Complex Atanassov's Intuitionistic Fuzzy Relation

Abstract

This paper presents distance measure between two complex Atanassov's intuitionistic fuzzy sets (CAIFSs). This distance measure is used to illustrate an application of CAIFSs in solving one of the most core application areas of fuzzy set theory, which is multiattributes decision-making (MADM) problems, in complex Atanassov's intuitionistic fuzzy realm. A new structure of relation between two CAIFSs, called complex Atanassov's intuitionistic fuzzy relation (CAIFR), is obtained. This relation is formally generalised from a conventional Atanassov's intuitionistic fuzzy relation, based on complex Atanassov's intuitionistic fuzzy sets, in which the ranges of values of CAIFR are extended to the unit circle in complex plane for both membership and nonmembership functions instead of [0, 1] as in the conventional Atanassov's intuitionistic fuzzy functions. Definition and some mathematical concepts of CAIFS, which serve as a foundation for the creation of complex Atanassov's intuitionistic fuzzy relation, are recalled. We also introduce the Cartesian product of CAIFSs and derive two properties of the product space. The concept of projection and cylindric extension of CAIFRs are also introduced. An example of CAIFR in real-life situation is illustrated in this paper. Finally, we introduce the concept of composition of CAIFRs.

1. Introduction

The idea of the concept of Atanassov’s intuitionistic fuzzy set (AIFS) was introduced by Atanassov [1], where he achieved his concept by adding the nonmembership term to the definition of fuzzy set (FS) that was given by Zadeh [2], while the fuzzy set has only one component, a membership function. It is well known that the range of each membership and nonmembership functions is limited to [0,1], where it belongs to the real numbers. In 2007, Fathi [3] introduced some operations on AIFSs called 𝑆-norm and 𝑇-norm. The question presented by Daniel Ramot and other researchers was that what will be the result if we change the codomain of the fuzzy sets to complex numbers? In 2002, Ramot et al. [4] introduced a new innovative concept which they called a complex fuzzy set (CFS). On the other hand, complex fuzzy set has been used to solve various problems, in particular the multiple periodic factor prediction problems [5]. A neurofuzzy system architecture implementing complex fuzzy rule was defined by Chen et al. [6], which is the first practical application of the concept of complex fuzzy logic.

Atanassov’s intuitionistic fuzzy set has several applications in many areas and has been used to solve multiple attributes decision making (MADM) problems, see [7–10]. Atanassov’s intuitionistic fuzzy set represents the information involving two or more answers of type: yes, no, I do not know, I am not sure, and so forth. Our purpose in this paper is to give an example of application that represents this type of information which is happening repeatedly over a period of time. In our previous conference paper [11], we introduced the concept of complex Atanassov’s intuitionistic fuzzy set. Complex Atanassov’s intuitionistic fuzzy set can represent the information on people’s decision which happens periodically. In addition, we also represent the uncertainty of complex-valued functions in many physical quantities, for instance, wave function in quantum mechanics, impedance in electrical engineering, complex amplitude, and so forth. We can also define the values of belongingness and nonbelongingness for any object in these complex-valued functions.

In the literature [4, 11], the novelty of using complex numbers appears in the phase term to get more range-valued membership functions to represent the problems with uncertainty and periodicity simultaneously. In addition, it indicates the uncertainty of complex-valued functions in many physical quantities, for instance, wave function in quantum mechanics, impedance in electrical engineering, complex amplitude, and so forth. In this paper, the novelty also appears in the phase term but for both membership and nonmembership functions in some inherent concepts, in contrast to CFS which is only characterized by membership function. The literature novelties of phase term are extended to appear in several prominent concepts under the case of CAIFS, for instance, distance measure, Cartesian product, relations, projection, cylindric extension, and so forth. The CFS [4] has only one additional phase term, but in CAIFS [11], we have two additional phase terms. This confers more range values to represent the uncertainty and periodicity semantics simultaneously, and to define the values of belongingness and nonbelongingness for any object in these complex-valued functions. Undoubtedly, the property of periodic nature of complex-valued memberships helps us to introduce the present concepts. In other words, the periodicity semantic can be denoted as a phase term in complex numbers, and the uncertainty semantic can be denoted as an amplitude term in complex numbers.

The difference between CAIFS and AIFS is that in our concept, CAIFS has an ability to represent the problems with Atanassov’s intuitionistic uncertainty and periodicity simultaneously. The phase term of CAIFS shows that at least in some cases a second dimension (periodic dimension) of membership is required, where the phase term is the distinguishing factor between AIFS and CAIFS. In this paper, we employ CAIFS structure in MADM problems to choose the best alternative by generalising the selection model from [12]. It is worth noting that one of the most useful properties of the phase term is the ability to convey Atanassov’s intuitionistic fuzzy information.

It should be noted that there are many researchers who combined complex numbers and fuzzy sets, such as Buckley [13], Nguyen et al. [14], and Zhang et al. [15, 16]. On the other hand, Ramot et al. [4] introduced a new approach that is absolutely different from them, where they extended the range of membership function to unit circle in the complex plane, unlike the others who limited the range to [0,1]. Ramot et al. [4] also added an additional term called the phase term to solve the enigma in translating some complex-valued functions on physical terms to human language and vice versa. In 2011, Jun et al. [5] employed complex fuzzy sets to represent the information with uncertainty and periodicity, where they introduced a product-sum aggregation operator- (PSAO-) based prediction (PSAOP) method to generate a solution of multiple periodic factor prediction (MPFP) problems.

Since 1965, the concept of fuzzy set theory was introduced by Zadeh [2]; numerous researchers did intensive studies on fuzzy relation (simply denoted by FR) such as Di Nola et al. [17, 18] and Bandler and Kohout [19]. According to Mendel [20] in 1995 “fuzzy relation represents a degree of presence or absence of association, interaction, or interconnectedness” between the elements of two or more sets. Later in 2002, complex fuzzy relation (CFR) was introduced by Ramot et al. [4], where “the CFRs represent both the degree of presence or absence of association, interaction, or interconnectedness, and the phase of association, interaction, or interconnectedness between the elements of two or more sets.”

Burillo and Bustince introduced the definition of Atanassov’s intuitionistic fuzzy relation and studied some of its properties [21–24]. In 2010, Atanassov’s intuitionistic fuzzy relation was reintroduced by Adam [25] based on the definition of Atanassov’s intuitionistic fuzzy relation given by Bustince and Burillo. Adam generalised fuzzy relation to Atanassov’s intuitionistic fuzzy relation by adding the nonmembership functions to the fuzzy relation structure. In this paper, we introduce CAIFRs by extending each value of the membership and nonmembership functions from real number, [0,1], to complex number, the unit circle. The main benefit expected to be gained from the introduction of the concept of CAIFR is the ability to denote both the presence and absence of association, interaction, or interconnectedness in one set of CAIFS instead of two sets as in CFS, where CAIFS has membership and nonmembership functions in the same set, whilst CFS contains only the membership functions and denotes the presence or absence of association, interaction, or interconnectedness.

2. Preliminaries

In this section, we recall some definitions and results that will be effectively employed in this paper.

Definition 1 (see [2]). A fuzzy set 𝐴 in a universe of discourse 𝑋 is characterized by a membership function 𝜇_𝐴(𝑥) that takes values in the interval [0,1].

Definition 2 (see [1]). An Atanassov’s intuitionistic fuzzy set (AIFS) 𝐴 in a nonempty set 𝑋 (a universe of discourse) is an object having the following form: 𝐴=𝑥,𝜇_𝐴(𝑥),𝛾_𝐴(𝑥)∶𝑥∈𝑋, (1) where the functions 𝜇_𝐴(𝑥)∶𝑋→[0,1],𝛾_𝐴(𝑥)∶𝑋→[0,1] denote the degree of membership and degree of nonmembership of each element 𝑥∈𝑋 to the set 𝐴, respectively, and 0≤𝜇_𝐴(𝑥)+𝛾_𝐴(𝑥)≤1 for all 𝑥∈𝑋.

Definition 3 (see [26]). The complement, union, and intersection of two AIFSs 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝛾_𝐴(𝑥)⟩∶𝑥∈𝑋} and 𝐵={⟨𝑥,𝜇_𝐵(𝑥),𝛾_𝐵(𝑥)⟩∶𝑥∈𝑋}, in a universe of discourse 𝑋, are defined as follows:(a)𝐴={⟨𝑥,𝛾_𝐴(𝑥),𝜇_𝐴(𝑥)⟩∶𝑥∈𝑋},(b)𝐴∪𝐵={⟨𝑥,max(𝜇_𝐴(𝑥),𝜇_𝐵(𝑥)),min(𝛾_𝐴(𝑥),𝛾_𝐵(𝑥))⟩∶𝑥∈𝑋}, (c)𝐴∩𝐵={⟨𝑥,min(𝜇_𝐴(𝑥),𝜇_𝐵(𝑥)),max(𝛾_𝐴(𝑥),𝛾_𝐵(𝑥))⟩∶𝑥∈𝑋}.

Definition 4 (see [4]). A complex fuzzy set (CFS) 𝐴, defined on a universe of discourse 𝑋, is characterized by membership functions 𝜇_𝐴(𝑥) that assign to any element 𝑥∈𝑋 a complex-valued grade of membership in 𝐴.
By definition, the values of 𝜇_𝐴(𝑥) may receive all lying within the unit circle in the complex plane and are thus of the form 𝜇_𝐴(𝑥)=𝑟_𝐴(𝑥)⋅𝑒^{𝑖𝜔𝐴(𝑥)}, where 𝑖=√−1, each of 𝑟_𝐴(𝑥) and 𝜔_𝐴(𝑥) is real-valued, and 𝑟_𝐴(𝑥)∈[0,1]. The CFS 𝐴 may be represented as the set of ordered pairs 𝐴={(𝑥,𝜇_𝐴(𝑥))∶𝑥∈𝑋}.

Definition 5 (see [25]). Let 𝐴₁,…,𝐴_𝑛 be Atanassov’s intuitionistic fuzzy sets in 𝑋₁,…,𝑋_𝑛, respectively. The Cartesian product 𝐴₁×⋯×𝐴_𝑛 is an Atanassov’s intuitionistic fuzzy set defined by 𝐴₁×⋯×𝐴_𝑛=𝑥₁,…,𝑥_𝑛,𝜇_{𝐴1×⋯×𝐴𝑛}𝑥₁,…,𝑥_𝑛,𝛾_{𝐴1×⋯×𝐴𝑛}𝑥₁,…,𝑥_𝑛∶𝑥₁,…,𝑥_𝑛∈𝑋₁,…,𝑋_𝑛, (2) where  𝜇_{𝐴1×⋯×𝐴𝑛}(𝑥₁,…,𝑥_𝑛)=min[𝜇_𝐴1(𝑥₁),…,𝜇_𝐴𝑛(𝑥_𝑛)], and  𝛾_{𝐴1×⋯×𝐴𝑛}(𝑥₁,…,𝑥_𝑛)=max[𝛾_𝐴1(𝑥₁),…,   𝛾_𝐴𝑛(𝑥_𝑛)].

Definition 6 (see [25]). Let 𝑄 be an Atanassov’s intuitionistic fuzzy relation in 𝑋₁×⋯×𝑋_𝑛, and let {𝑖₁,…,𝑖_𝑘} be a subsequence of {1,2,…,𝑛}. Then the projection of 𝑄 on 𝑋_𝑖1×⋯×𝑋_𝑖𝑘 is an Atanassov’s intuitionistic fuzzy relation 𝑄_𝑝 on 𝑋_𝑖1×⋯×𝑋_𝑖𝑘, defined by membership and nonmembership functions, 𝜇_𝑄𝑝𝑥_𝑖1,…,𝑥_𝑖𝑘=max_{𝑥𝑗1∈𝑋𝑗1,…,𝑥𝑗(𝑛−𝑘)∈𝑋𝑗(𝑛−𝑘)}𝜇_𝑄𝑥₁,…,𝑥_𝑛,𝛾_𝑄𝑝𝑥_𝑖1,…,𝑥_𝑖𝑘=min_{𝑥𝑗1∈𝑋𝑗1,…,𝑥𝑗(𝑛−𝑘)∈𝑋𝑗(𝑛−𝑘)}𝛾_𝑄𝑥₁,…,𝑥_𝑛, (3) where the {𝑥_𝑗1,𝑥_𝑗2,…,𝑥_{𝑗(𝑛−𝑘)}} is the complement of {𝑥_𝑖1,𝑥_𝑖2,…,𝑥_𝑖𝑘} with respect to {𝑥₁,𝑥₂,…,𝑥_𝑛}.

Definition 7 (see [25]). Let 𝑄 be an Atanassov’s intuitionistic fuzzy relation in 𝑋₁×⋯×𝑋_𝑛, and let {𝑖₁,…,𝑖_𝑘} be a subsequence of {1,2,…,𝑛}, then the cylindric extension of 𝑄_𝑝 to 𝑋₁×⋯×𝑋_𝑛 is an Atanassov’s intuitionistic fuzzy relation 𝑄_PE on 𝑋₁×⋯×𝑋_𝑛, defined by membership and nonmembership functions: 𝜇_𝑄PE𝑥₁,…,𝑥_𝑛=𝜇_𝑄𝑃𝑥_𝑖1,…,𝑥_𝑖𝑛,𝛾_𝑄PE𝑥₁,…,𝑥_𝑛=𝛾_𝑄𝑃𝑥_𝑖1,…,𝑥_𝑖𝑛. (4)

Definition 8 (see [4]). Let 𝑋 and 𝑌 be two sets. A complex fuzzy relation 𝑅 of 𝑋 and 𝑌, written 𝑅(𝑈,𝑉), is a complex fuzzy subset of the product space 𝑋×𝑌. The relation 𝑅(𝑋,𝑌) is characterized by the complex membership function 𝜇_𝑅(𝑥,𝑦), where 𝑥∈𝑋 and 𝑦∈𝑌, and 𝜇_𝑅(𝑥,𝑦) assigns to each pair (𝑥,𝑦) a complex-valued grade of membership to the set 𝑅(𝑋,𝑌). As always, 𝑅(𝑋,𝑌) may be represented as the set of ordered pairs: 𝑅(𝑋,𝑉)=𝑥,𝑦,𝜇_𝑅𝑥,𝑦∣𝑥,𝑦∈𝑋×𝑌. (5)
The values 𝜇_𝑅(𝑥,𝑦) may be took from the unit circle in the complex plane and are of the following form: 𝑟(𝑥)⋅𝑒^{𝑖𝜔(𝑥)}(𝑖=√−1),𝑟(𝑥) and 𝜔(𝑥) are both real-valued, with 𝑟(𝑥)∈[0,1].

The complex membership function is to be interpreted in the following manner:(i)𝑟(𝑥) represents a degree of presence or absence of association, interaction, or interconnectedness between the elements of 𝑋 and 𝑌;(ii)𝜔(𝑥) represents the phase of association, interaction, or interconnectedness between the elements of 𝑋 and 𝑌.

