Novel Generalized Divergence Measure on Intuitionistic Fuzzy Sets and Its Application

Abstract

In the present paper, we introduce a new parametric fuzzy divergence measure on intuitionistic fuzzy sets. Some properties of the proposed measure are also being studied. In addition, the application of the intuitionistic fuzzy divergence measure in decision making and consequently choosing the best medicines and treatment for the patients has also been discussed. There are some diseases for which vaccine is not available. In that case, we have devised a method to choose the best treatment for the patients based on the results of clinical trials.

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

Measuring the information has been one of the important topics of keen interest in the field of information theory because of its numerous applications in pattern recognition, decision making, etc. Shannon [1] was the first to argue that measure of information is essentially a measure of uncertainty, and he called it as measure of entropy. Based on probability distribution $P=p_1,p_2,\dots ,p_n$ in an experiment, he found that the information given by this experiment is $HP=-{\displaystyle {\sum }_{i=1}^n{p}_i\ln p_i}$ [1]. This formula for entropy given by Shannon is known as Shannon’s entropy. Kullback and Leibler [2] developed the corresponding divergence measure from probability distribution $P=p_1,p_2,\dots ,p_n$ to $Q=q_1,q_2,\dots ,q_n$ as $DP;Q={\displaystyle {\sum }_{i=1}^n{p}_i\ln p_i/q_i}$. After that, various other generalized measures of entropy were developed taking Shannon’s entropy as the base. Later on, in 1965, Zadeh [3] introduced the concept of fuzzy sets. Corresponding to Kullback and Leibler’s divergence measure [2], Bhandari and Pal [4] defined measures of fuzzy entropy and measure of fuzzy divergence. Corresponding to Harvada and Charvat [5], Hooda [6] presented a fuzzy divergence measure. Parkash et al. [7] proposed a fuzzy divergence measure corresponding to Ferreri’s probabilistic measure [8] of divergence. Thereafter, Hooda and Jain [9] offered a generalized fuzzy divergence measure. In recent years, Tomar and Ohlan [10–13] have extended fuzzy measures of information, divergence, and their generalizations. After this, Atanassov [14] proposed the concept of intuitionistic fuzzy set (IFS) which has broadened the idea of fuzzy set with the addition of degree of non-membership. As IFS theory is more efficient in taking care of the uncertainties in information, it is much better than the crisp set theory.

For this, Szmidt and Kacprzyk [15] proposed the set of axioms for the entropy under the IFS environment. Corresponding to De Luca and Termini’s fuzzy entropy measure [16], Vlachos and Sergiadis [17] extended their measure in the IFS environment. Thereafter, many researchers like Junjun et al. [18], Xia and Xu [19], Wei and Ye [20], Zhang and Jiang [21], and Hung and Yang [22] have defined the divergence measures for IFS, and the findings are applied in variety of fields. However, Wei and Ye [20] pointed out the downside of Vlachos and Sergiadis’ measure [17] and modified the measure of Vlachos and Sergiadis. Later on, Verma and Sharma [23] improvised the divergence measure of Wei and Ye. Harish Garg et al. [24] proposed parametric version of intuitionistic fuzzy divergence measure given by Verma and Sharma [23]. In 2016, Maheshwari and Srivastava [25] pointed that that the measures given by Vlachos and Sergiadis [17], Zhang and Jiang [21], Junjun et al. [18] do not satisfy the basic requirement of non-negativity of the intuitionistic fuzzy divergence measure. Ohlan [26] extended the idea of Fan and Xie [27] and proposed an intuitionistic fuzzy exponential divergence measure.

In the present study, we suggest a new intuitionistic fuzzy divergence measure. Afterwards, we examine its properties and provide an illustration of how it can help in choosing the best medical treatment for the patients.

2. Preliminaries

We start by reviewing some well-known concepts and definitions related to fuzzy set theory and intuitionistic fuzzy set theory. Since we use the proposed notions in applications, we provide all concepts for the finite universe X.

Definition 1.

Fuzzy set [3]: a fuzzy set defined on a finite universe of discourse $X=x_1{x}_2,\dots ,x_n$is given as$$\begin{array}{}\text{(1)}& R=x,{\mu }_{R}x:x\in X,\end{array}$$where ${\mu }_R$: X ⟶ [0, 1] is the membership function of R. The membership value${\mu }_{R}x$ gives the degree of the belongingness of x ∈ X in R when ${\mu }_{R}x$ is valued in [0, 1].

Atanassov [14] introduced the concept of intuitionistic fuzzy set (IFS) as the generalization of the concept of fuzzy set.

Definition 2.

