# Novel Generalized Divergence Measure on Intuitionistic Fuzzy Sets and Its Application

Authors: Priya Arora; V. P. Tomar
Journal: Advances in Fuzzy Systems
 (2021)

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## 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.


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## Body

## 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=p1,p2,…,pn in an experiment, he found that the information given by this experiment is HP=−∑i=1npilnpi [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=p1,p2,…,pn to Q=q1,q2,…,qn as DP;Q=∑i=1npilnpi/qi. 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 universeX.Definition 1.
Fuzzy set [3]: a fuzzy set defined on a finite universe of discourse X=x1x2,…,xnis given as(1)R=x,µRx:x∈X,where µR: X ⟶ [0, 1] is the membership function of R. The membership valueµRx gives the degree of the belongingness of x ∈ X in R when µRx 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=x1x2,…,xn is defined as(2)R=x,µRx,νRx:x∈X,where µR:X⟶0,1, νR:X⟶0,1, and 0≤µR+vR≤1,∀x∈X.
µRx denotes the degree of membership and νRx denotes the degree of non-membership of x in R. For each IFS R in X, we will call πRx=1−µRx−νRx = 1−µRx−νRx, the hesitation degree of x in R. It is obvious that 0≤πRx≤1 for each x∈X.
Atanassov [14] further introduced the set operations on IFSs as follows.
LetR and S ∈ IFS(X) be the family of all IFSs in the universe X, given by(3)R=x,µRx,νRx:x∈X,S=x,µSx,νSx:x∈X.(i)
R⊆S iff µRx≤µSx and νRx≥νSx,∀x∈X.(ii)R = S iff R⊆S and S⊆R.(iii)Rc=x,νRx,µRx|,x∈X.(iv)R∪S = {(x, max(µRx, µSx), min(νRx, νSx))|x ∈ X}.(v)R∩S = {(x, min(µRx, µSx), max(νRx, νSx))|x ∈ X}.Definition 3.
LetR and S be two intuitionistic fuzzy sets in X. A mapping D: IFS(X)×IFSX⟶R is a divergence measure for IFS if it satisfies the following axioms [17]:(D1)
D(R : S) ≥0.(D2)
D(R : S) = 0 if and only if R = S.(D3)
D(R : S) = D(Rc : Sc).
In 1967, Harvda and Charvat [5] defined the measure of divergence of a probability distribution P=p1p2,…,pn from another probability distribution Q=q1q2,…,qn as(4)DαP;Q=1α−1∑i=1npiαqi1−α−1,α>0,α≠1.
It is known as the generalized divergence of degreeα. Then, in 1959, Kullback and Leibler [2] proposed the following measure of symmetric divergence:(5)JαP:Q=1α−1∑i=1npiαqi1−α+qiαpi1−α−2,α>0,α≠1,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:(6)IαR:S=1α−1∑i=1nμRαxiμS1−αxi+1−µRxiα1−µSxi1−α−1,JαR:S=IαR:S+IαS:R.
Now, corresponding to Hooda [6], we define a new measure of intuitionistic fuzzy divergence.


## 3. New Parametric Divergence Measure on IFS

LetR and S be two IFSs defined on universal set X=x1x2,…,xn; then, a parametric intuitionistic fuzzy divergence measure based on parameter α is defined as follows:(7)DαIFSR:S=1α−1∑i=1nμRxi+1−νRxi2αμSxi+1−νSxi21−α+νRxi+1−μRxi2ανSxi+1−μSxi21−α−1,where α > 0, α≠1.SinceDαIFSR:S≠DαIFSS:R, the above proposed measure is not symmetric. In order to imbue the symmetry property, we define JαIFSR:S=DαIFSR:S+DαIFSS:R.(8)JαIFSR:S=1α−1∑i=1nμRxi+1−νRxi2αμSxi+1−νSxi21−α+νRxi+1−μRxi2ανSxi+1−μSxi21−α+μSxi+1−νSxi2αμRxi+1−νRxi21−α+νSxi+1−μSxi2ανRxi+1−μRxi21−α−2,where α > 0, α≠1.


