A robust correlation coefficient measure of complex intuitionistic fuzzy sets and their applications in decision-making

Abstract

The objective of this work is to present novel correlation coefficient measures for measuring the relationship between the two complex intuitionistic fuzzy sets (CIFSs). In the existing studies of fuzzy and its extension, the uncertainties present in the data are handled with the help of degrees of membership which are the subset of real numbers, and may lose some useful information and hence consequently affect on the decision results. An alternative to these, complex intuitionistic fuzzy set handles the uncertainties with the degrees whose ranges are extended from real subset to the complex subset with unit disc and hence handle the two-dimensional information in a single set. Thus, motivated by this, we develop correlation and weighted correlation coefficients under the CIFS environment in which pairs of the membership degrees represent the two-dimensional information. Also, some of the desirable properties of it are investigated. Further, based on these measures, a multicriteria decision-making approach is presented under the CIFS environment. Two illustrative examples are taken to demonstrate the efficiency of the proposed approach and validate it by comparing their results with the several existing approaches’ results.

1 Introduction

Multicriteria decision making (MCDM) process involves the analysis of a finite set of alternatives and ranking them in terms of how credible they are to decision-maker(s) when all the criteria is considered simultaneously. In this process, the rating values of each alternative include both precise data and experts’ subjective information. But, traditionally, it is assumed that the information provided by them are crisp in nature. However, due to the complexity of the system day-by-day, the real-life contains many MCDM problems where the information is either vague, imprecise or uncertain in nature. To deal with it, the theory of fuzzy set (FS) [1] or extended fuzzy sets such as intuitionistic fuzzy set (IFS) [2], interval-valued IFS (IVIFS) [3] are the most successful ones, which characterize the criterion values in terms of membership degrees. From their applications, researchers found that none of these models are able to represent the partial ignorance of the data and its fluctuations at a given phase of time during their execution. Furthermore, in our day-to-day life, uncertainty and vagueness which are present in the data occur concurrently with changes to the phase (periodicity) of the data. Thus, the existing theories are insufficient to consider this information and hence there is an information loss during the process. To overcome it, Ramot et al. [4] initiated a complex fuzzy set (CFS) in which the range of membership function is extended from real number to the complex number with the unit disc. As the complex fuzzy set considers only the membership degree, but doesn’t weight on the non-membership portion of the data entities, which likewise assume an equal part in assessing the object in the decision-making process. However, in the real world, it is regularly hard to express the estimation of the membership degree by an exact value in a fuzzy set. In such cases, it might be easier to depict vagueness and uncertainty in the real world using a 2-dimensional information instead of a single one. Consequently, an extension of the existing theories might be extremely valuable to depict the uncertainties because of his/her reluctant judgment in complex decision-making problems. For this reason, Alkouri and Salleh [5] extended the concept of CFS to complex intuitionistic fuzzy sets (CIFSs) by adding the degree of complex non-membership functions and studied their basic operations. Henceforth, a CIFS is a more generalized extension of the existing theories such as FSs, IFSs, CFSs. Clearly, the advantage of the CIFSs is that it can contain substantially more data to express the information.

Presently, correlation measure is one of the most important measures which will help not only in comparing one data entity with other but also show the extent of association between them and their direction. Also, CIFSs have powerful ability to model the imprecise and ambiguous information in real-world applications than the existing theories such as FSs, IFSs, and CFSs [6]. Therefore, keeping the advantages of this set and taking the importance of correlation measure, this paper presents the theory of the correlation coefficients among the CIFS. As per our knowledge, in the aforementioned studies, the correlation measures cannot be utilized to handle the CIFS information. Thus, in order to achieve it, we first define the informational energies and the covariance between the two CIFSs that involves both uncertainty and periodicity semantics. Then, based on these, we propose some correlation coefficients for CIFSs and investigate their properties. Further, some weighted correlation coefficients are proposed to address the situations where the element in the set is correlative. Furthermore, we propose a decision-making approach based on the proposed correlation coefficients for CIFSs. The feasibility, as well as superiority of the approach, has been demonstrated through two numerical examples.

To do so, the rest of the manuscript is summarized as follows. Section 2 is a literature review that presents papers on the correlation measures. In Section 3, we briefly present an overview related to the concepts of IFSs, CFSs, and CIFSs. In Section 4, we introduce correlation and weighted correlation coefficients for CIFSs and obtain some properties. In Section 5, we present a multicriteria decision-making approach based on the proposed correlation coefficients under CIFSs environment, where each element of the set is characterized by complex intuitionistic fuzzy numbers. In Section 6, two illustrative examples are presented to discuss the functionality of the proposed approach and compare their results with some of the existing approaches results in Section 7. Finally, Section 8 summarizes this study.

2 Related work

To process an uncertain and imprecise information during the decision-making process, numerable attempts have been made by different researchers in processing the information values using aggregation operators [7,8,9,10,11,12], information measures [13,14,15,16], score and accuracy functions [17, 18] under IFS, IVIFS environments. Among all these concepts, one of the significant ways to solve such type of problems by using the concept of correlation coefficient which provides us with the measurement of the dependence of the two variables. In statistical analysis, one of the important measures is correlation coefficients which give us an idea of the strength and direction of a linear relationship between the pairs of two variables. On the other hand, in fuzzy set theory, these measures determine the degree of the dependency between the two fuzzy sets. In that direction, Gerstenkorn and Manko [19] firstly introduced the concept of coefficient of correlation for measuring the interrelation of IFSs. Later on, Hong and Hwang [20] extended its concept into the probability spaces. Hung and Wu [21] studied the correlation coefficients by using the centroid method. Bustince and Burillo [22] extended the concept of correlation from IFS to IVIFS environment. Zeng and Li [23] presented a decision-making approach based on the correlation coefficients. Garg [24, 25] presented the correlation coefficients for Pythagorean fuzzy sets and intuitionistic multiplicative set respectively. Ye [26] presented the cosine similarity measures for IFSs. Garg [27] presented some improved cosine similarity measures for IFS and applied them to solve the decision-making problems. However, apart from these, some other kinds of the correlation coefficients [28,29,30] have been proposed by the different researchers and they applied them to solve the multicriteria decision-making problems.

The above measures and their corresponding approaches are widely used by the researchers, but from these studies, it has been analyzed the data under the FSs, IFSs or its generalizations is only able to handle the uncertainty and vagueness that exists in the data. But, simultaneously, none of these existing models are able to represent the partial ignorance of the data and its fluctuations, with changes to the phase (periodicity), at a given phase of time during their execution. To overcome it, Ramot et al. [4] initiated a complex fuzzy set (CFS) in which the range of membership function is extended from real number to the complex number with the unit disc. Ramot et al. [31] generalized traditional fuzzy logic to complex fuzzy logic in which the sets used in reasoning process are complex fuzzy sets, characterized by complex valued membership functions. Greenfield et al. [32] extended the concept of CFS by taking the grade of the membership function as an interval-number rather than single numbers. Furthermore, Alkouri and Salleh [5] extended the concepts of CFS to complex intuitionistic fuzzy sets (CIFSs) by adding the degree of complex non-membership functions and studied their basic operations. Alkouri and Salleh [33] introduced the concepts of complex intuitionistic fuzzy relation, composition, projections and proposed a distance measure between the two CIFSs. Rani and Garg [34] presented some series of distance measures under CIFS environment and their application in the decision- making process. Rani and Garg [35] presented power aggregation operators for different CIFSs. Kumar and Bajaj [36] proposed some distance and entropy measures for complex intuitionistic fuzzy soft sets.

In CIFS theory, membership and non-membership degrees are complex-valued and are represented in polar coordinates. The amplitude term corresponding to the membership (non-membership) degree gives the extent of belongings (not-belongings) of an object in a CIFS and the phase term associated with membership (non-membership) degree gives the additional information, generally related with periodicity. Since, in the existing IFSs theory, it is observed that there is only one parameter to represent the information which results in information loss in some instances. However, in day-to-day life, we come across complex natural phenomena where we need to add the second dimension to the expression of membership and non-membership grades. By introducing this second dimension, the complete information can be projected in one set, and hence loss of information can be avoided. For instance, suppose a certain company decides to set up biometric-based attendance devices (BBADs) in all of its offices spread all over the country. For this, the company consults an expert who gives the information regarding the two-dimensions namely, models of BBADs and their corresponding production dates of BBADs. The task of the company is to select the most optimal model of BBADs with its production date simultaneously. It is obviously seen that such type of problems cannot be modeled accurately by considering both the dimensions simultaneously using the traditional IFS theories. Thus, for such types of problem, there is a need to enhance the existing theories and hence a CIFS environment provides us with an efficient way to handle the two-step judgment scenarios in which an amplitude term may be employed to give a company’s decision regarding model of BBADs and the phase terms may be used to represent company’s decision regarding the production date of BBADs in the decision making process. Similarly, some other types of examples under CIFSs include the large amounts of data sets that are generated from medical research, as well as government databases for biometric and facial recognition, audio, and images etc. Henceforth, a CIFS is a more generalized extension of the existing theories such as FSs, IFSs, CFSs. Clearly, the advantage of the CIFSs is that it can contain substantially more data to express the information.

Therefore, keeping the advantages of this set and taking the importance of correlation measure, this paper presents the theory of the correlation coefficients among the CIFSs. Also, it is being computed that the several existing correlation measures can be easily obtained from the proposed measures.

3 Preliminaries

In this section, some basic concepts related to the IFSs, CFSs and CIFSs are reviewed over the universal set U≠ϕ.

Definition 1

[2, 9] An Atanassov’s Intuitionistic fuzzy set(IFS) S defined on U is an ordered pair given by

$$\begin{array}{@{}rcl@{}} S=\left\{\left( x,\mu_{S}(x),\nu_{S}(x)\right):x\in U\right\}, \end{array} $$

(1)

where μ_S,ν_S : U → [0,1] are real-valued membership and non-membership functions respectively such that μ_S(x) + ν_S(x) ≤ 1 for all x ∈ U. Also, π_S(x) = 1 − μ_S(x) − ν_S(x) is called the degree of hesitation of x to S. For convenience, this pair of membership and non-membership degrees is called as intuitionistic fuzzy number (IFN) and is denoted by S = (μ,ν), where 0 ≤ u,v ≤ 1 and u + v ≤ 1.

