Uncertainties are commonly present in optimization systems, and when they are considered in the design stage, the problem usually is called a robust optimization problem. Robust optimization problems can be treated as noisy optimization problems, as worst case minimization problems, or by considering the mean and standard deviation values of the objective and constraint functions. The worst case scenario is preferred when the effects of the uncertainties on the nominal solution are critical to the application under consideration. Based on this worst case scenario, we developed the [I]RMOEA (Interval Robust Multi-Objective Evolutionary Algorithm), a hybrid method that combines interval analysis techniques to deal with the uncertainties in a deterministic way and a multiobjective evolutionary algorithm. We introduce [I]RMOEA and illustrate it on three robust test functions based on the ZDT problems. The results show that [I]RMOEA is an adequate way of tackling robust optimization problems with evolutionary techniques taking advantage of the interval analysis framework.
a b s t r a c tThis paper introduces a method for solving multi-objective optimization problems in uncertain environment. When the uncertainty factors of the optimization problem can be included into the mathematical model, through bounded intervals, [I]RMOA (Interval Robust Multi-objective Algorithm) can find an enclosure of the robust Pareto frontier. In this approach, the robust Pareto solutions are the ones that have the best performance when the worst case scenario, characterized by the uncertainty parameters, is considered.[I]RMOA has some positive aspects: it does not require the calculus of derivatives; it has only two input parameters; it is a reliable tool for solving different robust optimization problems, which can be nonlinear and discontinuous or have nonconvex Pareto frontier, for instance. The success of the method depends only on the quality of the objective inclusion functions and the precision parameters. Its disadvantage lies in the fact that it requires high computational effort, when high-dimensional problems are considered or when a very accurate enclosure is needed. Two analytical robust test functions are proposed to be treated by [I]RMOA and to validate the results provided by a stochastic multi-objective optimization method. The results are satisfactory.
The objective of this research is to develop methodological tools to consider qualitative information in solving spatial problems with their particular application to the determination of generating sites for renewable energy systems. This locational problem has a complex nature and its solution has to be capable to take into account a wide range of considerations, as well as evaluations of a qualitative character. In this sense, the proposed process of elaborating decisions starts with studies related to the spatial criteria, which are modeled and processed within the Spatial Multicriteria Analysis. The relative importance of the spatial criteria is established by the involved experts on the basis of the construction, transformation, and processing of preference relations within the framework of the Analytic Hierarchy Process. The resulting decision maps are generated by aggregating the spatial criteria, applying the Ordered Weighted Averaging operator and the linguistic quantifiers. These maps permit one to highlight rational solution alternatives, which, if necessary, are evaluated, compared, chosen, prioritized, and/or ordered, applying additional spatial criteria (of quantitative as well as qualitative character), by means of techniques for preference modeling in a fuzzy environment (within the framework of so-called R X , models). The corresponding decision process permits one to take into account and process quantitative and qualitative spatial criteria as well as preferences and judgments of the experts. The paper results are illustrated by a Case Study related to the determination of the most suitable locational alternatives for renewable power plants in the Minas Gerais state, Brazil.
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