### 3.1. Validity Proof of the Defined Measure of Divergence

Theorem 1.
JαIFSR:S is a valid measure of directed divergence on IFS, i.e., it satisfies(9)JαIFSR:S≥0,JαIFSR:S=0,iffR=S,JαIFSR:S=JαIFSRc:Sc.Proof.
In [6], it is proved that IαR:S≥0 and vanishes only when R = S.
As(10)0≤μRxi+1−νRxi2≤1,the result holds for DαIFS also. Therefore,DαIFSR:S≥0, and henceJαIFSR:S≥0.
Similarly,JαIFSR:S=0 iff R = S.(11)JαIFSRc:Sc=1α−1∑i=1nνRxi+1−μRxi2ανSxi+1−μSxi21−α+μRxi+1−νRxi2αμSxi+1−νSxi21−α+νSxi+1−μSxi2ανRxi+1−μRxi21−α+μSxi+1−νSxi2αμRxi+1−νRxi21−α−2=JαIFSR:S.


### 3.2. Properties of the Proposed Intuitionistic Fuzzy Divergence Measure

For the proofs of the properties, we need to separateX into X1 and X2 such that(12)X1=xi|xi∈X,Rxi⊆Sxi,X2=xi|xi∈X,Rxi⊇Sxi.Therefore,for allxi∈X1, we have(13)µRxi≤µSxi,νRxi≥νSxi,and for allxi∈X2, we have(14)µRxi≥µSxi,νRxi≤νSxi.Theorem 2.
LetR, S, T be an IFS on universal set X = X=x1,x2,…,xn; then, the proposed measure given by (8) satisfies the following properties:(a)
JαIFSR∪S:R∩S=JαIFSR:S.(b)
JαIFSR:R∪S=JαIFSS:R∩S.(c)
JαIFSR:R∩S=JαIFSS:R∪S.(d)
JαIFSR:R∪S+JαIFSR:R∩S=JαIFSR:S.(e)
JαIFSR∪S:T≤JαIFSR:T+JαIFSS:T.(f)
JαIFSR∩S:T≤JαIFSR:T+JαIFSS:T.(g)
JαIFSR∩S:T+JαIFSR∪S:T=JαIFSR:T+JαIFSS:T.(h)
JαIFSR:Sc=JαIFSRc:S.Proof.
(a)
(15)JαIFSR∪S:R∩S=1α−1∑i=1nμR∪Sxi+1−νR∪Sxi2αμR∩Sxi+1−νR∩Sxi21−α+νR∪Sxi+1−μR∪Sxi2ανR∩Sxi+1−μR∩Sxi21−α+μR∩Sxi+1−νR∩Sxi2αμR∪Sxi+1−νR∪Sxi21−α+νR∩Sxi+1−μR∩Sxi2ανR∪Sxi+1−μR∪Sxi21−α,=1α−1∑xi∈X1μSxi+1−vSxi2αμRxi+1−vRxi21−α+vSxi+1−μSxi2αvRxi+1−μRxi21−α+μRxi+1−vRxi2αμSxi+1−vSxi21−α+vRxi+1−μRxi2αvSxi+1−μSxi21−α−2+∑xi∈X2μRxi+1−vRxi2αμSxi+1−vSxi21−α+vRxi+1−μRxi2αvSxi+1−μSxi21−α+μSxi+1−vSxi2αμRxi+1−vRxi21−α+vSxi+1−μSxi2αvRxi+1−μRxi21−α−2=JαIFSR:S.(b)
