DEVELOPMENT OF ALGORITHMS FOR SOME OPTIMIZATION PROBLEMS IN INTUITIONISTIC FUZZY ENVIRONMENT

Abstract

In real life problems such as in corporate or in industry, decision making is a universal process. The experts and the decision makers usually, have to su er with uncertainty as well as with hesitation, due to the complexity of the situations. The main reasons behind these complexities are lack of good communications with all involved persons, error in data, understanding of markets, unawareness of customers etc. So, the prediction of the parameters is a complex and challenging task. The classical methods encounter great di culty in dealing with uncertainty and complexity involved in such situations. In general, the parameters of an optimization problem are considered as crisp numbers. These crisp values are determined from past occurrences which are very uncertain, since the systems environment keep on changing. Therefore, some degree of uncertainty exists in such a determination. This led to the development of fuzzy set theory by Zadeh [86]. In order to handle the insu cient information, the fuzzy approach is used to model the problem and evaluate the optimal solution. Fuzzy set theory has been shown to be a useful tool to handle the situations in which the data are imprecise by attributing a degree to which a certain object belongs to a set. A fuzzy set is a generalization of an ordinary set in that it allows the degree of membership for each element to range over the unit interval [0, 1]. Thus, the membership function of a fuzzy set maps each element of the universe of discourse to its range space, which, in most cases, is assumed to be the unit interval. In many cases, the notion of representing the parameters by crisp numbers must be challenged. Using a fuzzy set approach, quantities are represented by membership functions, not by a single value. These membership functions represent the uncertainties involved in various parameters of the problem. Due to the capability of fuzzy sets to deal with qualitative data many fuzzy optimization methods have been proposed for the improvement of optimal i solution. Since the fuzzy set theory came into existence, many extensions of fuzzy sets also appeared over the time. One among these extensions is intuitionistic fuzzy set (IFS) which is playing an important role in decision making under uncertainty and gained popularity in recent years. The IFSs help more adequately to represent situations where decision makers abstain from expressing their assessment. In this way IFSs provide a richer tool to grasp imprecision and ambiguity than the conventional fuzzy sets. These characteristics of IFSs led to the extension of optimization methods in intuitionistic fuzzy environment. In real life, a person may assume that an object belongs to a set to a certain degree, but it is possible that he is not so sure about it. In other words, there may be hesitation or uncertainty about the membership degree. The main premise is that the parameters' demands across the problem are not xed, but are variable or uncertain. However, they are known to fall within a prescribed uncertainty set with some attributed degree. In fuzzy set theory, there is no means to incorporate this hesitation in the membership degree. To incorporate the hesitation in the membership degree, IFSs have been introduced. Here, in our study, the uncertainty with hesitation have been dealt with help of IFS theory of Attanassov [7]. The main objective of the thesis is to explain how di erent types of uncertainties and hesitation occur in real life optimization problems can be dealt with the help of intuitionistic fuzzy numbers. This thesis work is organized into ten chapters . The chapter wise summary of the thesis is as follows: Chapter 1 is introductory in nature. In this chapter, de nitions of various types of fuzzy and intuitionistic fuzzy numbers; and their mathematical operations have been introduced. To order these numbers a ranking function has been proposed. Ranking function is a method to transform a fuzzy or intuitionistic number into an equivalent real number. It also presents a brief review of work done in the eld of fuzzy and intuitionistic fuzzy optimization. In Chapter 2, an e cient algorithm is developed for solving intuitionistic fuzzy transportation problem of Type-1. The optimal solution obtained by the proposed approach has been compared with the solutions obtained by existing approaches. In Chapter 3, an algorithm has been developed for solving intuitionistic fuzzy transportaii tion problem of Type-2. The optimal solution obtained by the proposed approach has been compared with the solution obtained by existing approach. A real life transportation problem has been considered for study in intuitionistic fuzzy environment. In Chapter 4, an algorithm is developed to solve fully intuitionistic fuzzy transportation problems where all parameters, i.e, availabilities, demands and transportation costs have been taken as triangular intuitionistic fuzzy numbers. In Chapter 5, a generalized ranking function has been proposed which is used to develop a new algorithm for solving mixed intuitionisstic fuzzy transportation problem. Here the parameters of the problem include various types of uncertainties. In Chapter 6, various kinds of linear and nonlinear membership functions have been introduced and a new parabolic membership function has been proposed which are used to develop a procedure for solving multiobjective LPP in intuitionistic fuzzy environment. The procedure is also applied to solve multiobjective intuitionistic fuzzy transportation problem. In Chapter 7, a new method has been developed to solve intuitionistic fuzzy nonlinear programming problem. Modeling and optimization of manufacturing problems involving uncertainties and hesitations in parameters have been considered for study. Here, we have proposed and used a new nonlinear membership function. Numerical examples from manufacturing systems have been considered for illustration and compared with the existing approach. In Chapter 8, we have proposed various methods to solve multiobjective intuitionistic fuzzy nonlinear programming problem. By converting the problem into multiobjective LPP and then using Zimmerman's technique, 􀀀connective and minimum bounded sum operator various methods have been developed. Chapter 9 includes the study of fractional programming problems in intuitionistic fuzzy environment. Using the arithmetic operations on intuitionistic fuzzy numbers, we have developed an algorithm to solve intuitionistic fuzzy fractional programming problem. Here the parameters of the problem are taken as triangular intuitionistic fuzzy numbers. Finally, in Chapter 10, conclusions are drawn based on the present study and future research work is suggested in this direction.