# On Intuitionistic Fuzzy Entropy of Order-α

Authors: Rajkumar Verma; BhuDev Sharma
Journal: Advances in Fuzzy Systems
 (2014)

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## Abstract

Using the idea of Rènyi’s entropy, intuitionistic fuzzy entropy of order-α is proposed in the setting of intuitionistic fuzzy sets theory. This measure is a generalized version of fuzzy entropy of order-α proposed by Bhandari and Pal and intuitionistic fuzzy entropy defined by Vlachos and Sergiadis. Our study of the four essential and some other properties of the proposed measure clearly establishes the validity of the measure as intuitionistic fuzzy entropy. Finally, a numerical example is given to show that the proposed entropy measure for intuitionistic fuzzy set is reasonable by comparing it with other existing entropies.


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## Body

## 1. Introduction

In 1965, Zadeh [1] proposed the notion of fuzzy set (FS) to model nonstatistical imprecise or vague phenomena. Since then, the theory of fuzzy set has become a vigorous area of research in different disciplines such as engineering, artificial intelligence, medical science, signal processing, and expert systems. Fuzziness, a feature of uncertainty, results from the lack of sharp distinction of being or not being a member of a set; that is, the boundaries of the set under consideration are not sharply defined. A measure of fuzziness used and cited in the literature is fuzzy entropy, also first mentioned in 1968 by Zadeh [2]. In 1972, De Luca and Termini [3] first provided axiomatic structure for the entropy of fuzzy sets and defined an entropy measure of a fuzzy set based on Shannon’s entropy function [4]. Kaufmann [5] introduced a fuzzy entropy measure by a metric distance between the fuzzy set and that of its nearest crisp set. In addition, Yager [6] defined an entropy measure of a fuzzy set in terms of a lack of distinction between fuzzy set and its complement. In 1989, N. R. Pal and S. K. Pal [7] proposed fuzzy entropy based on exponential function to measure the fuzziness called exponential fuzzy entropy. Further, Bhandari and Pal [8] proposed fuzzy entropy of order-α corresponding to Rènyi entropy [9]. Recently, Verma and Sharma [10] have introduced a parametric generalized entropy measure for fuzzy sets called “exponential fuzzy entropy of order-α.”Atanassov [11, 12] introduced the concept of intuitionistic fuzzy set (IFS), which is a generalization of the notion of fuzzy set. The distinguishing characteristic of intuitionistic fuzzy set is that it assigns to each element a membership degree, a nonmembership degree, and the hesitation degree. Firstly, Burillo and Bustince [13] defined the entropy on intuitionistic fuzzy sets and on interval-valued fuzzy sets. Szmidt and Kacprzyk [14] used a different approach from that of Burillo and Bustince to introduce the entropy measure for IFS based on the geometric interpretation of IFS and the distances between them. Zeng and Li [15] considered the axioms of Szmidt and Kacprzyk using the notion of IVFSs and discussed the relationship between similarity measure and entropy. Hung and Yang [16] gave their axiomatic definition of entropy for IFSs by exploiting the concept of probability. Vlachos and Sergiadis [17] proposed a measure of intuitionistic fuzzy entropy and revealed an intuitive and mathematical connection between the notions of entropy for fuzzy set and intuitionistic fuzzy set. Zhang and Jiang [18] proposed intuitionistic (vague) fuzzy entropy by means of intersection and union of the membership degree and nonmembership degree of the intuitionistic (vague) fuzzy set.Having come to rather rigid measures, it is natural to see other possible measures and those having parametric generalization that covers existing measures and leads to many more. The parameters involved give flexibility in applications and their values can be determined from the data itself or experiment. In this paper, a new entropy measure for IFSs is defined. This is called “intuitionistic fuzzy entropy of order-α.”The paper is organized as follows. In Section2, some basic definitions related to fuzzy set theory and intuitionistic fuzzy set theory are briefly given. In Section 3, a new entropy measure called “intuitionistic fuzzy entropy of order-α” is proposed, and its axiomatic justification is established. Some mathematical properties of the proposed measure are also studied in this section. In Section 4, a numerical example is given to demonstrate the effectiveness of the proposed measure of intuitionistic fuzzy entropy with existing intuitionistic fuzzy entropies [13–18] by the comparison. Brief conclusions are presented in Section 5.


