# An Exponential Jensen Fuzzy Divergence Measure with Applications in Multiple Attribute Decision-Making

Authors: Rajesh Joshi; Satish Kumar
Journal: Mathematical Problems in Engineering
 (2018)

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## Abstract

A divergence measure plays an important role in discriminating two probability distributions and drawing inferences based on such discrimination. This communication introduces one such divergence measure based on Jensen inequality and exponential entropy introduced by Pal and Pal in the settings of probability theory. Further, the idea has been generalized to fuzzy sets to introduce a new fuzzy divergence measure. Besides establishing the validity, some of its major properties are also studied. At last, the application of proposed fuzzy divergence measure is given in strategic decision-making.


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## Body

## 1. Introduction

Uncertainty in real world is ubiquitous. Before the discovery of fuzzy sets by Zadeh [1], probability was the only way to measure the uncertainty. Introduction of fuzzy set (FS) theory by Zadeh [1] provided the researchers with an important tool to handle the uncertainty/fuzziness. Contrary to the classical definition of a set in which a particular either belongs to the set or not, in fuzzy set each member of the set is assigned a membership degree. Fuzzy entropy is an important concept related to FS theory. Fuzzy entropy quantifies the fuzziness arising due to the vagueness of being or not being the member of the set. It was Zadeh again who pioneered in the introduction of fuzzy entropy as weighted Shannon entropy [2]. The axiomatization of the concept of fuzzy entropy by De Luca and Termini [3] has caused several researchers to introduce fuzzy entropies from their viewpoints and gave their applications in distinct fields [4–6]. Recently, Joshi and Kumar [7, 8] have introduced two new fuzzy entropies and applied them in decision-making. Amongst various applications, one important application of fuzzy entropies has been found in determining the divergence between two distributions. A divergence measure is a measure which is used to quantify the distance between two distributions. To develop new fuzzy divergence measures and seek their applications in different areas is a hot topic amongst the researchers nowadays. Thus, many authors are contributing towards the development of fuzzy divergence measures and their applications in distinct fields like pattern recognition, image segmentation, etc. [9–18]. The above discussion highlights the importance of divergence measures based on fuzzy sets.Vagueness in scientific studies poses a problem. FSs introduced by Zadeh [1] have been proved to be more useful to characterizing such vagueness. Therefore, it is natural to develop such measures which can not only measure vagueness but also quantify the divergence between underlying characterizing FSs. The present communication is a sequel in this direction. Using the notion of Jensen-Shannon divergence proposed by Lin [19], a new concept called Jensen-Exponential divergence measure based on famous Jensen inequality and well-known exponential entropy introduced by Pal and Pal [5] has been put forward. One of the major benefits of using Jensen-Shannon divergence measure [19] is that we can give any weight-age to a probability distribution. While introducing the concept of exponential entropy by Pal and Pal [5], the authors claimed that exponential entropy has several advantages over logarithmic entropy like upper bound, etc. Exploring the properties of exponential entropy introduced by Pal and Pal [5], a new divergence measure has been proposed in this communication. The prime aims of introducing this manuscript are as follows: (1) to propose a new divergence measure based on exponential entropy introduced by Pal and Pal [5] and Jensen inequality; (2) to extend this idea to fuzzy settings to introduce a new fuzzy divergence measure; (3) to introduce a new multiple attribute decision-making method based on proposed fuzzy divergence measure; (4) to apply the proposed method in strategic decision-making problem.The paper is managed as follows: The contribution of earlier researchers to the field and the aim of the manuscript are given in Section1. Some basic concepts and definitions related to the topic under consideration are given in Section 2. In Section 3, a new divergence measure and its extension to fuzzy sets are given in Section 3. Besides this, the validity of proposed fuzzy divergence measures is also established in this section. Some major properties of proposed fuzzy divergence are discussed in Section 4. Section 5 is devoted to the application of proposed fuzzy divergence measure in decision-making. At last, the paper is summed up with “conclusions” in Section 6.The next section contains some relevant concepts and definitions necessary to understand this communication.


## 2. Preliminaries

Definition 1 (See [1]).
LetX=(g1,g2,…,gn) be a finite universe of discourse. A fuzzy set (FS) S in X is defined as(1)S=gi,μSgi∣gi∈X,where μS:X→[0,1] denotes the membership function and μS(gi) represents the membership degree or measure of belongingness of gi∈X to S.From here onwards,FS(X) will denote the set of all fuzzy sets defined on X.Now, we define some basic operations on FSs.Definition 2 (See [1]).
LetS,T∈FS(X). Then (1)
S⊆T if and only if μS(gi)≤μT(gi) for all gi∈X;(2)
S=T if and only if S⊆T and T⊆S;(3)
Sc={gi,1-μSgi∣gi∈X};(4)
S∪T=〈gimax(μS(gi),μT(gi))〉∣gi∈X;(5)
S∩T=〈gimin(μS(gi),μT(gi))〉∣gi∈X.