Intuitionistic fuzzy set (IFS) [14]: an intuitionistic fuzzy set (IFS) R on a finite universe of discourse $X=x_1{x}_2,\dots ,x_n$ is defined as$$\begin{array}{}\text{(2)}& R=x,{\mu }_{R}x,{\nu }_{R}x:x\in X,\end{array}$$where ${\mu }_R:X⟶\mathrm{0,1}$, ${\nu }_R:X⟶\mathrm{0,1}$, and $0\le {\mu }_R+v_R\le 1,\forall x\in X$.

${\mu }_{R}x$ denotes the degree of membership and ${\nu }_{R}x$ denotes the degree of non-membership of x in R. For each IFS R in X, we will call ${\pi }_{R}x=1-{\mu }_{R}x-{\nu }_{R}x$ = $1-{\mu }_{R}x-{\nu }_{R}x$, the hesitation degree of x in R. It is obvious that $0\le {\pi }_{R}x\le 1$ for each $x\in X$.

Atanassov [14] further introduced the set operations on IFSs as follows.

Let R and S ∈ IFS(X) be the family of all IFSs in the universe X, given by$$\begin{array}{c}\text{(3)}& R=x,{\mu }_{R}x,{\nu }_{R}x:x\in X,S=x,{\mu }_{S}x,{\nu }_{S}x:x\in X.\end{array}$$

(i) $R\subseteq S$ iff ${\mu }_{R}x\le {\mu }_{S}x$ and ${\nu }_{R}x\ge {\nu }_{S}x,\forall x\in X$.

  (ii) R = S iff $R\subseteq S$ and $S\subseteq R$.

  (iii)$R^c=x,{\nu }_{R}x,{\mu }_{R}x|,x\in X$.

  (iv)$R\cup S$ = {(x, max(${\mu }_{R}x$, ${\mu }_{S}x$), min(${\nu }_{R}x$, ${\nu }_{S}x$))|x ∈ X}.

  (v)$R\cap S$ = {(x, min(${\mu }_{R}x$, ${\mu }_{S}x$), max(${\nu }_{R}x$, ${\nu }_{S}x$))|x ∈ X}.

Definition 3.

Let R and S be two intuitionistic fuzzy sets in X. A mapping D: IFS(X)$\times \text{IFS}X⟶$R is a divergence measure for IFS if it satisfies the following axioms [17]:

(D1) D(R : S) $\ge 0$.

(D2) D(R : S) = 0 if and only if R = S.

(D3) D(R : S) = D($R^c$ : $S^c$).

In 1967, Harvda and Charvat [5] defined the measure of divergence of a probability distribution $P=p_1{p}_2,\dots ,p_n$ from another probability distribution $Q=q_1{q}_2,\dots ,q_n$ as$$\begin{array}{}\text{(4)}& D_{\alpha }P;Q=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{p}_i^{\alpha }{q}_i^{1-\alpha }-1},\hspace{1em}\alpha >0,\alpha \ne 1.\end{array}$$

It is known as the generalized divergence of degree α. Then, in 1959, Kullback and Leibler [2] proposed the following measure of symmetric divergence:$$\begin{array}{}\text{(5)}& J_{\alpha }P:Q=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{p}_i^{\alpha }{q}_i^{1-\alpha }+q_i^{\alpha }{p}_i^{1-\alpha }-2},\hspace{1em}\alpha >0,\alpha \ne 1,\end{array}$$which is also known as distance measure of degree α.

Corresponding to measures (4) and (5), Hooda [6] suggested the following measures of fuzzy divergence:$$\begin{array}{c}\text{(6)}& I_{\alpha }R:S=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{\mu }_R^{\alpha }{x}_i{\mu }_S^{1-\alpha }{x}_i+{1-{\mu }_R{x}_i}^{\alpha }{1-{\mu }_S{x}_i}^{1-\alpha }-1},J_{\alpha }R:S=I_{\alpha }R:S+I_{\alpha }S:R.\end{array}$$

Now, corresponding to Hooda [6], we define a new measure of intuitionistic fuzzy divergence.

3. New Parametric Divergence Measure on IFS

Let R and S be two IFSs defined on universal set $X=x_1{x}_2,\dots ,x_n$; then, a parametric intuitionistic fuzzy divergence measure based on parameter α is defined as follows:$$\begin{array}{c}\text{(7)}& D_{\alpha }^{\text{IFS}}R:S=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{1-\alpha }}+{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }-1,\end{array}$$where α > 0, α≠1.

Since $D_{\alpha }^{\text{IFS}}R:S\ne D_{\alpha }^{\text{IFS}}S:R$, the above proposed measure is not symmetric. In order to imbue the symmetry property, we define $J_{\alpha }^{\text{IFS}}R:S=D_{\alpha }^{\text{IFS}}R:S+D_{\alpha }^{\text{IFS}}S:R$.$$\begin{array}{c}\text{(8)}& J_{\alpha }^{\text{IFS}}R:S=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{1-\alpha }}+{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }-2,\end{array}$$where α > 0, α≠1.