Definition 9 (see [15]). Let 𝑋, 𝑌, and 𝑍 be universes, 𝐴 a complex fuzzy relation of 𝑋 and 𝑌, and 𝐵 a complex fuzzy relation of 𝑌and 𝑍. Then, we say a composition of 𝐴 and 𝐵, denoted by 𝐴∘𝐵, is a complex fuzzy relation of 𝑋 and 𝑍, is specified by the following function 𝜇_𝐴∘𝐵(𝑥,𝑧)=𝑟_𝐴∘𝐵(𝑥,𝑧)⋅𝑒^{𝑖arg𝐴∘𝐵(𝑥,𝑧)}=sup_𝑦∈𝑌min𝑟_𝐴𝑥,𝑦,𝑟_𝐵𝑦,𝑧⋅𝑒^{𝑖sup𝑦∈𝑌min[arg𝐴(𝑥,𝑦),arg𝐵(𝑦,𝑧)]}. (6)

Example 10 (see [15]). Let 𝐴()=𝑦𝑋,𝑌¹𝑦₂𝑥₁0.6𝑒^𝑖1.2𝜋0.8𝑒^𝑖1.6𝜋𝑥₂1.0𝑒^𝑖2𝜋0.5𝑒^𝑖𝜋,𝐵()=𝑧𝑌,𝑍¹𝑧₂𝑦₁0.6𝑒^𝑖1.2𝜋0.8𝑒^𝑖1.6𝜋𝑦₂1.0𝑒^𝑖2𝜋0.8𝑒^𝑖1.6𝜋 (7)
be two relational matrices. Then, to compute 𝐴∘𝐵(𝑋,𝑍), we should note that 𝑋×𝑍 has four elements: 𝐴∘𝐵=^{(𝑥1,𝑧1)(𝑥1,𝑧2)}_{(𝑥2,𝑧1)(𝑥2,𝑧2)}. Thus, our task is to determine the membership, 𝜇_𝐴∘𝐵, values for the four elements. For instance, we use the function in Definition 9 to determine the membership function of 𝐴∘𝐵(𝑥₁,𝑧₁), so we have 𝐴∘𝐵𝑥₁,𝑧₁=sup_𝑦∈𝑌min𝑟_𝐴𝑥₁,𝑦,𝑟_𝐵𝑦,𝑧₁⋅𝑒^{𝑖sup𝑦∈𝑌min(𝜔𝐴(𝑥1,𝑦),𝜔𝐵(𝑦,𝑧1))}=sup_𝑦∈𝑌min𝑟_𝐴𝑥₁,𝑦₁,𝑟_𝐵𝑦₁,𝑧₁,min𝑟_𝐴𝑥₁,𝑦₂,𝑟_𝐵𝑦₂,𝑧₁⋅𝑒^{sup𝑦∈𝑌{min[(𝜔𝑟𝐴(𝑥1,𝑦1)),𝜔𝑟𝐵(𝑦1,𝑧1)],min[(𝜔𝑟𝐴(𝑥1,𝑦2)),𝜔𝑟𝐵(𝑦2,𝑧1)]}}=sup_𝑦∈𝑌{min[0.6,0.6],min[0.8,1]}⋅𝑒^{𝑖sup𝑦∈𝑌{min[1.2𝜋,1.2𝜋],min[1.6𝜋,2𝜋]}}=sup_𝑦∈𝑌{0.6,0.8}⋅𝑒^{𝑖sup𝑦∈𝑌{1.2𝜋,1.6𝜋}}=0.8⋅𝑒^𝑖1.6𝜋. (8)
Analogously, we determine the elements (𝑥₁,𝑧₂), (𝑥₂,𝑧₁),  and  (𝑥₂,𝑧₂) therefore, the final relational matrix of 𝐴∘𝐵 is 𝐴∘𝐵=0.8𝑒^𝑖1.6𝜋0.6𝑒^𝑖1.2𝜋0.8𝑒^𝑖1.6𝜋0.8𝑒^𝑖1.6𝜋. (9)

Proposition 11 (see [15]). Let 𝑊, 𝑋, 𝑌, and 𝑍 be universes, 𝐴 a complex fuzzy relation of 𝑊 and 𝑋, 𝐵 a complex fuzzy relation of 𝑋 and 𝑌, and 𝐶 a complex fuzzy relation of 𝑌 and 𝑍. Then, 𝐴∘(𝐵∘𝐶)=(𝐴∘𝐵)∘𝐶.

Proposition 12 (see [15]). Let 𝑋, 𝑌, and 𝑍 be universes, 𝐴 and 𝐵 two complex fuzzy relations of 𝑋 and 𝑌,  and 𝐶 a complex fuzzy relation of  𝑌 and 𝑍. Then,(i)(𝐴∪𝐵)∘𝐶=(𝐴∘𝐶)∪(𝐵∘𝐶), (ii)If𝐴⊆𝐵, then𝐴∘𝐶⊆𝐵∘𝐶.

Definition 13 (see [15]). Let 𝐴 and 𝐵 be two complex fuzzy sets on 𝑋,𝜇_𝐴(𝑥)=𝑟_𝐴(𝑥)⋅𝑒^{𝑖arg𝐴(𝑥)} and 𝜇_𝐵(𝑥)=𝑟_𝐵(𝑥)⋅𝑒^{𝑖arg𝐵(𝑥)} their membership functions, respectively. We say that 𝐴 is greater than 𝐵, denoted by 𝐴⊇𝐵 or 𝐵⊆𝐴, if for any 𝑥∈𝑋,𝑟_𝐴(𝑥)≤𝑟_𝐵(𝑥), and arg_𝐴(𝑥)≤arg_𝐵(𝑥).

Definition 14 (see [11]). A complex Atanassov’s intuitionistic fuzzy set 𝐴, defined on a universe of discourse 𝑈, is characterized by membership and nonmembership functions 𝜇_𝐴(𝑥) and 𝛾_𝐴(𝑥), respectively, that assign to any element 𝑥∈𝑋 a complex-valued grade of both membership and nonmembership functions in 𝐴.
By definition, the values of 𝜇_𝐴(𝑥), 𝛾_𝐴(𝑥), and their sum may receive all lying within the unit circle in the complex plane and are of the form 𝜇_𝐴(𝑥)=𝑟_𝐴(𝑥)⋅𝑒^{𝑖𝜔𝜇𝐴(𝑥)} for membership function in 𝐴 and 𝛾_𝐴(𝑥)=𝑘_𝐴(𝑥)⋅𝑒^{𝑖𝜔𝛾𝐴(𝑥)} for nonmembership function in 𝐴, where 𝑖=√−1, each of 𝑟_𝐴(𝑥) and 𝑘_𝐴(𝑥) is real-valued and both belong to the closed unit interval [0,1] such that 0≤𝑟_𝐴(𝑥)+𝑘_𝐴(𝑥)≤1; also, 𝜔_𝜇A(𝑥) and 𝜔_𝛾𝐴(𝑥) are real-valued. We represent the CAIFS 𝐴 as 𝐴=𝑥,𝜇_𝐴(𝑥),𝛾_𝐴(𝑥)∶𝑥∈𝑋, (10) where 𝜇_𝐴(𝑥)∶𝑋→{𝑎∣𝑎∈𝐶,|𝑎|≤1},𝛾_𝐴(𝑥)∶𝑋→{𝑎^′∣𝑎^′∈𝐶,|𝑎^′|≤1} and |𝜇_𝐴(𝑥)+𝛾_𝐴(𝑥)|≤1.

Proposition 15 (see [11]). If 𝐴 is any CFS denoted by 𝐴={(𝑥,𝜇_𝐴(𝑥))∶𝑥∈𝑋}, then we can represent the set 𝐴 as a set of CAIFS as follows: 𝐴=𝑥,𝜇_𝐴(𝑥),𝛾_𝐴(𝑥)∶𝑥∈𝑋, (11) where the nonmembership of a CAIFS 𝐴 equals the complement of CFS 𝐴  (i.e., 𝛾_𝐴(𝑥)=𝜇_𝐴(𝑥)).

Definition 16 (see [11]). Let 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝛾_𝐴(𝑥)⟩∶𝑥∈𝑋} be a complex Atanassov’s intuitionistic fuzzy set. Define the complement of 𝐴, 𝑐(𝐴), as 𝑐(𝐴)=𝑥,𝛾_𝐴(𝑥),𝜇_𝐴(𝑥=)∶𝑥∈𝑋𝑥,𝑘_𝐴(𝑥)⋅𝑒^{𝑖(𝜔𝑘𝐴(𝑥))},𝑟_𝐴(𝑥)⋅𝑒^{𝑖(𝜔𝑟𝐴(𝑥))}∶𝑥∈𝑋, (12) where 𝜔_𝐴(𝑥)=𝜔_𝐴(𝑥), 2𝜋−𝜔_𝐴(𝑥), or 𝜔_𝐴(𝑥)+𝜋.

Definition 17 (see [11]). Let 𝐴 and 𝐵 be two CAIFSs on universe of discourse 𝑋, with complex-valued membership and nonmembership functions. The complex Atanassov’s intuitionistic fuzzy union of 𝐴 and 𝐵, denoted by 𝐴∪𝐵, is specified by the following function: 𝐸∶𝑎,𝑎^′∣𝑎,𝑎^′∈𝐶∶𝑎+𝑎^′≤1,|𝑎|≤1,𝑎^′×≤1𝑏,𝑏^′∣𝑏,𝑏^′∈𝐶∶𝑏+𝑏^′≤1,|𝑏|≤1,𝑏^′⟶≤1𝑑,𝑑^′∣𝑑,𝑑^′∈𝐶∶𝑑+𝑑^′≤1,|𝑑|≤1,𝑑^′≤1, (13) where 𝑎, 𝑏,  and  𝑑  and  𝑎^′, 𝑏^′,  and  𝑑^′ are the membership and nonmembership functions of 𝐴,𝐵, and 𝐴∪𝐵, respectively. 𝐸 assigns a complex value, 𝐸𝑎,𝑎^′,𝑏,𝑏^′=𝐸_𝜇(𝑎,𝑏),𝐸_𝛾𝑎^′,𝑏^′=𝜇_𝐴∪𝐵(𝑥),𝛾_𝐴∪𝐵(𝑥)=𝑑,𝑑^′,∀𝑥∈𝑋. (14)

The complex Atanassov’s intuitionistic fuzzy union function, 𝐸, must satisfy at least the following axiomatic requirements, for any 𝑎, 𝑏, 𝑐, 𝑑, 𝑎^′, 𝑏^′, 𝑐^′, and 𝑑^′∈{𝑥∶𝑥∈𝐶,|𝑥|≤1}.(i) Axiom 1: |𝐸_𝜇(𝑎,0)|=|𝑎| and |𝐸_𝛾(𝑎^′,1)|=|𝑎^′| (boundary condition).(ii) Axiom 2: 𝐸_𝜇(𝑎,𝑏)=𝐸_𝜇(𝑏,𝑎) and 𝐸_𝛾(𝑎^′,𝑏^′)=𝐸_𝛾(𝑏^′,𝑎^′) (commutative condition).(iii) Axiom 3: If |𝑏|≤|𝑑|, then |𝐸_𝜇(𝑎,𝑏)|≤|𝐸_𝜇(𝑎,𝑑)|, and if |𝑏^′|≤|𝑑^′|, then |𝐸_𝛾(𝑎^′,𝑏^′)|≤|𝐸_𝛾(𝑎^′,𝑑^′)| (monotonic condition).(iv) Axiom 4: 𝐸_𝜇(𝐸_𝜇(𝑎,𝑏),𝑐)=𝐸_𝜇(𝑎,𝐸_𝜇(𝑏,𝑐)) and 𝐸_𝛾(𝐸_𝛾(𝑎^′,𝑏^′),𝑐^′)=𝐸_𝛾(𝑎^′,𝐸_𝛾(𝑏^′,𝑐^′)) (associative condition). In some cases, it may be desirable that the following requirements are also satisfied.(v) Axiom 5: 𝐸 is a continuous function (continuity).(vi) Axiom 6: 1≥|𝐸_𝜇(𝑎,𝑎)|≥|𝑎| and 0≤|𝐸_𝛾(𝑎^′,𝑎^′)|≤|𝑎^′| (superidempotency).(vii) Axiom 7: |𝑎|≤|𝑐| and |𝑏|≤|𝑑|⇒|𝐸_𝜇(𝑎,𝑏)|≤|𝐸_𝜇(𝑐,𝑑)|. Also, |𝑎^′|≥|𝑐^′| and |𝑏^′|≥|𝑑^′|⇒|𝐸_𝛾(𝑎^′,𝑏^′)|≥|𝐸_𝛾(𝑐^′,𝑑^′)| (strict monotonicity).

We may represent the complex Atanassov’s intuitionistic fuzzy union in the following manner: 𝐴∪𝐵=𝑥,𝜇_𝐴∪𝐵(𝑥),𝛾_𝐴∪𝐵(𝑥)∶𝑥∈𝑋, (15) where 𝜇_𝐴∪𝐵(𝑥)=[𝑟_𝐴(𝑥)𝑆_𝜇𝑟_𝐵(𝑥)]⋅𝑒^{𝑖𝜔𝜇𝐴∪𝐵(𝑥)}  and  𝛾_𝐴∪𝐵(𝑥)=[𝑘_𝐴(𝑥)𝑆_𝛾𝑘_𝐵(𝑥)]⋅𝑒^{𝑖𝜔𝛾𝐴∪𝐵(𝑥)}.