Definition 2

[23] For two IFSs A = {(x,μ_A(x),ν_A(x)) : x ∈ U} and B = {(x,μ_B(x),ν_B(x)) : x ∈ U} defined on U = {x₁,x₂,…x_n}, [23] defined the correlation between them as

$$\begin{array}{@{}rcl@{}} C_{1}(A,B)=\frac{1}{n}\sum\limits_{j = 1}^{n} \left( \mu_{A}(x_{j}) \mu_{B}(x_{j})+\nu_{A}(x_{j}) \nu_{B}(x_{j})+\pi_{A}(x_{j}) \pi_{B}(x_{j})\right) \end{array} $$

(2)

and hence defined the correlation coefficient between A and B as:

$$\begin{array}{@{}rcl@{}} \rho_{1}(A,B)=\frac{C_{1}(A,B)}{\sqrt{C_{1}(A,A)\cdot C_{1}(B,B)}} \end{array} $$

(3)

and showed that it satisfies the following properties:

• 1. ρ₁(A,B) = ρ₁(B,A)

• 2. 0 ≤ ρ₁(A,B) ≤ 1

• 3. A = B ⇔ ρ₁(A,B) = 1

Ramot et al. [4] extended the theory of fuzzy set to the complex fuzzy set (CFS) by incorporating the phase angle into the analysis, which has been defined as follows:

Definition 3

[4] A CFS S defined on U is defined as a set of pairs given by

$$\begin{array}{@{}rcl@{}} S=\left\{\left( x,\mu_{S}(x)\right):x\in U\right\}, \end{array} $$

(4)

where μ_S is a membership function which can assign any element x ∈ U a complex valued grade of membership. The value of μ_S(x) lies in a unit circle in the complex plane and is of the form $\mu _{S}(x)=r_{S}(x) e^{iw_{r_{S}}(x)}$ where $i=\sqrt {-1}, r_{S}(x)\in [0,1]$ and $w_{r_{S}}(x)$ is real valued.

Later on, [5] extended the concept of CFS to complex intuitionistic fuzzy set (CIFS) by taking the degree of non-membership function into the analysis as follows:

Definition 4

[5] A CIFS S over U is defined as a set given by

$$\begin{array}{@{}rcl@{}} S=\left\{\left( x,\mu_{S}(x),\gamma_{S}(x)\right):x\in U\right\}, \end{array} $$

(5)

where μ_S : U →{a : a ∈ C,|a|≤ 1} and γ_S : U →{a : a ∈ C,|a|≤ 1} are complex valued membership and non-membership functions respectively given by:

$$\begin{array}{@{}rcl@{}} \mu_{S}(x)=r_{S}(x) e^{iw_{r_{S}}(x)}, \qquad \gamma_{S}(x)=k_{S}(x) e^{iw_{k_{S}}(x)}. \end{array} $$

Here r_S(x),k_S(x) ∈ [0,1] such that r_S(x) + k_S(x) ≤ 1. Also $w_{r_{S}}(x)$ and $w_{k_{S}}(x)$ are real valued which satisfy the conditions $w_{r_{S}}(x), w_{k_{S}}(x) \in [0,2\pi ]$ and $w_{r_{S}}(x)+w_{k_{S}}(x)\leq 2\pi$ for each x ∈ U. For the sake of convenience, we shall denote the set $\left \{(x,r_{S}(x) e^{iw_{r_{S}}(x)} \right ., \left . k_{S}(x) e^{iw_{k_{S}}(x)}): x \in U \right \}$ as $S=\left (r_{S}(x) e^{iw_{r_{S}}(x)},k_{S}(x) e^{iw_{k_{S}}(x)} \right )$.

4 Correlation coefficient for CIFSs

In this section, we propose some correlation coefficients for the CIFSs which can be applied in numerous engineering and scientific fields to rank the objects. For it, throughout this paper, we shall use U = {x₁,x₂,…x_n} as the universe of discourse.

Let $A=\left (r_{A}(x) e^{iw_{r_{A}}(x)},k_{A}(x) e^{iw_{k_{A}}(x)} \right )$ and $B=\left (r_{B}(x) e^{iw_{r_{B}}(x)},k_{B}(x) e^{iw_{k_{B}}(x)} \right )$ be two CIFSs defined on U. Then, the informational energies of two CIFSs A and B are defined as

$$\begin{array}{@{}rcl@{}} T(A) &=& \sum\limits_{j = 1}^{n} \left( {r_{A}^{2}}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right), \end{array} $$

(6)

$$\begin{array}{@{}rcl@{}} T(B) &=& \sum\limits_{j = 1}^{n} \left( {r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right). \end{array} $$

(7)

The correlation of the CIFSs A and B is defined as

$$\begin{array}{@{}rcl@{}} C(A,B) &=& \sum\limits_{j = 1}^{n} \left[ \begin{aligned} & r_{A}(x_{j})r_{B}(x_{j})+ \frac{1}{4\pi^{2}} w_{r_{A}}(x_{j})w_{r_{B}}(x_{j}) \\ & + k_{A}(x_{j})k_{B}(x_{j})+\frac{1}{4\pi^{2}}w_{k_{A}}(x_{j})w_{k_{B}}(x_{j}) \end{aligned} \right]. \end{array} $$

(8)

From Eq. (8), it is clearly seen that correlation of CIFSs satisfies the following properties:

• (P1) C(A,B) = C(B,A)

• (P2) C(A,A) = T(A)

Then, based on these, we defined the correlation coefficient between CIFSs A and B, as follows:

Definition 5

If $A=(r_{A}(x) e^{iw_{r_{A}}(x)},k_{A}(x) e^{iw_{k_{A}}(x)} )$ and $B=(r_{B}(x) e^{iw_{r_{B}}(x)},k_{B}(x) e^{iw_{k_{B}}(x)} )$ be two CIFSs defined on U, then the correlation coefficient between them is denoted by K₁(A,B) and is defined as

$$\begin{array}{@{}rcl@{}} K_{1}(A,B)&=& \frac{C(A,B)}{\sqrt{T(A)\times T(B)}} \\ &=& \frac{\sum\limits_{j = 1}^{n} \left( \begin{array}{ll} & r_{A}(x_{j})r_{B}(x_{j})+ \frac{1}{4\pi^{2}}\left( w_{r_{A}}(x_{j})w_{r_{B}}(x_{j})\right)\\ & +k_{A}(x_{j})k_{B}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{k_{A}}(x_{j})w_{k_{B}}(x_{j})\right) \end{array} \right)}{\left\{ \begin{array}{ll} & \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{A}^{2}}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right)} \\ &\times \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right)} \end{array} \right\}} \end{array} $$

(9)

Theorem 1

The correlation coefficientK₁between two CIFSs A and B defined on U satisfies the following properties:

• (P1) 0 ≤ K₁(A,B) ≤ 1.

• (P2) K₁(A,B) = K₁(B,A).

• (P3) K₁(A,B) = 1,ifA = B.

• (P4) IfA ⊆ B ⊆ Cthen,K₁(A,C) ≤ K₁(A,B) andK₁(A,C) ≤ K₁(B,C) for CIFS C defined on U.

Proof

Let $A=\left (r_{A}(x) e^{iw_{r_{A}}(x)}, k_{A}(x) e^{iw_{k_{A}}(x)} \right )$ and $B=\left (r_{B}(x) e^{iw_{r_{B}}(x)},k_{B}(x) e^{iw_{k_{B}}(x)} \right )$ be two CIFSs defined on U. Then, we have

• 1. The inequality K₁(A,B) ≥ 0 is obvious due to C(A,B) ≥ 0 is obtained from the Eq. (8). Now we shall prove K₁(A,B) ≤ 1. For it, based on the Definition 5, we get

  $$\begin{array}{@{}rcl@{}} K_{1}(A,B)&=& \frac{C(A,B)}{\sqrt{C(A,A)\times C(B,B)}} \\ &=& \frac{\sum\limits_{j = 1}^{n} \left( \begin{array}{ll} r_{A}(x_{j})r_{B}(x_{j})+ \frac{1}{4\pi^{2}}w_{r_{A}}(x_{j})w_{r_{B}}(x_{j}) \\ + k_{A}(x_{j})k_{B}(x_{j})+ \frac{1}{4\pi^{2}} w_{k_{A}}(x_{j})w_{k_{B}}(x_{j}) \end{array} \right)}{\left\{ \begin{array}{lll} & \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{A}^{2}}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right)} \\ &\times \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right)} \end{array} \right\}} \\ &=& \frac{ \left( \begin{array}{ll} & \sum\limits_{j = 1}^{n} r_{A}(x_{j})r_{B}(x_{j})+ \sum\limits_{j = 1}^{n}\left( \frac{w_{r_{A}}(x_{j})}{2\pi}\right)\left( \frac{w_{r_{B}}(x_{j})}{2\pi}\right) \\ & + \sum\limits_{j = 1}^{n}k_{A}(x_{j})k_{B}(x_{j}) +\sum\limits_{j = 1}^{n} \left( \frac{w_{k_{A}}(x_{j})}{2\pi}\right)\left( \frac{w_{k_{B}}(x_{j})}{2\pi}\right) \end{array} \right)} {\left\{ \begin{array}{ll} & \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{A}^{2}}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right)} \\ &\times \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right)} \end{array} \right\}} \end{array} $$