## 2. Preliminaries

In the following, some needed basic concepts and definitions related to fuzzy sets and intuitionistic fuzzy sets are introduced.Definition 1 (fuzzy set [1]).
A fuzzy setA~ in a finite universe of discourse X={x1,x2,…,xn} is given by
(1)A~={〈x,μA~(x)〉∣x∈X},
where μA~:X→[0,1] is the membership function of A~. The number μA~(x) describes the degree of belongingness of x∈X in A~.
De Luca and Termini [3] defined fuzzy entropy for a fuzzy set A~ corresponding to Shannon’s entropy [4] as
(2)H(A~)=-1n∑i=1n[μA~(xi)log⁡(μA~(xi))mmmmmc+(1-μA~(xi))log⁡(1-μA~(xi))].
Later, Bhandari and Pal [8] made a survey of information measures on fuzzy sets and proposed some new measures of fuzzy entropy. Corresponding to Rènyi entropy [9], they defined the following measure:
(3)Hα(A~)=1n(1-α)∑i=1nlog⁡[μA~α(xi)+(1-μA~(xi))α];α≠1,α>0. Atanassov [11, 12], as mentioned earlier, generalized Zadeh’s idea of fuzzy sets, by what is called intuitionistic fuzzy sets, defined as follows.Definition 2 (intuitionistic fuzzy set [11]).
An intuitionistic fuzzy setA in a finite universe of discourse X={x1,x2,…,xn} is given by
(4)A={〈x,μA(x),νA(x)〉∣x∈X},
where
(5)μA:X⟶[0,1],νA:X⟶[0,1]
with the condition
(6)0≤μA(x)+νA(x)≤1,∀x∈X.
In this definition numbers μA(x) and νA(x), respectively, denote thedegree of membership anddegree of nonmembership of x∈X to the set A.
For each IFSA in X, if πA(x)=1-μA(x)-νA(x), x∈X, then πA(x) represents the degree of hesitance of x∈X to the set A. πA(x) is also calledintuitionistic index.Obviously, whenπA(x)=0, that is, νA(x)=1-μA(x) for every x in X, then IFS set A becomes fuzzy set. Thus, fuzzy sets are the special cases of IFSs.Definition 3 (set operations on intuitionistic fuzzy sets [12]).
LetIFS(X) denote the family of all IFSs in the universe X, and let A,B∈IFS(X) given by
(7)A={〈x,μA(x),νA(x)〉∣x∈X},B={〈x,μB(x),νB(x)〉∣x∈X}.
Then usual set relations and operations are defined as follows:(i)
A⊆B if and only if μA(x)≤μB(x) and νA(x)≥νB(x) for all x∈X;(ii)
A=B if and only if A⊆B and B⊆A;(iii)
AC={〈x,νA(x),μA(x)〉∣x∈X};(iv)
A∩B={〈μA(x)∧μB(x)andνA(x)∨νB(x)〉∣x∈X};(v)
A∪B={〈μA(x)∨μB(x)andνA(x)∧νB(x)〉∣x∈X}.Szmidt and Kacprzyk [14] extended the axioms of De Luca and Termini [3] for proposing the entropy measure in the setting of IFSs.Definition 4 (see [14]).
An entropy onIFS(X) is a real-valued functional E:IFS(X)→[0,1], satisfying the following four axioms.(P1)
E(A)=0 if and only if A is a crisp set; that is, μA(xi)=0, νA(xi)=1 or μA(xi)=1, νA(xi)=0 for all xi∈X.(P2)
E(A)=1 if and only if μA(xi)=νA(xi) for all xi∈X.(P3)
E(A)≤E(B) if and only if A⊆B, that is, if μA(xi)≤μB(xi) and νA(xi)≥νB(xi), for μB(xi)≤νB(xi), or if μA(xi)≥μB(xi) and νA(xi)≤νB(xi), for μB(xi)≥νB(xi) for any xi∈X.(P4)
E(A)=E(AC).Corresponding to (2), the De Luca and Termini [3] entropy, Vlachos and Sergiadis [17] introduced a measure of intuitionistic fuzzy entropy as
(8)EVS(A)=-1n∑i=1n[μA(xi)log⁡μA(xi)+νA(xi)log⁡νA(xi)mmmmmc-(1-πA(xi))log⁡(1-πA(xi))mmmmmc-πA(xi)].
Throughout this paper, we will denote the set of all intuitionistic fuzzy sets in X by IFS(X) and by FS(X) the set of all fuzzy sets defined in X.With these background ideas and concepts, we, in the next section, introduce a new measure called “intuitionistic fuzzy entropy of order-α” for intuitionistic fuzzy sets, which has a parameter.