While the phase term for membership and nonmembership functions belongs to (0,2𝜋], we define 𝜔_{𝜇𝐴∪𝐵}(𝑥)=𝜔_𝐴∪𝐵(𝑥)=𝜔_{𝛾𝐴∪𝐵}(𝑥) with some forms that Buckley [13] presented to calculate 𝜔_𝐴∪𝐵(𝑥) as follows: (i)Sum: 𝜔_𝐴∪𝐵(𝑥)=𝜔_𝐴(𝑥)+𝜔_𝐵(𝑥).(ii)Max: 𝜔_𝐴∪𝐵(𝑥)=max(𝜔_𝐴(𝑥),𝜔_𝐵(𝑥)).(iii)Min: 𝜔_𝐴∪𝐵(𝑥)=min(𝜔_𝐴(𝑥),𝜔_𝐵(𝑥)).(iv)“Winner Takes All”: 𝜔_𝐴∪𝐵(𝑥)=𝜔_𝐴(𝑥)𝑟_𝐴>𝑟_𝐵,𝜔_𝐵(𝑥)𝑟_𝐴<𝑟_𝐵. (16)

Definition 18 (see [11]). Let 𝐴 and 𝐵 be two CAIFSs on universe of discourse 𝑋, with complex-valued membership and nonmembership functions. The complex Atanassov’s intuitionistic fuzzy intersection of 𝐴 and 𝐵, denoted by 𝐴∩𝐵, is specified by the following function: 𝑄∶𝑎,𝑎^′∣𝑎,𝑎^′∈𝐶∶𝑎+𝑎^′≤1,|𝑎|≤1,𝑎^′×≤1𝑏,𝑏^′∣𝑏,𝑏^′∈𝐶∶𝑏+𝑏^′≤1,|𝑏|≤1,𝑏^′⟶≤1𝑑,𝑑^′∣𝑑,𝑑^′∈𝐶∶𝑑+𝑑^′≤1,|𝑑|≤1,𝑑^′≤1, (17) where 𝑎, 𝑏,  and  𝑑  and  𝑎^′, 𝑏^′,  and 𝑑^′ are the membership and nonmembership functions of 𝐴, 𝐵, and 𝐴∩𝐵, respectively. 𝑄 assigns a complex value, 𝑄𝑎,𝑎^′,𝑏,𝑏^′=𝑄_𝜇(𝑎,𝑏),𝑄_𝛾𝑎^′,𝑏^′=𝜇_𝐴∩𝐵(𝑥),𝛾_𝐴∩𝐵(𝑥)=𝑑,𝑑^′,∀𝑥∈𝑋. (18)

The complex Atanassov’s intuitionistic fuzzy intersection function, 𝑄, must satisfy at least the following axiomatic requirements, for any 𝑎, 𝑏, 𝑐, 𝑑, 𝑎^′, 𝑏^′, 𝑐^′, and 𝑑^′∈{𝑥∶𝑥∈𝐶,|𝑥|≤1}.(i) Axiom 1: If |𝑏|=1, then  |𝑄_𝜇(𝑎,𝑏)|=|𝑎|, and if |𝑏^′|=0,then|𝐸_𝛾(𝑎^′,𝑏^′)|=|𝑎^′| (boundary condition).(ii) Axiom 2: 𝑄_𝜇(𝑎,𝑏)=𝑄_𝜇(𝑏,𝑎) and 𝑄_𝛾(𝑎^′,𝑏^′)=𝑄_𝛾(𝑏^′,𝑎^′) (commutative condition).(iii) Axiom 3: If |𝑏|≤|𝑑|, then |𝑄_𝜇(𝑎,𝑏)|≤|𝑄_𝜇(𝑎,𝑑)|, and if |𝑏^′|≤|𝑑^′|, then |𝑄_𝛾(𝑎^′,𝑏^′)|≤|𝑄_𝛾(𝑎^′,𝑑^′)| (monotonic condition).(iv) Axiom 4: 𝑄_𝜇(𝑄_𝜇(𝑎,𝑏),𝑐)=𝑄_𝜇(𝑎,𝑄_𝜇(𝑏,𝑐)) and 𝑄_𝛾(𝑄_𝛾(𝑎^′,𝑏^′),𝑐^′)=𝑄_𝛾(𝑎^′,𝑄_𝛾(𝑏^′,𝑐^′)) (associative condition). In some cases, it may be desirable that the following requirements are satisfied:(v) Axiom 5: 𝑄 is a continuous function (continuity).(vi) Axiom 6: 0≤|𝑄_𝜇(𝑎,𝑎)|≤|𝑎| and 1≥|𝑄_𝛾(𝑎^′,𝑎^′)|≥|𝑎^′| (superidempotency).(vii) Axiom 7: |𝑎|≤|𝑐| and |𝑏|≤|𝑑|⇒|𝑄_𝜇(𝑎,𝑏)|≤|𝑄_𝜇(𝑐,𝑑)|. Also, |𝑎^′|≥|𝑐^′| and |𝑏^′|≥|𝑑^′|⇒|𝑄_𝛾(𝑎^′,𝑏^′)|≥|𝑄_𝛾(𝑐^′,𝑑^′)| (strict monotonicity).

We may represent the complex Atanassov’s intuitionistic fuzzy intersection in the following manner: 𝐴∩𝐵=𝑥,𝜇_𝐴∩𝐵(𝑥),𝛾_𝐴∩𝐵(𝑥)∶𝑥∈𝑋, (19) where 𝜇_𝐴∩𝐵(𝑥)=[𝑟_𝐴(𝑥)𝑇_𝜇𝑟_𝐵(𝑥)]⋅𝑒^{𝑖𝜔𝜇𝐴∩𝐵(𝑥)}, 𝛾_𝐴∩𝐵(𝑥)=[𝑘_𝐴(𝑥)𝑇_𝛾𝑘_𝐵(𝑥)]⋅𝑒^{𝑖𝜔𝛾𝐴∩𝐵(𝑥)}.

We consider some forms to calculate the phase term, 𝜔_𝐴∩𝐵(𝑥), such that the same possible choices are given to calculate the 𝜔_𝐴∪𝐵(𝑥).

We can use any 𝑆-norm and 𝑇-norm that were introduced by Fathi [3] to demonstrate many examples on the amplitude terms on complex Atanassov’s intuitionistic fuzzy union and intersection, respectively, with the membership and nonmembership functions of 𝐴∪𝐵 and 𝐴∩𝐵 are given, respectively, as follows:

For CAIF union, 𝜇_𝐴∪𝐵(𝑥)=𝑟_𝐴(𝑥)𝑆_𝜇𝑟_𝐵(𝑥)⋅𝑒^{𝑖𝜔𝜇𝐴∪𝐵(𝑥)},𝛾_𝐴∪𝐵(𝑥)=𝑘_𝐴(𝑥)𝑆_𝛾𝑘_𝐵(𝑥)⋅𝑒^{𝑖𝜔𝛾𝐴∪𝐵(𝑥)}. (20) For CAIF intersection, 𝜇_𝐴∩𝐵(𝑥)=𝑟_𝐴(𝑥)𝑇_𝜇𝑟_𝐵(𝑥)⋅𝑒^{𝑖𝜔𝜇𝐴∩𝐵(𝑥)},𝛾_𝐴∩𝐵(𝑥)=𝑘_𝐴(𝑥)𝑇_𝛾𝑘_𝐵(𝑥)⋅𝑒^{𝑖𝜔𝛾𝐴∩𝐵(𝑥)}. (21)

Some examples of 𝑆-norm and 𝑇-norm are as follows.(1) The standard 𝑆-norm:  for any two AIFSs 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝛾_𝐴(𝑥)⟩∶𝑥∈𝑋} and 𝐵={⟨𝑥,𝜇_𝐵(𝑥),𝛾_𝐵(𝑥)⟩∶𝑥∈𝑋} in a universe of discourse 𝑋, then 𝐴∪𝐵 is given by 𝐴∪𝐵=𝑥,max𝜇_𝐴(𝑥),𝜇_𝐵(𝑥),min𝛾_𝐴(𝑥),𝛾_𝐵(𝑥)∶𝑥∈𝑋. (22) This union is called the basic AIF union or Atanassov union, and it is the smallest AIFS containing both 𝐴 and 𝐵.(2)Yager 𝑆-norm: 𝑆𝑎,𝑎^′,𝑏,𝑏^′=𝑆𝑆^𝜇_𝜔(𝑎,𝑏),𝑆^𝛾_𝜔𝑎^′,𝑏^′, (23) where 𝑆^𝜇_𝜔(𝑎,𝑏)=min(1,(𝑎^𝜔+𝑏^𝜔)^1/𝜔), and 𝑆^𝛾_𝜔(𝑎^′,𝑏^′)=1−min(1,((1−𝑎^′)^𝜔+(1−𝑏^′)^𝜔)^1/𝜔), with 𝜔∈(0,∞). (3)The standard 𝑇-norm:  for any two AIFSs 𝐴={⟨𝑥,𝜇_𝐴(𝑥),𝛾_𝐴(𝑥)⟩∶𝑥∈𝑋} and 𝐵={⟨𝑥,𝜇_𝐵(𝑥),𝛾_𝐵(𝑥)⟩∶𝑥∈𝑋} in a universe of discourse 𝑋, then 𝐴∩𝐵 is given by 𝐴∩𝐵=𝑥,min𝜇_𝐴(𝑥),𝜇_𝐵(𝑥),max𝛾_𝐴(𝑥),𝛾_𝐵(𝑥)∶𝑥∈𝑋. (24) This intersection is called the basic AIF intersection or Atanassov intersection, and it is the largest AIFS contained in both 𝐴 and 𝐵.(4)Yager 𝑇-norm: 𝑇𝑎,𝑎^′,𝑏,𝑏^′=𝑇^𝜇_𝜔(𝑎,𝑏),𝑇^𝛾_𝜔𝑎^′,𝑏^′, (25) where 𝑇^𝜇_𝜔(𝑎,𝑏)=1−min(1,((1−𝑎)^𝜔+(1−𝑏)^𝜔)^1/𝜔), and 𝑇^𝛾_𝜔(𝑎^′,𝑏^′)=min(1,((𝑎^′)^𝜔+(𝑏^′)^𝜔)^1/𝜔), with 𝜔∈(0,∞).

Definition 19 (see [15]). Let 𝐴 and 𝐵 be two complex fuzzy sets on 𝑋 and 𝑌, respectively, where 𝐴={⟨𝑥,𝜇_𝐴(𝑥)=𝑟_𝐴(𝑥)𝑒^{𝑖𝜔𝑟𝐴(𝑥)}⟩∶𝑥∈𝑋} and 𝐵={⟨𝑦,𝜇_𝐵(𝑦)=𝑟_𝐵(𝑦)𝑒^{𝑖𝜔𝑟𝐵(𝑦)}⟩∶𝑦∈𝑌}. The complex fuzzy union of 𝐴 and 𝐵 in different universe of discourse, denoted by 𝐴⊕𝐵, is specified by 𝐴⊕𝐵𝑥,𝑦,𝜇_𝐴⊕𝐵𝑥,𝑦∶𝑥,𝑦∈𝑋×𝑌, (26) where 𝜇_𝐴⊕𝐵(𝑥,𝑦)=𝑟_𝐴⊕𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑟𝐴⊕𝐵(𝑥,𝑦)}=max(𝑟_𝐴(𝑥),𝑟_𝐵(𝑦))𝑒^{𝑖max(𝜔𝑟𝐴(𝑥),𝜔𝑟𝐵(𝑦))}.

Definition 20 (see [15]). Let 𝐴 and 𝐵 be two complex fuzzy sets on 𝑋 and 𝑌, respectively, where 𝐴={⟨𝑥,𝜇_𝐴(𝑥)=𝑟_𝐴(𝑥)𝑒^{𝑖𝜔𝑟𝐴(𝑥)}⟩∶𝑥∈𝑋} and 𝐵={⟨𝑦,𝜇_𝐵(𝑦)=𝑟_𝐵(𝑦)𝑒^{𝑖𝜔𝑟𝐵(𝑦)}⟩∶𝑦∈𝑌}. The complex fuzzy intersection of 𝐴 and 𝐵 in different universe of discourse, denoted by 𝐴⊗𝐵, is specified by 𝐴⊗𝐵𝑥,𝑦,𝜇_𝐴⊗𝐵𝑥,𝑦∶𝑥,𝑦∈𝑋×𝑌, (27) where 𝜇_𝐴⊗𝐵(𝑥,𝑦)=𝑟_𝐴⊗𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑟𝐴⊗𝐵(𝑥,𝑦)}=min(𝑟_𝐴(𝑥),𝑟_𝐵(𝑦))𝑒^{𝑖min(𝜔𝑟𝐴(𝑥),𝜔𝑟𝐵(𝑦))}.

Definition 21 (see [15]). Let 𝐴 and 𝐵 be two complex fuzzy relations on 𝑋×𝑌, where 𝐴=𝑥,𝑦,𝜇_𝐴𝑥,𝑦=𝑟_𝐴𝑥,𝑦𝑒^{𝑖𝜔𝑟𝐴(𝑥,𝑦)}∶𝑥,𝑦∈𝑋×𝑌,𝐵=𝑥,𝑦,𝜇_𝐵𝑥,𝑦=𝑟_𝐵𝑥,𝑦𝑒^{𝑖𝜔𝑟𝐵(𝑥,𝑦)}∶𝑥,𝑦∈𝑋×𝑌. (28)
The complex fuzzy union relation of 𝐴 and 𝐵, denoted by 𝐴∪𝐵, is specified by 𝐴∪𝐵𝑥,𝑦,𝜇_𝐴∪𝐵𝑥,𝑦∶𝑥,𝑦∈𝑋×𝑌, (29) where 𝜇_𝐴∪𝐵(𝑥,𝑦)=𝑟_𝐴∪𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑟𝐴∪𝐵(𝑥,𝑦)}=max(𝑟_𝐴(𝑥,𝑦),𝑟_𝐵(𝑥,𝑦))𝑒^{𝑖max(𝜔𝑟𝐴(𝑥,𝑦),𝜔𝑟𝐵(𝑥,𝑦))}.

Definition 22 (see [15]). Let 𝐴 and 𝐵 be two complex fuzzy relations on 𝑋×𝑌, where 𝐴=𝑥,𝑦,𝜇_𝐴𝑥,𝑦=𝑟_𝐴𝑥,𝑦𝑒^{𝑖𝜔𝑟𝐴(𝑥,𝑦)}∶𝑥,𝑦∈𝑋×𝑌,𝐵=𝑥,𝑦,𝜇_𝐵𝑥,𝑦=𝑟_𝐵𝑥,𝑦𝑒^{𝑖𝜔𝑟𝐵(𝑥,𝑦)}∶𝑥,𝑦∈𝑋×𝑌. (30)
The complex fuzzy intersection relation of 𝐴 and 𝐵, denoted by 𝐴∩𝐵, is specified by 𝐴∩𝐵𝑥,𝑦,𝜇_𝐴∩𝐵𝑥,𝑦∶𝑥,𝑦∈𝑋×𝑌, (31) where 𝜇_𝐴∩𝐵(𝑥,𝑦)=𝑟_𝐴∩𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑟𝐴∩𝐵(𝑥,𝑦)}=min(𝑟_𝐴(𝑥,𝑦),𝑟_𝐵(𝑥,𝑦))𝑒^{𝑖min(𝜔𝑟𝐴(𝑥,𝑦),𝜔𝑟𝐵(𝑥,𝑦))}.
We call the relation 𝐼 an identical equal relation on 𝑋, if its membership is 𝜇_𝐼𝑥,𝑦=1if𝑥=𝑦0if𝑥≠𝑦. (32)

Definition 23 (see [15]). Let 𝐴∈CFS(𝑈×𝑈), if 𝜇_𝐴(𝑥,𝑥)=1  for  all  𝑥∈𝑋, then 𝐴 is a reflexive complex fuzzy relation.

Definition 24 (see [15]). Let 𝐴∈CFS(𝑋×𝑌), then 𝐴^𝑇∈CFS(𝑌×𝑋) is a replacement of 𝐴, where 𝜇_𝐴𝑇𝑦,𝑥=𝜇_𝐴𝑥,𝑦∀𝑥,𝑦∈𝑋×𝑌. (33)

Proposition 25 (see [15]). Let 𝐴 and 𝐵 be two complex fuzzy relations. Then, the following hold:(1)(𝐴^𝑇)^𝑇=𝐴,(2)(𝐴∪𝐵)^𝑇=𝐴^𝑇∪𝐵^𝑇 and  (𝐴∩𝐵)^𝑇=𝐴^𝑇∩𝐵^𝑇,(3)(𝐴∘𝐵)^𝑇=𝐵^𝑇∘𝐴^𝑇  and  (𝐴^𝑛)^𝑇=(𝐴^𝑇)^𝑛,(4)𝐴⊆𝐵  if  and  only  if  𝐵^𝑇⊆𝐴^𝑇.

Definition 26 (see [15]). Let 𝐴∈CFS(𝑋×𝑌). Then 𝐴 is a symmetrical complex Atanassov’s intuitionistic fuzzy relation if 𝜇_𝐴(𝑥,𝑦)=𝜇_𝐴𝑇(𝑥,𝑦)  for  all  (𝑥,𝑦)∈𝑋×𝑌.