3.1. Validity Proof of the Defined Measure of Divergence

Theorem 1.

$J_{\alpha }^{\text{IFS}}R:S$ is a valid measure of directed divergence on IFS, i.e., it satisfies$$\begin{array}{c}\text{(9)}& J_{\alpha }^{\text{IFS}}R:S\ge 0,J_{\alpha }^{\text{IFS}}R:S=0,\hspace{1em}\text{iff\hspace{0.17em}}R=S,\\ J_{\alpha }^{\text{IFS}}R:S=J_{\alpha }^{\text{IFS}}{R}^c:S^c.\end{array}$$

Proof.

In [6], it is proved that $I_{\alpha }R:S\ge 0$ and vanishes only when R = S.

As$$\begin{array}{}\text{(10)}& 0\le \frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}\le 1,\end{array}$$the result holds for $D_{\alpha }^{\text{IFS}}$ also. $\text{Therefore},D_{\alpha }^{\text{IFS}}R:S\ge 0$, and hence$J_{\alpha }^{\text{IFS}}R:S\ge 0$.

Similarly, $J_{\alpha }^{\text{IFS}}R:S=0$ iff R = S.$$\begin{array}{c}\text{(11)}& J_{\alpha }^{\text{IFS}}{R}^c:S^c=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }}+{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{1-\alpha }-2\\ =J_{\alpha }^{\text{IFS}}R:S.\end{array}$$

3.2. Properties of the Proposed Intuitionistic Fuzzy Divergence Measure

For the proofs of the properties, we need to separate X into X₁ and X₂ such that$$\begin{array}{c}\text{(12)}& X_1=x_i|x_i\in X,R{x}_i\subseteq S{x}_i,X_2=x_i|x_i\in X,R{x}_i\supseteq S{x}_i.\end{array}$$

Therefore, $\text{for\hspace{0.17em}all}{x}_i\in X_1$, we have$$\begin{array}{c}\text{(13)}& {\mu }_R{x}_i\le {\mu }_S{x}_i,{\nu }_R{x}_i\ge {\nu }_S{x}_i,\end{array}$$and $\text{for\hspace{0.17em}all}{x}_i\in X_2$, we have$$\begin{array}{c}\text{(14)}& {\mu }_R{x}_i\ge {\mu }_S{x}_i,{\nu }_R{x}_i\le {\nu }_S{x}_i.\end{array}$$

Theorem 2.

Let R, S, T be an IFS on universal set X = $X=x_1,x_2,\dots ,x_n$; then, the proposed measure given by (8) satisfies the following properties:

(a) ${\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}\cup \mathbf{S}:\mathbf{R}\cap \mathbf{S}={\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}:\mathbf{S}$.

(b) ${\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}:\mathbf{R}\cup \mathbf{S}={\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{S}:\mathbf{R}\cap \mathbf{S}$.

(c) ${\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}:\mathbf{R}\cap \mathbf{S}={\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{S}:\mathbf{R}\cup \mathbf{S}$.

(d) ${\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}:\mathbf{R}\cup \mathbf{S}+{\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}:\mathbf{R}\cap \mathbf{S}={\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}:\mathbf{S}$.

(e) ${\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}\cup \mathbf{S}:\mathbf{T}\le {\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}:\mathbf{T}+{\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{S}:\mathbf{T}$.

(f) ${\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}\cap \mathbf{S}:\mathbf{T}\le {\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}:\mathbf{T}+{\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{S}:\mathbf{T}$.

(g) ${\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}\cap \mathbf{S}:\mathbf{T}+{\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}\cup \mathbf{S}:\mathbf{T}={\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}:\mathbf{T}+{\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{S}:\mathbf{T}$.

(h) ${\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}\mathbf{R}:{\mathbf{S}}^{\mathbf{c}}={\mathbf{J}}_{\alpha }^{\mathbf{I}\mathbf{F}\mathbf{S}}{\mathbf{R}}^{\mathbf{c}}:\mathbf{S}$.

Proof.

