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Resumen:
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Mathematical Programming is a well-placed field of Operational Research to tackle problems as diverse as those that arise in Logistics and Disaster Management. The fundamental objective of Mathematical Programming is the selection of an optimal alternative that meets a series of restrictions. The criterion by which the alternatives are evaluated is traditionally only one (for example, minimizing cost), however it is also common for several objectives to want to be considered simultaneously, thus giving rise to the Multi-criteria Decision. If the conditions to be met by an alternative or the evaluation of said alternative depend on random (or unknown) factors, we are in an optimization context under uncertainty. In the first chapters of this thesis the fields of multicriteria decision and optimization with uncertainty are studied, in two applications in the context of sustainable development and disaster management. Optimization with uncertainty is introduced through an application to rural electrification. In rural areas, access to electricity through solar systems installed in consumers' homes is common. These systems have to be repaired when they fail, so the decision of how to size a maintenance network is affected by great uncertainty. A mathematical programming model is developed by treating uncertainty in an unexplained way, the objective of which is to obtain a maintenance network at minimum cost. This model is later used as a tool to obtain simple rules that can predict the cost of maintenance using little information. The model is validated using information from a real program implemented in Morocco. When studying Multicriteria Optimization it is considered a problem in forest fire management. To mitigate the effects of forest fires, it is common to modify forests, with what is known as fuel treatment. Through this practice, consisting of the controlled felling or burning of trees in selected areas, it is achieved that when fires inevitably occur, they are easier to control. Unfortunately, modifying the flora can affect the existing fauna, so it is sensible to look for solutions that improve the situation in the face of a fire but without great detriment to the existing species. In other words, there are several criteria to take into account when optimizing. A mathematical programming model is developed, which suggests which zones to burn and when, taking into account these competing criteria. This model is applied to a series of simulated realistic cases. The following is a theoretical study of the field of Multiobjective Stochastic Programming (MSP), in which problems that simultaneously have various criteria and uncertainty are considered. In this chapter, a new solution concept is developed for MSP problems with risk aversion, its properties are studied and a linear programming model capable of obtaining said solution is formulated. A computational study of the model is also carried out, applying it to a variation of the well-known backpack problem. Finally, the problem of controlled burning is studied again, this time considering the existing uncertainty as it is not possible to know with certainty how many controlled burns can be carried out in a year, due to the limited window of time in which these can be carried out. The problem is solved using the multi-criteria and stochastic methodology with risk aversion developed in the previous chapter. Finally, the resulting model is applied to a real case located in southern Spain.
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