## 3. A New Fuzzy Divergence Measure

### 3.1. Background

Let∆n={C=(c1,c2,…,cn);ci≥0,∑i=1nci=1}, n≥2, be a set of complete probability distributions. For some C∈∆n, the exponential entropy(2)HPPC=∑i=1ncie1-ci-1,proposed by Pal and Pal [5] is one of the well-known index of diversity. The concavity of measure (2) bifurcates the mixed distribution C+D/2 into two parts as follows:(3)HPPC+D2=12HPPC+HPPD+12JnPPC,D.The first term in (3), that is, HPP(C)+HPP(D)/2, is the average diversity within distributions C and D and the second term, that is,(4)JnPPC,D=-HPPC-HPPD-2-HPPC+D2,is called Exponential Jensen difference. Convexity of -HPP(C) implies the convexity of Jensen difference given by (4). Again, JnPP(C,D)=0 at C=D confirms that (4) is a natural measure of divergence [20, 21]. Besides this, the measure (4) also meets the intuitive requirement that average divergence between (C,D) and (E,F) is not less than their convex combination; that is, χ1HPP(C,D)+χ2HPP(E,F), where χ1,χ2≥0 satisfying χ1+χ2=1. The convex behavior of Jensen difference (4) is an additional feature of exponential entropy (2).The authors claim that (2) has some advantages over Shannon entropy [22] particularly in image processing. This is due to fact that entropy (2) has a fixed upper bound, that is, for an uniform distribution (1/n,1/n,…,1/n); limn→∞⁡H(C)=e-1 as compared to infinite limit (as n→∞) in case of Shannon entropy.Based on the concept of Jensen-Shannon divergence [19], corresponding to (2), for any two probability distributions C,D∈∆n with δ1 and δ2 as their respective weights satisfying δ1+δ2=1, we define a new divergence measure as(5)EJD^^C,D=∑i=1nδ1ci+δ2die1-δ1ci+δ2di+δ2ci+δ1die1-δ2ci+δ1di-cie1-ci-die1-di.To justify the existence, we now prove that proposed measure satisfies the following properties.Properties of E J D ^ ^ C , D. Some major properties of proposed divergence measure are(1)
EJD^^(C,D)≥0 with equality when C=D;(2)
EJD^^(C,D) is a convex function of C and D.Proof.
First we prove thatEJD^^(C,D)≥0.
Since (2) represents a concave function, therefore, for any C,D∈∆n, using Jensen’s inequality, we have(6)Hδ1C+δ2D≥δ1HC+δ2HD.This implies(7)∑i=1nδ1ci+δ2die1-δ1ci+δ2di≥δ1∑i=1ncie1-ci+δ2∑i=1ndie1-di.Similarly,(8)∑i=1nδ1di+δ2cie1-δ1di+δ2ci≥δ1∑i=1ndie1-di+δ2∑i=1ncie1-ci.Adding (7) and (8), we get EJD^^(C,D)≥0.
Now, we prove the convexity of (5).For this, we consider a function given by(9)fx,y=δ1x+δ2ye1-δ1x+δ2y-δ1xe1-x-δ2ye1-y.Definition 3.
The Hessian matrix of a functionf of two variables x and y is defined as(10)Hessianf=δ2fδx2δ2fδxδyδ2fδxδyδ2fδy2.The function f is said to be convex at any point in its domain if Hessian(f) is positive semidefinite and concave if Hessian(f) is negative semidefinite at that point.Now, differentiating (9) partially with respect to x and y, we get(11)δfδx=δ1e1-δ1x-δ2y1-δ1x-δ2y+δ1e1-xx-1,(12)δfδy=δ1e1-δ2x-δ1y1-δ2x-δ1y+e1-yy-1.To determine stationary point, we substitute δf/δx=0 and δf/δy=0. This gives x=y as a stationary point. Now, computing the Hessian matrix of f at x=y and using δ1+δ2=1, we have(13)Hessianf=e1-x2-xδ1δ21-1-11, which is positive semidefinite. This confirms the convex character of f.Extending the idea from probabilistic settings to fuzzy settings, in the next subsection, we introduce a new fuzzy divergence measure based on measure (5).


### 3.2. Definition

Definition 4.
LetX={g1,g2,…,gn} be a finite universe of discourse. For any S,T∈FS(X), we define a fuzzy divergence measure EJD^(S,T) as(14)EJD^S,T=1n∑i=1nδ1μSgi+δ2μTgie1-δ1μSgi+δ2μTgi+1-δ1μSgi-δ2μTgieδ1μSgi+δ2μTgi+δ1μTgi+δ2μSgie1-δ1μTgi+δ2μSgi+1-δ1μTgi-δ2μSgieδ1μTgi+δ2μSgi-μSgie1-μSgi+1-μSgieμSgi-μTgie1-μTgi+1-μTgieμTgi.