  Now, by using Cauchy-Schwarz inequality which states that, in Euclidean space Rⁿ with standard inner product, we have:

  $$\begin{array}{@{}rcl@{}} \left( \sum\limits_{j = 1}^{n} u_{j} v_{j}\right)^{2} \leq \left( \sum\limits_{j = 1}^{n} {u_{j}^{2}}\right) \left( \sum\limits_{j = 1}^{n} {v_{j}^{2}}\right) \end{array} $$

  where u = (u₁,u₂,…u_n) and v = (v₁,v₂,…v_n) ∈ Rⁿ and equality holds if and only if u and v are linearly dependent vectors. Therefore,

  $$\begin{array}{@{}rcl@{}} K_{1}(A,B) \leq \frac{\left( \begin{array}{ll} & \sqrt{\sum\limits_{j = 1}^{n} {r_{A}^{2}}(x_{j})}\sqrt{\sum\limits_{j = 1}^{n} {r_{B}^{2}}(x_{j})} + \sqrt{\sum\limits_{j = 1}^{n} \left( \frac{w_{r_{A}}(x_{j})}{2\pi}\right)^{2}}\sqrt{\sum\limits_{j = 1}^{n} \left( \frac{w_{r_{B}}(x_{j})}{2\pi}\right)^{2}} \\ & \sqrt{\sum\limits_{j = 1}^{n} {k_{A}^{2}}(x_{j})}\sqrt{\sum\limits_{j = 1}^{n} {k_{B}^{2}}(x_{j})} + \sqrt{\sum\limits_{j = 1}^{n} \left( \frac{w_{k_{A}}(x_{j})}{2\pi}\right)^{2}}\sqrt{\sum\limits_{j = 1}^{n} \left( \frac{w_{k_{B}}(x_{j})}{2\pi}\right)^{2}} \end{array} \right)}{\left\{ \begin{array}{ll} & \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{A}^{2}}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right)} \\ &\times \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right)} \end{array} \right\}} \end{array} $$

  By taking the notations, $\sum \limits _{j = 1}^{n} {r_{A}^{2}}(x_{j}) = a, \sum \limits _{j = 1}^{n} {r_{B}^{2}}(x_{j})$$= b, \sum \limits _{j = 1}^{n} {k_{A}^{2}}(x_{j}) = c, \sum \limits _{j = 1}^{n} {k_{B}^{2}}(x_{j}) = d, \sum \limits _{j = 1}^{n} \left (\frac {w_{r_{A}}(x_{j})}{2\pi }\right )^{2} \!=$$p, \sum \limits _{j = 1}^{n} \left (\frac {w_{r_{B}}(x_{j})}{2\pi }\right )^{2} = q, \sum \limits _{j = 1}^{n} \left (\frac {w_{k_{A}}(x_{j})}{2\pi }\right )^{2} = r$ and $\sum \limits _{j = 1}^{n} \left (\frac {w_{k_{B}}(x_{j})}{2\pi }\right )^{2}=s$, the above inequality reduces to

  $$\begin{array}{@{}rcl@{}} K_{1}(A,B) \leq \frac{\sqrt{ab}+\sqrt{cd}+\sqrt{pq}+\sqrt{rs}}{\sqrt{(a+c+p+r)(b+d+q+s)}} \end{array} $$

  Therefore,

  $$\begin{array}{@{}rcl@{}} \left( K_{1}(A,B)\right)^{2}-1 &\leq& \frac{\left( \sqrt{ab}+\sqrt{cd}+\sqrt{pq}+\sqrt{rs}\right)^{2}}{(a+c+p+r)(b+d+q+s)}-1 \\ &=& \frac{\left( \sqrt{ab}+\sqrt{cd}+\sqrt{pq}+\sqrt{rs}\right)^{2}-(a+c+p+r)(b+d+q+s)}{(a+c+p+r)(b+d+q+s)} \\ &=& \frac{\left( \begin{array}{ll} & ab+cd+pq+rs+ 2\sqrt{abcd}+ 2\sqrt{pqrs}+ 2\sqrt{abpq} + 2\sqrt{abrs} \\ & + 2\sqrt{cdpq}+ 2\sqrt{cdrs} -ab-ad-aq -as-cb-cd - cq\\ & -cs-pb -pd-pq-ps-rb-rd-rq-rs \end{array} \right)}{(a+c+p+r)(b+d+q+s)} \\ &=& - \left( \frac{ \begin{array}{ll} (\sqrt{ad}-\sqrt{bc})^{2} + (\sqrt{ps}-\sqrt{qr})^{2} + (\sqrt{aq}-\sqrt{bp})^{2} \\ +(\sqrt{as}-\sqrt{br})^{2} + (\sqrt{cq}-\sqrt{dp})^{2} + (\sqrt{cs}-\sqrt{dr})^{2} \end{array}} {(a+c+p+r)(b+d+q+s)} \right) \\ & \leq & 0 \end{array} $$

  Hence, ${K_{1}^{2}}(A,B) \leq 1$ which implies K₁(A,B) ≤ 1. So, 0 ≤ K₁(A,B) ≤ 1.

• 2. For any two CIFSs A and B, we have

  $$\begin{array}{@{}rcl@{}} K_{1}(A,B)&=& \frac{\sum\limits_{j = 1}^{n} \left( \begin{array}{ll} r_{A}(x_{j})r_{B}(x_{j})+\frac{1}{4\pi^{2}}w_{r_{A}}(x_{j})w_{r_{B}}(x_{j}) \\ + k_{A}(x_{j})k_{B}(x_{j})+ \frac{1}{4\pi^{2}}w_{k_{A}}(x_{j})w_{k_{B}}(x_{j}) \end{array} \right)}{\left\{ \begin{array}{ll} & \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{A}^{2}}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right)} \\ &\times \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right)} \end{array} \right\}} \\ \end{array} $$
  $$\begin{array}{@{}rcl@{}} &=& \frac{\sum\limits_{j = 1}^{n} \left( \begin{array}{ll} r_{B}(x_{j})r_{A}(x_{j})+\frac{1}{4\pi^{2}}w_{r_{B}}(x_{j})w_{r_{A}}(x_{j}) \\ + k_{B}(x_{j})k_{A}(x_{j})+ \frac{1}{4\pi^{2}}w_{k_{B}}(x_{j})w_{k_{A}}(x_{j}) \end{array} \right)}{\left\{ \begin{array}{ll} & \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right)} \\ &\times \sqrt{\sum\limits_{j = 1}^{n} \left( {r_{A}^{2}}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right)} \end{array} \right\}} \\ &=& K_{1}(B,A) \end{array} $$

• 3. If A = B this implies that r_A(x_j) = r_B(x_j), k_A(x_j) = k_B(x_j), $w_{r_{A}}(x_{j})=w_{r_{B}}(x_{j})$ and $w_{k_{A}}(x_{j})=w_{k_{B}}(x_{j})$ for all j and thus from Eq. (9), it follows that K₁(A,B) = 1.

• 4. Geometrically, if A ⊆ B ⊆ C, then the angle between A and C should be larger than the angle between B and C for any element x_j and cos𝜃 is decreasing function within interval $[0, \frac {\pi }{2}]$. Therefore, K₁(A,C) ≤ K₁(A,B) and K₁(A,C) ≤ K₁(B,C).

Hence, the theorem holds. □

Example 1

Let U={x₁,x₂,x₃} be the universal set and A = { (x₁,0.6e^i2π(0.7), 0.2e^i2π(0.2)), (x₂,0.7e^i2π(0.5),0.3e^i2π(0.4)), (x₃,0.5e^i2π(0.4),0.4e^i2π(0.1))},B = {(x₁,0.5e^i2π(0.6), 0.1e^i2π(0.2)), (x₂,0.7e^i2π(0.4),0.1e^i2π(0.4)),(x₃,0.6e^i2π(0.5), 0.3e^i2π(0.4) ) } are two CIFSs defined on the universal set U. By applying Eq. (6), we obtain the informational energy of A as:

$$\begin{array}{@{}rcl@{}} T(A) &=& \sum\limits_{j = 1}^{n} \left( {r_{A}^{2}}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right)\\ &=& (0.6)^{2}+(0.2)^{2}+\frac{1}{4\pi^{2}}\left[\left( 2\pi\times 0.7\right)^{2}+\left( 2\pi\times 0.2\right)^{2}\right]\\ \end{array} $$

$$\begin{array}{@{}rcl@{}} &&+ (0.7)^{2}+(0.3)^{2} +\frac{1}{4\pi^{2}}\left[\left( 2\pi\times 0.5\right)^{2}+\left( 2\pi\times 0.4\right)^{2}\right] \\ &&+(0.5)^{2}+(0.4)^{2}+\frac{1}{4\pi^{2}}\left[\left( 2\pi\times 0.4 \right)^{2}+\left( 2\pi\times 0.1 \right)^{2}\right] \\ &=& 0.36 + 0.04 + 0.49 + 0.04 + 0.49 + 0.09 + 0.25 \\ && + 0.16 + 0.25 + 0.16 + 0.16 + 0.01 \\ &=& 2.5 \end{array} $$

Similarly, the informational energy of CIFS B is:

$$\begin{array}{@{}rcl@{}} T(B) &=& \sum\limits_{j = 1}^{n} \left( {r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right) \\ &=& (0.5)^{2}+(0.1)^{2}+\frac{1}{4\pi^{2}}\left[\left( 2\pi\times 0.6\right)^{2}+\left( 2\pi\times 0.2\right)^{2}\right] \\ &&+ (0.7)^{2}+(0.1)^{2} + \frac{1}{4\pi^{2}}\left[\left( 2\pi\times 0.4 \right)^{2}+\left( 2\pi\times 0.4\right)^{2} \right] \\ &&+ (0.6)^{2}+(0.3)^{2}+\frac{1}{4\pi^{2}}\left[\left( 2\pi\times 0.5\right)^{2}+\left( 2\pi\times 0.4\right)^{2}\right] \\ &=& 0.25 + 0.01 + 0.36 + 0.04 + 0.49 + 0.01 + 0.16 \\ && + 0.16 + 0.36 + 0.09 + 0.25 + 0.16 \\ &=& 2.34 \end{array} $$

