Métodos Fuzzy aplicados ao estudo de problemas de programação linear

Abstract

The objective of this work is to study Linear Programming Problems (LPP) and apply the algebra and the α-levels of fuzzy numbers in these problems with parameters and variables being real numbers or fuzzy numbers. The first problem studied is a voltage divider circuit that is modeled to determine the values of the centered resistors, so that the impedance of the voltage divider resistor is minimal. Three cases are analyzed for the LPP components: real numbers, type-1 fuzzy numbers and type-2 fuzzy sets. The first case was considered to validate the other two cases. Another analysis was carried out to evaluate the benefits and restrictions in the use of algebra and the α-levels of fuzzy numbers in an affine method of interior points, the Primal - Affine Algorithm (APA), in determining the optimal solution of a LPP with parameters and variables being real numbers. The modified algorithm, Primal – Affin Fuzzy Algorithm (APAF), used triangular fuzzy numbers for each LPP variable and the algebra and α-levels of these numbers. The results obtained by APAF are better with each iteration and APAF requires a smaller number of iterations to reach the optimal value compared to APA, considering the absolute error. This work also demonstrates that a type-2 fuzzy set is a useful and insightful way to model optimization problems under uncertainty.Toward this objective, a new representation of interval-valued fuzzy sets based on constraint functions, called generalized fuzzy intervals, is developed for the analysis of possibilistic optimization problems in which the parameters are generalizations of interval-valued fuzzy numbers. The aim of this study is to represent interval-valued fuzzy sets in a way that possibilistic optimization models solved with generalized fuzzy intervals result in increased information about the risk associated with proposed actions that might be taken based on a solution strategy. This methodology uses the penalty method to reduce a possibilistic LPP into a nonlinear optimization problem, whose results are compared with the resolution of the LPP by α-levels whose solutions are defuzzified via centroid and, possibility and necessity, taking as reference the solution of the deterministic LPP. An application of a supplemental diet is presented to an individual who has been diagnosed with nutritional deficiencies, with varying amounts of nutrients relative to their preferred foods, mathematically described by interval-valued fuzzy number. The results obtained in solving the possibilistic optimization problem using the penalty method allow the individual to determine the best diet to meet their daily nutrient needs at the lowest cost compared to the other methods presented.