## 3. Intuitionistic Fuzzy Entropy of Order-α

Definition 5.
The intuitionistic fuzzy entropy of order-α for intuitionistic fuzzy set A is defined as
(9)Eα(A)=1n(1-α)∑i=1nlog⁡[21-απA(xi)(μAα(xi)+νAα(xi))mmmmmmmmmmc×(μA(xi)+νA(xi))1-αmmmmmmmmmmc+21-απA(xi)],α>0,α≠1.Theorem 6.
TheEα(A) measure in (9) of the intuitionistic fuzzy entropy of order-α is an entropy measure for IFSs; that is, it satisfies all the axioms given in Definition 4 above.Proof.
(P1) Let A be a crisp set with membership values being either 0 or 1 for all xi∈X. Then from (9) we obtain that
(10)Eα(A)=0.Next, if Eα(A)=0, that is,(11)1n(1-α)∑i=1nlog⁡[(μAα(xi)+νAα(xi))(μA(xi)+νA(xi))1-αmmmmmmmmcc+21-απA(xi)(μA(xi)+νA(xi))1-α]=0
or
(12)(μAα(xi)+νAα(xi))(μA(xi)+νA(xi))1-α+21-απA(xi)=1∀xi∈X(μAα(xi)+νAα(xi))(μA(xi)+νA(xi))1-α-21-α(μA(xi)+νA(xi))=1-21-α∀xi∈X(μA(xi)+νA(xi))×[(μAα(xi)+νAα(xi))(μA(xi)+νA(xi))α-21-α]=1-21-α,∀xi∈X.
Since α>0, α≠1, therefore (12) will hold only if μA(xi)=0, νA(xi)=1, or μA(xi)=1, νA(xi)=0, for all xi∈X.
Hence,Eα(A)=0 if and only if μA(xi)=0, νA(xi)=1, or μA(xi)=1, νA(xi)=0, for all xi∈X. This proves (P1).
(P2) First let μA(xi)=νA(xi) for all xi∈X. Then from (9) we have
(13)Eα(A)=1n(1-α)∑i=1nlog⁡[(μA(xi)+μA(xi))1-α(μAα(xi)+μAα(xi))mmmmmmmmmmc×(μA(xi)+μA(xi))1-αmmmmmmmmmmc+21-α(1-2μA(xi))(μA(xi)+μA(xi))1-α]=1n(1-α)∑i=1nlog⁡[22-αμA(xi)+21-α-22-αμA(xi)]=1n(1-α)×n(1-α)=1.
Next, let Eα(A)=1; that is,
(14)∑i=1nlog⁡[(μAα(xi)+νAα(xi))(μA(xi)+νA(xi))1-αmmmmc+21-απA(xi)(μA(xi)+νA(xi))1-α]=n(1-α)
or
(15)log⁡[(μAα(xi)+νAα(xi))(μA(xi)+νA(xi))1-αmmcc+21-απA(xi)(μA(xi)+νA(xi))1-α]=(1-α)∀xi∈X
or
(16)[(μAα(xi)+νAα(xi))(μA(xi)+νA(xi))1-αc+21-α(1-μA(xi)-νA(xi))(μA(xi)+νA(xi))1-α]=21-α∀xi∈X
or
(17)(μA(xi)+νA(xi))×[μAα(xi)+νAα(xi)2-(μA(xi)+νA(xi)2)α]=0.mmmmmmmmmmmmmmmmmmmmmmmc∀xi∈X.
Equation (17) will hold if
(18)μA(xi)+νA(xi)=0⟹μA(xi)=νA(xi)=0∀xi∈X
or
(19)[μAα(xi)+νAα(xi)2-(μA(xi)+νA(xi)2)α]=0.
Now consider the following function:
(20)f(z)=zαwherez∈[0,1].
Differentiating (20) with respect to z, we get
(21)f′(z)=αzα-1,f′′(x)=α(α-1)zα-2.
Since f′′(z)>0 when α>1 and f′′(z)<0 when α<1, f(z) is convex (∪) or concave (∩) function according to α>1 or α<1.
Therefore, for any two pointsz1 and z2 in [0,1], the following inequalities hold:
(22)f(z1)+f(z2)2-f(z1+z22)≥0forα>1,f(z1)+f(z2)2-f(z1+z22)≤0forα<1
with the equality holding in (22) only for z1=z2. Therefore, from (18), (19), and (22), we conclude that (17) holds only if μA(xi)=νA(xi) for all xi∈X.