### 3.3. Justification

Now, a very natural question that comes in mind is about the validation of the proposed measure. We answer this query by showing that the proposed measure satisfies (1) nonnegativity and (2) convexity.Proof.
We start with proving the nonnegativity of (14).
Using (6), we have(15)∑i=1nδ1μSgi+δ2μTgie1-δ1μSgi+δ2μTgi+δ11-μSgi+δ21-μTgieδ1μSgi+δ2μTgi≥∑i=1nδ1μSgie1-μSgi+1-μSgieμSgi+δ2μTgie1-μTgi+1-μTgieμTgi.Similarly,(16)∑i=1nδ1μTgi+δ2μSgie1-δ1μTgi+δ2μSgi+δ11-μTgi+δ21-μSgieδ1μTgi+δ2μSgi≥∑i=1nδ1μTgie1-μTgi+1-μTgieμTgi+δ2μSgie1-μSgi+1-μSgieμSgi.Adding (15) and (16), we have EJD^(S,T)≥0.
SinceμS(gi)+(1-μS(gi))=1 and μT(gi)+(1-μT(gi))=1 for all S,T∈FS(X) and gi∈X, therefore, convexity of (14) follows directly from the convexity of (5).
This, justifies the existence of (14).In the next Section, we study some major properties ofEJD^(S,T).


## 3.1. Background

Let∆n={C=(c1,c2,…,cn);ci≥0,∑i=1nci=1}, n≥2, be a set of complete probability distributions. For some C∈∆n, the exponential entropy(2)HPPC=∑i=1ncie1-ci-1,proposed by Pal and Pal [5] is one of the well-known index of diversity. The concavity of measure (2) bifurcates the mixed distribution C+D/2 into two parts as follows:(3)HPPC+D2=12HPPC+HPPD+12JnPPC,D.The first term in (3), that is, HPP(C)+HPP(D)/2, is the average diversity within distributions C and D and the second term, that is,(4)JnPPC,D=-HPPC-HPPD-2-HPPC+D2,is called Exponential Jensen difference. Convexity of -HPP(C) implies the convexity of Jensen difference given by (4). Again, JnPP(C,D)=0 at C=D confirms that (4) is a natural measure of divergence [20, 21]. Besides this, the measure (4) also meets the intuitive requirement that average divergence between (C,D) and (E,F) is not less than their convex combination; that is, χ1HPP(C,D)+χ2HPP(E,F), where χ1,χ2≥0 satisfying χ1+χ2=1. The convex behavior of Jensen difference (4) is an additional feature of exponential entropy (2).The authors claim that (2) has some advantages over Shannon entropy [22] particularly in image processing. This is due to fact that entropy (2) has a fixed upper bound, that is, for an uniform distribution (1/n,1/n,…,1/n); limn→∞⁡H(C)=e-1 as compared to infinite limit (as n→∞) in case of Shannon entropy.Based on the concept of Jensen-Shannon divergence [19], corresponding to (2), for any two probability distributions C,D∈∆n with δ1 and δ2 as their respective weights satisfying δ1+δ2=1, we define a new divergence measure as(5)EJD^^C,D=∑i=1nδ1ci+δ2die1-δ1ci+δ2di+δ2ci+δ1die1-δ2ci+δ1di-cie1-ci-die1-di.To justify the existence, we now prove that proposed measure satisfies the following properties.Properties of E J D ^ ^ C , D. Some major properties of proposed divergence measure are(1)
EJD^^(C,D)≥0 with equality when C=D;(2)
EJD^^(C,D) is a convex function of C and D.Proof.
First we prove thatEJD^^(C,D)≥0.
Since (2) represents a concave function, therefore, for any C,D∈∆n, using Jensen’s inequality, we have(6)Hδ1C+δ2D≥δ1HC+δ2HD.This implies(7)∑i=1nδ1ci+δ2die1-δ1ci+δ2di≥δ1∑i=1ncie1-ci+δ2∑i=1ndie1-di.Similarly,(8)∑i=1nδ1di+δ2cie1-δ1di+δ2ci≥δ1∑i=1ndie1-di+δ2∑i=1ncie1-ci.Adding (7) and (8), we get EJD^^(C,D)≥0.
Now, we prove the convexity of (5).For this, we consider a function given by(9)fx,y=δ1x+δ2ye1-δ1x+δ2y-δ1xe1-x-δ2ye1-y.Definition 3.
The Hessian matrix of a functionf of two variables x and y is defined as(10)Hessianf=δ2fδx2δ2fδxδyδ2fδxδyδ2fδy2.The function f is said to be convex at any point in its domain if Hessian(f) is positive semidefinite and concave if Hessian(f) is negative semidefinite at that point.Now, differentiating (9) partially with respect to x and y, we get(11)δfδx=δ1e1-δ1x-δ2y1-δ1x-δ2y+δ1e1-xx-1,(12)δfδy=δ1e1-δ2x-δ1y1-δ2x-δ1y+e1-yy-1.To determine stationary point, we substitute δf/δx=0 and δf/δy=0. This gives x=y as a stationary point. Now, computing the Hessian matrix of f at x=y and using δ1+δ2=1, we have(13)Hessianf=e1-x2-xδ1δ21-1-11, which is positive semidefinite. This confirms the convex character of f.Extending the idea from probabilistic settings to fuzzy settings, in the next subsection, we introduce a new fuzzy divergence measure based on measure (5).