(a) $$\begin{array}{c}\text{(15)}& J_{\alpha }^{\text{IFS}}R\cup S:R\cap S=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{\frac{{\mu }_{R\cup S}{x}_i+1-{\nu }_{R\cup S}{x}_i}{2}}^{\alpha }{\frac{{\mu }_{R\cap S}{x}_i+1-{\nu }_{R\cap S}{x}_i}{2}}^{1-\alpha }}+{\frac{{\nu }_{R\cup S}{x}_i+1-{\mu }_{R\cup S}{x}_i}{2}}^{\alpha }{\frac{{\nu }_{R\cap S}{x}_i+1-{\mu }_{R\cap S}{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_{R\cap S}{x}_i+1-{\nu }_{R\cap S}{x}_i}{2}}^{\alpha }{\frac{{\mu }_{R\cup S}{x}_i+1-{\nu }_{R\cup S}{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\nu }_{R\cap S}{x}_i+1-{\mu }_{R\cap S}{x}_i}{2}}^{\alpha }{\frac{{\nu }_{R\cup S}{x}_i+1-{\mu }_{R\cup S}{x}_i}{2}}^{1-\alpha },\\ =\frac{1}{\alpha -1}{\displaystyle \sum _{x_i\in X_1}{\frac{{\mu }_S{x}_i+1-v_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-v_R{x}_i}{2}}^{1-\alpha }}\\ +{\frac{v_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{v_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_R{x}_i+1-v_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-v_S{x}_i}{2}}^{1-\alpha }\\ +{\frac{v_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{v_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }-2\\ +{\displaystyle \sum _{x_i\in X_2}{\frac{{\mu }_R{x}_i+1-v_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-v_S{x}_i}{2}}^{1-\alpha }}\\ +{\frac{v_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{v_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_S{x}_i+1-v_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-v_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{v_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{v_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }-2\\ =J_{\alpha }^{\text{IFS}}R:S.\end{array}$$

(b) $$\begin{array}{c}\text{(16)}& J_{\alpha }^{\text{IFS}}R:R\cup S=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_{R\cup S}{x}_i+1-{\nu }_{R\cup S}{x}_i}{2}}^{1-\alpha }}+{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{{\nu }_{R\cup S}{x}_i+1-{\mu }_{R\cup S}{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_{R\cup S}{x}_i+1-{\nu }_{R\cup S}{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\nu }_{R\cup S}{x}_i+1-{\mu }_{R\cup S}{x}_i}{2}}^{\alpha }{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }-2,\\ \text{(17)}& =\frac{1}{\alpha -1}{\displaystyle \sum }_{x_i\in X_1}{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{1-\alpha }+{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }-2+{\displaystyle \sum _{x_i\in X_2}0},\\ \text{(18)}& J_{\alpha }^{\text{IFS}}S:R\cap S=\frac{1}{\alpha -1}{\displaystyle \sum }_{x_i\in X_1}{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{1-\alpha }+{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }-2+{\displaystyle \sum _{x_i\in X_2}0}.\end{array}$$

  From (17) and (18),$$\begin{array}{}\text{(19)}& J_{\alpha }^{\text{IFS}}R:R\cup S=J_{\alpha }^{\text{IFS}}S:R\cap S.\end{array}$$

(c) $$\begin{array}{c}\text{(20)}& J_{\alpha }^{\text{IFS}}R:R\cap S=\frac{1}{\alpha -1}{\displaystyle \sum _{x_i\in X_1}0}+{\displaystyle \sum _{x_i\in X_2}{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{1-\alpha }}+{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }-2\\ =J_{\alpha }^{IFS}S:R\cup S.\end{array}$$

(d) Adding (17) and (20), we get$$\begin{array}{}\text{(21)}& J_{\alpha }^{\text{IFS}}R:R\cup S+J_{\alpha }^{\text{IFS}}R:R\cap S=J_{\alpha }^{\text{IFS}}R:S.\end{array}$$

(e) It suffices to show that$$\begin{array}{c}\text{(22)}& J_{\alpha }^{\text{IFS}}R:T+J_{\alpha }^{\text{IFS}}S:T-J_{\alpha }^{\text{IFS}}R\cup S:T\ge 0,J_{\alpha }^{\text{IFS}}R:T=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_T{x}_i+1-{\nu }_T{x}_i}{2}}^{1-\alpha }}\\ +{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{{\nu }_T{x}_i+1-{\mu }_T{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_T{x}_i+1-{\nu }_T{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\nu }_T{x}_i+1-{\mu }_T{x}_i}{2}}^{\alpha }{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }-2,\\ J_{\alpha }^{\text{IFS}}S:T=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_T{x}_i+1-{\nu }_T{x}_i}{2}}^{1-\alpha }}\\ +{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{{\nu }_T{x}_i+1-{\mu }_T{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_T{x}_i+1-{\nu }_T{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\nu }_T{x}_i+1-{\mu }_T{x}_i}{2}}^{\alpha }{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }-2,\\ J_{\alpha }^{\text{IFS}}R:T+J_{\alpha }^{\text{IFS}}S:T-J_{\alpha }^{\text{IFS}}R\cup S:T\\ =\frac{1}{\alpha -1}{\displaystyle \sum _{x_i\in X_2}{\frac{{\mu }_S{x}_i+1-v_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_T{x}_i+1-v_T{x}_i}{2}}^{1-\alpha }}\\ +{\frac{v_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{v_T{x}_i+1-{\mu }_T{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_T{x}_i+1-v_T{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-v_S{x}_i}{2}}^{1-\alpha }\\ {\frac{v_T{x}_i+1-{\mu }_T{x}_i}{2}}^{\alpha }{\frac{v_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }-2\\ +{\displaystyle \sum _{x_i\in }{\frac{{\mu }_R{x}_i+1-v_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_T{x}_i+1-v_T{x}_i}{2}}^{1-\alpha }}\\ +{\frac{v_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{v_T{x}_i+1-{\mu }_T{x}_i}{2}}^{1-\alpha }\\ ={\frac{{\mu }_T{x}_i+1-v_T{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-v_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{v_T{x}_i+1-{\mu }_T{x}_i}{2}}^{\alpha }{\frac{v_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }-2\\ \ge 0.\end{array}$$