(P3) In order to show that (9) satisfies (P3), it suffices to prove that the function
(23)g(x,y)=1(1-α)log⁡[(xα+yα)(x+y)1-αmmmmmmcm+21-α(1-x-y)(x+y)1-α],
where x,y∈[0,1], is an increasing function with respect to x and decreasing with respect to y. Taking the partial derivatives of g with respect to x and y, respectively, yields
(24)∂g(x,y)∂x=(1-α)(x+y)-α(xα+yα)+α(x+y)1-αxα-1-21-α(1-α)(x+y)1-α(xα+yα)+21-α(1-x-y),(25)∂g(x,y)∂y=(1-α)(x+y)-α(xα+yα)+α(x+y)1-αyα-1-21-α(1-α)(x+y)1-α(xα+yα)+21-α(1-x-y).For critical point of g, we set ∂g(x,y)/∂x=0 and ∂g(x,y)/∂y=0. This gives(26)x=y.
Also, from (24) and (26), we have
(27)∂g(x,y)∂x≥0whenx≤y,0<α<1as  also  forα>1,∂g(x,y)∂x≤0whenx≥y,0<α<1as  also  forα>1,
for any x,y∈[0,1]. Thus g(x,y) is increasing with respect to x for x≤y and decreasing when x≥y.
Similarly, we see that(28)∂g(x,y)∂y≤0,whenx≤y,0<α<1as  also  forα>1,∂g(x,y)∂y≥0,whenx≥y,0<α<1asalsoforα>1.
Let us now consider two sets A,B∈IFS(X), with A⊆B. Assume that the finite universe of discourse X={x1,x2,…,xn} is partitioned into two disjoint sets X1 and X2 with X1∪X2=X.
Let us further suppose that allxi∈X1 are dominated by the condition
(29)μA(xi)≤μB(xi)≤νB(xi)≤νA(xi),
while for all xi∈X2 satisfy
(30)μA(xi)≥μB(xi)≥νB(xi)≥νA(xi).
Then from the monotonicity of the function g and (9), we obtain that Eα(A)≤Eα(B) when A⊆B.
(P4) It is clear that AC={〈x,νA(x),μA(x)〉∣x∈X} for xi∈X; that is
(31)μAC(xi)=νA(xi),νAC(xi)=μA(xi).
Then, from (9), we straightforwardly have
(32)Eα(A)=Eα(AC).
Hence, Eα(A) is a measure of intuitionistic fuzzy entropy.
This proves the theorem.The proposed intuitionistic fuzzy entropy of order-α satisfies the following additional properties.Theorem 7.
LetA and B be two intuitionistic fuzzy sets defined in X={x1,x2,…,xn}, where A={〈xi,μA(xi),νA(xi)〉∣xi∈X}, B={〈xi,μB(xi),νB(xi)〉∣xi∈X}, such that they satisfy for any xi∈X either A⊆B or A⊇B; then one has
(33)Eα(A∪B)+Eα(A∩B)=Eα(A)+Eα(B).Proof.
Let us separateX into two parts X1 and X2, where
(34)X1={xi∈X:A⊆B},X2={xi∈X:A⊇B}.
That is, for all xi∈X1,
(35)μA(xi)≤μB(xi),νA(xi)≥νB(xi)
and, for all xi∈X2,
(36)μA(xi)≥μB(xi),νA(xi)≤νB(xi).
From (9), we then have