Definition 27 (see [15]). Let 𝐴∈CFS(𝑋×𝑋). Then 𝐴 is a transitive complex fuzzy relation if 𝐴∘𝐴⊆𝐴.

3. Distance Measure on Complex Atanassov’s Intuitionistic Fuzzy Sets

In 1962, Hausdorff introduced the concept of distance as a measure of similarity or difference between sets [27]. Distance measure is very important because the “spatial distribution” of sets is fixed with respect to selected reference set [28]. Therefore, distance measure is used in significant real-life application, for instance, reasoning and decision making [28–40], fuzzy number ranking, and information retrieval in databases [41]. Also, in [28], Yang and Chiclana proved that the 3D distance of AIFS (i.e., the distance measured with three parameters membership, nonmembership, and hesitation functions) is useless and is not necessary because two-dimensional distance functions (membership and nonmembership functions) have already provided a simple and concise expression of the distance between two AIFSs. All distance measures in fuzzy set and intuitionistic fuzzy set are dealing with real-valued parameter, membership, and/or nonmembership functions, which only represented the uncertainty semantic. In the case of CAIFS, these parameters are complex-valued parameters and have two terms, uncertainty and periodicity semantics, where the uncertainty semantic is analogous to the one in traditional FS and/or AIFS. Therefore, the priority now is to introduce a distance measure that may involve both uncertainty and periodic semantics in one structure to confer the concepts of CFS and CAIFS a simple and concise expression. In application 1, we illustrate a kind of MADM problem with an additional factor, the periodic factors (i.e., MADM problems in CIF fields).

In this section, we generalise a distance measure on CAIFSs to implement a model which is utilized to select the best product with multiple attributes based on two dimensions. One of them is the periodic dimension which is considered as a critical influence to take a suitable decision. Song et al. [12] introduced a model to select a proper supplier based on distance measure of AIFS. They presented the definition of distance between two AIFSs. Here, we generalise their definition to CAIFSs and use this new approach of distance to find out the best periodic product from several selected products. This is a kind of MADM problem with an additional factor, the periodic factor (i.e., MADM problems in CIF fields).

Definition 28. If 𝐴 and 𝐵 are CAIFSs in a universe of discourse 𝑋, where 𝐴={(𝑥,𝑟_𝐴(𝑥)𝑒^{𝑖𝜔𝑟𝐴(𝑥)},𝑘_𝐴(𝑥)𝑒^{𝑖𝜔𝑘𝐴(𝑥)})}, and 𝐵={(𝑥,𝑟_𝐵(𝑥)𝑒^{𝑖𝜔𝑟𝐵(𝑥)},𝑘_𝐵(𝑥)𝑒^{𝑖𝜔𝑘𝐵(𝑥)})}, then(1)𝐴⊂𝐵 if and only if 𝑟_𝐴(𝑥)<𝑟_𝐵(𝑥)  and  𝑘_𝐴(𝑥)>𝑘_𝐵(𝑥), for amplitude terms and the phase terms (arguments) 𝜔_𝑟𝐴(𝑥)<𝜔_𝑟𝐵(𝑥) and 𝜔_𝑘𝐴(𝑥)>𝜔_𝑘𝐵(𝑥), for all 𝑥∈𝑋.(2)𝐴=𝐵 if and only if 𝑟_𝐴(𝑥)=𝑟_𝐵(𝑥)  and  𝑘_𝐴(𝑥)=𝑘_𝐵(𝑥), for amplitude terms and the phase terms (arguments) 𝜔_𝑟𝐴(𝑥)=𝜔_𝑟𝐵(𝑥) and 𝜔_𝑘𝐴(𝑥)=𝜔_𝑘𝐵(𝑥), for all 𝑥∈𝑋.
Let CIFS(𝑋) be the set of all complex Atanassov’s intuitionistic fuzzy sets on 𝑋.

Definition 29. A distance of complex Atanassov’s intuitionistic fuzzy sets is a function 𝑑∶CIFS(𝑋)×CIFS(𝑋)→[0,1] for any 𝐴,𝐵, and 𝐶∈CIFS(𝑋), satisfying the following properties:(D1)0≤𝑑(𝐴,𝐵)≤1.(D2)𝑑(𝐴,𝐵)=0 if and only if 𝐴=𝐵.(D3)𝑑(𝐴,𝐵)=𝑑(𝐵,𝐴). (D4) If 𝐴⊂𝐵⊂𝐶 and 𝐴,𝐵,𝐶∈CIFS(𝑋), then 𝑑(𝐴,𝐶)>𝑑(𝐴,𝐵),𝑑(𝐴,𝐶)>𝑑(𝐵,𝐶).
We introduce a function 𝑑∶CIFS(𝑋)×CIFS(𝑋)→[0,1] between CAIFSs 𝐴 and 𝐵, defined as follows: 𝑑()=1𝐴,𝐵2∑^𝑛_𝑖=1𝑤_𝑖×^𝑛_𝑖=1𝑤_𝑖𝛼₁⋅𝑟_𝐴𝑥_𝑖−𝑟_𝐵𝑥_𝑖+𝛽₁⋅𝑘_𝐴𝑥_𝑖−𝑘_𝐵𝑥_𝑖+𝜎₁⋅max𝑟_𝐴𝑥_𝑖−𝑟_𝐵𝑥_𝑖,𝑘_𝐴𝑥_𝑖−𝑘_𝐵𝑥_𝑖+1𝛼2𝜋²⋅𝜔_𝑟𝐴𝑥_𝑖−𝜔_𝑟𝐵𝑥_𝑖+𝛽₂⋅𝜔_𝑘𝐴𝑥_𝑖−𝜔_𝑘𝐵𝑥_𝑖+𝜎₂⋅max𝜔_𝑟𝐴𝑥_𝑖−𝜔_𝑟𝐵𝑥_𝑖,𝜔_𝑘𝐴𝑥_𝑖−𝜔_𝑘𝐵𝑥_𝑖, (34) where 𝛼₁,  𝛽₁,  𝜎₁,  𝛼₂,  𝛽₂, and 𝜎₂∈[0,1], 𝛼₁+𝛽₁+𝜎₁=1 and 𝛼₂+𝛽₂+𝜎₂=1. 𝑤_𝑖∈[0,1], 𝑖∈{1,2,…,𝑚}.

Theorem 30. The function 𝑑(𝐴,𝐵) defined in (34) is a distance measure of complex Atanassov’s intuitionistic fuzzy set between two CAIFSs 𝐴 and 𝐵 in 𝑋.

Proof. (1) By definition of CAIFS, we have for all 𝑥_𝑖 in 𝑋, where 𝑖=1,2,…,𝑛, each of 𝑟_𝐴(𝑥_𝑖),𝑟_𝐵(𝑥_𝑖),𝑘_𝐴(𝑥_𝑖),  and  𝑟_𝐵(𝑥_𝑖) lies in the interval [0,1]. Also each of 𝜔_𝑟𝐴(𝑥_𝑖),𝜔_𝑟𝐵(𝑥_𝑖),𝜔_𝑘𝐴(𝑥_𝑖),  and  𝜔_𝑘𝐵(𝑥_𝑖) lies in the interval [0,2𝜋]. So it is easy to conclude that each of the following lies between 0 and 1: |𝑟_𝐴(𝑥_𝑖)−𝑟_𝐵(𝑥_𝑖)|, and |𝑘_𝐴(𝑥_𝑖)−𝑘_𝐵(𝑥_𝑖)|. Also, |𝜔_𝑟𝐴(𝑥_𝑖)−𝜔_𝑟𝐵(𝑥_𝑖)| and |𝜔_𝑘𝐴(𝑥_𝑖)−𝜔_𝑘𝐵(𝑥_𝑖)| lie between 0 and 2𝜋.
So we have 𝑑()≤1𝐴,𝐵2∑^𝑛_𝑖=1𝑤_𝑖×^𝑛_𝑖=1𝑤_𝑖𝛼₁⋅1+𝛽₁⋅1+𝜎₁()+1⋅max1,1𝛼2𝜋²⋅2𝜋+𝛽₂⋅2𝜋+𝜎₂(=1⋅max2𝜋,2𝜋)2∑^𝑛_𝑖=1𝑤_𝑖^𝑛_𝑖=1𝑤_𝑖𝛼₁+𝛽₁+𝜎₁+𝛼₂+𝛽₂+𝜎₂. (35) Since 𝛼₁+𝛽₁+𝜎₁=1 and 𝛼₂+𝛽₂+𝜎₂=1, we have ()≤10≤𝑑𝐴,𝐵2∑^𝑛_𝑖=1𝑤_𝑖^𝑛_𝑖=12𝑤_𝑖=1. (36)
(2) By Definition 28, it is easy to see that 𝑑(𝐴,𝐵) satisfies the second and third conditions of Definition 29.
(3) By using Definition 28, 1≥𝑘_𝐴𝑥_𝑖≥𝑘_𝐵𝑥_𝑖≥𝑘_𝐶(𝑥)≥0,2𝜋≥𝜔_𝑘𝐴𝑥_𝑖≥𝜔_𝑘𝐵𝑥_𝑖≥𝜔_𝑘𝐶𝑥_𝑖≥0. (37)
Then, we can conclude that 𝑟_𝐴𝑥_𝑖−𝑟_𝐵𝑥_𝑖≤𝑟_𝐴𝑥_𝑖−𝑟_𝐶𝑥_𝑖,𝜔_𝑟𝐴𝑥_𝑖−𝜔_𝑟𝐵𝑥_𝑖≤𝜔_𝑟𝐴𝑥_𝑖−𝜔_𝑟𝐶𝑥_𝑖,𝑘_𝐴𝑥_𝑖−𝑘_𝐵𝑥_𝑖≥𝑘_𝐴𝑥_𝑖−𝑘_𝐶𝑥_𝑖,𝜔_𝑘𝐴𝑥_𝑖−𝜔_𝑘𝐵𝑥_𝑖≥𝜔_𝑘𝐴𝑥_𝑖−𝜔_𝑘𝐶𝑥_𝑖. (38)
So, 𝑑(𝐴,𝐵)≤𝑑(𝐴,𝐶).
Analogously, we can get 𝑑(𝐵,𝐶)≤𝑑(𝐴,𝐶).
From (82), (83), and (84), we conclude that 𝑑(𝐴,𝐵) is a distance measure between CAIFSs 𝐴 and 𝐵.

In the following example, Application 31, we apply a selection model from [12]. Also, we generalise the selection operation to be covered by CAIFS to get the desired car or the best alternative for a company to select a car from four models of cars with its production date simultaneously. So we have a two-dimensional problem, namely, model of cars and production date of the cars.

Application 31. Suppose Company 𝑋 decides to purchase cars from a carmaker 𝑌. The carmaker 𝑌 offers Company 𝑋 some information on four models of cars with different production dates for each model. So Company 𝑋 has four models (Car₁, Car₂, Car₃, and Car₄) to select with its production date simultaneously. The company’s team of analysts agreed that five attributes should be considered. They are 𝐴₁: reliability, 𝐴₂: maximum payload, 𝐴₃: purchasing cost, 𝐴₄: maximum speed, and 𝐴₅: durability. But these attributes will be affected and changed if the production date is different for the same model of cars. The decision made by the team will also depend on their knowledge and experience. So, we may get “yes,” “no,” “I do not know,” or “I am not sure” as an answer to the selection of the desired cars. Consequently, as illustrated above, the best way to represent this sort of information is by using CAIFS (i.e., represent the information on people’s decision which happens periodically), in which with each new product model of car, the consumers have different opinions and mentalities. To be more explicit, assume that the company’s team of analysts has suggested an ideal car, before getting the characteristic information from carmaker 𝑌. The target of the team is to select a suitable car listed by carmaker 𝑌 that is most likely to be the ideal car. Then, every analyst in the selection team gives each car’s attribute a score 1 or −1 to indicate whether the car is suitable for the attribute or not and gives a score 0 when he is not sure of the model and production date of car.
For instance, suppose the team of analysts feels that at least 70% of them believe that the ideal model car is suitable at the first attribute; and not more than 10% of the analysts believe that the ideal model car is poor at the first attribute, in which this process is utilized to calculate the amplitude terms for both membership and nonmembership functions respectively, in CAIFS. The phase terms that present production date for first attribute of ideal car can be given as follows: if the team of analysts thought that at least 80% of them believe that the ideal production date of car is suitable at the first attribute; and not more than 15% of them believe that the ideal production date of car is poor. So the ideal car’s first attribute can be presented as (0.7𝑒^{𝑖2𝜋(0.8)},0.1𝑒^{𝑖2𝜋(0.15)}). In this way, all data can be obtained in the form of CAIFS, where both amplitude and phase terms can represent information of Atanassov’s intuitionistic uncertainty (see Table 1).
To continue our example, we recall a proper model to get a suitable car that is most likely to be the ideal car (see [12]).
Let 𝑑(Car_𝑗,I.C.) be a distance measure between complex Atanassov’s intuitionistic fuzzy sets Car_𝑗 and I.C. So, the model can be introduced as follows: 𝑀=Car_𝑗∣𝑑Car_𝑗,I.C.=min^𝑛_𝑖=1𝑑Car_1,𝑖,I.C_𝑖.,…,_𝑛𝑖=1𝑑Car_𝑘,𝑖,I.C_𝑖.,𝑗=1,…,𝑘,𝑖=1,…,𝑛. (39)
Suppose the team of analysts gives the weight for each attribute as follows.
Let 𝑤₁=0.2, 𝑤₂=0.1, 𝑤₃=0.5, 𝑤₄=0.7, and 𝑤₅=0.5 present the weight for each attribute. Let 𝛼₁=0.2, 𝛽₁=0.5, and  𝜎₁=0.3 be the weight for the amplitude term, while 𝛼₂=0.4,  𝛽₂=0.4,  and  𝜎₂=0.2 be the weight for the phase term.

Table 1 

Car data and the ideal car.

