An Integrated Interval-Valued Intuitionistic Fuzzy Vague Set and Their Linguistic Variables

Abstract

Interval-valued intuitionistic fuzzy sets and vague sets are two sets that are commonly used in expressing vague and uncertain information. Despite their commonality, these two sets are characterized by different memberships. Interval-valued intuitionistic fuzzy sets are characterized by membership and non-membership with the sum of these memberships being less or equal than one. On the other hand, vague sets are characterized by either truth or false memberships with no clear-cut values of their sum. However, these two sets have shown a great extent of superiority in dealing with uncertain and imprecise information despite differences in the memberships. Therefore, it is a worthy effort to develop a new generic set by combining these two sets: interval-valued intuitionistic fuzzy sets and vague sets. In this paper, the notion of interval-valued intuitionistic fuzzy vague sets (IVIFVS) is proposed where membership and non-membership of interval-valued intuitionistic fuzzy sets are combined with truth membership and false membership of vague sets. To complement this new definition, several mathematical propositions, algebraic relations and arithmetic operations of IVIFVS are presented. Moreover, a new linguistic variable of IVIFVS with five linguistic scales that are normally used in solving multi-criteria decision-making problems is developed. Some examples are provided to illustrate the conceptual definitions and linguistic variables of the new sets.