(37)Eα(A∪B)=1n(1-α)∑i=1nlog⁡[(μA∪B(xi)+νA∩B(xi))1-α(μA∪Bα(xi)+νA∩Bα(xi))mmmmmmmmcmcc×(μA∪B(xi)+νA∩B(xi))1-αmmmmmmmmcmcc+21-α(1-μA∪B(xi)-νA∩B(xi))(μA∪B(xi)+νA∩B(xi))1-α]=1n(1-α)[{∑x∈X1log⁡[(μB(xi)+νB(xi))1-α(μBα(xi)+νBα(xi))mmmmmmmmmmmmcc×(μB(xi)+νB(xi))1-αmmmmmmmmmmmmcc+21-α(1-μB(xi)-νB(xi))(μB(xi)+νB(xi))1-α]∑x∈X1log⁡[(μBα(xi)+νBα(xi))(μB(xi)+νB(xi))1-α}mmmmcmccm+{∑x∈X2log⁡[(μA(xi)+νA(xi))1-α(μAα(xi)+νAα(xi))mmmmmmmmmmmmmmm×(μA(xi)+νA(xi))1-αmmmmmmmmmmmmmmm+21-α(1-μA(xi)mmmmmmmmmmmmmmmmcmmm-νA(xi))(μA(xi)+νA(xi))1-α]∑x∈X2log⁡[(μAα(xi)+νAα(xi))(μA(xi)+νA(xi))1-α}],Eα(A∩B)=1n(1-α)∑i=1nlog⁡[(μA∩B(xi)+νA∪B(xi))1-α(μA∩Bα(xi)+νA∪Bα(xi))mmmmmmmmmmc×(μA∩B(xi)+νA∪B(xi))1-αmmmmmmmmmmc+21-α(1-μA∩B(xi)-νA∪B(xi))(μA∩B(xi)+νA∪B(xi))1-α]=1n(1-α)[{∑x∈X1log⁡[(μA(xi)+νA(xi))1-α(μAα(xi)+νAα(xi))mmmmmmmmmmmmc×(μA(xi)+νA(xi))1-αmmmmmmmmmmmmc+21-α(1-μA(xi)-νA(xi))(μA(xi)+νA(xi))1-α]∑x∈X1log⁡[(μAα(xi)+νAα(xi))(μA(xi)+νA(xi))1-α}mmmmmmm+{∑x∈X2log⁡[(μB(xi)+νB(xi))1-α(μBα(xi)+νBα(xi))mmmmmmmmmmmmmm×(μB(xi)+νB(xi))1-αmmmmmmmmmmmmmm+21-α(1-μB(xi)mmmmmmmmmmmmmmmmmcm-νB(xi))(μB(xi)+νB(xi))1-α]∑x∈X2log⁡[(μBα(xi)+νBα(xi))(μB(xi)+νB(xi))1-α}].
Now from (37), we get
(38)Eα(A∪B)+Eα(A∩B)=Eα(A)+Eα(B).
This proves the theorem.Corollary 8.
For anyA∈IFS(X) and AC the complement of intuitionistic fuzzy set A,
(39)Eα(A)=Eα(AC)=Eα(A∪AC)=Eα(A∩AC).Theorem 9.
E
α
(
A
) attains the maximum value when the set is most intuitionistic fuzzy set and the minimum value when the set is crisp set; moreover, maximum and minimum values are independent of α.Proof.
It has already been proved thatEα(A) is maximum if and only if A is most intuitionistic fuzzy set, that is, μA(xi)=νA(xi), for all xi∈X, and minimum when A is a crisp set. So, it is enough to prove that the maximum and minimum values are independent of α.
LetA be the most intuitionistic fuzzy set; that is, μA(xi)=νA(xi), for all xi∈X. Then
(40)Eα(A)=1n(1-α)∑i=1nlog⁡[(νA(xi)+νA(xi))1-α(νAα(xi)+νAα(xi))mmmmmmmmmmcc×(νA(xi)+νA(xi))1-αmmmmmmmmmmcc+21-α(1-νA(xi)-νA(xi))(νA(xi)+νA(xi))1-α]=1n(1-α)∑i=1nlog⁡[21-α]=1,
which is independent of α.
On the other hand, ifA is a crisp set, that is, μA(xi)=1 and νA(xi)=0 or μA(xi)=0 and νA(xi)=1, for all xi∈X, then Eα(A)=0 for any value of α.
This proves the theorem.Particular and Limiting Cases
(1)
Whenα→1, then measure in (9) reduces to measure in (3).
(2)
It may be noticed that if an intuitionistic fuzzy set is an ordinary fuzzy set, that is, for allxi∈X, νA(xi)=1-μA(xi), then the intuitionistic fuzzy entropy of order-α reduces to fuzzy entropy of order-α [13].In the next section, we consider an example to demonstrate the performance of proposed intuitionistic fuzzy entropy of order-α by comparing with other existing measures of intuitionistic fuzzy entropy.