## 3.2. Definition

Definition 4.
LetX={g1,g2,…,gn} be a finite universe of discourse. For any S,T∈FS(X), we define a fuzzy divergence measure EJD^(S,T) as(14)EJD^S,T=1n∑i=1nδ1μSgi+δ2μTgie1-δ1μSgi+δ2μTgi+1-δ1μSgi-δ2μTgieδ1μSgi+δ2μTgi+δ1μTgi+δ2μSgie1-δ1μTgi+δ2μSgi+1-δ1μTgi-δ2μSgieδ1μTgi+δ2μSgi-μSgie1-μSgi+1-μSgieμSgi-μTgie1-μTgi+1-μTgieμTgi.


## 3.3. Justification

Now, a very natural question that comes in mind is about the validation of the proposed measure. We answer this query by showing that the proposed measure satisfies (1) nonnegativity and (2) convexity.Proof.
We start with proving the nonnegativity of (14).
Using (6), we have(15)∑i=1nδ1μSgi+δ2μTgie1-δ1μSgi+δ2μTgi+δ11-μSgi+δ21-μTgieδ1μSgi+δ2μTgi≥∑i=1nδ1μSgie1-μSgi+1-μSgieμSgi+δ2μTgie1-μTgi+1-μTgieμTgi.Similarly,(16)∑i=1nδ1μTgi+δ2μSgie1-δ1μTgi+δ2μSgi+δ11-μTgi+δ21-μSgieδ1μTgi+δ2μSgi≥∑i=1nδ1μTgie1-μTgi+1-μTgieμTgi+δ2μSgie1-μSgi+1-μSgieμSgi.Adding (15) and (16), we have EJD^(S,T)≥0.
SinceμS(gi)+(1-μS(gi))=1 and μT(gi)+(1-μT(gi))=1 for all S,T∈FS(X) and gi∈X, therefore, convexity of (14) follows directly from the convexity of (5).
This, justifies the existence of (14).In the next Section, we study some major properties ofEJD^(S,T).


## 4. Properties of Proposed Divergence Measure

Measure (14) possesses the following properties.Theorem 5.
ForS,T,R∈FS(X),(1)
EJD^(S,T)=EJD^(T,S);(2)
EJD^(S,T)=0 if and only if S=T;(3)
EJD^(S,Sc)=0 if and only if μSc(gi)=1-μS(gi) for all gi∈X;(4)
EJD^(S,S∪T)=EJD^(S∩T,T)≤EJD^(S,T);(5)
EJD^(S∪T,S∩T)=EJD^(S∩T,S∪T)=EJD^(S,T);(6)
EJD^(S∩T,S∪T)=EJD^(T,S);(7)
EJD^(S,S∪T)+EJD^(S,S∩T)=EJD^(S,T);(8)
EJD^(T,S∪T)+EJD^(T,S∩T)=EJD^(T,S);(9)
EJD^(S∪T,R)≤EJD^(S,R)+EJD^(T,R);(10)
EJD^(S∩T,R)≤EJD^(S,R)+EJD^(T,R);(11)
EJD^(S∪T,R)+EJD^(S∩T,R)=EJD^(S,R)+EJD^(T,R).(12)
EJD^(S,T)=EJD^(Sc,Tc);(13)
EJD^(S,Tc)=EJD^(Sc,T);(14)
EJD^(S,T)+EJD^(Sc,T)=EJD^(Sc,Tc)+EJD^(S,Tc), where Sc and Tc are the complements of FSs S and T, respectively.Proof.
Proofs of above properties have been provided in the Appendix.In the next section, we give an application of proposed divergence measure in multiple attribute decision-making problem (MADM) with the help of an illustrative example.


