Risk management for mathematical optimization under uncertainty

We present a general multistage stochastic mixed 0-1 problem where the uncertainty appears everywhere in the objective function, constraints matrix and right-hand-side. The uncertainty is represented by a scenario tree that can be a symmetric or a nonsymmetric one. The stochastic model is converted in a mixed 0-1 Deterministic Equivalent Model in compact representation. Due to the difficulty of the problem, the solution offered by the stochastic model has been traditionally obtained by optimizing the objective function expected value (i.e., mean) over the scenarios, usually, along a time horizon. This approach (so named risk neutral) has the inconvenience of providing a solution that ignores the variance of the objective value of the scenarios and, so, the occurrence of scenarios with an objective value below the expected one. Alternatively, we present several approaches for risk averse management, namely, a scenario immunization strategy, the optimization of the well known Value-at-Risk (VaR) and several variants of the Conditional Value-at-Risk strategies, the optimization of the expected mean minus the weighted probability of having a "bad" scenario to occur for the given solution provided by the model, the optimization of the objective function expected value subject to stochastic dominance constraints (SDC) for a set of profiles given by the pairs of threshold objective values and either bounds on the probability of not reaching the thresholds or the expected shortfall over them, and the optimization of a mixture of the VaR and SDC strategies.

Two-Stage Stochastic Optimization. An Application in the Third Sector

[en] It is known that most of the problems applied in the real life present uncertainty. In the rst part of the dissertation, basic concepts and properties of the Stochastic Programming have been introduced to the reader, also known as Optimization under Uncertainty. Moreover, since stochastic programs are complex to compute, we have presented some other models such as wait-and-wee, expected value and the expected result of using expected value. The expected value of perfect information and the value of stochastic solution measures quantify how worthy the Stochastic Programming is, with respect to the other models. In the second part, it has been designed and implemented with the modeller GAMS and the optimizer CPLEX an application that optimizes the distribution of non-perishable products, guaranteeing some nutritional requirements with minimum cost. It has been developed within Hazia project, managed by Sortarazi association and associated with Food Bank of Biscay and Basic Social Services of several districts of Biscay.