Bounds in multi-horizon stochastic programs
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Bounds in multi-horizon stochastic programs Francesca Maggioni1
· Elisabetta Allevi2 · Asgeir Tomasgard3
© Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, we present bounds for multi-horizon stochastic optimization problems, a class of problems introduced in Kaut et al. (Comput Manag Sci 11:179–193, 2014) relevant in many industry-life applications typically involving strategic and operational decisions on two different time scales. After providing three general mathematical formulations of a multi-horizon stochastic program, we extend the definition of the traditional Expected Value problem and Wait-and-See problem from stochastic programming in a multi-horizon framework. New measures are introduced allowing to quantify the importance of the uncertainty at both strategic and operational levels. Relations among the solution approaches are then determined and chain of inequalities provided. Numerical experiments based on an energy planning application are finally presented. Keywords Bounds · Multi-horizon · Stochastic programs · Energy · Strategic decisions · Operational decisions
The idea of this work was originated by a discussion with Marida Bertocchi. The authors would like to express their gratitude for being an exceptional colleague and friend. This work is dedicated to her memory.
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Francesca Maggioni [email protected] Elisabetta Allevi [email protected] Asgeir Tomasgard [email protected]
1
Department of Management, Economics and Quantitative Methods, University of Bergamo, Via dei Caniana 2, 24127 Bergamo, Italy
2
Department of Economics and Management, Brescia University, Contrada S. Chiara 50, 25122 Brescia, Italy
3
Department of Industrial Economics and Technology Management, Norwegian University of Science and Technology, Høgskoleringen 1, 7491 Trondheim, Norway
123
Annals of Operations Research
1 Introduction Many real-life problems of industries with large capital investments and infrastructure planning, need to combine decisions at long and short-term time scales, which are both typically affected by uncertainty. Long-term uncertainty includes for example costs for infrastructure elements with time horizons of many years. Short-term uncertainty comprises for instance daily variations in demand or prices. In optimization models, this uncertainty can be handled by applying stochastic programming methodology, where the uncertain parameters are represented by discrete values in scenario trees for possible future realizations of the parameters. Computational tractability is usually of great concern in such models, because variables and constraints are duplicated for each scenario in the scenario tree. If all short-term variations are to be included, the scenario tree and, the optimization model will become intractable due to the exponential growth in problem size. In order to cope with this issue, an alternative formulation of the problem that combines the two time scales has been introduced in Hellemo et al. (2012), Kaut et
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