## 5. Application of Proposed Fuzzy Divergence Measure in Decision-Making

In recent years, fuzzy divergence measures have been extensively applied by many researchers in many fields like in image thresholding, bioinformatics, etc. [5, 8–11]. In this paper, we apply the proposed fuzzy divergence measure in strategic decision-making. In strategic decision-making problems, we have a set of strategies say Λ1,Λ2,…,Λn and another set of inputs say Ω1,Ω2,…,Ωm. Our aim is to find a particular strategy which if applied with a specific input produces optimum results. Degree of effectiveness of a particular strategy corresponding to specific input can be yielded by the method suggested by Liu and Wang [23] as follows.Consider a team comprised ofN-number of experts. Team members are expected to respond in the form “yes” if he/she supports a particular strategy coupled with a specific input and “no” in case he/she does not feel the particular strategy to be applied with a specific input. Let the number of experts who support the strategy Λj;(j=1,2,…,n) to be applied with input “Ωi;(i=1,2,…,m)” be denoted by “nyes(i,j)”. Then, the effectiveness of strategy “Λj” with input Ωi’ can be calculated as(17)μΩiΛj=nyesi,jN.Now, let the degree of effectiveness of Λj’s (j=1,2,…,n) at uniform inputs be denoted by fuzzy set P given by(18)P=Λj,μPΛj;j=1,2,…,n; where μP(Λj) denotes the membership degree satisfying 0≤μP(Λj)≤1.Further, let the degree of effectiveness ofΛj’s j=1,2,…,n at particular inputs be denoted by the fuzzy set Q given by(19)Q=Λj,μQΛj;j=1,2,…n.