(f) The proof of (f) is similar to (e).

(g) $$\begin{array}{c}\text{(23)}& J_{\alpha }^{\text{IFS}}R\cap S:T=\frac{1}{\alpha -1}{\displaystyle \sum _{x_i\in X_1}{\frac{{\mu }_R{x}_i+1-v_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_T{x}_i+1-v_T{x}_i}{2}}^{1-\alpha }}\\ +{\frac{v_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{v_T{x}_{ii}+1-{\mu }_T{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_T{x}_i+1-v_T{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-v_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{v_T{x}_i+1-{\mu }_T{x}_i}{2}}^{\alpha }{\frac{v_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }-2\\ +{\displaystyle \sum _{x_i\in X_2}{\frac{{\mu }_S{x}_i+1-v_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_T{x}_i+1-v_T{x}_i}{2}}^{1-\alpha }}\\ +{\frac{v_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{v_T{x}_i+1-{\mu }_T{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_T{x}_i+1-v_T{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-v_S{x}_i}{2}}^{1-\alpha }\\ +{\frac{v_T{x}_i+1-{\mu }_T{x}_i}{2}}^{\alpha }{\frac{v_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }-2.\end{array}$$

  Add$$\begin{array}{}\text{(24)}& J_{\alpha }^{\text{IFS}}R\cap S:T+J_{\alpha }^{\text{IFS}}R\cup S:T=J_{\alpha }^{\text{IFS}}R:T+J_{\alpha }^{\text{IFS}}S:T.\end{array}$$

(h) $$\begin{array}{c}\text{(25)}& J_{\alpha }^{\text{IFS}}R:S^c=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{\alpha }{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }}+{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{\alpha }{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }-2,\\ J_{\alpha }^{\text{IFS}}{R}^c:S=\frac{1}{\alpha -1}{\displaystyle \sum _{i=1}^n{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{\alpha }{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{1-\alpha }}\\ +{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{\alpha }{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\mu }_S{x}_i+1-{\nu }_S{x}_i}{2}}^{\alpha }{\frac{{\nu }_R{x}_i+1-{\mu }_R{x}_i}{2}}^{1-\alpha }\\ +{\frac{{\nu }_S{x}_i+1-{\mu }_S{x}_i}{2}}^{\alpha }{\frac{{\mu }_R{x}_i+1-{\nu }_R{x}_i}{2}}^{1-\alpha }-2\\ \therefore J_{\alpha }^{\text{IFS}}R:S^c=J_{\alpha }^{\text{IFS}}{R}^c:S.\end{array}$$

4. Drawbacks of Other Measures of Divergence for IFS

Vlachos and Sergiadis [17] defined the following measure of divergence:$$\begin{array}{}\text{(26)}& D_{1}R:S={\displaystyle \sum _{i=1}^n{\mu }_R{x}_i\ln \frac{{\mu }_R{x}_i}{1/2{\mu }_R{x}_i+{\mu }_S{x}_i}+{\nu }_R{x}_i\ln \frac{{\nu }_R{x}_i}{1/2{\nu }_R{x}_i+{\nu }_S{x}_i}}.\end{array}$$

Junjun et al. [18] defined$$\begin{array}{}\text{(27)}& D_{2}R:S={\displaystyle \sum _{i=1}^n{\pi }_R{x}_i\ln \frac{{\pi }_R{x}_i}{1/2{\pi }_R{x}_i+{\pi }_S{x}_i}+{\Delta }_R{x}_i\ln \frac{{\Delta }_R{x}_i}{1/2{\Delta }_R{x}_i+{\Delta }_S{x}_i}},\end{array}$$where ${\mathrm{\Delta }}_R{x}_i={\mu }_R{x}_i-{\nu }_R{x}_i$.

Example 1.