By using Eq. (8), the correlation between CIFSs A and B is computed as:

$$\begin{array}{@{}rcl@{}} C(A,B) &=& \sum\limits_{j = 1}^{n} \left( \begin{array}{cc} r_{A}(x_{j})r_{B}(x_{j}) + \frac{1}{4\pi^{2}} w_{r_{A}}(x_{j})w_{r_{B}}(x_{j}) \\ +k_{A}(x_{j})k_{B}(x_{j}) + \frac{1}{4\pi^{2}} w_{k_{A}}(x_{j})w_{k_{B}}(x_{j}) \end{array} \right) \\ &=& 0.6 \times 0.5 + 0.2 \times 0.1+\frac{1}{4\pi^{2}}\left( 2\pi(0.7) \times 2\pi(0.6)+ 2\pi(0.2) \times 2\pi(0.2)\right)\\ &&+ 0.7 \times 0.7 + 0.3 \times 0.1+\frac{1}{4\pi^{2}}\left( 2\pi(0.5) \times 2\pi(0.4)+ 2\pi(0.4)\times 2\pi(0.4)\right)\\ &&+ 0.5 \times 0.6 + 0.4 \times 0.3+\frac{1}{4\pi^{2}}\left( 2\pi(0.4) \times 2\pi(0.5)+ 2\pi(0.1) \times 2\pi(0.4)\right) \\ &=& 0.30 + 0.02 + 0.42 + 0.04 + 0.49 + 0.03 + 0.20 + 0.16 + 0.30 \\ && + 0.12 + 0.20 + 0.04 \\ &=& 2.32 \end{array} $$

Thus, the correlation coefficient between A and B is given by Eq. (9) as

$$\begin{array}{@{}rcl@{}} K_{1}(A,B)&=& \frac{C(A,B)}{\sqrt{T(A)\times T(B)}} \\ &=& \frac{2.32}{\sqrt{2.5 \times 2.34}}= 0.9592 \end{array} $$

Definition 6

Let $A=\left (r_{A}(x) e^{iw_{r_{A}}(x)}, k_{A}(x) e^{iw_{k_{A}}(x)} \right )$ and $B=\left (r_{B}(x) e^{iw_{r_{B}}(x)}, k_{B}(x) e^{iw_{k_{B}}(x)} \right )$ be two CIFSs defined on U. Then, the correlation coefficient, denoted by K₂, is defined as

$$\begin{array}{@{}rcl@{}} K_{2}(A,B)&=& \frac{C(A,B)}{\max \{T(A),T(B)\}} \\ &=& \frac{\sum\limits_{j = 1}^{n} \left( \begin{array}{ll} r_{A}(x_{j})r_{B}(x_{j}) +\frac{1}{4\pi^{2}} \left( w_{r_{A}}(x_{j})w_{r_{B}}(x_{j})\right)\\ +k_{A}(x_{j})k_{B}(x_{j}) +\frac{1}{4\pi^{2}} \left( w_{k_{A}}(x_{j})w_{k_{B}}(x_{j})\right) \end{array} \right)} {\max \left\{ \begin{array}{ll} \sum\limits_{j = 1}^{n} \left( {r_{A}^{2}}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right), \\ \sum\limits_{j = 1}^{n} \left( {r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right) \end{array} \right\}} \qquad \end{array} $$

(10)

Theorem 2

The correlation coefficient of two CIFSs A and B, as defined in Eq. (10), satisfies the following properties:

• (P1) 0 ≤ K₂(A,B) ≤ 1.

• (P2) K₂(A,B) = K₂(B,A).

• (P3) K₂(A,B) = 1 ifA = B.

• (P4) IfA ⊆ B ⊆ Cthen,K₂(A,C) ≤ K₂(A,B) andK₂(A, C) ≤ K₂(B, C) for any CIFS C defined on U.

Proof

Since A,B ∈CIFSs, then $0\leq r_{A}(x_{j}), k_{A}(x_{j})\leq 1, 0\leq w_{r_{A}}, w_{k_{A}}\leq 2\pi$ and r_A(x_j) + k_A(x_j) ≤ 1 and $w_{r_{A}}+w_{k_{A}}\leq 2\pi$ for all x_j ∈ U and thus from Eq. (10), we can obtain the inequality K₂(A,B) ≥ 0. The inequality K₂(A,B) ≤ 1 can be proven directly by using the well-known Cauchy-Schwarz inequality:

$$\begin{array}{@{}rcl@{}} \sum\limits_{j = 1}^{n} a_{j} b_{j} \leq \sqrt{\left( {\sum}_{j = 1}^{n} {a_{j}^{2}}\right) \cdot \left( {\sum}_{j = 1}^{n} {b_{j}^{2}}\right)} \end{array} $$

(11)

with equality if and only if the two vectors a = (a₁, a₂,…, a_n) and b = (b₁,b₂,…,b_n) are linearly dependent.

In fact, by Eq. (11), we have

$$\begin{array}{@{}rcl@{}} \sum\limits_{j = 1}^{n} a_{j} b_{j} &\leq& \sqrt{\left( {\sum}_{j = 1}^{n} {a_{j}^{2}}\right) \cdot \left( {\sum}_{j = 1}^{n} {b_{j}^{2}}\right)} \leq \sqrt{\left( \max\left\{{\sum}_{j = 1}^{n} {a_{j}^{2}}, {\sum}_{j = 1}^{n} {b_{j}^{2}}\right\}\right)^{2}} \\ &=& \max\left\{{\sum}_{j = 1}^{n} {a_{j}^{2}}, {\sum}_{j = 1}^{n} {b_{j}^{2}}\right\} \end{array} $$

and from Eq. (10), it follows that 0 ≤ K₂(A,B) ≤ 1 which completes the proof of (P1). In addition, by Eq. (10), we have

$$\begin{array}{@{}rcl@{}} K_{2}(A,B)&=& \frac{\sum\limits_{j = 1}^{n} \left( \begin{array}{ll} r_{A}(x_{j})r_{B}(x_{j}) +\frac{1}{4\pi^{2}} \left( w_{r_{A}}(x_{j})w_{r_{B}}(x_{j})\right)\\ +k_{A}(x_{j})k_{B}(x_{j}) +\frac{1}{4\pi^{2}} \left( w_{k_{A}}(x_{j})w_{k_{B}}(x_{j})\right) \end{array} \right)} {\max \left\{ \begin{array}{ll} \sum\limits_{j = 1}^{n} \left( {r_{A}^{2}}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right), \\ \sum\limits_{j = 1}^{n} \left( {r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right) \end{array} \right\}} \\ &=& \frac{\sum\limits_{j = 1}^{n} \left( \begin{array}{ll} r_{B}(x_{j})r_{A}(x_{j}) +\frac{1}{4\pi^{2}} \left( w_{r_{B}}(x_{j})w_{r_{A}}(x_{j})\right)\\ +k_{B}(x_{j})k_{A}(x_{j}) +\frac{1}{4\pi^{2}} \left( w_{k_{B}}(x_{j})w_{k_{A}}(x_{j})\right) \end{array} \right)} {\max \left\{ \begin{array}{ll} \sum\limits_{j = 1}^{n} \left( {r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right), \\ \sum\limits_{j = 1}^{n} \left( {r_{A}^{2}}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right) \end{array} \right\}} \\ &=& K_{2}(B,A) \end{array} $$

Thus, (P2) also holds. Similarly, we can complete the proofs of (P3) and (P4).

Hence, the theorem holds. □

From Definition 5 and Definition 6, we observe that the correlation coefficient defined by Eq. (9) uses the geometric mean of the informational energies of the CIFSs A and B, and the correlation coefficient defined by Eq. (10) applies the maximum between them. For the optimistic decision makers, they tend to use the correlation coefficient defined by Eq. (9). Contrary to the optimistic decision makers, the pessimistic decision makers tend to apply the correlation coefficient defined by Eq. (10).

In the above defined formulas for calculating coefficient of correlation, equal importance is given to all the elements of the universal set. But in real life situations, this may not be always possible. Some elements in the universal set are more important than the others. So we must take into account the proper weightage given to the various elements of the universal set. In the following, we propose a weighted correlation coefficient between CIFSs. Let ξ = (ξ₁,ξ₂,…ξ_n)^T be the weight vector corresponding to the elements x_j (j = 1,2,…,n) with ξ_j > 0 and $\sum \limits _{j = 1}^{n} \xi _{j}= 1$. Then, we extend the above defined correlation coefficients K₁ and K₂ to weighted correlation coefficients, K₃ and K₄ respectively, as follows:

$$\begin{array}{@{}rcl@{}} K_{3}(A,B)&=&\frac{C_{w}(A,B)}{\sqrt{T_{w}(A)\times T_{w}(B)}} \\ &=& \frac{\sum\limits_{j = 1}^{n} \left( \xi_{j} \left[ \begin{array}{llll} r_{A}(x_{j})r_{B}(x_{j}) +\frac{1}{4\pi^{2}}\left( w_{r_{A}}(x_{j})w_{r_{B}}(x_{j})\right) \\ +k_{A}(x_{j})k_{B}(x_{j})+ \frac{1}{4\pi^{2}}\left( w_{k_{A}}(x_{j})w_{k_{B}}(x_{j})\right) \end{array} \right] \right)} {\left\{ \begin{array}{ll} \sqrt{\sum\limits_{j = 1}^{n} \left( \xi_{j}\left[r_{A}^{2}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right]\right)} \\ \times \sqrt{\sum\limits_{j = 1}^{n} \left( \xi_{j}\left[{r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right]\right)} \end{array}\right\}} \end{array} $$

(12)

and

$$\begin{array}{@{}rcl@{}} K_{4}(A,B)&=&\frac{C_{w}(A,B)}{\max\{T_{w}(A), T_{w}(B)\}} \\ &=& \frac{\sum\limits_{j = 1}^{n} \left( \xi_{j} \left[ \begin{array}{ll} r_{A}(x_{j})r_{B}(x_{j}) +\frac{1}{4\pi^{2}}\left( w_{r_{A}}(x_{j})w_{r_{B}}(x_{j})\right) \\ +k_{A}(x_{j})k_{B}(x_{j})+ \frac{1}{4\pi^{2}}\left( w_{k_{A}}(x_{j})w_{k_{B}}(x_{j})\right) \end{array} \right] \right)}{\max\left\{ \begin{array}{ll} \sum\limits_{j = 1}^{n} \left( \xi_{j}\left[r_{A}^{2}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right]\right), \\ \sum\limits_{j = 1}^{n} \left( \xi_{j}\left[{r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right]\right) \end{array}\right\}} \qquad \end{array} $$

(13)

It can be easily verified that, if ξ = (1/n,1/n,…,1/n)^T then Eqs. (12) and (13) reduce to the correlation coefficients given in Eqs. (9) and (10) respectively. Further, can be deduced that the correlation coefficients K₃ and K₄ between CIFSs A and B also satisfies the property of 0 ≤ K₃(A,B) ≤ 1 and 0 ≤ K₄(A,B) ≤ 1.