### 5.1. A New MADM Method

TOPSIS (Technique of Order Preference by Similarity to Ideal Solutions) method is a well-known technique employed for solving multiple attribute decision-making (MADM) problems. In this technique, an option nearest to the positive ideal solution and farthest from the negative ideal solution is chosen as most suitable option. Using the concept of TOPSIS initially introduced by Xu and Yager [24], we now propose a new MADM method to solve the above strategic decision-making problem. The procedural steps of proposed method are listed as follows:(1)
Determine the best strategy as follows:(20)Λ~z=maxμPΛj;j=1,…,n.(2)
Construct the fuzzy decision matrix as(21)Λ1Λ2…ΛnΩ1s11s12…s1nΩ2s21s22…s2n……………Ωmsm1sm2…smnandU=[u1,u2,…,un]. The number “(sij)1≤i≤m,1≤j≤n” in the above matrix denotes the degree of effectiveness of strategy Λj corresponding to input Ωi using (17).(3)
Determine the normalized fuzzy decision matrixnij as(22)nij=sij∑i=1msij2,i=1,2,…,m;j=1,2,…,n.(4)
Determine the weighted normalized fuzzy decision matrixwij as(23)wij=uinij,i=1,2,…,m;j=1,2,…,n.whereU=1,1,…,1 denote the weight matrix and uj is the weight of jth strategy.(5)
Determine the positive ideal solution (Ω+) and negative ideal solution (Ω-) as(24)Ω+=w1+,w2+,…,wm+,(25)Ω-=w1-,w2-,…,wm-,wherewi+=maxj(wij), wi-=minj(wij), i=1,2,…,m, and j=1,2,…,n.(6)
Determine the separation of each alternativeΩi(i=1,2,…,m) from Ω+ and Ω-, respectively, by using (14). Let these be denoted by EJD^(Ωi,Ω+) and EJD^(Ωi,Ω-).(7)
Determine the relative closeness coefficients of each alternativeΩi(i=1,2,…,m) from Ω+ and Ω- as(26)Si=EJD^Ωi,Ω-EJD^Ωi,Ω-+EJD^Ωi,Ω+,i=1,2,…,m.(8)
Rank the alternatives according to relative closeness coefficients obtained in Step (7).Therefore, if the strategyΛ~z coupled with input Ωt is applied, then we get optimal output.An Illustrative Example. This example is adapted from Joshi and Kumar [8].Consider an example of a company that wants to launch a new product in the market. For an effective marketing of the product, company has conducted a survey to know the opinion of people about the product before its launching. The whole compiled views are divided into five categories called inputs denoted byΩi;i=1,2,3,4,5. Depending upon these views, company’s think tank has prepared five strategies to launch the product in the market denoted by Λj;j=1,2,3,4,5. Now, our aim is to find a particular strategy which if coupled with a specific input returns maximum gains. As a part of full-proof success of the product in the market, company has constituted a committee of 100 experts (in real world problem, no. of experts may increase or decrease). Tables 1 and 2 represent the degrees of effectiveness of different strategies at uniform inputs and at particular inputs.Table 1
Efficiencies ofΛj’s at uniform inputs.
μ P ( Λ 1 ) μ P ( Λ 2 ) μ P ( Λ 3 ) μ P ( Λ 4 ) μ P ( Λ 5 ) .7 .3 .8 .5 .6Table 2
Efficiencies of strategies at particular inputs.
Ω j μ Q ( Λ 1 ) μ Q ( Λ 2 ) μ Q ( Λ 3 ) μ Q ( Λ 4 ) μ Q ( Λ 5 ) Ω 1 .5 .2 .6 .9 .4  Ω 2 .4 .6 .8 .3 .6  Ω 3 .8 .3 .5 .7 .2  Ω 4 .6 .4 .2 .8 .9  Ω 5 .9 .5 .3 .4 .7Now, we employ the proposed measure (14) to compute the above example using TOPSIS method defined above.(1)
Applying Step (1) in Table1, the best strategy so obtained is given by Λ3.(2)
The calculated fuzzy decision matrix using Step (2) is given in Table2.(3)
The computed weighted normalized fuzzy decision matrix using (22) and (23) is shown in Table 3.(4)
Computed values of positive ideal solution(Ω+) and negative ideal solution (Ω-) using Step (5) are given in Table 4.(5)
Separation measures ofΩi’s from Ω+ and Ω- using (14) are displayed in Table 5.(6)
Calculated values of relative closeness coefficients using Step (7) are presented in Table6.Table 3
Normalized/weighted normalized fuzzy decision matrix.
Λ 1 Λ 2 Λ 3 Λ 4 Λ 5 Ω 1 .3356 .2108 .5108 .6082 .2933  Ω2 .2685 .6325 .6810 .2027 .4399  Ω3 .5369 .3162 .4256 .4730 .1466  Ω4 .4027 .4216 .1703 .5406 .6599  Ω5 .6040 .5270 .2554 .2703 .5133Table 4
Fuzzy positive and negative ideal solutions.
Ω+ .6082 .6810 .5369 .6599 .6040  Ω- .2108 .2027 .1466 .1703 .2554Table 5
DistancesΩi’s from Ω+ and Ω-.
E J D ^ ( Ω i , Ω + ) E J D ^ ( Ω i , Ω - ) δ1=.1,δ2=.9 δ1=.5,δ2=.5 δ1=.1,δ2=.9 δ1=.5,δ2=.5 .0179 .0494 .0158 .0436  .0170 .0469 .0233 .0643  .0180 .0497 .0140 .0388  .0119 .0329 .0177 .0489  .0119 .0329 .0160 .0442Table 6
Coefficients of closeness and ranking.
Alternative (Inputs) δ 1 = . 1 ,   δ 2 = . 9 Ranking δ 1 = . 5 ,   δ 2 = . 5 Ranking Ω1 .4688 IV .4687 IV  Ω2 .5782 II .5781 II  Ω3 .4375 V .4383 V  Ω4 .5980 I .5978 I  Ω5 .5735 III .5731 IIIArranging the alternatives according to values of relative closeness coefficients in descending order, the preferential sequence so obtained is given by the following.Forδ1=.1,δ2=.9;Ω4>Ω2>Ω5>Ω3>Ω1.Forδ1=.5,δ2=.5;Ω4>Ω2>Ω5>Ω1>Ω3.Thus, for different values ofδ1 and δ2, the best alternative Ω4 remains unaltered.Now, we compute the above Example by fuzzy MOORA (Multiobjective Optimization on the basis of Ratio Analysis) method suggested by Brauers and Zavadskas [25] for the sake of comparison. This method refers to a matrix of responses of alternatives to objectives, to which ratios are applied. The procedural steps are briefed as follows.Fuzzy MOORA Method [25]. Steps (1)-(4) of fuzzy MOORA method are same as Steps (2)-(4) of the proposed method. The remaining steps are as follows:(5)
Compute the overall performance indexEJD^(Ω+,Ω-) for all Ωi’s; i=1,2,3,4,5. The calculated values are given in Table 7.(6)
Rank theΩi’s i=1,2,3,4,5 according to the values of EJD^(Ω+,Ω-) obtained in Step (5) in descending order. The ranking results so obtained for different values of δ1 and δ2 are shown in Table 7.Table 7
Computed ranks using MOORA method.
E J D ^ ( Ω + , Ω - ) Alternative Inputs δ 1 = . 1 ,   δ 2 = . 9 Ranking δ 1 = . 5 ,   δ 2 = . 5 Ranking Ω1 .0359 III .0993 III  Ω2 .0521 II .1436 II  Ω3 .0353 IV .0977 IV  Ω4 .0548 I .1512 I  Ω5 .0275 V .0760 VThe ranking order of alternatives is as follows:Forδ1=.1,δ2=.9; Ω4>Ω2>Ω1>Ω3>Ω5.Forδ1=.5,δ2=.5; Ω4>Ω2>Ω1>Ω3>Ω5.Thus, the best alternative, that is,Ω4, remains unaltered for different values of δ1 and δ2.


### 5.2. A Comparative Analysis

On comparing the output obtained by using proposed method with that of fuzzy MOORA method proposed by Brauers and Zavadskas [25], we find that the best alternative generated by two methods coincide. But there is a big difference in computational procedures adopted in two methods. In fuzzy MOORA method, the best alternative is decided on the basis of overall performance score values of positive ideal solution and negative ideal solution. In this way, each attribute may not get its due weight-age during the selection of the best alternative, whereas, in proposed method, the best alternative is decided on the basis of relative closeness coefficients; that is, each attribute is given due weight-age in decision-making process [8]. Secondly, in proposed method there is a particular strategy with a specific input whereas this is not the case with fuzzy MOORA method. Thus, the performance of proposed method is considerably good.


