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.
Notes
Building the set MancsN(M,N) we need a temporary variable (tmp) to ensure a correct assignment of tuple (m, n).
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Acknowledgements
The research of Sebastiano Vitali was supported by the Czech Science Foundation project GAČR No. 19-28231X and by MIUR-ex60% 2020 sci.resp. Sebastiano Vitali. The research was partially supported by MIUR-ex60% 2019 and by MIUR-ex60% 2020 sci.resp. Vittorio Moriggia. Ruth Dominguez was partly supported by Univ. of Bergamo STaRS 2019 - Second Tranche and Univ. of Bergamo STaRS 2020 - Action 2.
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Vitali, S., Domínguez, R. & Moriggia, V. Comparing stage-scenario with nodal formulation for multistage stochastic problems. 4OR-Q J Oper Res 19, 613–631 (2021). https://doi.org/10.1007/s10288-020-00462-x
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DOI: https://doi.org/10.1007/s10288-020-00462-x
Keywords
- Power generation capacity expansion
- Stage scenario formulation
- Nodal formulation
- Multistage planning
- Power systems
- Stochastic optimization
- GAMS