To identify the ideal car from the information given in Table 1, we substitute the data into distance formula (Definition 29) to evaluate the values between each car (Car_𝑗,𝑗=1,2,3,4) and the ideal car (I.C.): 𝑑Car₁=,I.C._5𝑖=1𝑑Car_1,𝑖,I.C_𝑖.=1()×2⋅2.9_5𝑖=1𝑤_𝑖𝛼₁⋅𝑟_𝐴𝑥_𝑖−𝑟_𝐵𝑥_𝑖+𝛽₁⋅𝑘_𝐴𝑥_𝑖−𝑘_𝐵𝑥_𝑖+𝜎₁⋅max𝑟_𝐴𝑥_𝑖−𝑟_𝐵𝑥_𝑖,𝑘_𝐴𝑥_𝑖−𝑘_𝐵𝑥_𝑖+1𝛼2𝜋²⋅𝜔_𝑟𝐴𝑥_𝑖−𝜔_𝑟𝐵𝑥_𝑖+𝛽₂⋅𝜔_𝑘𝐴𝑥_𝑖−𝜔_𝑘𝐵𝑥_𝑖+𝜎₂⋅max𝜔_𝑟𝐴𝑥_𝑖−𝜔_𝑟𝐵𝑥_𝑖,𝜔_𝑘𝐴𝑥_𝑖−𝜔_𝑘𝐵𝑥_𝑖. (40) For the first attribute 𝐴₁, (𝑖=1), the distance is given as follows: 𝑑Car_1,1,I.C₁.=1()||||(+12⋅2.90.2(0.2⋅0.4−0.7+0.5⋅0.3−0.1+0.3⋅max0.3,0.2))|()(|()((()(=12𝜋0.4⋅0.32𝜋−0.52𝜋)|+0.4⋅0.22𝜋−0.32𝜋)|+0.2⋅max0.22𝜋,0.12𝜋))()[].2⋅2.90.082 (41)

Analogously, we calculate the distances for Car₁ for each attribute, 𝑑Car_1,2,I.C₂.=1()[]2⋅2.90.054,𝑑Car_1,3,I.C₃.=1()[]2⋅2.90.05,𝑑Car_1,4,I.C₄.=1()[]2⋅2.90.112,𝑑Car_1,5,I.C₅.=1()[].2⋅2.90.305 (42)

Then, 𝑑Car₁=1,I.C.()×[]2⋅2.90.082+0.054+0.05+0.112+0.305=0.104. (43)

Analogously, we calculate the distance values for 𝑑(Car₂,I.C.)=0.842, 𝑑(Car₃,I.C.)=0.538, and 𝑑(Car₄,I.C.)=0.796. Therefore, 𝑑(Car₁,I.C.)=0.104 is the smallest value, so Car₁ is the desired car.

Note 1. Different weights of the criteria lead to different results. A company needs to set proper weights for the criteria, in order to get the proper result.
If we have the same model of cars but with different production date for the first attribute, surely the team of analysts will give us different values from those mentioned in the table above for the first attribute, because every year, the carmakers produce the same model of cars but with a slight improvement and difference. Even though the changes are slightly improved in the new models of cars, people’s acceptance and appreciation may get changed to buy the new production model.

4. Complex Atanassov’s Intuitionistic Fuzzy Relation

Fuzzy logics and systems open immense applications in engineering, medical, physics, automobiles, defense and security, internet and computer security, and other fields. It is a meaningful topic to introduce the concept of CAIF logics and systems which give us the ability to extend the realm of real-life applications in the CAIF environment. So, we develop the concept of complex Atanassov’s intuitionistic fuzzy relations and compositions before introducing the concept of CAIF logics and systems.

The relation 𝑅 between the sets 𝑋₁,…,𝑋_𝑛 is a subset of the Cartesian product 𝑋₁×⋯×𝑋_𝑛. If we denote the relation above by 𝑅(𝑋₁,…,𝑋_𝑛), then 𝑅𝑋₁,…,𝑋_𝑛⊂𝑋₁×⋯×𝑋_𝑛. (44)

Firstly, we should introduce the Cartesian product between complex Atanassov’s intuitionistic fuzzy sets. In general, the Cartesian product between several CAIFSs is given as follows.

Definition 32. Let 𝐴₁,…,𝐴_𝑛 be complex Atanassov’s intuitionistic fuzzy sets in 𝑋₁,…,𝑋_𝑛, respectively. The Cartesian product 𝐴₁×⋯×𝐴_𝑛 is a complex intuitionistic fuzzy set defined by 𝐴₁×⋯×𝐴_𝑛=𝑥₁,…,𝑥_𝑛,𝜇_{𝐴1×⋯×𝐴𝑛}𝑥₁,…,𝑥_𝑛,𝛾_{𝐴1×⋯×𝐴𝑛}𝑥₁,…,𝑥_𝑛∶𝑥₁,…,𝑥_𝑛∈𝑋₁×⋯×𝑋_𝑛, (45) where 𝜇_{𝐴1×⋯×𝐴𝑛}𝑥₁,…,𝑥_𝑛=min𝜇_𝐴1𝑥₁,…,𝜇_𝐴𝑛𝑥_𝑛⋅𝑒^{𝑖min(𝜔𝜇𝐴1(𝑥1),…,𝜔𝜇𝐴𝑛(𝑥𝑛))},𝛾_{𝐴1×⋯×𝐴𝑛}𝑥₁,…,𝑥_𝑛=max𝛾_𝐴1𝑥₁,…,𝛾_𝐴𝑛𝑥_𝑛⋅𝑒^{𝑖max(𝜔𝛾𝐴1(𝑥1),…,𝜔𝛾𝐴𝑛(𝑥𝑛))}. (46) In the case of two complex Atanassov’s intuitionistic fuzzy sets 𝐴 and 𝐵 in 𝑋 and 𝑌, respectively, we have =𝐴×𝐵𝑥,𝑦,𝜇_𝐴×𝐵𝑥,𝑦,𝛾_𝐴×𝐵𝑥,𝑦∶𝑥,𝑦∈𝑋×𝑌, (47) where 𝜇_𝐴×𝐵𝑥,𝑦=min𝜇_𝐴(𝑥),𝜇_𝐵𝑦⋅𝑒^{𝑖min(𝜔𝜇𝐴(𝑥),𝜔𝜇𝐵(𝑦))},𝛾_𝐴×𝐵𝑥,𝑦=max𝛾_𝐴(𝑥),𝛾_𝐵𝑦⋅𝑒^{𝑖max(𝜔𝛾𝐴(𝑥),𝜔𝛾𝐵(𝑦))}. (48)
A special case is obtained when we generate a relation between two sets 𝑋  and  𝑌. So, we present a definition of a complex Atanassov’s intuitionistic fuzzy relation as follows.

Definition 33. A complex Atanassov’s intuitionistic fuzzy relation 𝑅(𝑋,𝑌) between CAIFSs 𝑈  and  𝑉 is a subset of the product space 𝑋×𝑌, where the complex membership and nonmembership functions of the relation 𝑅(𝑋,𝑌) are characterized by 𝜇_𝑅(𝑥,𝑦) and 𝛾_𝑅(𝑥,𝑦), respectively, that assign each pair to each (𝑥,𝑦)∈𝑋×𝑌 a complex-valued grade of membership and nonmembership functions to the set 𝑅(𝑋,𝑌), where 𝑥∈𝑋 and 𝑦∈𝑌. We write the relation 𝑅(𝑋,𝑌) as 𝑅(𝑋,𝑌)=𝑥,𝑦,𝜇_𝑅𝑥,𝑦,𝛾_𝑅𝑥,𝑦∣𝑥,𝑦∈𝑋×𝑌. (49)
The values 𝜇_𝑅(𝑥,𝑦) and 𝛾_𝑅(𝑥,𝑦) may receive lie within the unit circle in the complex plane and are of the following form: 𝑟(𝑥)⋅𝑒^{𝑖𝜔𝜇(𝑥)}, and 𝑘(𝑥)⋅𝑒^{𝑖𝜔𝛾(𝑥)}(𝑖=√−1), where 𝑟(𝑥),𝑘(𝑥),𝜔_𝜇(𝑥), and 𝜔_𝛾(𝑥) are all real-valued, with 𝑟(𝑥),𝑘(𝑥)∈[0,1], and  0≤𝑟(𝑥)+𝑘(𝑥)≤1.
The membership and nonmembership functions in a CAISR represent both the degree of presence and absence of association, interaction, or interconnectedness, and the phase of association, interaction, or interconnectedness between the elements of two or more sets. Under the assertion that any of the complex fuzzy sets can be represented as a CAIFS (see [11, Proposition 2.3]), we rewrite Example 5 of Ramot et al. [4, page 183] in terms of CAIFS to demonstrate the utilization of CAIFR in real life.

Example 34. Let 𝑋 be the set of financial indicators or indexes of the American economy. Possible elements of this set are unemployment rate, inflation, interest rates, growth rate, GDP, Dow-Jones industrial average, and so forth. Let 𝑌 be the set of financial indicators of the Japanese economy. Let the complex fuzzy relation 𝑅(𝑋,𝑌) represent the relation of influence of American financial indexes on Japanese financial indexes: “𝑦 is influenced by 𝑥,” where 𝑥∈𝑋 and 𝑦∈𝑌.
The membership and nonmembership functions for the relation 𝑅(𝑋,𝑌),𝜇(𝑋,𝑌), and 𝛾(𝑋,𝑌) are complex-valued, with an amplitude term and a phase term. The amplitude terms of membership and nonmembership functions indicate the degree of influence and noinfluence of an American financial index on a Japanese financial index, respectively. An amplitude term with a value close to one implies a large degree of influence, while a value close to zero suggests small to no influence. The phase terms of membership and nonmembership functions indicate the “phase” of influence and no-influence, or time lag that characterizes the influence and no influence of an American index on a Japanese index. Thus, the phase terms represent the time that elapses before the influence and no-influence of a certain occurrence in an American financial indicator is evident in a Japanese counterpart.
Consider for example, 𝜇_𝑅(GrowthRate,Export) and 𝛾_𝑅(GrowthRate,Export), that is, the grade of membership and nonmembership associated with the statement “American growth rate influences and noinfluences Japanese export, respectively.” The values of 𝜇_𝑅 and 𝛾_𝑅 may be calculated from available economic statistics using a variety of methods (genetic algorithms, neural nets, etc.) or obtained from an expert. In this example, the latter of the two options is considered.
Suppose an expert was to state that “the influence of American growth rate on Japanese export is large, and the effect of a decline or increase in American growth is evident in Japanese export in three–five months.” If 𝑅(𝑋,𝑌) was a traditional Atanassov’s intuitionistic fuzzy relation, values of about 0.8 and 0.1 would be selected for the grade of membership and nonmembership, respectively, and all information regarding the time frame of the interaction between these two economic parameters would be lost. However, 𝑅(𝑋,𝑌) is a complex Atanassov’s intuitionistic fuzzy relation, 𝜇_𝑅(GrowthRate,Export) and 𝛾_𝑅(GrowthRate,Export) thus can be assigned a complex value which incorporates all of the information provided by the expert.
Assume 𝑅(𝑋,𝑌) measures interactions between American and Japanese financial indicators in the limited time frame of 12 months. The following value may, therefore, be attributed to 𝜇_𝑅(GrowthRate,Export) and 𝛾_𝑅(GrowthRate,Export)∶𝜇_𝑅GrowthRate,Export=0.8⋅𝑒^{(4/12)2𝜋𝑖},𝛾_𝑅GrowthRate,Export=0.1⋅𝑒^{(4/12)2𝜋𝑖}. (50)
Note that the amplitude terms selected are 0.8 and 0.1, similar to the grade of membership and nonmembership of a traditional Atanassov’s intuitionistic fuzzy set. The phase term is chosen to be 4 as an average of “three–five months,” normalized by 12 months—the maximum timeframe the relation is designed to take into account for both membership phase and nonmembership phase functions.

Definition 35. Let 𝐴 and 𝐵 be two complex Atanassov’s intuitionistic fuzzy sets on 𝑋 and 𝑌, respectively, where 𝐴=𝑥,𝜇_𝐴(𝑥)=𝑟_𝐴(𝑥)𝑒^{𝑖𝜔𝑟𝐴(𝑥)},𝛾_𝐴(𝑥)=𝑘_𝐴(𝑥)𝑒^{𝑖𝜔𝑘𝐴(𝑥)}∶𝑥∈𝑋,𝐵=𝑦,𝜇_𝐵𝑦=𝑟_𝐵𝑦𝑒^{𝑖𝜔𝑟𝐵(𝑦)},𝛾_𝐵𝑦=𝑘_𝐵𝑦𝑒^{𝑖𝜔𝑘𝐵(𝑦)}∶𝑦∈𝑌. (51)
The complex Atanassov’s intuitionistic fuzzy union of 𝐴 and 𝐵 in different universe of discourse, denoted by 𝐴⊕𝐵, is specified by 𝐴⊕𝐵𝑥,𝑦,𝜇_𝐴⊕𝐵𝑥,𝑦,𝛾_𝐴⊕𝐵𝑥,𝑦∶𝑥,𝑦∈𝑋×𝑌, (52) where 𝜇_𝐴⊕𝐵(𝑥,𝑦)=𝑟_𝐴⊕𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑟𝐴⊕𝐵(𝑥,𝑦)}=max(𝑟_𝐴(𝑥),𝑟_𝐵(𝑦))𝑒^{𝑖max(𝜔𝑟𝐴(𝑥),𝜔𝑟𝐵(𝑦))} and 𝛾_𝐴⊕𝐵(𝑥,𝑦)=𝑘_𝐴⊕𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑘𝐴⊕𝐵(𝑥,𝑦)}=min(𝑘_𝐴(𝑥),𝑘_𝐵(𝑦))𝑒^{𝑖min(𝜔𝑘𝐴(𝑥),𝜔𝑘𝐵(𝑦))}.

Example 36. Let 𝐴=0.5𝑒^𝑖1.3𝜋,0.4𝑒^𝑖0.5𝜋,−11.0𝑒^𝑖1.5𝜋,0.0𝑒^𝑖0.5𝜋0,0.7𝑒^𝑖0.3𝜋,0.2𝑒^𝑖1.5𝜋1,𝐵=0.8𝑒^𝑖1.1𝜋,0.2𝑒^𝑖0.7𝜋,−10.4𝑒^𝑖0.9𝜋,0.4𝑒^𝑖𝜋0,0.7𝑒^𝑖0.9𝜋,0.2𝑒^𝑖0.4𝜋1, (53) then 𝐴⊕𝐵=0.8𝑒^𝑖1.3𝜋,0.2𝑒^𝑖0.5𝜋(),−1,−10.5𝑒^𝑖1.3𝜋,0.4𝑒^𝑖0.5𝜋(),−1,00.7𝑒^𝑖1.3𝜋,0.2𝑒^𝑖0.4𝜋(),−1,11.0𝑒^𝑖1.5𝜋,0.0𝑒^𝑖0.5𝜋(),0,−11.0𝑒^𝑖1.5𝜋,0.0𝑒^𝑖0.5𝜋(),0,00.7𝑒^𝑖0.9𝜋,0.2𝑒^𝑖1.0𝜋(),1,00.8𝑒^𝑖1.1𝜋,0.2𝑒^𝑖0.7𝜋(),1,−10.7𝑒^𝑖0.9𝜋,0.2𝑒^𝑖1.0𝜋(),1,00.7𝑒^𝑖0.9𝜋,0.2𝑒^𝑖0.4𝜋().1,1 (54)

Definition 37. Let 𝐴 and 𝐵 be two complex Atanassov’s intuitionistic fuzzy sets on 𝑋 and 𝑌, respectively, where 𝐴=𝑥,𝜇_𝐴(𝑥)=𝑟_𝐴(𝑥)𝑒^{𝑖𝜔𝑟𝐴(𝑥)},𝛾_𝐴(𝑥)=𝑘_𝐴(𝑥)𝑒^{𝑖𝜔𝑘𝐴(𝑥)}∶𝑥∈𝑋,𝐵=𝑦,𝜇_𝐵𝑦=𝑟_𝐵𝑦𝑒^{𝑖𝜔𝑟𝐵(𝑦)},𝛾_𝐵𝑦=𝑘_𝐵𝑦𝑒^{𝑖𝜔𝑘𝐵(𝑦)}∶𝑦∈𝑌. (55)
The complex Atanassov’s intuitionistic fuzzy intersection of 𝐴 and 𝐵 in different universe of discourse, denoted by 𝐴⊗𝐵, is specified by 𝐴⊗𝐵𝑥,𝑦,𝜇_𝐴⊗𝐵𝑥,𝑦,𝛾_𝐴⊗𝐵𝑥,𝑦∶𝑥,𝑦∈𝑋×𝑌, (56) where 𝜇_𝐴⊗𝐵(𝑥,𝑦)=𝑟_𝐴⊗𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑟𝐴⊗𝐵(𝑥,𝑦)}=min(𝑟_𝐴(𝑥),𝑟_𝐵(𝑦))𝑒^{𝑖min(𝜔𝑟𝐴(𝑥),𝜔𝑟𝐵(𝑦))} and 𝛾_𝐴⊗𝐵(𝑥,𝑦)=𝑘_𝐴⊗𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑘𝐴⊗𝐵(𝑥,𝑦)}=max(𝑘_𝐴(𝑥),𝑘_𝐵(𝑦))𝑒^{𝑖max(𝜔𝑘𝐴(𝑥),𝜔𝑘𝐵(𝑦))}.

