Intuitionistic fuzzy-valued neutrosophic multi-sets and numerical applications to classification

Abstract

A single-valued neutrosophic multi-set is characterized by a sequence of truth membership degrees, a sequence of indeterminacy membership degrees and a sequence of falsity membership degrees. Nature of a single-valued neutrosophic multi-set allows us to consider multiple information in the truth, indeterminacy and falsity memberships which is pretty useful in multi-criteria group decision making. In this paper, we consider sequences of intuitionistic fuzzy values instead of numbers to define the concept of intuitionistic fuzzy-valued neutrosophic multi-set. In this manner, such a set gives more powerful information. We also present some set theoretic operations and a partial order for intuitionistic fuzzy-valued neutrosophic sets and provide some algebraic operations between intuitionistic fuzzy-valued neutrosophic values. Then, we develop two types of weighted aggregation operators with the help of intuitionistic fuzzy t-norms and t-conorms. By considering some well-known additive generators of ordinary t-norms, we give the Algebraic weighted arithmetic and geometric aggregation operators and the Einstein weighted arithmetic and geometric aggregation operators that are the particular cases of the weighted aggregation operators defined via general t-norms and t-conorms. We also define a simplified neutrosophic valued similarity measure and we use a score function for simplified neutrosophic values to rank similarities of intuitionistic fuzzy-valued neutrosophic multi-values. Finally, we give an algorithm to solve classification problems using intuitionistic fuzzy-valued neutrosophic multi-values and proposed aggregation operators and we apply the theoretical part of the paper to a real classification problem.

An information measure based extended VIKOR method in intuitionistic fuzzy valued neutrosophic value setting for multi-criteria group decision making

“…Definition 2.2. [24] Let Y = {u 1 , ..., u n } be a finite set. An IFVNMS defined on Y is given with…”

“…where T A , I A and F A are the intuitionistic fuzzy valued truth, indeterminacy and falsity membership degrees, respectively, i.e., [24] have presented an algebraic weighted arithmetic aggregation operator and an algebraic weighted geometric aggregation operator for IFVNMVs. By simplifying these operators for sequence length 1, we consider the weighted arithmetic operator and weighted geometric operator…”

“…We have compared the performance of the stated algorithm with the existing measures such as similarity measure [24] and the score function [51].…”

“…The concept has various generalizations such as single-valued neutrosophic set (SVNS) [19], interval neutrosophic set [20], neutrosophic cubic set [21], single-valued neutrosophic linguistic set [22], simplified neutrosophic set (SNS) [23] etc. nver et al [24] has defined the concepts of intuitionistic fuzzy valued neutrosophic multi-set (IFVNMS) and intuitionistic fuzzy valued neutrosophic multi-value (IFVNMV).…”

“…They have defined some arithmetic operations for IFVNMVs and have proposed a weighted arithmetic and a weighted geometric aggregation operator. In [24] the authors studying in multi-set setting. In this paper, we take the sequence length equal to 1 for each criterion.…”