A New Ranking Approach for Interval Valued Intuitionistic Fuzzy Sets and its Application in Decision Making

1. Introduction

Atanassov (1986) developed the theory of Intuitionistic fuzzy set (IFS), a generalised notion of fuzzy set theory (FST). In FST, to each element of the universe of discourse a degree of membership between 0 and 1 is assigned and the degree of non-membership is considered as complement to one of the membership degree. On the other hand, IFS does not imply that the non-membership degree is always the complement of the membership degree. Instead, it characterised some hesitation degree between membership and non-membership degrees. Nowadays, IFS theory is being paid more attention for the uncertainty modelling problem and applied in a wide range of areas, such as, decision making, medical diagnosis, fuzzy optimization, pattern recognition, etc. Gau and Buehrer (1993) studied the notion of vague set, which is also a generalization of fuzzy sets. Bustince and Burillo (1996) established that the concept vague set is also equivalent to IFS theory. In modern years, intuitionistic fuzzy sets theory has been broadly used in decision-making theory. The notion of interval-valued fuzzy sets (IVFSs) was studied by Turksen (1986) and Gorzaleczany (1987). Atanassov and Gargov (1989) presented that the interval valued intuitionistic fuzzy set is a generalization concept of the IFS theory. The used of aggregation operators and the ranking methods of interval valued intuitionistic fuzzy numbers plays an important role concerning intuitionistic fuzzy decision-making problems. Real number set is the well-ordered set that is linearly ordered by or ; but these inequalities cannot work here as in real numbers. Thus, a clear comparison is not possible among the fuzzy numbers as fuzzy numbers (FNs) are represented with the help of distribution and overlapping of the fuzzy numbers occurred with each other. A general order ranking of the fuzzy numbers is defined as a ranking function , where represents the set of all fuzzy numbers. Thus, a precise ranking process is necessary for fuzzy numbers in fuzzy decision-making problems. Hence, ranking of fuzzy numbers and intuitionistic fuzzy numbers (IFNs) came into picture as a core problem in fuzzy set theory and a lot of effort is made for an efficient ranking process.