Example 38. Using the CAIFS 𝐴 and 𝐵 as in Example 36, we have 𝐴⊗𝐵=0.5𝑒^𝑖1.1𝜋,0.4𝑒^𝑖0.7𝜋(),−1,−10.4𝑒^𝑖0.9𝜋,0.4𝑒^𝑖1.0𝜋(),−1,00.5𝑒^𝑖0.9𝜋,0.4𝑒^𝑖0.5𝜋(),−1,10.8𝑒^𝑖1.1𝜋,0.2𝑒^𝑖0.7𝜋(),0,−10.4𝑒^𝑖0.9𝜋,0.4𝑒^𝑖1.0𝜋(),0,00.7𝑒^𝑖0.9𝜋,0.2𝑒^𝑖0.5𝜋(),0,10.7𝑒^𝑖0.3𝜋,0.2𝑒^𝑖1.5𝜋(),1,−10.4𝑒^𝑖0.3𝜋,0.4𝑒^𝑖1.5𝜋(),1,00.7𝑒^𝑖0.3𝜋,0.2𝑒^𝑖1.5𝜋().1,1 (57)

Definition 39. Let 𝐴 and 𝐵 be two complex Atanassov’s intuitionistic fuzzy relations on 𝑋×𝑌, where 𝐴=𝑥,𝑦,𝜇_𝐴𝑥,𝑦=𝑟_𝐴𝑥,𝑦𝑒^{𝑖𝜔𝑟𝐴(𝑥,𝑦)},𝛾_𝐴𝑥,𝑦=𝑘_𝐴𝑥,𝑦𝑒^{𝑖𝜔𝑘𝐴(𝑥,𝑦)}∶𝑥,𝑦∈𝑋×𝑌,𝐵=𝑥,𝑦,𝜇_𝐵𝑥,𝑦=𝑟_𝐵𝑥,𝑦𝑒^{𝑖𝜔𝑟𝐵(𝑥,𝑦)},𝛾_𝐵𝑥,𝑦=𝑘_𝐵𝑥,𝑦𝑒^{𝑖𝜔𝑘𝐵(𝑥,𝑦)}∶𝑥,𝑦∈𝑋×𝑌. (58)
The complex Atanassov’s intuitionistic fuzzy union relation of 𝐴 and 𝐵, denoted by 𝐴∪𝐵, is specified by 𝐴∪𝐵𝑥,𝑦,𝜇_𝐴∪𝐵𝑥,𝑦,𝛾_𝐴∪𝐵𝑥,𝑦∶𝑥,𝑦∈𝑋×𝑌, (59) where 𝜇_𝐴∪𝐵(𝑥,𝑦)=𝑟_𝐴∪𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑟𝐴∪𝐵(𝑥,𝑦)}=max(𝑟_𝐴(𝑥,𝑦),𝑟_𝐵(𝑥,𝑦))𝑒^{𝑖max(𝜔𝑟𝐴(𝑥,𝑦),𝜔𝑟𝐵(𝑥,𝑦))} and 𝛾_𝐴∪𝐵(𝑥,𝑦)=𝑘_𝐴∪𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑘𝐴∪𝐵(𝑥,𝑦)}=min(𝑘_𝐴(𝑥,𝑦),𝑘_𝐵(𝑥,𝑦))𝑒^{𝑖min(𝜔𝑘𝐴(𝑥,𝑦),𝜔𝑘𝐵(𝑥,𝑦))}.

Example 40. Let 𝑋={𝑎,𝑏} and 𝑌={1,2}. Let 𝐴=0.5𝑒^𝑖1.3𝜋,0.4𝑒^𝑖0.5𝜋(),𝑎,10.7𝑒^𝑖0.3𝜋,0.2𝑒^𝑖1.5𝜋(),𝑎,21.0𝑒^𝑖1.5𝜋,0.0𝑒^𝑖0.5𝜋(),𝑏,10.8𝑒^𝑖0.7𝜋,0.1𝑒^𝑖1.1𝜋(),𝑏,2𝐵=0.8𝑒^𝑖1.1𝜋,0.2𝑒^𝑖0.7𝜋(),𝑎,10.7𝑒^𝑖0.9𝜋,0.2𝑒^𝑖0.4𝜋(),𝑎,20.4𝑒^𝑖0.9𝜋,0.4𝑒^𝑖𝜋(),𝑏,10.1𝑒^𝑖𝜋,0.7𝑒^𝑖0.6𝜋().𝑏,2 (60) Then, 𝐴∪𝐵=0.8𝑒^𝑖1.3𝜋,0.2𝑒^𝑖0.5𝜋(),𝑎,10.7𝑒^𝑖0.9𝜋,0.2𝑒^𝑖0.4𝜋(),𝑎,21.0𝑒^𝑖1.5𝜋,0.0𝑒^𝑖0.5𝜋(),𝑏,10.8𝑒^𝑖1.0𝜋,0.1𝑒^𝑖0.6𝜋().𝑏,2 (61)

Definition 41. Let 𝐴 and 𝐵 be two complex Atanassov’s intuitionistic fuzzy relations on 𝑋×𝑌, where 𝐴=𝑥,𝑦,𝜇_𝐴𝑥,𝑦=𝑟_𝐴𝑥,𝑦𝑒^{𝑖𝜔𝑟𝐴(𝑥,𝑦)},𝛾_𝐴𝑥,𝑦=𝑘_𝐴𝑥,𝑦𝑒^{𝑖𝜔𝑘𝐴(𝑥,𝑦)}∶𝑥,𝑦∈𝑋×𝑌,𝐵=𝑥,𝑦,𝜇_𝐵𝑥,𝑦=𝑟_𝐵𝑥,𝑦𝑒^{𝑖𝜔𝑟𝐵(𝑥,𝑦)},𝛾_𝐵𝑥,𝑦=𝑘_𝐵𝑥,𝑦𝑒^{𝑖𝜔𝑘𝐵(𝑥,𝑦)}∶𝑥,𝑦∈𝑋×𝑌. (62)
The complex Atanassov’s intuitionistic fuzzy intersection relation of 𝐴 and 𝐵, denoted by𝐴∩𝐵, is specified by 𝐴∩𝐵𝑥,𝑦,𝜇_𝐴∩𝐵𝑥,𝑦,𝛾_𝐴∩𝐵𝑥,𝑦∶𝑥,𝑦∈𝑋×𝑌, (63) where 𝜇_𝐴∩𝐵(𝑥,𝑦)=𝑟_𝐴∩𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑟𝐴∩𝐵(𝑥,𝑦)}=min(𝑟_𝐴(𝑥,𝑦),𝑟_𝐵(𝑥,𝑦))𝑒^{𝑖min(𝜔𝑟𝐴(𝑥,𝑦),𝜔𝑟𝐵(𝑥,𝑦))} and 𝛾_𝐴∩𝐵(𝑥,𝑦)=𝑘_𝐴∩𝐵(𝑥,𝑦)𝑒^{𝑖𝜔𝑘𝐴∩𝐵(𝑥,𝑦)}=max(𝑘_𝐴(𝑥,𝑦),𝑘_𝐵(𝑥,𝑦))𝑒^{𝑖max(𝜔𝑘𝐴(𝑥,𝑦),𝜔𝑘𝐵(𝑥,𝑦))}.

Example 42. Using the CAIFS 𝐴 and 𝐵 as in Example 40, we have 𝐴∩𝐵=0.8𝑒^𝑖1.3𝜋,0.2𝑒^𝑖0.5𝜋(),𝑎,10.7𝑒^𝑖0.9𝜋,0.2𝑒^𝑖0.4𝜋(),𝑎,21.0𝑒^𝑖1.5𝜋,0.0𝑒^𝑖0.5𝜋(),𝑏,10.8𝑒^𝑖1.0𝜋,0.1𝑒^𝑖0.6𝜋().𝑏,2 (64)
Following Zhang et al. [15], we call the relation 𝐼 an identical equal relation on 𝑋, if its membership and nonmembership functions are defined as follows: 𝜇_𝐼𝑥,𝑦=1𝑒^𝑖2𝜋=1if𝑥=𝑦0𝑒^𝑖0𝜋𝛾=0if𝑥≠𝑦,_𝐼𝑥,𝑦=0𝑒^𝑖0𝜋=0if𝑥=𝑦1𝑒^𝑖2𝜋=1if𝑥≠𝑦. (65)

Definition 43. Let 𝐴∈CAIFS(𝑋×𝑋). If 𝜇_𝐴(𝑥,𝑥)=1 and 𝛾_𝐴(𝑥,𝑥)=0  for  all  𝑥∈𝑈, then 𝐴 is a reflexive complex Atanassov’s intuitionistic fuzzy relation.

Definition 44. Let 𝐴∈CIFS(𝑋×𝑌). Then 𝐴^𝑇∈CIFS(𝑌×𝑋) is a replacement of 𝐴, where 𝜇_𝐴𝑇𝑦,𝑥=𝜇_𝐴,𝛾𝑥,𝑦_𝐴𝑇𝑦,𝑥=𝛾_𝐴,∀𝑥,𝑦𝑥,𝑦∈𝑋×𝑌. (66)

Proposition 45. If 𝐴 and 𝐵 are two complex Atanassov’s intuitionistic fuzzy relations, then the following hold:(1)(𝐴^𝑇)^𝑇=𝐴,(2)(𝐴∪𝐵)^𝑇=𝐴^𝑇∪𝐵^𝑇  and  (𝐴∩𝐵)^𝑇=𝐴^𝑇∩𝐵^𝑇,(3)(𝐴∘𝐵)^𝑇=𝐵^𝑇∘𝐴^𝑇  and (𝐴^𝑛)^𝑇=(𝐴^𝑇)^𝑛,(4)𝐴⊆𝐵  if  and  only  if  𝐵^𝑇⊆𝐴^𝑇.

Proof. Trivial.

Definition 46. Let 𝐴∈CIFS(𝑋×𝑌). Then 𝐴 is a symmetrical complex Atanassov’s intuitionistic fuzzy relation if 𝜇_𝐴𝑥,𝑦=𝜇_𝐴𝑇,𝛾𝑥,𝑦_𝐴𝑥,𝑦=𝛾_𝐴𝑇,∀𝑥,𝑦𝑥,𝑦∈𝑋×𝑌. (67)

Definition 47. Let 𝐴∈CIFS(𝑋×𝑋). Then 𝐴 is a transitive complex Atanassov’s intuitionistic fuzzy relation if 𝐴∘𝐴⊆𝐴.

5. Projection and Cylindric Extension of Complex Atanassov’s Intuitionistic Fuzzy Relations

In this section, we introduce the concepts of projection and cylindric extension as generalizations of projection and cylindric extension, respectively, for AIFRs.

Definition 48. Let 𝑄 be a complex Atanassov’s intuitionistic fuzzy relation on 𝑋₁×⋯×𝑋_𝑛, and let {𝑖₁,…,𝑖_𝑘} be a subsequence of {1,…,𝑛}. Then the projection of 𝑄 on 𝑋_𝑖1×⋯×𝑋_𝑖𝑘 is a complex Atanassov’s intuitionistic fuzzy relation 𝑄_𝑝 on 𝑋_𝑖1×⋯×𝑋_𝑖𝑘, defined by membership and nonmembership functions: 𝜇_𝑄𝑝𝑥_𝑖1,…,𝑥_𝑖𝑘=max_{𝑥𝑗𝑖∈𝑋𝑗1,…,𝑥𝑗(𝑛−𝑘)∈𝑋𝑗(𝑛−𝑘)}𝑟_𝑄𝑥₁,…,𝑥_𝑛⋅𝑒^{𝑖max𝑥𝑗𝑖𝑗1𝑗𝑗∈𝑋,…,𝑥(𝑛−𝑘)∈𝑋(𝑛−𝑘)𝜔𝑟𝑄(𝑥1,…,𝑥𝑛)},𝛾_𝑄𝑝𝑥_𝑖1,…,𝑥_𝑖𝑘=min_{𝑥𝑗𝑖∈𝑋𝑗1,…,𝑥𝑗(𝑛−𝑘)∈𝑋𝑗(𝑛−𝑘)}𝑘_𝑄𝑥₁,…,𝑥_𝑛⋅𝑒^{𝑖min𝑥𝑗𝑖𝑗1𝑗𝑗∈𝑋,…,𝑥(𝑛−𝑘)∈𝑋(𝑛−𝑘)𝜔𝑘𝑄(𝑥1,…,𝑥𝑛)}, (68) where the {𝑥_𝑗1,…,𝑥_{𝑗(𝑛−𝑘)}} is the complement of {𝑥_𝑖1,…,𝑥_𝑖𝑘} with respect to {𝑥₁,…,𝑥_𝑛}.

