DEVELOPMENT OF ALGORITHMS FOR SOME OPTIMIZATION PROBLEMS IN INTUITIONISTIC AND INTERVAL VALUED INTUITIONISTIC FUZZY ENVIRONMENTS

Abstract

In real world problems such as in corporate or in industry, decision making is a universal process. The experts and the decision makers (DMs) usually, have to suffer with uncertainty as well as with hesitation due to the complexity of situations. The main reasons behind these complexities are error in data, unawareness of customers, lack of good communications with all involved persons etc. So, the anticipation of parameters is a complex and challenging project. The classical methods come across great difficulty in managing with uncertainty and complexity involved in such situations. In general, the parameters of an optimization problem are taken into consideration 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 (FS) theory by Zadeh [115]. In order to handle the insufficient 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 a classical 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 carries each element of the universe of discourse to its range space, which, in most cases, is assumed to be the unit interval [0,1]. In many cases, the notion of representing the parameters by crisp numbers must be challenged. Using FS approach, quantities are represented by membership functions, instead of a particular value. These membership functions represent the uncertainties involved in various parameters of the problem. During the last decades, FS theory played an important role in modelling uncertain optimization problems. Zimmermann[117] showed that the solution of fuzzy linear programming problems (FLPPs) are always efficient. Since the FS theory came into existence, many extensions of fuzzy sets also appeared over the time, e.g., intuitionistic fuzzy set (IFS) proposed by Atanassov[13], interval-valued fuzzy set (IVFS) proposed by Gorzalczany[45] etc.IFS helps more adequately to represent situations where DMs abstain from expressing their assessment. In this way IFSs provide a richer tool to grasp imprecision and ambiguity than the conventional FSs. In real world, a person may assume that an object lies inside a set to a certain degree. However it is possible that he is not always so positive about it. So, in this situation there may be some hesitation or uncertainty about the membership degree. In FS theory, there is no manner to include this hesitation in the membership degree. To deal such difficulties in FS theory, IFS has been introduced. In real life, many people consider that assigning an exact number to an expert’s opinion is too restrictive, and that the assignment of an interval of values is more realistic. Keeping these things in mind, IVFS [46] has been introduced.The degree of membership in an IVFS lies in an interval [a; b] [0; 1] while the degree of membership in an IFS is a real number in [0,1]. In this way, IVFSs provide a richer tool to grasp imprecision and ambiguity than the IFSs. IVFSs play an important role in decision making under uncertainty and gained popularity in recent years. The main objective of the thesis is to explain how different types of uncertainties and hesitation occurring in real life optimization problems can be dealt with the help of intuitionistic fuzzy numbers (IFNs) and interval valued intuitionistic fuzzy numbers (IVIFNs). This thesis work is organized into six chapters. The chapter wise summary of the thesis is as follows: Chapter 1 is introductory in nature. In this chapter, definitions of various types of fuzzy, intuitionistic fuzzy and interval valued 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, intuitionistic fuzzy and interval valued intuitionistic fuzzy number into an equivalent real number. It also presents a brief review of research work in the area of fuzzy, intuitionistic fuzzy and interval valued intuitionistic fuzzy optimization. In Chapter 2, an algorithms is developed to solve fully intuitionistic fuzzy transportation problem where all parameters are taken as triangular intuitionistic fuzzy numbers. The optimal solution obtained by the proposed method are compared with the solution obtained by existing method. In Chapter 3, an algorithm is developed to solve fuzzy transportation problem where availabilities and demands are taken as real numbers but costs are interval valued triangular intuitionistic fuzzy numbers. The optimal solution obtained by the proposed method are compared with the solution obtained by existing methods. In Chapter 4, Various types of linear and non-linear membership functions are introduced and a new parabolic membership function is proposed which are used to develop an algorithm for solving multi-objective linear programming problem in interval valued intuitionistic fuzzy environment. Chapter 5 includes the study of fractional programming problems in interval valued intuitionistic fuzzy environment. Using the arithmetic operations on interval valued intuitionistic fuzzy numbers, we have developed an algorithm to solve interval valued intuitionistic fractional programming problem. Here, the parameters of the problem are taken as interval valued triangular intuitionistic fuzzy numbers. Finally, in Chapter 6, Conclusions are drawn based on the present study and the scope for future research work is suggested.