Theorem 3

Let A and B be two CIFSs defined on U. Ifξ = (ξ₁,ξ₂,…,ξ_n)^Tbe the weight vector corresponding tox_j,(j = 1,2,…,n) withξ_j > 0 and$\sum \limits _{j = 1}^{n} \xi _{j}= 1$then the weightedcorrelation coefficientK₃(A,B) between the two CIFSs A and B defined in Eq. (12), satisfies the following properties:

• (P1) 0 ≤ K₃(A,B) ≤ 1

• (P2) K₃(A,B) = K₃(B,A)

• (P3) K₃(A,B) = 1 ifA = B

• (P4) IfA ⊆ B ⊆ Cthen,K₃(A,C) ≤ K₃(A,B) andK₃(A,C) ≤ K₃(B,C) for any CIFS C defined on U.

Proof

The properties (P2)-(P4) are straightforward, so we omit to proof here. Now, we shall proof only (P1) property. For it, let $A=(r_{A}(x) e^{iw_{r_{A}}(x)}, k_{A}(x) e^{iw_{k_{A}}(x)} )$ and $B=(r_{B}(x) e^{iw_{r_{B}}(x)}, k_{B}(x) e^{iw_{k_{B}}(x)} )$ be two CIFSs defined on U. From the Eq. (12), it is clearly seen that, K₃(A,B) ≥ 0. So, we will prove only K₃(A,B) ≤ 1.

$$\begin{array}{@{}rcl@{}} K_{3}(A,B)&=&\frac{C_{w}(A,B)}{\sqrt{T_{w}(A)\times T_{w}(B)}} \\ &=& \frac{\sum\limits_{j = 1}^{n} \left( \xi_{j} \left[ \begin{array}{ll} r_{A}(x_{j})r_{B}(x_{j}) +\frac{1}{4\pi^{2}}\left( w_{r_{A}}(x_{j})w_{r_{B}}(x_{j})\right) \\ +k_{A}(x_{j})k_{B}(x_{j})+ \frac{1}{4\pi^{2}}\left( w_{k_{A}}(x_{j})w_{k_{B}}(x_{j})\right) \end{array} \right] \right)}{\left\{ \begin{array}{ll} \sqrt{\sum\limits_{j = 1}^{n} \left( \xi_{j}\left[r_{A}^{2}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right]\right)} \\ \times \sqrt{\sum\limits_{j = 1}^{n} \left( \xi_{j}\left[{r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right]\right)} \end{array}\right\}} \\ &=& \frac{\left( \begin{array}{ll} \sum\limits_{j = 1}^{n} \left( \sqrt{\xi_{j}} r_{A}(x_{j})\right)\left( \sqrt{\xi_{j}} r_{B}(x_{j})\right) + \sum\limits_{j = 1}^{n} \frac{1}{4\pi^{2}}\left( \sqrt{\xi_{j}} w_{r_{A}}(x_{j})\right)\left( \sqrt{\xi_{j}} w_{r_{B}}(x_{j})\right) \\ + \sum\limits_{j = 1}^{n} \left( \sqrt{\xi_{j}} k_{A}(x_{j})\right)\left( \sqrt{\xi_{j}} k_{B}(x_{j})\right) + \sum\limits_{j = 1}^{n} \frac{1}{4\pi^{2}}\left( \sqrt{\xi_{j}} w_{k_{A}}(x_{j})\right) \left( \sqrt{\xi_{j}} w_{k_{B}}(x_{j})\right) \end{array} \right)}{\left\{ \begin{array}{ll} \sqrt{\sum\limits_{j = 1}^{n} \left( \xi_{j}\left[r_{A}^{2}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right]\right)} \\ \times \sqrt{\sum\limits_{j = 1}^{n} \left( \xi_{j}\left[{r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right]\right)} \end{array}\right\}} \end{array} $$

By using Cauchy-Schwarz inequality, we obtain

$$\begin{array}{@{}rcl@{}} K_{3}(A,B) &\leq & \frac{\left( \begin{array}{ll} \sqrt{\sum\limits_{j = 1}^{n} \xi_{j} {r_{A}^{2}}(x_{j})}\sqrt{\sum\limits_{j = 1}^{n} \xi_{j} {r_{B}^{2}}(x_{j})} + \sqrt{\sum\limits_{j = 1}^{n} \xi_{j} \left( \frac{ w_{r_{A}}(x_{j})}{2\pi}\right)^{2}}\sqrt{\sum\limits_{j = 1}^{n} \xi_{j} \left( \frac{w_{r_{B}}(x_{j})}{{2\pi}}\right)^{2}} \\ + \sqrt{\sum\limits_{j = 1}^{n} \xi_{j} {k_{A}^{2}}(x_{j})} \sqrt{\sum\limits_{j = 1}^{n} \xi_{j} {k_{B}^{2}}(x_{j})}+ \sqrt{\sum\limits_{j = 1}^{n} \xi_{j} \left( \frac{w_{k_{A}}(x_{j})}{2\pi}\right)^{2}} \sqrt{\sum\limits_{j = 1}^{n} \xi_{j} \left( \frac{w_{k_{B}}(x_{j})}{2\pi}\right)^{2}} \end{array} \right)}{\left\{ \begin{array}{ll} \sqrt{\sum\limits_{j = 1}^{n} \left( \xi_{j}\left[r_{A}^{2}(x_{j})+{k_{A}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{A}}^{2}(x_{j})+w_{k_{A}}^{2}(x_{j})\right)\right]\right)} \\ \times \sqrt{\sum\limits_{j = 1}^{n} \left( \xi_{j}\left[{r_{B}^{2}}(x_{j})+{k_{B}^{2}}(x_{j})+\frac{1}{4\pi^{2}}\left( w_{r_{B}}^{2}(x_{j})+w_{k_{B}}^{2}(x_{j})\right)\right]\right)} \end{array}\right\}} \end{array} $$

By using the following notations

$$\begin{array}{@{}rcl@{}} && \sum\limits_{j = 1}^{n}\xi_{j}{r_{A}^{2}}(x_{j})=a, \sum\limits_{j = 1}^{n}\xi_{j}{r_{B}^{2}}(x_{j})=b, \sum\limits_{j = 1}^{n} \xi_{j} \left( \frac{w_{r_{A}}(x_{j})}{2\pi}\right)^{2}=p, \sum\limits_{j = 1}^{n}\xi_{j}{k_{A}^{2}}(x_{j})=c, \\ && \sum\limits_{j = 1}^{n} \xi_{j}{k_{B}^{2}}(x_{j})=d, \sum\limits_{j = 1}^{n} \xi_{j} \left( \frac{w_{r_{B}}(x_{j})}{2\pi}\right)^{2}=q, \sum\limits_{j = 1}^{n} \xi_{j} \left( \frac{w_{k_{A}}(x_{j})}{2\pi}\right)^{2}=r, \sum\limits_{j = 1}^{n} \xi_{j} \left( \frac{w_{k_{B}}(x_{j})}{2\pi}\right)^{2}=s, \end{array} $$

we can reduce the above inequality to

$$\begin{array}{@{}rcl@{}} K_{3}(A,B) \leq \frac{\sqrt{ab}+\sqrt{cd}+\sqrt{pq}+\sqrt{rs}}{\sqrt{(a+c+p+r)(b+d+q+s)}} \end{array} $$

Therefore,

$$\begin{array}{@{}rcl@{}} \left( K_{3}(A,B)\right)^{2}-1 &\leq& \frac{\left( \sqrt{ab}+\sqrt{cd}+\sqrt{pq}+\sqrt{rs}\right)^{2}}{(a+c+p+r)(b+d+q+s)}-1 \\ &=& \frac{\left( \sqrt{ab}+\sqrt{cd}+\sqrt{pq}+\sqrt{rs}\right)^{2}-(a+c+p+r)(b+d+q+s)}{(a+c+p+r)(b+d+q+s)} \\ &=& \frac{ \left( \begin{array}{ll} ab+cd+pq+rs+ 2\sqrt{abcd}+ 2\sqrt{pqrs}+ 2\sqrt{abpq}\\ + 2\sqrt{abrs} + 2\sqrt{cdpq}+ 2\sqrt{cdrs} -ab-ad-aq-as -cb \\ -cd-cq-cs-pb-pd-pq-ps-rb-rd-rq-rs \end{array} \right)} {(a+c+p+r)(b+d+q+s)} \\ &=& -\left( \frac{\begin{array}{l}{(\sqrt{ad}-\sqrt{bc})^{2}+(\sqrt{ps}-\sqrt{qr})^{2}+(\sqrt{aq}-\sqrt{bp})^{2}}\\ {+(\sqrt{as}-\sqrt{br})^{2}+(\sqrt{cq}-\sqrt{dp})^{2}+(\sqrt{cs}-\sqrt{dr})^{2}}\end{array}}{(a+c+p+r)(b+d+q+s)}\right) \\ & \leq & 0 \end{array} $$

which implies that K₃(A,B) ≤ 1. Hence, 0 ≤ K₃(A,B) ≤ 1. □

Theorem 4

The correlation coefficient of two CIFSs A and B, as defined in Eq. (13) i.e.,K₄,satisfies the same properties as those in Theorem 2.

