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Inequalities for Stochastic Linear Programming Problems

Author

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  • Albert Madansky

    (The RAND Corporation)

Abstract

Consider a linear-programming problem in which the "right-hand side" is a random vector whose expected value is known and where the expected value of the objective function is to be minimized. An approximate solution is often found by replacing the "right-hand side" by its expected value and solving the resulting linear programming problem. In this paper conditions are given for the equality of the expected value of the objective function for the optimal solution and the value of the objective function for the approximate solution; bounds on these values are also given. In addition, the relation between this problem and a related problem, where one makes an observation on the "right-hand side" and solves the (nonstochastic) linear programming problem based on this observation, is discussed.

Suggested Citation

  • Albert Madansky, 1960. "Inequalities for Stochastic Linear Programming Problems," Management Science, INFORMS, vol. 6(2), pages 197-204, January.
    • Handle: RePEc:inm:ormnsc:v:6:y:1960:i:2:p:197-204
    DOI: 10.1287/mnsc.6.2.197
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    Cited by:

    1. D. Kuhn, 2009. "Convergent Bounds for Stochastic Programs with Expected Value Constraints," Journal of Optimization Theory and Applications, Springer, vol. 141(3), pages 597-618, June.
    2. Guo, Jian-Xin & Zhu, Kaiwei & Tan, Xianchun & Gu, Baihe, 2021. "Low-carbon technology development under multiple adoption risks," Technological Forecasting and Social Change, Elsevier, vol. 172(C).
    3. Douglas Alem & Pedro Munari & Marcos Arenales & Paulo Ferreira, 2010. "On the cutting stock problem under stochastic demand," Annals of Operations Research, Springer, vol. 179(1), pages 169-186, September.
    4. Crean, Jason & Parton, Kevin & Mullen, John & Jones, Randall, 2013. "Representing climatic uncertainty in agricultural models – an application of state-contingent theory," Australian Journal of Agricultural and Resource Economics, Australian Agricultural and Resource Economics Society, vol. 57(3).
    5. Osman Y. Özaltın & Oleg A. Prokopyev & Andrew J. Schaefer & Mark S. Roberts, 2011. "Optimizing the Societal Benefits of the Annual Influenza Vaccine: A Stochastic Programming Approach," Operations Research, INFORMS, vol. 59(5), pages 1131-1143, October.
    6. Guo, Penghui & Zhu, Jianjun, 2023. "Capacity reservation for humanitarian relief: A logic-based Benders decomposition method with subgradient cut," European Journal of Operational Research, Elsevier, vol. 311(3), pages 942-970.
    7. Douglas T. Gardner & J. Scott Rogers, 1999. "Planning Electric Power Systems Under Demand Uncertainty with Different Technology Lead Times," Management Science, INFORMS, vol. 45(10), pages 1289-1306, October.
    8. Francesca Maggioni & Elisabetta Allevi & Marida Bertocchi, 2014. "Bounds in Multistage Linear Stochastic Programming," Journal of Optimization Theory and Applications, Springer, vol. 163(1), pages 200-229, October.
    9. Bistline, John E., 2015. "Electric sector capacity planning under uncertainty: Climate policy and natural gas in the US," Energy Economics, Elsevier, vol. 51(C), pages 236-251.
    10. Mestre, Ana Maria & Oliveira, Mónica Duarte & Barbosa-Póvoa, Ana Paula, 2015. "Location–allocation approaches for hospital network planning under uncertainty," European Journal of Operational Research, Elsevier, vol. 240(3), pages 791-806.
    11. Zhuang, Jifang & Gabriel, Steven A., 2008. "A complementarity model for solving stochastic natural gas market equilibria," Energy Economics, Elsevier, vol. 30(1), pages 113-147, January.
    12. Steftcho P. Dokov & David P. Morton, 2005. "Second-Order Lower Bounds on the Expectation of a Convex Function," Mathematics of Operations Research, INFORMS, vol. 30(3), pages 662-677, August.
    13. Francesca Maggioni & Elisabetta Allevi & Marida Bertocchi, 2016. "Monotonic bounds in multistage mixed-integer stochastic programming," Computational Management Science, Springer, vol. 13(3), pages 423-457, July.
    14. Laureano Escudero & Araceli Garín & María Merino & Gloria Pérez, 2007. "The value of the stochastic solution in multistage problems," TOP: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 15(1), pages 48-64, July.
    15. David P. Morton & R. Kevin Wood, 1999. "Restricted-Recourse Bounds for Stochastic Linear Programming," Operations Research, INFORMS, vol. 47(6), pages 943-956, December.
    16. Elise D. Miller-Hooks & Hani S. Mahmassani, 2000. "Least Expected Time Paths in Stochastic, Time-Varying Transportation Networks," Transportation Science, INFORMS, vol. 34(2), pages 198-215, May.
    17. Jirutitijaroen, Panida & Kim, Sujin & Kittithreerapronchai, Oran & Prina, José, 2013. "An optimization model for natural gas supply portfolios of a power generation company," Applied Energy, Elsevier, vol. 107(C), pages 1-9.
    18. de Boer, Sanne V. & Freling, Richard & Piersma, Nanda, 2002. "Mathematical programming for network revenue management revisited," European Journal of Operational Research, Elsevier, vol. 137(1), pages 72-92, February.
    19. İ. Esra Büyüktahtakın, 2022. "Stage-t scenario dominance for risk-averse multi-stage stochastic mixed-integer programs," Annals of Operations Research, Springer, vol. 309(1), pages 1-35, February.
    20. Sheng-I Chen & Delvinia Su, 2022. "A multi-stage stochastic programming model of lot-sizing and scheduling problems with machine eligibilities and sequence-dependent setups," Annals of Operations Research, Springer, vol. 311(1), pages 35-50, April.
    21. Bomze, Immanuel M. & Gabl, Markus & Maggioni, Francesca & Pflug, Georg Ch., 2022. "Two-stage stochastic standard quadratic optimization," European Journal of Operational Research, Elsevier, vol. 299(1), pages 21-34.
    22. Francesca Maggioni & Elisabetta Allevi & Asgeir Tomasgard, 2020. "Bounds in multi-horizon stochastic programs," Annals of Operations Research, Springer, vol. 292(2), pages 605-625, September.
    23. Munoz, F.D. & Hobbs, B.F. & Watson, J.-P., 2016. "New bounding and decomposition approaches for MILP investment problems: Multi-area transmission and generation planning under policy constraints," European Journal of Operational Research, Elsevier, vol. 248(3), pages 888-898.
    24. Xuecheng Yin & İ. E. Büyüktahtakın, 2021. "A multi-stage stochastic programming approach to epidemic resource allocation with equity considerations," Health Care Management Science, Springer, vol. 24(3), pages 597-622, September.
    25. Lee, Chia-Yen, 2018. "Mixed-strategy Nash equilibrium in data envelopment analysis," European Journal of Operational Research, Elsevier, vol. 266(3), pages 1013-1024.

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