Example 49. Let 𝑋={2,3,4} and 𝑌={𝑎,𝑏} be two universes of discourse, and let 𝑄 be a complex Atanassov’s intuitionistic fuzzy relation on 𝑋×𝑌 defined by 𝑄=(2,𝑎),0.3⋅𝑒^𝑖0.9𝜋,0.5⋅𝑒^𝑖0.3𝜋,(3,𝑎),0.1⋅𝑒^𝑖1.5𝜋,0.7⋅𝑒^𝑖𝜋,(4,𝑎),0.5⋅𝑒^𝑖1.7𝜋,0.4⋅𝑒^𝑖0.5𝜋,(2,𝑏),0.5⋅𝑒^𝑖0.7𝜋,0.5⋅𝑒^𝑖2𝜋,(3,𝑏),0.3⋅𝑒^𝑖0.2𝜋,0.4⋅𝑒^𝑖1.7𝜋,(4,𝑏),0.3⋅𝑒^𝑖1.5𝜋,0.4⋅𝑒^𝑖1.3𝜋. (69)
We compute the projection of CAIFR 𝑄 on 𝑋 and 𝑌, denoted by 𝑄_𝑋 and 𝑄_𝑌 respectively. The projection 𝑄_𝑋, is given by 𝜇_𝑄𝑝(𝑥)=max_𝑦∈𝑌𝑟_𝑄𝑥,𝑦⋅𝑒^{𝑖max𝑦∈𝑌𝜔𝑟𝑄(𝑥,𝑦)},𝛾_𝑄𝑝(𝑥)=min_𝑦∈𝑌𝑘_𝑄𝑥,𝑦⋅𝑒^{𝑖min𝑦∈𝑌𝜔𝑘𝑄(𝑥,𝑦)}.𝑄_𝑋(𝑥)=2,∨(0.3,0.5)𝑒^{𝑖∨(0.9𝜋,0.7𝜋)},∧(0.5,0.5)𝑒^{𝑖∧(0.3𝜋,2𝜋)},3,∨(0.1,0.3)𝑒^{𝑖∨(1.5𝜋,0.2𝜋)},∧(0.7,0.4)𝑒^{𝑖∧(𝜋,1.7𝜋)},4,∨(0.5,0.3)𝑒^{𝑖∨(1.7𝜋,1.5𝜋)},∧(0.4,0.4)𝑒^{𝑖∧(0.5𝜋,1.3𝜋)}.=2,0.5𝑒^𝑖0.9𝜋,0.5𝑒^𝑖0.3𝜋,3,0.3𝑒^𝑖1.5𝜋,0.4𝑒^𝑖𝜋,4,0.5𝑒^𝑖1.7𝜋,0.4𝑒^𝑖0.5𝜋. (70)
The projection 𝑄_𝑌 is given by 𝜇_𝑄𝑌𝑦=max_𝑥∈𝑋𝑟_𝑄𝑥,𝑦⋅𝑒^{𝑖max𝑥∈𝑋𝜔𝑟𝑄(𝑥,𝑦)},𝛾_𝑄𝑌(𝑥)=min_𝑥∈𝑋𝑘_𝑄𝑥,𝑦⋅𝑒^{𝑖minx∈X𝜔𝑘𝑄(𝑥,𝑦)}.𝑄_𝑌𝑦=𝑎,∨(0.3,0.1,0.5)𝑒^{𝑖∨(0.9𝜋,1.5𝜋,1.7𝜋)},∧(0.5,0.7,0.4)𝑒^{𝑖∧(0.3𝜋,𝜋,0.5𝜋)},𝑏,∨(0.5,0.3,0.3)𝑒^{𝑖∨(0.7𝜋,0.2𝜋,1.5𝜋)},∧(0.5,0.4,0.4)𝑒^{𝑖∧(2𝜋,1.7𝜋,1.3𝜋)}.=𝑎,0.5𝑒^𝑖1.7𝜋,0.4𝑒^𝑖0.3𝜋,𝑏,0.5⋅𝑒^𝑖1.5𝜋,0.4𝑒^𝑖1.3𝜋. (71)

Definition 50. Let 𝑄 be a complex Atanassov’s intuitionistic fuzzy relation on 𝑋₁×⋯×𝑋_𝑛 and let {𝑖₁,…,𝑖_𝑘} be a subsequence of {1,…,𝑛}. The cylindric extension of 𝑄_𝑃 to 𝑋₁×⋯×𝑋_𝑛, is a complex Atanassov’s intuitionistic fuzzy relation 𝑄_PE on 𝑋₁×⋯×𝑋_𝑛, defined by membership and nonmembership functions: 𝜇_𝑄PE𝑥₁,…,𝑥_𝑛=𝑟_𝑄𝑃𝑥_𝑖1,…,𝑥_𝑖𝑘⋅𝑒^{𝑖𝜔𝑟𝑄𝑃(𝑥𝑖1,…,𝑥𝑖𝑘)},𝛾_𝑄PE𝑥₁,…,𝑥_𝑛=𝑘_𝑄𝑃𝑥_𝑖1,…,𝑥_𝑖𝑘⋅𝑒^{𝑖𝜔𝑘𝑄𝑃(𝑥𝑖1,…,𝑥𝑖𝑘)}. (72)

Example 51. We consider Example 49 to find the cylindric extensions of 𝑄_𝑋 and 𝑄_𝑌. The cylindric extension of 𝑄_𝑋 is a complex Atanassov’s intuitionistic fuzzy relation 𝑄_XE given by 𝜇_𝑄XE(𝑥,𝑦)=𝜇_𝑄𝑋(𝑥), and 𝛾_𝑄XE(𝑥,𝑦)=𝛾_𝑄𝑋(𝑥). Then, 𝑄_XE=(2,𝑎),0.5𝑒^𝑖0.9𝜋,0.5𝑒^𝑖0.3𝜋,(3,𝑎),0.3𝑒^𝑖1.5𝜋,0.4𝑒^𝑖𝜋,(4,𝑎),0.5𝑒^𝑖1.7𝜋,0.4𝑒^𝑖0.5𝜋,(2,𝑏),0.5𝑒^𝑖0.9𝜋,0.5𝑒^𝑖0.3𝜋,(3,𝑏),0.3𝑒^𝑖1.5𝜋,0.4𝑒^𝑖𝜋,(4,𝑏),0.5𝑒^𝑖1.7𝜋,0.4𝑒^𝑖0.5𝜋. (73)
The cylindric extension of 𝑄_𝑌 is a complex Atanassov’s intuitionistic fuzzy relation 𝑄_YE given by 𝜇_𝑄YE(𝑥,𝑦)=𝜇_𝑄𝑌(𝑦), and 𝛾_𝑄YE(𝑥,𝑦)=𝛾_𝑄𝑌(𝑦). Thus, 𝑄_YE=(2,𝑎),0.5𝑒^𝑖1.7𝜋,0.4𝑒^𝑖0.3𝜋,(3,𝑎),0.5𝑒^𝑖1.7𝜋,0.4𝑒^𝑖0.3𝜋,(4,𝑎),0.5𝑒^𝑖1.7𝜋,0.4𝑒^𝑖0.3𝜋,(2,𝑏),0.5𝑒^𝑖1.5𝜋,0.4𝑒^𝑖1.3𝜋,(3,𝑏),0.5𝑒^𝑖1.5𝜋,0.4𝑒^𝑖1.3𝜋,(4,𝑏),0.5𝑒^𝑖1.5𝜋,0.4𝑒^𝑖1.3𝜋. (74)

6. Composition of Complex Atanassov’s Intuitionistic Fuzzy Relations

In this section, we introduce the definition of composition of complex Atanassov’s intuitionistic fuzzy relations. Some properties of composition of CAIFRs are obtained. The composition of complex Atanassov’s intuitionistic fuzzy relations is introduced as in the following.

Definition 52. Let 𝑋, 𝑌, and 𝑍 be universes, 𝐴 a complex Atanassov’s intuitionistic fuzzy relation of 𝑋 and 𝑌, and 𝐵 a complex Atanassov’s intuitionistic fuzzy relation of 𝑌 and 𝑍. The composition of 𝐴 and 𝐵, denoted by 𝐴∘𝐵, is a complex Atanassov’s intuitionistic fuzzy relation of 𝑋 and 𝑍, specified by the following functions: 𝜇_𝐴∘𝐵(𝑥,𝑧)=𝑟_𝐴∘𝐵(𝑥,𝑧)⋅𝑒^{𝑖𝜔𝑟𝐴∘𝐵(𝑥,𝑧)}=sup_𝑦∈𝑌min𝑟_𝐴𝑥,𝑦,𝑟_𝐵𝑦,𝑧⋅𝑒^{𝑖sup𝑦∈𝑌min(𝜔𝑟𝐴(𝑥,𝑦),𝜔𝑟𝐵(𝑦,𝑧))},𝛾_𝐴∘𝐵(𝑥,𝑧)=𝑟_𝐴∘𝐵𝑥,𝑦⋅𝑒^{𝑖𝜔𝑘𝐴∘𝐵(𝑥,𝑧)}=inf_𝑦∈𝑌max𝑘_𝐴𝑥,𝑦,𝑘_𝐵𝑦,𝑧⋅𝑒^{𝑖inf𝑦∈𝑌max(𝜔𝑘𝐴(𝑥,𝑦),𝜔𝑘𝐵(𝑦,𝑧))}. (75)

Example 53. Let 𝐾, 𝑆, 𝐽, 𝑃, and 𝐿 be name of cities, and let 𝑋={𝐾,𝑆}, 𝑌={𝐾,𝐽} and 𝑊={𝑃,𝐿} be three sets of these cities. Then, the Cartesian product of 𝑋 and 𝑌 is the set 𝑋×𝑌={(𝐾,𝐾),(𝐾,𝐽),(𝑆,𝐾),(𝑆,𝐽)}, and the Cartesian product of 𝑌 and 𝑍 is the set 𝑌×𝑍={(𝐾,𝑃),(𝐾,𝐿),(𝐽,𝑃),(𝐽,𝐿)}. For example, let 𝐴(𝑋,𝑌) be a relation called “the first element (city) is colder than the second element (city) in winter, summer, autumn, or spring,” and let 𝐵(𝑌,𝑍) be a relation called “the first element (city) is more attractive than the second element (city) in winter, summer, autumn or spring.” The relations 𝐴(𝑋,𝑌) and 𝐵(𝑌,𝑍) can be presented by the following relational matrices: 𝐴()=𝐾𝑋,𝑌𝐾𝐽0.9𝑒^{𝑖(0.3/24)𝜋},0𝑒^{𝑖(3/24)𝜋}0.7𝑒^{𝑖(1.5/24)𝜋},0.3𝑒^{𝑖(5/24)𝜋}𝑆0.6𝑒^{𝑖(1/24)𝜋},0.3𝑒^{𝑖(1.8/24)𝜋}0.3𝑒^{𝑖(3/24)𝜋},0.7𝑒^{𝑖(1/24)𝜋},𝐵()=𝐾𝑌,𝑍𝑃𝐿0.2𝑒^{𝑖(3/24)𝜋},0.7𝑒^{𝑖(3/24)𝜋}0.1𝑒^{𝑖(12/24)𝜋},0.8𝑒^{𝑖(5/24)𝜋}𝐽0.1𝑒^{𝑖(15/24)𝜋},0.9𝑒^{𝑖(5/24)𝜋}0.2𝑒^{𝑖(3/24)𝜋},0.6𝑒^{𝑖(2/24)𝜋}. (76)
We compute the composite relational matrix, denoted by 𝐴∘𝐵(𝑋,𝑍). We should note that 𝑋×𝑍 has four elements: (𝐾,𝑃), (𝐾,𝐿), (𝑆,𝑃),  and  (𝑆,𝐿). Thus, our task is to determine the membership, 𝜇_𝐴∘𝐵, and nonmembership, 𝛾_𝐴∘𝐵, values for the four elements above. We use Definition 52 to determine the membership and nonmembership of 𝐴∘𝐵(𝐾,𝑃). Thus, we have 𝜇_𝐴∘𝐵(𝐾,𝑃)=sup_𝑦∈𝑉min𝑟_𝐴(𝐾,𝐾),𝑟_𝐵(,𝑟𝐾,𝑃)min_𝐴(𝐾,𝐽),𝑟_𝐵(𝐽,𝑃)⋅𝑒^{sup𝑦∈𝑉{min[(𝜔𝑟𝐴(𝐾,𝐾)),𝜔𝑟𝐵(𝐾,𝑃)],min[(𝜔𝑟𝐴(𝐾,𝐽)),𝜔𝑟𝐵(𝐽,𝑃)]}}=sup_𝑦∈𝑉{min[0.9,0.2],min[0.7,0.1]}⋅𝑒^{𝑖sup𝑦∈𝑉{min[(0.3/24)𝜋,(3/24)𝜋],min[(1.5/24)𝜋,(15/24)𝜋]}}=sup_𝑦∈𝑉{0.2,0.1}𝑒^{𝑖sup𝑦∈𝑉{(0.3/24)𝜋,(1.5/24)𝜋}}=0.2𝑒^{𝑖(1.5/24)𝜋}. (77)
Analogously, we have 𝛾_𝐴∘𝐵(𝐾,𝑃)=inf_𝑦∈𝑉max𝑘_𝐴(𝐾,𝐾),𝑘_𝐵(,𝑘𝐾,𝑃)max_𝐴(𝐾,𝐽),𝑘_𝐵(𝐽,𝑃)⋅𝑒^{inf𝑦∈𝑉{max[(𝜔𝑘𝐴(𝐾,𝐾)),𝜔𝑘𝐵(𝐾,𝑃)],max[(𝜔𝑘𝐴(𝐾,𝐽)),𝜔𝑘𝐵(𝐽,𝑃)]}}=inf_𝑦∈𝑉{max[0,0.7],max[0.3,0.9]}⋅𝑒^{𝑖inf𝑦∈𝑉{max[(3/24)𝜋,(3/24)𝜋],max[(5/24)𝜋,(5/24)𝜋]}}=inf_𝑦∈𝑉{0.7,0.9}𝑒^{𝑖inf𝑦∈𝑉{(3/24)𝜋,(5/24)𝜋]}}=0.7𝑒^{𝑖(3/24)𝜋}. (78)
Analogously, we determine the elements (𝐾,𝐿),(𝑆,𝑃),  and  (𝑆,𝐿). Thus, the final relational matrix of 𝐴∘𝐵 is ()=𝐾𝐴∘𝐵𝑋,𝑍𝑃𝐿0.2𝑒^{𝑖(15/24)𝜋},0.7𝑒^{𝑖(3/24)𝜋}0.2𝑒^{𝑖(3/24)𝜋},0.6𝑒^{𝑖(5/24)𝜋}𝑆0.2𝑒^{𝑖(3/24)𝜋},0.7𝑒^{𝑖(5/24)𝜋}0.2𝑒^{𝑖(3/24)𝜋},0.7𝑒^{𝑖(2/24)𝜋}. (79)

Note 2. In Example 53 the amplitude terms represent the belongingness and nonbelongingness to the set of cold and attractiveness, and the phase terms represent the belongingness and nonbelongingness to the phase of season.

Max-Min and Max Product Compositions(1)The max-min composition of CAIFRs between 𝐴(𝑋,𝑌) and 𝐵(𝑌,𝑍) is a CAIFR 𝐴∘𝐵 in 𝑋×𝑍, defined by the membership and nonmembership functions 𝜇_𝐴∘𝐵(𝑥,𝑧)=max_𝑦∈𝑉min𝑟_𝐴𝑥,𝑦,𝑟_𝐴𝑦,𝑧⋅𝑒^{𝑖max𝑦∈𝑉min(arg𝜇𝐴(𝑥,𝑦),arg𝜇𝐵(𝑦,𝑧))},𝛾_𝐴∘𝐵(𝑥,𝑧)=min_𝑦∈𝑉max𝑟_𝐴𝑥,𝑦,𝑟_𝐵𝑦,𝑧⋅𝑒^{𝑖min𝑦∈𝑉max(arg𝛾𝐴(𝑥,𝑦),arg𝛾𝐵(𝑦,𝑧))}. (80)(2)The max product composition of CAIFRs between 𝐴(𝑋,𝑌) and 𝐵(𝑌,𝑍) is a CAIFR 𝐴∘𝐵 in 𝑋×𝑍, defined by the membership and nonmembership functions: 𝜇_𝐴∘𝐵(𝑥,𝑧)=max_𝑦∈𝑉𝑟_𝐴𝑥,𝑦⋅𝑟_𝐵𝑦,𝑧⋅𝑒^{𝑖(1/2𝜋)max𝑦∈𝑉(arg𝜇𝐴(𝑥,𝑦)⋅arg𝜇𝐵(𝑦,𝑧))},𝜇_𝐴∘𝐵(𝑥,𝑧)=min_𝑦∈𝑉𝑟_𝐴𝑥,𝑦⋅𝑟_𝐵𝑦,𝑧⋅𝑒^{𝑖(1/2𝜋)min𝑦∈𝑉(arg𝛾𝐴(𝑥,𝑦)⋅arg𝛾𝐵(𝑦,𝑧))}. (81)

Proposition 54. Let 𝑊, 𝑋, 𝑌, and 𝑍 be universes, 𝐴 a complex Atanassov’s intuitionistic fuzzy relation of 𝑊 and 𝑋, 𝐵 a complex Atanassov’s intuitionistic fuzzy relation of 𝑋 and 𝑌, and 𝐶 a complex Atanassov’s intuitionistic fuzzy relation of 𝑌 and 𝑍. Then 𝐴∘(𝐵∘𝐶)=(𝐴∘𝐵)∘𝐶.