Consider the IFS given by$$\begin{array}{c}\text{(28)}& R=x_1,0.44,0.385,x_2,0.43,0.39,x_3,0.42,0.38,S=x_1,0.34,0.48,x_2,0.37,0.46,x_3,0.38,0.45.\end{array}$$

Corresponding to R and S, we get$$\begin{array}{c}\text{(29)}& D_{1}R:S=-0.00712,D_{2}R:S=-\mathrm{0.04547.}\end{array}$$

Therefore, the measures given by (26) and (27) do not satisfy the axiom of non-negativity.

5. Application

A lot of uncertainty and fuzziness is associated with most of the decisions in medical science. There are various diseases like migraine, depression, and many viral diseases for which vaccine is not available or 100 percent cure is not available. One of the current viral diseases which is known to affect more than 50,00,000 lives in the whole world is COVID-19. On 11^th March 2020, the WHO officially declared the outbreak of COVID-19 as pandemic. Still, no vaccine or 100 percent effective medicine is invented for its cure. In such cases, doctors have to choose the medicines already available in the market. For example, in the case of COVID-19, medicines already available are remdesivir, favipiravir, and hydroxychloroquine. Now, the following question arises: which medicine or treatment is most effective among the available ones? This kind of decision-making problem can be solved by IFS theory. IFS theory makes it possible to define the medical information in terms of intuitionistic fuzzy sets and thereby apply the decision-making process discussed below to choose the best treatment or medicine.

Now, we describe the decision-making process in intuitionistic fuzzy sets.

Let M = {M₁, M₂, M₃.....M_p} be the set of p alternatives to be examined under the set of q criterion set N = {N₁, N₂, N₃......N_q}. Then, we take following steps to choose the best alternative.

Step 1.

Constructing the decision-making matrix: let ${\mu }_{ij}$ denote the degree of the alternative $M_i$ satisfying the criteria $N_j$ and ${\nu }_{ij}$ denote the degree of the alternative $M_i$ not satisfying the criteria $N_j$. Then, the decision-making matrix is a $p\times q$ matrix whose entries $x_{ij}$ are$$\begin{array}{c}\text{(30)}& {\mu }_{ij},{\nu }_{ij},\hspace{1em}i=\mathrm{1,2},\dots ,p,j=\mathrm{1,2},\dots ,q,\hspace{1em}\text{where\hspace{0.17em}}{\mu }_{ij}{\nu }_{ij}\in \mathrm{0,1}\text{\hspace{0.17em}and\hspace{0.17em}}{\mu }_{ij}+{\nu }_{ij}\le 1.\end{array}$$

Step 2.

Computing the ideal alternative: find the ideal alternative $M^{\ast }$ as$$\begin{array}{}\text{(31)}& M^{\ast }={\mu }_1^{\ast },{\nu }_1^{\ast },{\mu }_2^{\ast },{\nu }_2^{\ast },\dots ,{\mu }_q^{\ast },{\nu }_q^{\ast },\end{array}$$where ${\mu }_j^{\ast }={\max }_{1\le i\le p}{\mu }_{ij}$ and ${\upsilon }_j^{\ast }={\min }_{1\le i\le q}{\nu }_{ij}$.

Step 3.

Calculating the value of proposed divergence measure: find $J_{\alpha }^{IFS}{M}_i:M^{\ast }$ using the formula$$\begin{array}{c}\text{(32)}& J_{\alpha }^{IFS}{M}_i:M^{\ast }=\frac{1}{\alpha -1}{\displaystyle \sum _{j=1}^q{\frac{{\mu }_{ij}{x}_{ij}+1-{\nu }_{ij}{x}_{ij}}{2}}^{\alpha }{\frac{{\mu }_{j^{\ast }}{x}_{ij}+1-{\nu }_{j^{\ast }}{x}_{ij}}{2}}^{1-\alpha }}+{\frac{{\nu }_{ij}{x}_{ij}+1-{\mu }_{ij}{x}_{ij}}{2}}^{\alpha }{\frac{{\nu }_{j^{\ast }}{x}_{ij}+1-{\mu }_{j^{\ast }}{x}_{ij}}{2}}^{1-\alpha }\\ +{\frac{{\mu }_{j^{\ast }}{x}_{ij}+1-{\nu }_{j^{\ast }}{x}_{ij}}{2}}^{\alpha }{\frac{{\mu }_{ij}{x}_{ij}+1-{\nu }_{ij}{x}_{ij}}{2}}^{1-\alpha }\\ +{\frac{{\nu }_{j^{\ast }}{x}_{ij}+1-{\mu }_{j^{\ast }}{x}_{ij}}{2}}^{\alpha }{\frac{{\nu }_{ij}{x}_{ij}+1-{\mu }_{ij}{x}_{ij}}{2}}^{1-\alpha }-2,\hspace{1em}\forall i=\mathrm{1,2},\dots ,p.\end{array}$$

Step 4.

