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Monotonic bounds in multistage mixed-integer stochastic programming

Author

Listed:
  • Francesca Maggioni

    () (Bergamo University)

  • Elisabetta Allevi

    () (Brescia University)

  • Marida Bertocchi

    (Bergamo University)

Abstract

Multistage stochastic programs bring computational complexity which may increase exponentially with the size of the scenario tree in real case problems. For this reason approximation techniques which replace the problem by a simpler one and provide lower and upper bounds to the optimal value are very useful. In this paper we provide monotonic lower and upper bounds for the optimal objective value of a multistage stochastic program. These results also apply to stochastic multistage mixed integer linear programs. Chains of inequalities among the new quantities are provided in relation to the optimal objective value, the wait-and-see solution and the expected result of using the expected value solution. The computational complexity of the proposed lower and upper bounds is discussed and an algorithmic procedure to use them is provided. Numerical results on a real case transportation problem are presented.

Suggested Citation

  • 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.
  • Handle: RePEc:spr:comgts:v:13:y:2016:i:3:d:10.1007_s10287-016-0254-5
    DOI: 10.1007/s10287-016-0254-5
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    References listed on IDEAS

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    1. Albert Madansky, 1960. "Inequalities for Stochastic Linear Programming Problems," Management Science, INFORMS, vol. 6(2), pages 197-204, January.
    2. Francesca Maggioni & Stein Wallace, 2012. "Analyzing the quality of the expected value solution in stochastic programming," Annals of Operations Research, Springer, vol. 200(1), pages 37-54, November.
    3. Willem Klein Haneveld & Maarten van der Vlerk, 1999. "Stochastic integer programming:General models and algorithms," Annals of Operations Research, Springer, vol. 85(0), pages 39-57, January.
    4. 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.
    5. Guglielmo Lulli & Suvrajeet Sen, 2004. "A Branch-and-Price Algorithm for Multistage Stochastic Integer Programming with Application to Stochastic Batch-Sizing Problems," Management Science, INFORMS, vol. 50(6), pages 786-796, June.
    6. Maarten Vlerk, 2010. "Convex approximations for a class of mixed-integer recourse models," Annals of Operations Research, Springer, vol. 177(1), pages 139-150, June.
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    Citations

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    Cited by:

    1. Bertazzi, Luca & Maggioni, Francesca, 2018. "A stochastic multi-stage fixed charge transportation problem: Worst-case analysis of the rolling horizon approach," European Journal of Operational Research, Elsevier, vol. 267(2), pages 555-569.
    2. Francesca Maggioni & Matteo Cagnolari & Luca Bertazzi, 2019. "The value of the right distribution in stochastic programming with application to a Newsvendor problem," Computational Management Science, Springer, vol. 16(4), pages 739-758, October.
    3. Gambella, Claudio & Maggioni, Francesca & Vigo, Daniele, 2019. "A stochastic programming model for a tactical solid waste management problem," European Journal of Operational Research, Elsevier, vol. 273(2), pages 684-694.
    4. Ching-Hui Tang, 2018. "Two-stage stochastic modeling of transportation outsourcing plans for transshipment centers," 4OR, Springer, vol. 16(1), pages 67-94, March.
    5. Mahmutoğulları, Ali İrfan & Çavuş, Özlem & Aktürk, M. Selim, 2018. "Bounds on risk-averse mixed-integer multi-stage stochastic programming problems with mean-CVaR," European Journal of Operational Research, Elsevier, vol. 266(2), pages 595-608.
    6. Audrius Kabašinskas & Francesca Maggioni & Kristina Šutienė & Eimutis Valakevičius, 2019. "A multistage risk-averse stochastic programming model for personal savings accrual: the evidence from Lithuania," Annals of Operations Research, Springer, vol. 279(1), pages 43-70, August.
    7. Kevin Ryan & Shabbir Ahmed & Santanu S. Dey & Deepak Rajan & Amelia Musselman & Jean-Paul Watson, 2020. "Optimization-Driven Scenario Grouping," INFORMS Journal on Computing, INFORMS, vol. 32(3), pages 805-821, July.
    8. Francesca Maggioni & Florian A. Potra & Marida Bertocchi, 2017. "A scenario-based framework for supply planning under uncertainty: stochastic programming versus robust optimization approaches," Computational Management Science, Springer, vol. 14(1), pages 5-44, January.

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