Proof. Let 𝐴∘(𝐵∘𝐶)={⟨(𝑤,𝑧),𝜇_{𝐴∘(𝐵∘𝐶)}(𝑤,𝑧),𝛾_{𝐴∘(𝐵∘𝐶)}(𝑤,𝑧)⟩∶(𝑤,𝑧)∈𝑊×𝑍}, and (𝐴∘𝐵)∘𝐶={⟨(𝑤,𝑧),𝜇_{(𝐴∘𝐵)∘𝐶}(𝑤,𝑧),𝛾_{(𝐴∘𝐵)∘𝐶}(𝑤,𝑧)⟩∶(𝑤,𝑧)∈𝑊×𝑍}. To prove the equality 𝐴∘(𝐵∘𝐶)=(𝐴∘𝐵)∘𝐶, we have to show that 𝜇_{𝐴∘(𝐵∘𝐶)}=𝜇_{(𝐴∘𝐵)∘𝐶} and 𝛾_{𝐴∘(𝐵∘𝐶)}=𝛾_{(𝐴∘𝐵)∘𝐶}. By Definition 52, we have 𝜇_{𝐴∘(𝐵∘𝐶)}()=𝑤,𝑧sup_𝑥∈𝑋min𝑟_𝐴(𝑤,𝑥),𝑟_{(𝐵∘𝐶)}(𝑥,𝑧)⋅𝑒^{𝑖sup𝑥∈𝑋min[𝜔𝜇𝐴(𝑤,𝑥),(𝜔𝜇(𝐵∘𝐶)(𝑥,𝑧))]}=sup_𝑥∈𝑋min𝑟_𝐴(𝑤,𝑥),sup_𝑦∈𝑌min𝑟_𝐵𝑥,𝑦,𝑟_𝐶𝑦,𝑧⋅𝑒^{𝑖sup𝑥∈𝑋min[𝜔𝜇𝐴(𝑤,𝑥),(sup𝑦∈𝑌min(𝜔𝜇𝐵(𝑥,𝑦),𝜔𝜇𝐶(𝑦,𝑧)))]}=sup_{𝑥∈𝑋,𝑦∈𝑌}min𝑟_𝐴(𝑤,𝑥),𝑟_𝐵𝑥,𝑦,𝑟_𝐶𝑦,𝑧⋅𝑒^{𝑖sup𝑥∈𝑋,𝑦∈𝑌min[𝜔𝜇𝐴(𝑤,𝑥),𝜔𝜇𝐵(𝑥,𝑦),𝜔𝜇𝐶(𝑦,𝑧)]}=sup_𝑦∈𝑌minsup_𝑥∈𝑋min𝑟_𝐴(𝑤,𝑥),𝑟_𝐵𝑥,𝑦,𝑟_𝐶𝑦,𝑧⋅𝑒^{𝑖sup𝑦∈𝑌min[(sup𝑥∈𝑋min(𝜔𝜇𝐴(𝑤,𝑥),𝜔𝜇𝐵(𝑥,𝑦))),𝜔𝜇𝐶(𝑦,𝑧)]}=sup_𝑦∈𝑌min𝑟_{(𝐴∘𝐵)}𝑤,𝑦,𝑟_𝐶𝑦,𝑧⋅𝑒^{𝑖sup𝑦∈𝑌min[(𝜔𝜇(𝐴∘𝐵)(𝑤,𝑦)),𝜔𝜇𝐶(𝑦,𝑧)]}=𝜇_{(𝐴∘𝐵)∘𝐶}(𝑤,𝑧). (82)
Also, 𝛾_{𝐴∘(𝐵∘𝐶)}()=𝑤,𝑧inf_𝑥∈𝑋max𝑘_𝐴(𝑤,𝑥),𝑘_{(𝐵∘𝐶)}(𝑥,𝑧)⋅𝑒^{𝑖inf𝑥∈𝑋max[𝜔𝛾𝐴(𝑤,𝑥),(𝜔𝛾(𝐵∘𝐶)(𝑥,𝑧))]}=inf_𝑥∈𝑋max𝑘_𝐴(𝑤,𝑥),inf_𝑦∈𝑌max𝑘_𝐵𝑥,𝑦,𝑘_𝐶𝑦,𝑧⋅𝑒^{𝑖inf𝑥∈𝑋max[𝜔𝛾𝐴(𝑤,𝑥),(inf𝑦∈𝑌max(𝜔𝛾𝐵(𝑥,𝑦),𝜔𝛾𝐶(𝑦,𝑧)))]}=inf_{𝑥∈𝑋,𝑦∈𝑌}𝑘_𝐴(𝑤,𝑥),𝑘_𝐵𝑤,𝑦,𝑘_𝐶𝑦,𝑧⋅𝑒^{𝑖inf𝑥∈𝑋,𝑦∈𝑌max[𝜔𝛾𝐴(𝑤,𝑥),𝜔𝛾𝐵(𝑤,𝑦),𝜔𝛾𝐶(𝑦,𝑧)]}=inf_𝑦∈𝑌maxinf_𝑥∈𝑋max𝑘_𝐴(𝑤,𝑥),𝑘_𝐵,𝑘𝑥,𝑦_𝐶𝑦,𝑧⋅𝑒^{𝑖inf𝑦∈𝑌max}=inf_𝑦∈𝑌max𝑘_{(𝐴∘𝐵)}𝑤,𝑦,𝑘_𝐶𝑦,𝑧⋅𝑒^{𝑖inf𝑦∈𝑌max[(𝜔𝛾(𝐴∘𝐵)(𝑤,𝑦)),𝜔𝛾𝐶(𝑦,𝑧)]}=𝛾_{(𝐴∘𝐵)∘𝐶}(𝑤,𝑧). (83)
So we have 𝜇_{𝐴∘(𝐵∘𝐶)}=𝜇_{(𝐴∘𝐵)∘𝐶} from (82), and we have 𝛾_{𝐴∘(𝐵∘𝐶)}=𝛾_{(𝐴∘𝐵)∘𝐶} from (83). Therefore, the proof is completed.

Proposition 55. Let 𝑋, 𝑌, and 𝑍 be universes, 𝐴 and 𝐵 two complex Atanassov’s intuitionistic fuzzy relations of 𝑋 and 𝑌, and 𝐶 a complex Atanassov’s intuitionistic fuzzy relation of 𝑍 and 𝑋. Then,(i)𝐶∘(𝐴∪𝐵)=(𝐶∘𝐴)∪(𝐶∘𝐵),(ii)𝐶∘(𝐴∩𝐵)⊆(𝐶∘𝐴)∩(𝐶∘𝐵).

Proof. (i) To prove the equality 𝐶∘(𝐴∪𝐵)=(𝐶∘𝐴)∪(𝐶∘𝐵), we have to show that 𝜇_{𝐶∘(𝐴∪𝐵)}=𝜇_{(𝐶∘𝐴)∪(𝐶∘𝐵)} and 𝛾_{𝐶∘(𝐴∪𝐵)}=𝛾_{(𝐶∘𝐴)∪(𝐶∘𝐵)}. By Definitions 52, 17, and 18 we have 𝜇_{𝐶∘(𝐴∪𝐵)}=supmin𝑟_𝐶,𝑟_{(𝐴∪𝐵)}⋅𝑒^{𝑖supmin[𝜔𝜇𝐶,(𝜔𝜇(𝐴∪𝐵))]}=supmin𝑟_𝐶,max𝑟_𝐴,𝑟_𝐵⋅𝑒^{𝑖supmin[𝜔𝜇𝐶,(max(𝜔𝜇𝐴,𝜔𝜇𝐵))]}=maxsupmin𝑟_𝐶,𝑟_𝐴,supmin𝑟_𝐶,𝑟_𝐵⋅𝑒^{𝑖max[(supmin(𝜔𝜇𝐶,𝜔𝜇𝐴)),(supmin(𝜔𝜇𝐶,𝜔𝜇𝐵))]}=𝜇_{(𝐶∘𝐴)∪(𝐶∘𝐵)}, (84) Also, 𝛾_{𝐶∘(𝐴∪𝐵)}=infmax𝑘_𝐶,𝑘_{(𝐴∪𝐵)}⋅𝑒^{infmax[𝜔𝛾𝐶,(𝜔𝛾(𝐴∪𝐵))]}=infmax𝑘_𝐶,min𝑘_𝐴,𝑘_𝐵⋅𝑒^{infmax[𝜔𝛾𝐶,(min(𝜔𝛾𝐴,𝜔𝛾𝐵))]}=mininfmax𝑘_𝐶,𝑘_𝐴,infmax𝑘_𝐶,𝑘_𝐵⋅𝑒^{min[(infmax(𝜔𝛾𝐶,𝜔𝛾𝐴)),inf(max(𝜔𝛾𝐶,𝜔𝛾𝐵))]}=𝛾_{(𝐶∘𝐴)∪(𝐶∘𝐵)}. (85)
Therefore, by (84) and (85), proof of (i) is completed.
(ii) To prove the subsets 𝐶∘(𝐴∩𝐵)⊆(𝐶∘𝐴)∩(𝐶∘𝐵), we have to show that 𝑟_{𝐶∘(𝐴∩𝐵)}≤𝑟_{(𝐶∘𝐴)∩(𝐶∘𝐵)} and Arg_{𝜇𝐶∘(𝐴∩𝐵)}≥Arg_{𝜇(𝐶∘𝐴)∩(𝐶∘𝐵)} for the membership function and 𝑘_{𝐶∘(𝐴∩𝐵)}≤𝑘_{(𝐶∘𝐴)∩(𝐶∘𝐵)} and Arg_{𝛾𝐶∘(𝐴∩𝐵)}≤Arg_{𝛾(𝐶∘𝐴)∩(𝐶∘𝐵)} for the nonmembership function. By Definitions 28, 52, and 18, we have 𝑟_{𝐶∘(𝐴∩𝐵)}=supmin𝑟_𝐶,min𝑟_𝐴,𝑟_𝐵=sup𝑟_𝑗𝑟_{(𝐶∘𝐴)∩(𝐶∘𝐵)}=minsupmin𝑟_𝐶,𝑟_𝐴,supmin𝑟_𝐶,𝑟_𝐵=minsup𝑟_𝑗, (86) where 𝑗=𝐴, 𝐵, or 𝐶 for any of the following cases: 𝑟_𝐴≤𝑟_𝐵≤𝑟_𝐶,𝑟_𝐵≤𝑟_𝐴≤𝑟_𝐶,𝑟_𝐶≤𝑟_𝐵≤𝑟_𝐴,𝑟_𝐶≤𝑟_𝐴≤𝑟_𝐵,𝑟_𝐴≤𝑟_𝐶≤𝑟_𝐵,𝑟_𝐵≤𝑟_𝐶≤𝑟_𝐴. (87)
So, we conclude that 𝑟_{𝐶∘(𝐴∩𝐵)}=sup𝑟_𝑗≥minsup𝑟_𝑗=𝑟_{(𝐶∘𝐴)∩(𝐶∘𝐵)}. (88)
To prove the nonmembership parts, we use similar steps as for the membership term. Therefore, this completes the proof of (ii).

Corollary 56. Let 𝑋, 𝑌, and 𝑍 be universes, 𝐴_𝑡(𝑡∈𝑇) complex Atanassov’s intuitionistic fuzzy relations of 𝑋 and 𝑌, and 𝐶 a complex Atanassov’s intuitionistic fuzzy relation of 𝑍 and 𝑋, where 𝑇 is an arbitrary index set. Then,(i)𝐶∘(∪_𝑡∈𝑇𝐴_𝑡)=∪_𝑡∈𝑇(𝐶∘𝐴_𝑡), (ii)𝐶∘(∩_𝑡∈𝑇𝐴_𝑡)⊆∩_𝑡∈𝑇(𝐶∘𝐴_𝑡).

Proof. This is straightforward from Proposition 55.

Proposition 57. Let 𝑋, 𝑌, and 𝑍 be universes, 𝐴 and 𝐵 two complex Atanassov’s intuitionistic fuzzy relations of 𝑋 and 𝑌, and 𝐶 a complex fuzzy relation of 𝑍 and 𝑋. If 𝐴⊆𝐵, then(i)𝐶∘𝐴⊆𝐶∘𝐵.

Proof. This is straightforward from Proposition 55 and Definitions 52 and 28.

7. Conclusions

In this research, a new model deals with multi-attribute decision-making problems by representing the type of Atanassov’s intuitionistic fuzzy information which is happening repeatedly over a period of time. This generalised model (Application 31) is easy to realize and appropriate with human recognition. The difference between this model and Atanassov’s intuitionistic fuzzy model in [12] is that in our approach, complex Atanassov’s intuitionistic fuzzy set represents the problems with Atanassov’s intuitionistic fuzzy uncertainty and periodicity simultaneously, in which the phase degree, of membership and nonmembership have precious and pivotal values. Besides, the phase term can represent both the fuzzy information and precise/accurate time. So, CAIFSs can be useful and applicable in many areas, such as decision-making processes, to select the best alternative.

The development of complex Atanassov’s intuitionistic fuzzy relation is important as it forms a basis for the derivation of complex Atanassov’s intuitionistic fuzzy logic. Complex Atanassov’s intuitionistic fuzzy relation is formally generalised from a conventional Atanassov’s intuitionistic fuzzy relation, based on complex Atanassov’s intuitionistic fuzzy sets, in which the ranges of values of CAIFR are extended to the unit circle in complex plane instead of [0,1] as in the conventional Atanassov’s intuitionistic fuzzy functions. We have also introduced the Cartesian product of CAIFSs and derived two properties of the product space. The concepts of projection, cylindric extension, and composition of CAIFRs are also introduced. Finally, an example of CAIFR in real life situation is illustrated in this paper.

8. Future Research

In future research, it is a meaningful topic to introduce the concept of CAIF logics and systems based on CAIF relation, which gives the ability to extend the realm of real-life applications in the CAIF environment. The concepts of logics and systems open for us immense applications in engineering, medical, physics, automobiles, defense and security, internet and computer security, and other fields. We can also introduce δ-equalities of complex Atanassov’s intuitionistic fuzzy relations.

Acknowledgment

The work presented in this paper was supported by Universiti Kebangsaan Malaysia under Grant no. UKM-DLP-2011-038.