## 5.1. A New MADM Method

TOPSIS (Technique of Order Preference by Similarity to Ideal Solutions) method is a well-known technique employed for solving multiple attribute decision-making (MADM) problems. In this technique, an option nearest to the positive ideal solution and farthest from the negative ideal solution is chosen as most suitable option. Using the concept of TOPSIS initially introduced by Xu and Yager [24], we now propose a new MADM method to solve the above strategic decision-making problem. The procedural steps of proposed method are listed as follows:(1)
Determine the best strategy as follows:(20)Λ~z=maxμPΛj;j=1,…,n.(2)
Construct the fuzzy decision matrix as(21)Λ1Λ2…ΛnΩ1s11s12…s1nΩ2s21s22…s2n……………Ωmsm1sm2…smnandU=[u1,u2,…,un]. The number “(sij)1≤i≤m,1≤j≤n” in the above matrix denotes the degree of effectiveness of strategy Λj corresponding to input Ωi using (17).(3)
Determine the normalized fuzzy decision matrixnij as(22)nij=sij∑i=1msij2,i=1,2,…,m;j=1,2,…,n.(4)
Determine the weighted normalized fuzzy decision matrixwij as(23)wij=uinij,i=1,2,…,m;j=1,2,…,n.whereU=1,1,…,1 denote the weight matrix and uj is the weight of jth strategy.(5)
Determine the positive ideal solution (Ω+) and negative ideal solution (Ω-) as(24)Ω+=w1+,w2+,…,wm+,(25)Ω-=w1-,w2-,…,wm-,wherewi+=maxj(wij), wi-=minj(wij), i=1,2,…,m, and j=1,2,…,n.(6)
Determine the separation of each alternativeΩi(i=1,2,…,m) from Ω+ and Ω-, respectively, by using (14). Let these be denoted by EJD^(Ωi,Ω+) and EJD^(Ωi,Ω-).(7)
Determine the relative closeness coefficients of each alternativeΩi(i=1,2,…,m) from Ω+ and Ω- as(26)Si=EJD^Ωi,Ω-EJD^Ωi,Ω-+EJD^Ωi,Ω+,i=1,2,…,m.(8)
Rank the alternatives according to relative closeness coefficients obtained in Step (7).Therefore, if the strategyΛ~z coupled with input Ωt is applied, then we get optimal output.An Illustrative Example. This example is adapted from Joshi and Kumar [8].Consider an example of a company that wants to launch a new product in the market. For an effective marketing of the product, company has conducted a survey to know the opinion of people about the product before its launching. The whole compiled views are divided into five categories called inputs denoted byΩi;i=1,2,3,4,5. Depending upon these views, company’s think tank has prepared five strategies to launch the product in the market denoted by Λj;j=1,2,3,4,5. Now, our aim is to find a particular strategy which if coupled with a specific input returns maximum gains. As a part of full-proof success of the product in the market, company has constituted a committee of 100 experts (in real world problem, no. of experts may increase or decrease). Tables 1 and 2 represent the degrees of effectiveness of different strategies at uniform inputs and at particular inputs.Table 1
Efficiencies ofΛj’s at uniform inputs.
μ P ( Λ 1 ) μ P ( Λ 2 ) μ P ( Λ 3 ) μ P ( Λ 4 ) μ P ( Λ 5 ) .7 .3 .8 .5 .6Table 2
Efficiencies of strategies at particular inputs.
Ω j μ Q ( Λ 1 ) μ Q ( Λ 2 ) μ Q ( Λ 3 ) μ Q ( Λ 4 ) μ Q ( Λ 5 ) Ω 1 .5 .2 .6 .9 .4  Ω 2 .4 .6 .8 .3 .6  Ω 3 .8 .3 .5 .7 .2  Ω 4 .6 .4 .2 .8 .9  Ω 5 .9 .5 .3 .4 .7Now, we employ the proposed measure (14) to compute the above example using TOPSIS method defined above.(1)
Applying Step (1) in Table1, the best strategy so obtained is given by Λ3.(2)
The calculated fuzzy decision matrix using Step (2) is given in Table2.(3)
The computed weighted normalized fuzzy decision matrix using (22) and (23) is shown in Table 3.(4)
Computed values of positive ideal solution(Ω+) and negative ideal solution (Ω-) using Step (5) are given in Table 4.(5)
Separation measures ofΩi’s from Ω+ and Ω- using (14) are displayed in Table 5.(6)
Calculated values of relative closeness coefficients using Step (7) are presented in Table6.Table 3
Normalized/weighted normalized fuzzy decision matrix.
Λ 1 Λ 2 Λ 3 Λ 4 Λ 5 Ω 1 .3356 .2108 .5108 .6082 .2933  Ω2 .2685 .6325 .6810 .2027 .4399  Ω3 .5369 .3162 .4256 .4730 .1466  Ω4 .4027 .4216 .1703 .5406 .6599  Ω5 .6040 .5270 .2554 .2703 .5133Table 4