(16)JαIFSR:R∪S=1α−1∑i=1nμRxi+1−νRxi2αμR∪Sxi+1−νR∪Sxi21−α+νRxi+1−μRxi2ανR∪Sxi+1−μR∪Sxi21−α+μR∪Sxi+1−νR∪Sxi2αμRxi+1−νRxi21−α+νR∪Sxi+1−μR∪Sxi2ανRxi+1−μRxi21−α−2,(17)=1α−1∑xi∈X1μRxi+1−νRxi2αμSxi+1−νSxi21−α+νRxi+1−μRxi2ανSxi+1−μSxi21−α+μSxi+1−νSxi2αμRxi+1−νRxi21−α+νSxi+1−μSxi2ανRxi+1−μRxi21−α−2+∑xi∈X20,(18)JαIFSS:R∩S=1α−1∑xi∈X1μSxi+1−νSxi2αμRxi+1−νRxi21−α+νSxi+1−μSxi2ανRxi+1−μRxi21−α+μRxi+1−νRxi2αμSxi+1−νSxi21−α+νRxi+1−μRxi2ανSxi+1−μSxi21−α−2+∑xi∈X20.From (17) and (18),(19)JαIFSR:R∪S=JαIFSS:R∩S.(c)
(20)JαIFSR:R∩S=1α−1∑xi∈X10+∑xi∈X2μRxi+1−νRxi2αμSxi+1−νSxi21−α+νRxi+1−μRxi2ανSxi+1−μSxi21−α+μSxi+1−νSxi2αμRxi+1−νRxi21−α+νSxi+1−μSxi2ανRxi+1−μRxi21−α−2=JαIFSS:R∪S.(d)
Adding (17) and (20), we get(21)JαIFSR:R∪S+JαIFSR:R∩S=JαIFSR:S.(e)
It suffices to show that(22)JαIFSR:T+JαIFSS:T−JαIFSR∪S:T≥0,JαIFSR:T=1α−1∑i=1nμRxi+1−νRxi2αμTxi+1−νTxi21−α+νRxi+1−μRxi2ανTxi+1−μTxi21−α+μTxi+1−νTxi2αμRxi+1−νRxi21−α+νTxi+1−μTxi2ανRxi+1−μRxi21−α−2,JαIFSS:T=1α−1∑i=1nμSxi+1−νSxi2αμTxi+1−νTxi21−α+νSxi+1−μSxi2ανTxi+1−μTxi21−α+μTxi+1−νTxi2αμSxi+1−νSxi21−α+νTxi+1−μTxi2ανSxi+1−μSxi21−α−2,JαIFSR:T+JαIFSS:T−JαIFSR∪S:T=1α−1∑xi∈X2μSxi+1−vSxi2αμTxi+1−vTxi21−α+vSxi+1−μSxi2αvTxi+1−μTxi21−α+μTxi+1−vTxi2αμSxi+1−vSxi21−αvTxi+1−μTxi2αvSxi+1−μSxi21−α−2+∑xi∈μRxi+1−vRxi2αμTxi+1−vTxi21−α+vRxi+1−μRxi2αvTxi+1−μTxi21−α=μTxi+1−vTxi2αμRxi+1−vRxi21−α+vTxi+1−μTxi2αvRxi+1−μRxi21−α−2≥0.(f)
The proof of (f) is similar to (e).(g)
(23)JαIFSR∩S:T=1α−1∑xi∈X1μRxi+1−vRxi2αμTxi+1−vTxi21−α+vRxi+1−μRxi2αvTxii+1−μTxi21−α+μTxi+1−vTxi2αμRxi+1−vRxi21−α+vTxi+1−μTxi2αvRxi+1−μRxi21−α−2+∑xi∈X2μSxi+1−vSxi2αμTxi+1−vTxi21−α+vSxi+1−μSxi2αvTxi+1−μTxi21−α+μTxi+1−vTxi2αμSxi+1−vSxi21−α+vTxi+1−μTxi2αvSxi+1−μSxi21−α−2.Add(24)JαIFSR∩S:T+JαIFSR∪S:T=JαIFSR:T+JαIFSS:T.(h)
(25)JαIFSR:Sc=1α−1∑i=1nμRxi+1−νRxi2ανSxi+1−μSxi21−α+νRxi+1−μRxi2αμSxi+1−νSxi21−α+νSxi+1−μSxi2αμRxi+1−νRxi21−α+μSxi+1−νSxi2ανRxi+1−μRxi21−α−2,JαIFSRc:S=1α−1∑i=1nνRxi+1−μRxi2αμSxi+1−νSxi21−α+μRxi+1−νRxi2ανSxi+1−μSxi21−α+μSxi+1−νSxi2ανRxi+1−μRxi21−α+νSxi+1−μSxi2αμRxi+1−νRxi21−α−2∴JαIFSR:Sc=JαIFSRc:S.


