Is ‘Fuzzy Theory’ An Appropriate Tool for Large Size Decision Problems?

Abstract

This chapter presents a review work in brief of the work [11] which is on a recently unearthed domain of the intuitionistic fuzzy set theory of Atanassov [1-8]. The most useful soft computing set theories [17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 28, 29, 31, 32] being used to solve the real life decision making problems are: fuzzy set theory, intuitionistic fuzzy set theory (vague sets are nothing but intuitionistic fuzzy sets, justified and reported by many authors), i–v fuzzy set theory, i-v intuitionistic fuzzy set theory, L-fuzzy set theory, type-2 fuzzy set theory, and also rough set theory, soft set theory, etc. While facing a decision making problem, the concerned decision maker in many cases choose one or more of these soft computing set theories by his own choice. Corresponding to each element x of all the universes involved in the decision problem, the value of µ(x) is proposed by the concerned decision maker by his best possible judgment. In real life situation, most of the decision making problems are of large size in the sense of the number of universes and the number of elements in the universes. For example, the populations in Big Data Statistics, be it R-Statistics or NR-Statistics [10], are all about big data; and decision analysis in many such cases involve the application of various soft-computing tools. But there arises a question: Is ‘Fuzzy Theory’ an appropriate tool for solving large size decision problems? In the work [11] a rigorous amount of mathematical analysis, logical analysis and justifications have been made to answer this question, introducing the ‘Theory of CIFS’ (Cognitive Intuitionistic Fuzzy System). In this chapter we revisit the mathematical analysis of [11] in brief, and discuss only the important issues of the ‘Theory of CIFS’ presented in [11]. Many of the decision problems are solved in computers using fuzzy numbers. It is observed that the existing notion of triangular fuzzy numbers and trapezoidal fuzzy numbers are having major drawbacks to the decision makers while solving problems using computer programs or softwares, the issue which is also discussed in this chapter.