Ranking the alternatives: give ranking to all the alternatives. The alternative whose $J_{\alpha }^{IFS}{M}_i:M^{\ast }$ is minimum will be considered as the best alternative with rank 1.

We demonstrate the application of proposed divergence method in choosing the best medical treatment for the patients.

Suppose a person is suffering from anxiety. A doctor tried different treatments $T_1$,$T_2,T_3,T_4{T}_1,T_2,T_3,T_4$ on 4 different subjects $A_1$,$A_2,A_3,A_4$ and noted the results of clinical trials (see Table 1).

Based on this normalized intuitionistic fuzzy decision matrix, we find the ideal alternative$$\begin{array}{}\text{(33)}& M^{\ast }=0.9,0.1,0.8,0.1,0.7,0.2,0.8,0.2.\end{array}$$

Now, we calculate the value of divergence of each treatment $T_i$ from the ideal alternative $M^{\ast }$ for different values of α.

According to Table 2, $T_4$ is the best treatment for the patients of anxiety. The order of preference of treatments is as follows:$$\begin{array}{}\text{(34)}& T_4\succ T_2\succ T_1\succ T_3.\end{array}$$

Table 1

Normalized intuitionistic fuzzy decision matrix.

	$A_1$	$A_2$	$A_3$	$A_4$
$T_1$	<0.7, 0.2>	<0.8, 0.1>	<0.5, 0.3>	<0.6, 0.3>
$T_2$	<0.9, 0.1>	<0.8, 0.2>	<0.5, 0.3>	<0.8, 0.2>
$T_3$	<0.6, 0.4>	<0.5, 0.3>	<0.6, 0.3>	<0.5, 0.4>
$T_4$	<0.8, 0.1>	<0.7, 0.2>	<0.7, 0.2>	<0.8, 0.2>

Table 2

Values of ${\mathbf{J}}_{\alpha }^{\mathbf{IFS}}{\mathbf{T}}_{\mathbf{i}}:{\mathbf{M}}^{\ast }$ for different values of α.

	α = 0.5	α = 2	α = 3	α = 4	α = 5	Rank
$J_{\alpha }^{\text{IFS}}{T}_1:M^{\ast }$	0.1903	0.8233	1.4254	2.3755	6.606	3rd
$J_{\alpha }^{\text{IFS}}{T}_2:M^{\ast }$	0.0605	0.2490	0.3928	0.5651	0.7817	2nd
$J_{\alpha }^{\text{IFS}}{T}_3:M^{\ast }$	0.5948	2.8656	6.0696	13.8161	35.305	4th
$J_{\alpha }^{\text{IFS}}{T}_4:M^{\ast }$	0.0432	0.1791	0.2866	0.4209	0.5980	1st

6. Conclusion

In this paper, we have proposed a new parametric intuitionistic fuzzy divergence measure for IFS with its proof of validity. The proposed measure is found to satisfy various properties and does not assume any negative value as in case of many existing divergence measures. The proposed divergence measure also has application in decision making and thereby deciding the best medical treatment for the patients. Furthermore, the parameter provides flexibility in criteria for decision making.

References

[1] C. E. Shannon, "A mathematical theory of communication," Bell System Technical Journal, vol. 27 no. 3, pp. 379-423, DOI: 10.1002/j.1538-7305.1948.tb01338.x, 1948.

[2] S. Kullback, R. A. Leibler, "On information and sufficiency," The Annals of Mathematical Statistics, vol. 22 no. 1, pp. 79-86, DOI: 10.1214/aoms/1177729694, 1951.

[3] L. A. Zadeh, "Fuzzy sets," Information and Control, vol. 8 no. 3, pp. 338-353, DOI: 10.1016/s0019-9958(65)90241-x, 1965.

[4] D. Bhandari, N. R. Pal, "Some new information measures for fuzzy sets," Information Science, vol. 67, pp. 204-228, DOI: 10.1016/0020-0255(93)90073-u, 1993.

[5] J. H. Harvda, F. Charvat, "Quantification method of classification processes - concept of structural α - entropy," Kybernetika, vol. 3, pp. 30-35, 1967.

[6] D. S. Hooda, "On generalized measures of fuzzy entropy," Mathematica Slovaca, vol. 54, pp. 315-325, 2004.

[7] O. Parkash, P. K. Sharma, S. Kumar, "Two new measures of fuzzy divergence and their properties," Sultan Qaboos University Journal for Science (SQUJS), vol. 11, pp. 69-77, DOI: 10.24200/squjs.vol11iss0pp69-77, 2006.

