Comparing stage-scenario with nodal formulation for multistage stochastic problems
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Comparing stage-scenario with nodal formulation for multistage stochastic problems Sebastiano Vitali1,2
· Ruth Domínguez3
· Vittorio Moriggia2
Received: 21 March 2020 / Revised: 21 March 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract To solve real life problems under uncertainty in Economics, Finance, Energy, Transportation and Logistics, the use of stochastic optimization is widely accepted and appreciated. However, the nature of stochastic programming leads to a conflict between adaptability to reality and tractability. To formulate a multistage stochastic model, two types of formulations are typically adopted: the so-called stage-scenario formulation named also formulation with explicit non-anticipativity constraints and the so-called nodal formulation named also formulation with implicit non-anticipativity constraints. Both of them have advantages and disadvantages. This work aims at helping the scholars and practitioners to understand the two types of notation and, in particular, to reformulate with the nodal formulation a model that was originally defined with the stage-scenario formulation presenting this implementation in the algebraic language GAMS. In addition, this work presents an empirical analysis applying the two formulations both without any further decomposition to perform a fair comparison. In this way, we show that the difficulties to implement the model with the nodal formulation are somehow reworded making the problem tractable without any decomposition algorithm. Still, we remark that in some other applications the stage-scenario formulation could be more helpful to understand the structure of the problem since it allows to relax the non-anticipativity constraints. Keywords Power generation capacity expansion · Stage scenario formulation · Nodal formulation · Multistage planning · Power systems · Stochastic optimization · GAMS Mathematics Subject Classification 90C15 · 90C05 · 90C06 · 97M50
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Vittorio Moriggia [email protected]
1
Department of Mathematics and Physics, Charles University, Prague, Czech Republic
2
Department of Management, Economics and Quantitative Methods, University of Bergamo, Bergamo, Italy
3
Department of Electrical Engineering, University of Castilla – La Mancha, Toledo, Spain
123
S. Vitali et al.
1 Introduction Real world decision-making problems are typically solved under uncertainty. In operation research literature, one of the most useful tool proposed to represent and solve problems under uncertainty is stochastic optimization. Stochastic optimization allows to determine the optimal strategy in a decision-making problem considering the impact of the possible realizations of the uncertain parameters over the objective function value, see Birge and Louveaux (2011) and Powell (2014). If decisions are made in a sequence of time stages, the process can be formulated as a multistage stochastic programming problem, see Dupaˇcová et al. (2002), Shapiro et al. (2009), Pflug and Pichler (2014). Multistage stochastic optimiza
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