Study on Similarity Measures of Fuzzy Numbers Aggregation Operators in Fuzzy Systems and Decision Making

Abstract

In many cases imprecision plays an important role in information representation in real processes where increase in precision would otherwise become unmanageable. Among the several methods available Fuzzy Set Theory is the most adopted method to handle the imprecision. Aggregation and fusion of information are major problems in various fields of pattern recognition, image processing and decision making. Making decisions is undoubtedly one of the most fundamental activities of human beings. The objective of fuzzy decision is to obtain a decision that is optimum. The similarity of fuzzy numbers and aggregation operators characterizes the imprecision built-in in a decision making problem. Comparison of two or more objects/alternatives is a fundamental property for many decision making model. In this work the main ingredients namely the similarity measure and aggregation operators required for decision making are studied to obtain a more realistic decision process. Indeed, identifying a robust measure and defining a suitable aggregation operator, contributes significantly to the important goal of improving the practice of decision analysis. An important concept of fuzzy sets in decision making is fuzzy numbers. Fuzzy numbers can very well represent the shape of the objects. Consequently measuring the degree of similarity between the fuzzy numbers plays an important role in fuzzy decision making, information fusion, pattern recognition etc. A similarity measure for generalized fuzzy numbers is proposed in this thesis work and the similarity measure is applied to various decision making problems. The fuzzy similarity measure is developed by integrating the concept of centre of gravity (COG) points and fuzzy difference of distance of points of fuzzy numbers. It turned out that in real situations the newly proposed similarity measure gives more intuitively appealing results. Study of appropriate antecedent connector model is an important criterion for a fuzzy system to produce proper output otherwise the decision output of the system is a failure. For neuro-fuzzy logic systems that needs to be optimized during tuning procedure requires the firing degree to be separable. Tuning methods like back propagation algorithm involves the partial derivatives of the membership functions appearing in the objective functions. Two different aggregation operators namely, compensatory operators and soft ordered weighted averaging operator are taken for study in this work. Various t-norms and t-conorms are applied for these operators and tested for the usability of the models. It is observed that only ‘multiplicative compensatory and’ and ‘additive compensatory and’ with t- norm as product and t- conorm as bounded sum are usable for a nonsingleton fuzzy logic systems. Aggregation operators serve as a tool for combining various degrees of membership into one numerical value and are used routinely in decision making problem. The decision making activity considered in this thesis is a multiperson decision making model, the situations in which the decision maker is required to choose the best alternative from a set of explicitly available alternatives. In decision making problems procedures have been established to combine opinions about alternatives related to different points of view. The question of defining weights is the kernel of the problem. In this work generalized mixture operator with linear and exponential weight generating function are taken and a comparative analysis is made on the performance of two different aggregation operators, ordered weighted aggregation operator and generalized mixture operator. An extensive analysis revealed that generalized mixture operator is very sensitive especially in situations where preferences expressed by the experts are conflicting. For a multipurpose decision making problem two classical methods are available in literature, Analytic Hierarchy process (AHP) and fuzzy majority selection scheme. In AHP reciprocal multiplicative preference relation is taken as the preference representation and in fuzzy majority based selection fuzzy preference relation is taken as uniform representation element. A new transformation function is constructed for a multiplicative preference relation and fuzzy preference relation. In decision making, in order to avoid misleading solutions the study of consistency of preference relation becomes very important aspect. The consistency property of the newly defined function is studied. The newly defined transformation function transforms a reciprocal multiplicative preference relation into reciprocal fuzzy preference relation. The function retains the ordering of the alternatives as obtained by fuzzy majority scheme and AHP. The flexibility of applying this transformation function in implementing fuzzy majority based selection scheme is observed. Also the important aspect of the maintenance of the reciprocity property of the transformation function is verified.