## 3.1. Validity Proof of the Defined Measure of Divergence

Theorem 1.
JαIFSR:S is a valid measure of directed divergence on IFS, i.e., it satisfies(9)JαIFSR:S≥0,JαIFSR:S=0,iffR=S,JαIFSR:S=JαIFSRc:Sc.Proof.
In [6], it is proved that IαR:S≥0 and vanishes only when R = S.
As(10)0≤μRxi+1−νRxi2≤1,the result holds for DαIFS also. Therefore,DαIFSR:S≥0, and henceJαIFSR:S≥0.
Similarly,JαIFSR:S=0 iff R = S.(11)JαIFSRc:Sc=1α−1∑i=1nνRxi+1−μRxi2ανSxi+1−μSxi21−α+μRxi+1−νRxi2αμSxi+1−νSxi21−α+νSxi+1−μSxi2ανRxi+1−μRxi21−α+μSxi+1−νSxi2αμRxi+1−νRxi21−α−2=JαIFSR:S.


## 3.2. Properties of the Proposed Intuitionistic Fuzzy Divergence Measure

For the proofs of the properties, we need to separateX into X1 and X2 such that(12)X1=xi|xi∈X,Rxi⊆Sxi,X2=xi|xi∈X,Rxi⊇Sxi.Therefore,for allxi∈X1, we have(13)µRxi≤µSxi,νRxi≥νSxi,and for allxi∈X2, we have(14)µRxi≥µSxi,νRxi≤νSxi.Theorem 2.
LetR, S, T be an IFS on universal set X = X=x1,x2,…,xn; then, the proposed measure given by (8) satisfies the following properties:(a)
JαIFSR∪S:R∩S=JαIFSR:S.(b)
JαIFSR:R∪S=JαIFSS:R∩S.(c)
JαIFSR:R∩S=JαIFSS:R∪S.(d)
JαIFSR:R∪S+JαIFSR:R∩S=JαIFSR:S.(e)
JαIFSR∪S:T≤JαIFSR:T+JαIFSS:T.(f)
JαIFSR∩S:T≤JαIFSR:T+JαIFSS:T.(g)
JαIFSR∩S:T+JαIFSR∪S:T=JαIFSR:T+JαIFSS:T.(h)
JαIFSR:Sc=JαIFSRc:S.Proof.
(a)
(15)JαIFSR∪S:R∩S=1α−1∑i=1nμR∪Sxi+1−νR∪Sxi2αμR∩Sxi+1−νR∩Sxi21−α+νR∪Sxi+1−μR∪Sxi2ανR∩Sxi+1−μR∩Sxi21−α+μR∩Sxi+1−νR∩Sxi2αμR∪Sxi+1−νR∪Sxi21−α+νR∩Sxi+1−μR∩Sxi2ανR∪Sxi+1−μR∪Sxi21−α,=1α−1∑xi∈X1μSxi+1−vSxi2αμRxi+1−vRxi21−α+vSxi+1−μSxi2αvRxi+1−μRxi21−α+μRxi+1−vRxi2αμSxi+1−vSxi21−α+vRxi+1−μRxi2αvSxi+1−μSxi21−α−2+∑xi∈X2μRxi+1−vRxi2αμSxi+1−vSxi21−α+vRxi+1−μRxi2αvSxi+1−μSxi21−α+μSxi+1−vSxi2αμRxi+1−vRxi21−α+vSxi+1−μSxi2αvRxi+1−μRxi21−α−2=JαIFSR:S.(b)
