 669-686.
    14. Escudero, Laureano F. & Landete, Mercedes & Rodríguez-Chía, Antonio M., 2011. "Stochastic set packing problem," European Journal of Operational Research, Elsevier, vol. 211(2), pages 232-240, June.

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