Multi-criteria decision making process using complex cubic interval valued intuitionistic fuzzy set

This paper presents a new notion of complex cubic interval valued intuitionistic fuzzy set (CCIVIFS) which is an extension from the innovative concept of a cubic interval valued intuitionistic fuzzy set (CIVFS). The novelty of CCIVIFS is to achieve more range of values with the combination of interval-valued membership, interval-valued non-membership and fuzzy membership. We define some basic operations namely complement, union, intersection and notions of α˜ -internal, b˜ -internal, α˜ -external and b˜ -external complex cubic interval valued intuitionistic fuzzy set are introduced. Also P-union, P-intersection, R-union and R-intersection of α˜ -internal and b˜ -external complex cubic IVIF sets are discussed. Furthermore, a group decision-making method is discussed to illustrate the applicability and validity of the proposed approach.


Introduction
Zadeh [7] introduced the concept of fuzzy set and studied its properties. Since then many articles on fuzzy sets appeared highlighting the importance of the concept and its applications to logic, set theory, group theory, real analysis, measure theory, topology, etc. In 1975, Zadeh [15] introduced a new notion called interval-valued fuzzy subsets, where the values of the membership function are closed interval of numbers instead of a number. In 1999, Molodstov [4] introduced the concept of soft sets and discussed the fundamental results of the new theory. To deal with it, the theory of fuzzy set [7] or extended fuzzy sets such as intuitionistic fuzzy set [2], interval-valued intuitionistic fuzzy set [3] are the most successful ones, which characterize the parameter values in terms of membership degree. Under these environments, several researchers have paid great attention and successfully applied the concept of these theories to many practical areas. Jun [8] introduced the concept of cubic set which is a combination of interval-valued fuzzy set and fuzzy set and investigated several properties of cubic sets. In 2017, Chinnadurai and Barkavi [10][11][12][13] introduced a new concept of cubic soft matrix and studied its properties. In 2002, Ramot et. al., [5] introduced the concept of complex fuzzy set whose range is expanded to a unit cricle in a complex plane. Madad Khan et. al., [14] introduced complex fuzzy soft matrices and discussed some of its operations. Alkouri and salleh [1] extended the concept of complex fuzzy set to complex intuitionistic fuzzy set(CIFS) by adding the degree of non-membership and defined the basic operations such as union, intersection, complement etc. Then Rani and Gare [6] extended the concept of complex interval-valued intuitionistic fuzzy sets and operation on aggregation. Jun and Zun [9] introduced the concept of cubic interval-valued intuitionistic fuzzy sets and its applications. In this paper, we define some operations on complex cubic interval-valued intuitionistic fuzzy sets. We discuss α-internal, β-internal, α-external and β-external complex cubic intervalvalued intuitionistic fuzzy sets. Also P-union, P-intersection, R-union and R-intersection of α-internal and α-external complex cubic intuitionistic fuzzy sets are introduced. Finally, a real life application is discussed to show the reliability of the tool.

Preliminaries
In this section first we review some basic concepts and definition. Definition 2.1 [7] A fuzzy set A on a universe U is a mapping A : U → [0, 1] , for all u ∈ U . For any u ∈ U , A(u) denotes the membership degree of u in A. The class of fuzzy sets defined on U is denoted by F(U ).

Definition 2.2[2]
A generalization of the notions of intuitionistic fuzzy set(IFS) where the functions µ A : E → L and ν A : E → L define the degree of membership and the degree of nonmembership of the element x ∈ E to A ⊂ E, respectively the function µ A and ν A should satisfy the condition: Let a set E be a fixed. An interval-valued intuitionistic fuzzy sets(IVIFS) A over E is an object Let X be a nonempty set. By a cubic interval-valued intuitionistic fuzzy set in X we mean a structure A = { x, µ(x), A(x) |x ∈ X} in which µ(x) is a fuzzy set in X and A(x) is an intervalvalued intuitionistic fuzzy set in X . Definition 2.5 [6] Let U be the universe of discourse. A complex interval-valued intuitionistic fuzzy sets defined on U is a set given by represent the degree of lower and upper bound of the membership and nonmembership which aer defined as µ −

Complex cubic interval valued intuitionistic fuzzy set(CCIVIFS)
In this section we define complex cubic interval-valued intuitionistic fuzzy sets and investigate αinternal, β-internal, α-external and β-external complex cubic interval-valued intuitionistic fuzzy sets. Definition:3.1 Let U be the universe of discourse. A Complex cubic interval valued intuitionistic fuzzy set [CCIVIFS] defined on U is a set given by, represent the degree of lower and upper bound of the membership and non-membership which are defined as, α − k (x) = . The fuzzy amplitude terms r f k ∈ [0, 1] and the fuzzy phase term θ f r k (x) ∈ [0, 2π] for all x ∈ U . Therefore, mathematically [CCIVIFS] A k defined on U can be represented as, ..., u m be a finite universal set and E = e 1 , e 2 , ...., e n be a finite set of parameters. Let A ⊆ E. Then complex cubic interval valued intuitionistic fuzzy set A k can be expressed in matrix form as then it is β− internal and so internal. Also if A k is β− external, then it is α− external and so external. Proof: Straight forward.
be a CCIVIFS in non-empty set U in which the right end point of membership value(or non-membership value) is equal to the left end point of non-membership (or membership) for all x ∈ U If we define µ f k (x) by Given an CCIVIFS sets A k in U, the complement of A k is denoted by (A k ) c and is defined as followes, (ii) R-order: Example:3.2 (i) P-order: Let A 1 , A 2 be two CCIVIFSs. Then P-order of (ii) R-order: Let A 1 , A 2 be two CCIVIFS. Then R-order of Let , µ f 2 (x) be two CCIVIFSs . Then the union of A 1 and A 2 is, for all x ∈ U. and intersection of A 1 , A 2 is, for all x ∈ U. Definition:3.7 of a CCIVIFS in U,(k = 1, 2, 3, .., n) we define for all x ∈ U. (1)(P-union) Proof: It is similar to the proof of theorem 3.3 Theorem:3.5 is a set of α− internal CCIVIFS in U. Then the P-union and the P-intersection of for all x ∈ U (k = 1, 2, 3, ..., n). It follows that Then the following example R-union and R-intersection of α− internal CCIVIFSs need not be α− internal. Let A 1 , A 2 be two CCIVIFSs.
We provide a condition for the R-union of two α− internal CCIVIFS to be α− internal .  We predict from Table 1 that the region a 3 is the most suitable for well drilling.

Conclusion
In this paper, we have introduced the concept of complex cubic interval valued intuitionistic fuzzy set and matrix to deal with uncertainty. Also, a decision making method has been proposed to demonstrate the reliability and validity of the tool.