(16)JαIFSR:R∪S=1α−1∑i=1nμRxi+1−νRxi2αμR∪Sxi+1−νR∪Sxi21−α+νRxi+1−μRxi2ανR∪Sxi+1−μR∪Sxi21−α+μR∪Sxi+1−νR∪Sxi2αμRxi+1−νRxi21−α+νR∪Sxi+1−μR∪Sxi2ανRxi+1−μRxi21−α−2,(17)=1α−1∑xi∈X1μRxi+1−νRxi2αμSxi+1−νSxi21−α+νRxi+1−μRxi2ανSxi+1−μSxi21−α+μSxi+1−νSxi2αμRxi+1−νRxi21−α+νSxi+1−μSxi2ανRxi+1−μRxi21−α−2+∑xi∈X20,(18)JαIFSS:R∩S=1α−1∑xi∈X1μSxi+1−νSxi2αμRxi+1−νRxi21−α+νSxi+1−μSxi2ανRxi+1−μRxi21−α+μRxi+1−νRxi2αμSxi+1−νSxi21−α+νRxi+1−μRxi2ανSxi+1−μSxi21−α−2+∑xi∈X20.From (17) and (18),(19)JαIFSR:R∪S=JαIFSS:R∩S.(c)
(20)JαIFSR:R∩S=1α−1∑xi∈X10+∑xi∈X2μRxi+1−νRxi2αμSxi+1−νSxi21−α+νRxi+1−μRxi2ανSxi+1−μSxi21−α+μSxi+1−νSxi2αμRxi+1−νRxi21−α+νSxi+1−μSxi2ανRxi+1−μRxi21−α−2=JαIFSS:R∪S.(d)
Adding (17) and (20), we get(21)JαIFSR:R∪S+JαIFSR:R∩S=JαIFSR:S.(e)
It suffices to show that(22)JαIFSR:T+JαIFSS:T−JαIFSR∪S:T≥0,JαIFSR:T=1α−1∑i=1nμRxi+1−νRxi2αμTxi+1−νTxi21−α+νRxi+1−μRxi2ανTxi+1−μTxi21−α+μTxi+1−νTxi2αμRxi+1−νRxi21−α+νTxi+1−μTxi2ανRxi+1−μRxi21−α−2,JαIFSS:T=1α−1∑i=1nμSxi+1−νSxi2αμTxi+1−νTxi21−α+νSxi+1−μSxi2ανTxi+1−μTxi21−α+μTxi+1−νTxi2αμSxi+1−νSxi21−α+νTxi+1−μTxi2ανSxi+1−μSxi21−α−2,JαIFSR:T+JαIFSS:T−JαIFSR∪S:T=1α−1∑xi∈X2μSxi+1−vSxi2αμTxi+1−vTxi21−α+vSxi+1−μSxi2αvTxi+1−μTxi21−α+μTxi+1−vTxi2αμSxi+1−vSxi21−αvTxi+1−μTxi2αvSxi+1−μSxi21−α−2+∑xi∈μRxi+1−vRxi2αμTxi+1−vTxi21−α+vRxi+1−μRxi2αvTxi+1−μTxi21−α=μTxi+1−vTxi2αμRxi+1−vRxi21−α+vTxi+1−μTxi2αvRxi+1−μRxi21−α−2≥0.(f)
The proof of (f) is similar to (e).(g)
(23)JαIFSR∩S:T=1α−1∑xi∈X1μRxi+1−vRxi2αμTxi+1−vTxi21−α+vRxi+1−μRxi2αvTxii+1−μTxi21−α+μTxi+1−vTxi2αμRxi+1−vRxi21−α+vTxi+1−μTxi2αvRxi+1−μRxi21−α−2+∑xi∈X2μSxi+1−vSxi2αμTxi+1−vTxi21−α+vSxi+1−μSxi2αvTxi+1−μTxi21−α+μTxi+1−vTxi2αμSxi+1−vSxi21−α+vTxi+1−μTxi2αvSxi+1−μSxi21−α−2.Add(24)JαIFSR∩S:T+JαIFSR∪S:T=JαIFSR:T+JαIFSS:T.(h)
(25)JαIFSR:Sc=1α−1∑i=1nμRxi+1−νRxi2ανSxi+1−μSxi21−α+νRxi+1−μRxi2αμSxi+1−νSxi21−α+νSxi+1−μSxi2αμRxi+1−νRxi21−α+μSxi+1−νSxi2ανRxi+1−μRxi21−α−2,JαIFSRc:S=1α−1∑i=1nνRxi+1−μRxi2αμSxi+1−νSxi21−α+μRxi+1−νRxi2ανSxi+1−μSxi21−α+μSxi+1−νSxi2ανRxi+1−μRxi21−α+νSxi+1−μSxi2αμRxi+1−νRxi21−α−2∴JαIFSR:Sc=JαIFSRc:S.


