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Monotonic bounds in multistage mixed-integer stochastic programming

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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.

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Acknowledgments

The authors thank the anonymous referees on an earlier version of the paper for their helpful comments. The work has been supported by Bergamo and Brescia University grants 2014–2015.

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Authors and Affiliations

  1. Department of Management, Economics and Quantitative Methods, Bergamo University, Via dei Caniana 2, 24127, Bergamo, Italy

    Francesca Maggioni & Marida Bertocchi

  2. Department of Economics and Management, Brescia University, Contrada S. Chiara 50, 25122, Brescia, Italy

    Elisabetta Allevi

Authors
  1. Francesca Maggioni
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  2. Elisabetta Allevi
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  3. Marida Bertocchi
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Corresponding author

Correspondence to Francesca Maggioni.

Appendix

Appendix

Table 2 Percentage deviation from RP of lower bounds MEGSO(1, R) (see fourth column), with R fixed scenarios \(R=1,\ldots ,53\) (second column) and cardinality of each subproblem \(R+1\) (first column)
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Table 3 Percentage deviation from RP of lower bounds MEGSO(2, R) (see fourth column), with R fixed scenarios \(R=1,\ldots ,52\) (second column) and cardinality of each subproblem \(R+2\) (first column)
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Table 4 Percentage deviation from RP of lower bounds MEGSO(3, R) (see fourth column), with R fixed scenarios \(R=1,\ldots ,51\) (second column) and cardinality of each subproblem \(R+1\) (first column)
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Table 5 Percentage deviation from RP of lower bounds MEGSO(k, 40) (see fourth column) and MEGS(k, 40) (see fifth column) with 40 fixed scenarios (second column) and k free scenarios where \(k=1,\ldots ,14\)
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Table 6 Percentage deviation from RP of upper bounds MEGS(1, R) (see fourth column), with R fixed scenarios \(R=1,\ldots ,53\) (second column) and cardinality of each subproblem \(R+1\) (first column)
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Table 7 Percentage deviation from RP of the expected value problem EV and of the Expected result at stage t by using the expected value solution \(EEV^t\)
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Maggioni, F., Allevi, E. & Bertocchi, M. Monotonic bounds in multistage mixed-integer stochastic programming. Comput Manag Sci 13, 423–457 (2016). https://doi.org/10.1007/s10287-016-0254-5

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