Fuzzy positive and negative ideal solutions.
Ω+ .6082 .6810 .5369 .6599 .6040  Ω- .2108 .2027 .1466 .1703 .2554Table 5
DistancesΩi’s from Ω+ and Ω-.
E J D ^ ( Ω i , Ω + ) E J D ^ ( Ω i , Ω - ) δ1=.1,δ2=.9 δ1=.5,δ2=.5 δ1=.1,δ2=.9 δ1=.5,δ2=.5 .0179 .0494 .0158 .0436  .0170 .0469 .0233 .0643  .0180 .0497 .0140 .0388  .0119 .0329 .0177 .0489  .0119 .0329 .0160 .0442Table 6
Coefficients of closeness and ranking.
Alternative (Inputs) δ 1 = . 1 ,   δ 2 = . 9 Ranking δ 1 = . 5 ,   δ 2 = . 5 Ranking Ω1 .4688 IV .4687 IV  Ω2 .5782 II .5781 II  Ω3 .4375 V .4383 V  Ω4 .5980 I .5978 I  Ω5 .5735 III .5731 IIIArranging the alternatives according to values of relative closeness coefficients in descending order, the preferential sequence so obtained is given by the following.Forδ1=.1,δ2=.9;Ω4>Ω2>Ω5>Ω3>Ω1.Forδ1=.5,δ2=.5;Ω4>Ω2>Ω5>Ω1>Ω3.Thus, for different values ofδ1 and δ2, the best alternative Ω4 remains unaltered.Now, we compute the above Example by fuzzy MOORA (Multiobjective Optimization on the basis of Ratio Analysis) method suggested by Brauers and Zavadskas [25] for the sake of comparison. This method refers to a matrix of responses of alternatives to objectives, to which ratios are applied. The procedural steps are briefed as follows.Fuzzy MOORA Method [25]. Steps (1)-(4) of fuzzy MOORA method are same as Steps (2)-(4) of the proposed method. The remaining steps are as follows:(5)
Compute the overall performance indexEJD^(Ω+,Ω-) for all Ωi’s; i=1,2,3,4,5. The calculated values are given in Table 7.(6)
Rank theΩi’s i=1,2,3,4,5 according to the values of EJD^(Ω+,Ω-) obtained in Step (5) in descending order. The ranking results so obtained for different values of δ1 and δ2 are shown in Table 7.Table 7
Computed ranks using MOORA method.
E J D ^ ( Ω + , Ω - ) Alternative Inputs δ 1 = . 1 ,   δ 2 = . 9 Ranking δ 1 = . 5 ,   δ 2 = . 5 Ranking Ω1 .0359 III .0993 III  Ω2 .0521 II .1436 II  Ω3 .0353 IV .0977 IV  Ω4 .0548 I .1512 I  Ω5 .0275 V .0760 VThe ranking order of alternatives is as follows:Forδ1=.1,δ2=.9; Ω4>Ω2>Ω1>Ω3>Ω5.Forδ1=.5,δ2=.5; Ω4>Ω2>Ω1>Ω3>Ω5.Thus, the best alternative, that is,Ω4, remains unaltered for different values of δ1 and δ2.


## 5.2. A Comparative Analysis

On comparing the output obtained by using proposed method with that of fuzzy MOORA method proposed by Brauers and Zavadskas [25], we find that the best alternative generated by two methods coincide. But there is a big difference in computational procedures adopted in two methods. In fuzzy MOORA method, the best alternative is decided on the basis of overall performance score values of positive ideal solution and negative ideal solution. In this way, each attribute may not get its due weight-age during the selection of the best alternative, whereas, in proposed method, the best alternative is decided on the basis of relative closeness coefficients; that is, each attribute is given due weight-age in decision-making process [8]. Secondly, in proposed method there is a particular strategy with a specific input whereas this is not the case with fuzzy MOORA method. Thus, the performance of proposed method is considerably good.


## 6. Concluding Remarks

In this communication, we have successfully introduced an exponential divergence measure based on the concept of Jensen-Shannon divergence measure. Further, the proposed divergence measure has been extended to fuzzy sets to introduce a new fuzzy divergence measure. The validation of proposed divergence measure is well established along with a discussion on some major properties. In the end, the application of proposed dissimilarity measure is given in decision-making. The proposed fuzzy divergence can be further generalized to intuitionistic fuzzy settings. This work is under consideration and will be reported somewhere else.



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