## 4. Drawbacks of Other Measures of Divergence for IFS

Vlachos and Sergiadis [17] defined the following measure of divergence:(26)D1R:S=∑i=1nμRxilnμRxi1/2μRxi+μSxi+νRxilnνRxi1/2νRxi+νSxi.Junjun et al. [18] defined(27)D2R:S=∑i=1nπRxilnπRxi1/2πRxi+πSxi+ΔRxilnΔRxi1/2ΔRxi+ΔSxi,where ΔRxi=μRxi−νRxi.Example 1.
Consider the IFS given by(28)R=x1,0.44,0.385,x2,0.43,0.39,x3,0.42,0.38,S=x1,0.34,0.48,x2,0.37,0.46,x3,0.38,0.45.
Corresponding toR and S, we get(29)D1R:S=−0.00712,D2R:S=−0.04547.
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 11th 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.LetM = {M1, M2, M3.....Mp} be the set of p alternatives to be examined under the set of q criterion set N = {N1, N2, N3......Nq}. Then, we take following steps to choose the best alternative.Step 1.
Constructing the decision-making matrix: letμij denote the degree of the alternative Mi satisfying the criteria Nj and νij denote the degree of the alternative Mi not satisfying the criteria Nj. Then, the decision-making matrix is a p×q matrix whose entries xij are(30)μij,νij,i=1,2,…,p,j=1,2,…,q,whereμijνij∈0,1andμij+νij≤1.Step 2.
Computing the ideal alternative: find the ideal alternativeM∗ as(31)M∗=μ1∗,ν1∗,μ2∗,ν2∗,…,μq∗,νq∗,where μj∗=max1≤i≤pμij and υj∗=min1≤i≤qνij.Step 3.
Calculating the value of proposed divergence measure: findJαIFSMi:M∗ using the formula(32)JαIFSMi:M∗=1α−1∑j=1qμijxij+1−νijxij2αμj∗xij+1−νj∗xij21−α+νijxij+1−μijxij2ανj∗xij+1−μj∗xij21−α+μj∗xij+1−νj∗xij2αμijxij+1−νijxij21−α+νj∗xij+1−μj∗xij2ανijxij+1−μijxij21−α−2,∀i=1,2,…,p.Step 4.
Ranking the alternatives: give ranking to all the alternatives. The alternative whoseJαIFSMi:M∗ 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 treatmentsT1,T2,T3,T4T1,T2,T3,T4 on 4 different subjects A1,A2,A3,A4 and noted the results of clinical trials (see Table 1).
Based on this normalized intuitionistic fuzzy decision matrix, we find the ideal alternative(33)M∗=0.9,0.1,0.8,0.1,0.7,0.2,0.8,0.2.
Now, we calculate the value of divergence of each treatmentTi from the ideal alternative M∗ for different values of α.
According to Table2, T4 is the best treatment for the patients of anxiety. The order of preference of treatments is as follows:(34)T4≻T2≻T1≻T3.Table 1
Normalized intuitionistic fuzzy decision matrix.
A1A2A3A4T1<0.7, 0.2><0.8, 0.1><0.5, 0.3><0.6, 0.3>T2<0.9, 0.1><0.8, 0.2><0.5, 0.3><0.8, 0.2>T3<0.6, 0.4><0.5, 0.3><0.6, 0.3><0.5, 0.4>T4<0.8, 0.1><0.7, 0.2><0.7, 0.2><0.8, 0.2>Table 2
Values ofJαIFSTi:M∗ for different values of α.
α = 0.5α = 2α = 3α = 4α = 5RankJαIFST1:M∗0.19030.82331.42542.37556.6063rdJαIFST2:M∗0.06050.24900.39280.56510.78172ndJαIFST3:M∗0.59482.86566.069613.816135.3054thJαIFST4:M∗0.04320.17910.28660.42090.59801st


## 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.



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*Source: 5544993-2021-07-23.xml*