[8] C. Ferreri, "Hyperentropy and related heterogeneity divergence and information measures," Statistica, vol. 40 no. 2, pp. 155-168, 1980.

[9] D. S. Hooda, D. Jain, "The generalized fuzzy measures of directed divergence, total ambiguity and information improvement," Investigations in Mathematical Sciences, vol. 2, pp. 239-260, 2012.

[10] V. P Tomar, A. Ohlan, "Two new parametric generalized R − norm fuzzy information measures," International Journal of Computer Applications, vol. 93 no. 13, pp. 22-27, 2014.

[11] V. P Tomar, A. Ohlan, "Sequence of fuzzy divergence measures and inequalities," AMO-Advanced Modeling and Optimization, vol. 16 no. 2, pp. 439-452, 2014.

[12] V. P Tomar, A. Ohlan, "Sequence of inequalities among fuzzy mean difference divergence measures and their applications," SpringerPlus, vol. 3 no. 623, pp. 623-720, DOI: 10.1186/2193-1801-3-623, 2014.

[13] V. P Tomar V, "New parametric generalized exponential fuzzy divergence measure," Journal of Uncertainty Analysis and Applications, vol. 2 no. 1,DOI: 10.1186/s40467-014-0024-2, 2014.

[14] K. T. Atanassov, "Intuitionistic fuzzy sets," Fuzzy Sets and Systems, vol. 20 no. 1, pp. 87-96, DOI: 10.1016/s0165-0114(86)80034-3, 1986.

[15] E. Szmidt, J. Kacprzyk, "Entropy for intuitionistic fuzzy sets," Fuzzy Sets and Systems, vol. 118 no. 3, pp. 467-477, DOI: 10.1016/s0165-0114(98)00402-3, 2001.

[16] A. De Luca, S. Termini, "A definition of a nonprobabilistic entropy in the setting of fuzzy sets theory," Information and Control, vol. 20 no. 4, pp. 301-312, DOI: 10.1016/s0019-9958(72)90199-4, 1972.

[17] I. K. Vlachos, G. D. Sergiadis, "Intuitionistic fuzzy information-applications to pattern recognition," Pattern Recognition Letters, vol. 28 no. 2, pp. 197-206, DOI: 10.1016/j.patrec.2006.07.004, 2007.

[18] M. Junjun, Y. Dengbao, W. Cuicui, "A novel cross-entropy and entropy measures of IFSs and their applications," Knowledge-Based Systems, vol. 48, pp. 37-45, 2013.

[19] M. Xia, Z. Xu, "Entropy/cross entropy-based group decision making under intuitionistic fuzzy environment," Information Fusion, vol. 13 no. 1, pp. 31-47, DOI: 10.1016/j.inffus.2010.12.001, 2012.

[20] P. Wei, J. Ye, "Improved intuitionistic fuzzy cross-entropy and its application to pattern recognition," Proceedings of the International Conference on Intelligent Systems and Knowledge Engineering, pp. 114-116, .

[21] Q.-S. Zhang, S.-Y. Jiang, "A note on information entropy measures for vague sets and its applications," Information Sciences, vol. 178 no. 21, pp. 4184-4191, DOI: 10.1016/j.ins.2008.07.003, 2008.

[22] W.-L. Hung, M.-S. Yang, "On the J-divergence of intuitionistic fuzzy sets with its application to pattern recognition," Information Sciences, vol. 178 no. 6, pp. 1641-1650, DOI: 10.1016/j.ins.2007.11.006, 2008.

[23] R. K. Verma, B. D. Sharma, "On generalized intuitionistic fuzzy divergence (relative information) and their properties," Journal of Uncertain Systems, vol. 6, pp. 308-320, 2012.

[24] H. Harish Garg, N. Nikunj Aggarwal, A. Tripathi, "A novel generalized parametric directed divergence measure of intuitionistic fuzzy sets with its application," Annals of Fuzzy Mathematics and Informatics, vol. 13 no. 6, pp. 703-727, DOI: 10.30948/afmi.2017.13.6.703, 2017.

[25] S. Maheshwari, A. Srivastava, "Study on divergence measures for intuitionistic fuzzy sets and its application in medical diagnosis," Journal of Applied Analysis & Computation, vol. 6 no. 3, pp. 772-789, DOI: 10.11948/2016050, 2016.

[26] A. Ohlan, "Intuitionistic fuzzy exponential divergence: application in multi-attribute decision making," Journal of Intelligent & Fuzzy Systems, vol. 30 no. 3, pp. 1519-1530, DOI: 10.3233/ifs-151859, 2016.

[27] J. Fan, W. Xie, "Distance measure and induced fuzzy entropy," Fuzzy Sets and Systems, vol. 104 no. 2, pp. 305-314, DOI: 10.1016/s0165-0114(99)80011-6, 1999.