Proof

The proof is similar to Theorem 2, so we omit here. □

5 MCDM approach based on the proposed correlation coefficients

In this section, we utilize the proposed correlation coefficients of CIFSs to present the multicriteria decision making method.

For a multi criteria decision-making with complex intuitionistic fuzzy information, assume that there are m different alternatives denoted by A₁,A₂,…,A_m which have to be evaluated under the set of the n criteria denoted by C₁,C₂,…,C_n. Assume that an expert is invited to evaluate these alternatives under the set of each criteria. Further, assume that the importance of each criteria is considered in the form of the weight vector as ξ = (ξ₁,ξ₂,…,ξ_n)^T such that ξ_q > 0 and $\sum \limits _{q = 1}^{n} \xi _{q} = 1$. Consider an expert provide the values under the CIFS environment and these values can be considered as a complex intuitionistic fuzzy element. The rating values corresponding to each alternative are represented in the form of CIFS A_p(p = 1,2,…,m) as follows

$$\begin{array}{@{}rcl@{}} A_{p} \!\!&=&\!\! \left\{\!\left( \!C_{q}, r_{pq}(C_{q})e^{iw_{r_{pq}}(C_{q})}\!,k_{pq}(C_{q})e^{iw_{k_{pq}}(C_{q})}\!\right)\!\!\mid\! q = 1,2, \ldots, n\!\right\}\!;\\ p\!\!&=&\!\!1,2, \ldots, m. \end{array} $$

where r_pq(C_q) ∈ [0,1] represent the satisfaction degree of the alternative A_p towards the criteria C_q and k_pq(C_q) represent the possible degree of the rejection for the alternative A_p under the criteria C_q. For convenience, we denote this CIFS by $\alpha _{pq}=\left (r_{pq}e^{iw_{r_{pq}}}, k_{pq}e^{iw_{k_{pq}}}\right )$ where $r_{pq}, k_{pq}\in [0,1], w_{r_{pq}}, w_{k_{pq}}\in [0,2\pi ]$ and $r_{pq}+k_{pq}\leq 1 , w_{r_{pq}} + w_{k_{pq}}\leq 2\pi$ for p = 1,2,…,m;q = 1,2,…,n and call it as complex intuitionistic fuzzy numbers (CIFNs). Then, we utilize the following steps based on the proposed correlation coefficients for solving the MCDM problems under the CIFSs environment.

• Step 1: Construct the ideal reference set to find the best alternative in the decision set whose rating values are taken under the CIFSs environment. We denote such reference set by B.

• Step 2: Construct the decision matrix based on the collective information of the alternatives A_p(p = 1,2,…,m) under the set of criteria C_q(q = 1,2,…,n) as provided by an expert in terms of CIFNs $\alpha _{pq}=\left (r_{pq}e^{iw_{r_{pq}}}, k_{pq}e^{iw_{k_{pq}}}\right )$. We denote such matrix as D = (α_pq)_m×n which can be represented as

  $$\begin{array}{@{}rcl@{}} D = \begin{array}{cccc} C_{1} & C_{2} & {\ldots} & C_{n} \\ \begin{array}{cc} A_{1} \\ A_{2} \\ {\vdots} \\ A_{m} \end{array} \left( \begin{array}{ccccc} \alpha_{11} & \alpha_{12} & {\ldots} & \alpha_{1n} \\ \alpha_{21} & \alpha_{22} & {\ldots} & \alpha_{2n} \\ {\vdots} & {\vdots} & {\ddots} & {\vdots} \\ \alpha_{m1} & \alpha_{m2} & {\ldots} & \alpha_{mn} \end{array}\right) \end{array} \end{array} $$

• Step 3: Calculate the correlation coefficient between the alternatives A_p(p = 1,2,…,m) and the reference set B by using either K₁ or K₂ or by using weighted correlation coefficients K₃ or K₄, to compute the degree of the relationship among the alternatives.

• Step 4: Rank the alternatives based on the index values of correlation coefficients as obtained from argmax{K}. As larger the value of correlation coefficients, the better is the alternative A_p(p = 1,2,…,m).

6 Illustrative examples

In order to demonstrate the above mentioned approach based on correlation coefficients, we present two illustrative examples which are described as follows.

6.1 Example 1: Decision-making problem

An earthquake of 7.8 magnitude, also called as Gorkha earthquake, racked Nepal on 25 April 2015 at a depth of approximately 15km and lasted nearly fifty seconds and its epicenter was about 21 miles east southest of Lamjung and 48 miles northwest of Kathmandu and its focus was 9.3 miles underground and it destroyed thousands of houses across many districts of the country with entire villages flattened especially near the epicenter. An aftershock occurred on 12 May 2015 in Nepal which heightened the fears and tensions among the affected people. The two earthquakes together resulted in many damages, economic losses in almost 35 districts out of which 5 regions were severely affected namely: A₁ : Lalitpur, A₂ : Kathmandu, A₃ : Gorkha, A₄ : Bhaktapur and A₅ : Makwanpur. An earthquake relief camp decided to help victims of the earthquake in these five different regions A_p(p = 1,2,…,5) of the affected area with a different intensity of an earthquake. The panel decided that they will plan their budget by considering the four basic needs of victims, considered as criterion, namely C₁(Food), C₂(Shelter), C₃(Clothes) and C₄(Medical requirements) and decided to allocate the budget firstly to the most affected region so that by initial efforts only, a large strata of people get relief. The weight vector corresponding to these basic needs is taken as ξ = (0.30,0.25,0.15,0.30)^T. Clearly, according to the intensity of the earthquake, the basic needs of victims will be affected and changed. The target of this problem is to find the most affected region out of A₁,A₂,…,A₅ so as to allocate the proper budget and all the necessary facilities to them. To achieve this, we utilize the developed approach to rank the regions and the best one(s) can be found by implementing the steps of the proposed approach as follows:

• Step 1: Assume that an expert gives their preference in terms of maximum possible needs of all the five regions over the each basic need C_q(q = 1, 2, 3, 4) as a complex intuitionistic fuzzy set. The rating values of this set, denoted by B and called as a “reference set” in order to evaluate the given five regions are summarized as below:

  $$\begin{array}{@{}rcl@{}} B = \left\{ \begin{array}{ll} & \left( C_{1}, 0.7e^{i2\pi(0.5)}, 0.1e^{i2\pi(0.3)}\right), \left( C_{2},0.4e^{i2\pi(0.6)}, 0.5e^{i2\pi(0.2)}\right), \\ & \left( C_{3}, 0.5e^{i2\pi(0.5)}, 0.3e^{i2\pi(0.1)}\right), \left( C_{4},0.8e^{i2\pi(0.7)}, 0.2e^{i2\pi(0.1)}\right) \end{array} \right\} \end{array} $$

• Step 2: An expert evaluates each region A_p(p = 1, 2,…,5) individually and estimate the requirements under the set of criteria C_q(q = 1,2,3,4). Their rating values towards each region is expressed in terms of CIFNs whose values are summarized as follows.

  $$\begin{array}{@{}rcl@{}} D = \begin{array}{ccccc} C_{1} & C_{2} & C_{3} & C_{4} \\ \begin{array}{ll} A_{1} \\ A_{2} \\ A_{3} \\ A_{4} \\ A_{5} \end{array} \left( \begin{array}{ccccc} \left( 0.6e^{i2\pi(0.7)},0.1e^{i2\pi(0.2)}\right) & \left( 0.9e^{i2\pi(0.8)},0.1e^{i2\pi(0.1)}\right) & \left( 0.5e^{i2\pi(0.4)},0.3e^{i2\pi(0.4)}\right) & \left( 0.6e^{i2\pi(0.4)},0.2e^{i2\pi(0.1)}\right) \\ \left( 0.4e^{i2\pi(0.2)},0.3e^{i2\pi(0.1)}\right) & \left( 0.5e^{i2\pi(0.3)},0.1e^{i2\pi(0.1)}\right) & \left( 0.6e^{i2\pi(0.4)},0.2e^{i2\pi(0.3)}\right) & \left( 0.8e^{i2\pi(0.6)},0.1e^{i2\pi(0.2)}\right) \\ \left( 0.7e^{i2\pi(0.7)},0.1e^{i2\pi(0.2)}\right) & \left( 0.4e^{i2\pi(0.6)},0.3e^{i2\pi(0.1)}\right) & \left( 0.7e^{i2\pi(0.7)},0.1e^{i2\pi(0.1)}\right) & \left( 0.6e^{i2\pi(0.5)},0.3e^{i2\pi(0.4)}\right) \\ \left( 0.7e^{i2\pi(0.6)},0.3e^{i2\pi(0.3)}\right) & \left( 0.4e^{i2\pi(0.9)},0.2e^{i2\pi(0.1)}\right) & \left( 0.7e^{i2\pi(0.7)},0.2e^{i2\pi(0.3)}\right) & \left( 0.5e^{i2\pi(0.3)},0.3e^{i2\pi(0.6)}\right) \\ \left( 0.2e^{i2\pi(0.8)},0.5e^{i2\pi(0.1)}\right) & \left( 0.7e^{i2\pi(0.3)},0.3e^{i2\pi(0.3)}\right) & \left( 0.6e^{i2\pi(0.5)},0.1e^{i2\pi(0.3)}\right) & \left( 0.6e^{i2\pi(0.5)},0.3e^{i2\pi(0.4)}\right) \end{array}\right) \end{array} \end{array} $$

• Step 3a: By applying the correlation coefficient K₁ as given in Eq. (9) between the alternatives A_p(p = 1, 2, 3, 4, 5) and the reference set B, we can obtain their measurement values as K₁(A₁,B) = 0.8936, K₁(A₂,B) = 0.9068,K₁(A₃,B) = 0.9424, K₁(A₄,B) = 0.8830 and K₁(A₅,B) = 0.8450. On the other hand, by utilizing the correlation coefficient K₂ given in Eq. (10), their corresponding measurement values are K₂(A₁,B) = 0.8722, K₂(A₂,B) = 0.7522, K₂(A₃,B) = 0.9316,K₂(A₄,B) = 0.8228 and K₂(A₅,B) = 0.8251.

