<i>N</i>-Cubic sets and aggregation operators

Abstract

The generalization of different types of fuzzy sets is on the way since the discovery of fuzzy sets by Zadeh 1965 based on mathematical logic. Some of them are, the interval-valued fuzzy sets, N-fuzzy sets and interval-valued N-fuzzy sets, where N-fuzzy sets are used to the co-domain [-1, 0] for the membership function. Cubic sets are the combination of interval-valued fuzzy sets and fuzzy sets presented in 2012 by Jun et al. Since the cubic sets considered only the positive aspects of a certain thing in many physical problems and ignore the negative aspects totally. So to cover the negative part the need was felt to extend the concept of cubic sets along the co-domain [-1, 0]. Thus in this paper, we initiate a new theory by mixing the cubic sets with N-fuzzy sets and define the new notion of N-cubic sets. We are inspired by the hybrid nature of the cubic sets which can handle positive characteristics of a certain thing in a much better way as compared to the fuzzy sets. Whereas, our case will be more helpful in order to characterize the negative characteristics of a certain thing in a more affective way as compared to N-fuzzy sets. Motivating from the idea of cubic sets, we initiate to combine the interval-valued N-fuzzy sets and N-fuzzy sets to produce a new class of cubic sets known as N-cubic sets based on mathematical logics. We define N-cubic sets and define many different operations like P-union (resp., P-intersection) R-union (resp., R-intersection) and complement. The notions of external and internal N-cubic sets and related operations and properties are investigated. We discuss some algebraic properties of N-cubic sets and define some aggregation operators which will be helpful in order to find more algebraic and practical applications in the future. At the end we present a N-cubic-decision making Problem by using the developed procedure.