• Step 3b: If we assign the weight vector ξ = (0.30, 0.25, 0.15, 0.30)^T to the criteria, then by utilizing a weighted correlation coefficient K₃ as given in Eq. (12) to compute the measurement values between the alternatives A_p(p = 1,2,…, 5) and set B, we get K₃(A₁,B) = 0.8965, K₃(A₂,B) = 0.9087, K₃(A₃,B) = 0.9439, K₃(A₄,B) = 0.8747 and K₃(A₅,B) = 0.8351. Similarly, by using correlation coefficient K₄, we get their corresponding results are K₄(A₁,B) = 0.8920, K₄(A₂,B) = 0.7379, K₄(A₃,B) = 0.9299, K₄(A₄,B) = 0.8429 and K₄(A₅,B) = 0.8058.

• Step 4: From these computed measurement values, we conclude that the ranking order of the regions A_p(p = 1,2,…,5) is A₃ ≻ A₂ ≻ A₁ ≻ A₄ ≻ A₅, where “≻” stands for “preferred to” when K₁ correlation coefficient index has been used while A₃ ≻ A₁ ≻ A₅ ≻ A₄ ≻ A₂ when K₂ index has been used. Similarly, the ranking order of the region by considering the weight factor into the account is A₃ ≻ A₂ ≻ A₁ ≻ A₄ ≻ A₅ and A₃ ≻ A₁ ≻ A₄ ≻ A₅ ≻ A₂ respectively, when either the correlation coefficient K₃ or K₄ is utilized. It is observed from this analysis that the ranking order of the region is different for the different indices. However, the best alternative i.e., the most affected area remains same (A₃) while the worst changes according to the optimistic to pessimistic behavior. Hence, based on the behavior of the decision makers’ toward the ranking order related to optimistic and pessimistic behavior, they can choose the desired one accordingly. For instance, related to an optimistic decision maker’s behavior, they tend to prefer A₄ over A₅ due to ranking order A₄ ≻ A₅ while a pessimistic attitude will choose towards the region, they tend to choose A₅ over A₄ regions to allocate the funds due to A₅ ≻ A₄. Therefore, based on the attitude behavioral characteristic of the decision maker, they can use the best and worst region to allocate the budget.

6.2 Example 2: Medical diagnosis

Consider a decision-making problem with respect to the medical diagnosis which consists of a set of four diseases Q = {Q₁(Viral fever), Q₂(Malaria), Q₃(Typhoid), Q₄(Stomach Problem)} and a set of symptoms S = {s₁(Temperature), s₂(HeadAche), s₃(Stomach Pain), s₄(Cough)}. Assume that the patient P, with respect to all the symptoms, has been evaluated by an expert in order to find which diseases are patient affected by the most. So, the target of this decision-making problem is to diagnose the disease of the patient P among Q₁,Q₂,Q₃,Q₄. For it, we utilized the steps of the proposed approach to obtain the suitable ranking of the diagnoses and are summarized as follows:

• Step 1: The patient P is treated as a reference set and an expert gave their preferences with respect to all the symptoms in terms of CIFSs and is represented by the following set:

  $$\begin{array}{@{}rcl@{}} P = \left\{ \begin{array}{ll} & \left( s_{1},0.8e^{i2\pi(0.6)},0.1e^{i2\pi(0.2)}\right), \left( s_{2}, 0.9e^{i2\pi(0.7)}, 0.1e^{i2\pi(0.2)}\right), \\ & \left( s_{3}, 0.7e^{i2\pi(0.8)}, 0.2e^{i2\pi(0.1)}\right), \left( s_{4}, 0.6e^{i2\pi(0.5)},0.2e^{i2\pi(0.4)}\right) \end{array}\right\} \end{array} $$

• Step 2: The rating values of each diagnosis Q_p(p = 1,2,3,4) are expressed by a doctor (called as an expert) under the set of symptoms s_q(q = 1, 2, 3, 4) and are summarized as a complex intuitionistic fuzzy decision matrix D as

  $$\begin{array}{@{}rcl@{}} D = \begin{array}{ccccc} s_{1} & s_{2} & s_{3} & s_{4} \\ \begin{array}{ll}Q_{1} \\ Q_{2} \\ Q_{3} \\ Q_{4} \end{array} \left( \begin{array}{cccc} \left( 0.8e^{i2\pi(0.7)},0.1e^{i2\pi(0.2)}\right) & \left( 0.9e^{i2\pi(0.6)},0.1e^{i2\pi(0.2)}\right) & \left( 0.7e^{i2\pi(0.8)},0.2e^{i2\pi(0.1)}\right) & \left( 0.8e^{i2\pi(0.7)},0.2e^{i2\pi(0.1)}\right) \\ \left( 0.6e^{i2\pi(0.4)},0.1e^{i2\pi(0.5)}\right) & \left( 0.4e^{i2\pi(0.9)},0.5e^{i2\pi(0.1)}\right) & \left( 0.5e^{i2\pi(0.5)},0.3e^{i2\pi(0.3)}\right) & \left( 0.4e^{i2\pi(0.9)},0.5e^{i2\pi(0.1)}\right) \\ \left( 0.3e^{i2\pi(0.8)},0.3e^{i2\pi(0.1)}\right) & \left( 0.8e^{i2\pi(0.3)},0.1e^{i2\pi(0.6)}\right) & \left( 0.7e^{i2\pi(0.6)},0.2e^{i2\pi(0.2)}\right) & \left( 0.2e^{i2\pi(0.7)},0.8e^{i2\pi(0.2)}\right) \\ \left( 0.5e^{i2\pi(0.3)},0.4e^{i2\pi(0.6)}\right) & \left( 0.3e^{i2\pi(0.1)},0.6e^{i2\pi(0.3)}\right) & \left( 0.8e^{i2\pi(0.3)},0.1e^{i2\pi(0.5)}\right) & \left( 0.1e^{i2\pi(0.3)},0.6e^{i2\pi(0.5)}\right) \end{array}\right) \end{array} \end{array} $$

• Step 3a: By applying the correlation coefficient K₁ between the set Q_p(p = 1,2,3,4) and the patient P, we get their measurement values are K₁(Q₁,P) = 0.9800,K₁(Q₂,P) = 0.8582,K₁(Q₃,P) = 0.8446 and K₁(Q₄,P) = 0.7037. On the other hand, if we utilize the correlation coefficient K₂, then their corresponding measurement values are K₂(Q₁,P) = 0.9412,K₂(Q₂,P) = 0.8109,K₃(Q₃,P) = 0.8132 and K₄(Q₄,P) = 0.5923.

• Step 3b: If we assign the weightage to the set of symptoms s_q(q = 1,2,3,4) as ξ = (0.30, 0.20, 0.10, 0.40)^T then by applying the weighted correlation coefficient K₃ and K₄, we get their respectively measurement values are K₃(Q₁,P) = 0.9696,K₃(Q₂,P) = 0.8486,K₃(Q₃,P) = 0.8008,K₃(Q₄,P) = 0.6980 and K₄(Q₁,P) = 0.9015,K₄(Q₂,P) = 0.8433,K₄(Q₃,P) = 0.7935,K₄(Q₄,P) = 0.5969.

• Step 4: Based on the optimal values of the diseases, we conclude that its ranking order is Q₁ ≻ Q₂ ≻ Q₃ ≻ Q₄ when the K₁ correlation coefficient index has been used, while Q₁ ≻ Q₃ ≻ Q₂ ≻ Q₄ when K₂ index has been used. From this analysis, it is concluded that the patient P suffer from the Q₁ diseases. Further, from these ranking orders, we observe that when decision maker utilize the K₁ correlation coefficient by keeping his mind towards the optimistic view, then the second most diseases affected to the patient is Q₂. On the other hand, if the decision maker’s attitude towards the diseases is pessimistic in nature, they will tend towards Q₃ to be the second most diseases affected to the patient P. Similarly, we get the ranking order of the diseases affected to the patient corresponding to the utilization of K₃ and K₄ as Q₁ ≻ Q₂ ≻ Q₃ ≻ Q₄.

7 Comparative analysis

In this section, we compare the performance of the proposed measures with some of the existing approaches under the CIFS as well as IFS environment. The detailed analysis of the above considered examples is explained as below.

7.1 Comparative studies of Example 1 under CIFS environment

In order to compare the proposed approach results with some of the existing approaches [33, 34] under the CIFS environment, an analysis has been conducted for the considered data. The results corresponding to these approaches are summarized as follows:

• (i) If we utilize the distance measure, denoted by d₁, as proposed by [33] to the considered data, then corresponding to each region the measurement values from the reference set B are d₁(A₁,B) = 0.1817, d₁(A₂,B) = 0.1917, d₁(A₃,B) = 0.1400, d₁(A₄,B) = 0.2167 and d₁(A₅,B) = 0.2600. Since d₁(A₃,B) < d₁(A₁,B) < d₁(A₂,B) < d₁(A₄,B) < d₁(A₅,B) and hence the ranking order of the given regions is A₃ ≻ A₁ ≻ A₂ ≻ A₄ ≻ A₅. Thus, it has been computed that that A₃ is the most affected region.

• (ii) If we utilize the weighted Euclidean distance measure, denoted by d₂, as defined by [34] to the considered problem, then the measurement values of each region are computed as d₂(A₁,B) = 0.1871,d₂(A₂,B) = 0.1803,d₂(A₃,B) = 0.1374,d₂(A₄,B) = 0.2086 and d₂(A₅,B) = 0.2225. Since d₂(A₃,B) < d₂(A₂,B) < d₂(A₁,B) < d₂(A₄,B) < d₂(A₅,B) and hence ranking order of the region is A₃ ≻ A₂ ≻ A₁ ≻ A₄ ≻ A₅. Therefore, we conclude that A₃ is again the most affected region.

From these comparative studies, it is concluded that the best region obtained from the proposed measure coincides with the existing measures hence it validates the feasibility of the approach under the CIFS environment.

7.2 Comparative studies of Example 1 under IFS environment

In order to compare the proposed approach results with the results obtained under IFSs environment, we conducted an analysis by executing some of the existing approaches [23, 26, 28, 29] to the considered data. Since IFS is a special case of the CIFS, so we firstly convert the CIFS environment data into the IFS environment data by setting phase term corresponding to each criteria to 0 in every CIFN. Then, based on these existing approaches, we conduct the following analysis to compute the most affected region(s) under the IFS environment.

• (i) If we apply the correlation coefficient, denoted by ρ₁, as defined by [23] to the considered problem, then we get the measurement value of each region is ρ₁(A₁,B) = 0.8740,ρ₁(A₂,B) = 0.8874,ρ₁(A₃,B) = 0.9442,ρ₁(A₄,B) = 0.8822 and ρ₁(A₅,B) = 0.8262. Since ρ₁(A₃,B) > ρ₁(A₂,B) > ρ₁(A₄,B) > ρ₁(A₁,B) > ρ₁(A₅,B) and hence we conclude that A₃ is the most affected region.

• (ii) If we apply the correlation coefficient, denoted by ρ₂, as defined by [26] on the reduced data, then we get ρ₂(A₁,B) = 0.8808,ρ₂(A₂,B) = 0.9106,ρ₂(A₃,B) = 0.9551,ρ₂(A₄,B) = 0.9246 and ρ₂(A₅,B) = 0.8250. Since, ρ₂(A₃,B) > ρ₂(A₄,B) > ρ₂(A₂,B) > ρ₂(A₁,B) > ρ₂(A₅,B) and hence ranking order is A₃ ≻ A₄ ≻ A₂ ≻ A₁ ≻ A₅. Thus, we conclude that A₃ is the most affected region which coincides with the proposed one.

• (iii) If we utilize correlation coefficient (ρ₃) as presented by [28], then the corresponding indices to each region are computed as ρ₃(A₁,B) = − 0.4603,ρ₃(A₂,B) = 0.0000,ρ₃(A₃,B) = 0.5198,ρ₃(A₄,B) = 0.1143 and ρ₃(A₅,B) = − 0.6336. Since, ρ₃(A₃,B) > ρ₃(A₄,B) > ρ₃(A₂,B) > ρ₃(A₁,B) > ρ₃(A₅,B) and hence again the most affected region is A₃ and it coincides with the proposed measure results.

• (iv) By utilizing the similarity measure [29], denoted by ρ₄, on the considered data, we get their measurement values are ρ₄(A₁,B) = 0.8221,ρ₄(A₂,B) = 0.8527,ρ₄(A₃,B) = 0.8827,ρ₄(A₄,B) = 0.8492 and ρ₄(A₅,B) = 0.7562. From it, we get the ranking order of the regions is A₃ ≻ A₂ ≻ A₄ ≻ A₁ ≻ A₅ and hence we conclude that the most affected region is again A₃.

Thus, from this analysis, it has been clearly seen that the results computed by the existing approaches coincides with the proposed one and hence it supports the proposed results.

7.3 Comparative studies of Example 2 under CIFS environment

To compare the performance of the proposed approach with some of the existing approaches under the CIFSs environment, an analysis has been done and their results are summarized as follows.

• (i) By applying the approach of [33] using distance measures, denoted by d₁ to the considered data, we get the measurement values of each disease as d₁(Q₁,P) = 0.0967,d₁(Q₂,P) = 0.2717,d₁(Q₃,P) = 0.2867 and d₁(Q₄,P) = 0.3550. From these values, we observed that d₁(Q₁,P) < d₁(Q₂,P) < d₁(Q₃,P) < d₁(Q₄,P) and hence conclude that the patient P suffers from disease Q₁.

• (ii) By utilizing the distance measure (d₂) as proposed by [34] to the considered problem, then the measurement values for each disease are computed as d₂(Q₁,P) = 0.1194,d₂(Q₂,P) = 0.2291,d₂(Q₃,P) = 0.2669 and d₂(Q₄,P) = 0.3004. Since measurement value of Q₁ is minimum among all these and hence we conclude that patient P suffers from disease Q₁ which again coincides with the proposed measure results.

7.4 Comparative studies of Example 2 under IFS environment

In order to validate the efficiency of the proposed approach under the IFS environment, we conducted an analysis based on some of the existing correlation coefficients [23, 26, 28, 29]. The results corresponding to its are summarized as follows.

• (i) If we utilize correlation coefficient (ρ₁) as proposed by [23], then their measurement values for each diagnosis are summarized as ρ₁(Q₁,P) = 0.9856,ρ₁(Q₂,P) = 0.8461,ρ₁(Q₃,P) = 0.7959 and ρ₁(Q₄,P) = 0.7258. From it, we conclude that ρ₁(Q₁,P) > ρ₁(Q₂,P) > ρ₁(Q₃,P) > ρ₁(Q₄,P) and hence the patient P suffers from disease Q₁.

• (ii) By applying the correlation coefficient(ρ₂) as defined by [26] to the considered data then we get ρ₂(Q₁,P) = 0.9912, ρ₂(Q₂,P) = 0.8585, ρ₂(Q₃,P) = 0.7265 and ρ₂(Q₄,P) = 0.6645. Thus ρ₂(Q₁,P) > ρ₂(Q₂,P) > ρ₂(Q₃,P) > ρ₂(Q₄,P). From it, we conclude that patient P suffers from disease Q₁.

• (iii) If we apply the correlation coefficient(ρ₃) as proposed by [28] to the data, then the measurement values are obtained as ρ₃(Q₁,P) = 0.8485,ρ₃(Q₂,P) = 0.1907,ρ₃(Q₃,P) = 0.6608 and ρ₃(Q₄,P) = − 0.0690. Thus ρ₃(Q₁,P) > ρ₃(Q₃,P) > ρ₃(Q₂,P) > ρ₃(Q₄,P) which implies that the ranking order of Q_p(p = 1,2,3,4) is Q₁ ≻ Q₃ ≻ Q₂ ≻ Q₄. From it, we conclude that patient P suffers from disease Q₁.

• (iv) On applying the similarity measure (ρ₄) as defined by [29] on the considered information, then we get ρ₄(Q₁,P) = 0.9642,ρ₄(Q₂,P) = 0.7394,ρ₄(Q₃,P) = 0.7725 and ρ₄(Q₄,P) = 0.6538. As, ρ₄(Q₁,P) > ρ₄(Q₃,P) > ρ₄(Q₂,P) > ρ₄(Q₄,P) and hence we conclude that patient P suffers from disease Q₁.

Thus, from the above analysis, it has been seen that results computed by the proposed approach coincide with the existing approaches which validates the feasibility of the proposed approach.

7.5 Advantages of the proposed approach

From the existing studies and the proposed measures, we address the following merits of the proposed method to solve the decision-making problem under the CIFS environment.

• (i) A complex intuitionistic fuzzy set is a generalization of the existing studies such as complex fuzzy sets [4], intuitionistic fuzzy sets [2], fuzzy set [1] by considering much more information related to an object during the process and to handle the two-dimensional information in a single set. For instance, CIFS contains information (both the membership and non-membership degrees are complex valued) with amplitude and phase terms than the CFS (contains only complex valued membership degree), IFS (with a real-valued membership and non-membership degrees and only considered amplitude term), FS (with only crisp membership degrees with amplitude term only). Thus, the proposed correlation coefficients under CIFSs environment are more generalized than the existing correlation coefficients [19,20,21,22,23,24,25,26,27,28,29,30].

• (ii) It is revealed from the present study that the correlation coefficients under IFSs, FSs [19,20,21,22,23,24,25,26,27,28,29,30] are the special cases of the proposed measures. Thus, the proposed correlation coefficients can be equivalently utilized to solve the MCDM problem under these existing environment by setting phase term to be zero while the existing measures [19, 21,22,23, 26, 27] are unable to solve the problems under the environment considered in the present paper.

• (iii) The major advantages of the proposed decision-making approach are to consider the much more information to access the alternative to reduce the information loss. Further, the correlation coefficients based on the optimistic and pessimistic with or without weighting factor will help the decision maker to select the best alternative(s) more accurately. In other words, we can say that the proposed correlation coefficients will give the various choices to the decision makers based on their optimistic and pessimistic behavior towards the decision-making process.

8 Conclusion

In this article, an attempt has been made to present different kinds of the coefficients of correlation for decision-making process under the complex intuitionistic fuzzy set environment. Earlier, various existing coefficients of correlation have been defined under the IFSs environment where the range of their corresponding membership and non-membership degrees is the subset of the real numbers. But this condition has been relaxed in the present manuscript by considering the CIFSs where the ranges of the membership degrees are extended from the real numbers to the complex numbers with the unit disc. Therefore, considered environment models the information in a better way for time-periodic problems and has the ability to handle two-dimensional information in the single set. Keeping these points in view, under this environment, we present various coefficients of correlation and studied their properties in detail. Further, based on them, a decision-making approach is presented to find the best alternative in the CIFSs environment. Two numerical examples are taken for illustrating the developed approach and their results are compared with some of the existing correlation measures to show the validity of it. From the studies, we conclude that the proposed approach can be efficiently used in decision-making problems where two-dimensional information is clubbed in a single set. Also, it is observed that the existing correlation measures under the IFSs environment can be taken as a special case of the proposed measure. In the future, we will extend the study of CIFS to present some aggregation operators and the results of this paper to complex interval-valued IFS, type-2 fuzzy set, Pythagorean fuzzy set, multiplicative fuzzy set [37,38,39,40].