## 4. Numerical Example

LetA={(xi,μA(xi),νA(xi))∣xi∈X} be an IFS in X={x1,x2,…,xn}. For any positive real number n, De et al. [19] defined the IFS An as follows:
(41)An={(xi,(μA(xi))n,1-(1-νA(xi))n)∣xi∈X}.
Using the above operation, they also defined the concentration and dilation of A given by
(42)Concentration:  CON(A)=A2,Dilation:  DIL(A)=A1/2.
Like fuzzy sets, CON(A) and DIL(A) may be treated as “very (A)” and “more or less (A),” respectively.Example 1.
Let us consider an IFSA in X={6,7,8,9,10}, defined by De et al. [19], as
(43)A={(6,0.1,0.8),(7,0.3,0.5),(8,0.5,0.4),mm(9,0.9,0.0),(10,1.0,0.0)}.
Using the operation defined in (41), we can generate the following IFSs:
(44)A1/2,A2,A3,A4.
By taking into account the characterization of linguistic variables, we regarded A as “LARGE” in X. Using the above operation, we have
(45)A1/2={(6,0.3162,0.5528),(7,0.5477,0.2929),mm(8,0.7071,0.2254),(9,0.9487,0.0),mm(10,1.0,0.0)},A2={(6,0.0100,0.9600),(7,0.0900,0.7500),mm(8,0.2500,0.6400),(9,0.8100,0.000),mm(10,1.0,0.0)},A3={(6,0.0010,0.9920),(7,0.0270,0.8750),mm(8,0.1250,0.7840),(9,0.7290,0.000),mm(10,1.0,0.0)},A4={(6,0.0001,0.9984),(7,0.0081,0.9375),mm(8,0.0625,0.8704),(9,0.6591,0.000),mm(10,1.0,0.0)}.
The levels represented by the above intuitionistic fuzzy sets are described as follows:A1/2 may be treated as “More or less LARGE,”A2 may be treated as “Very LARGE,”A3 may be treated as “Quite very LARGE,”A4 may be treated as “Very very LARGE.”Now we use these IFSs to compare the intuitionistic fuzzy entropy of order-α and other existing measures of intuitionistic fuzzy entropy.For the purpose of comparison, we first mention here some entropy measures for intuitionistic fuzzy sets defined by various researchers asBurillo and Bustince’s entropy [13]:
(46)EBB(A)=1n∑i=1nπA(xi),
Zeng and Li’s entropy [15]:
(47)EZL(A)=1-1n∑i=1n|μA(xi)-νA(xi)|,
Szmidt and Kacprzyk’s entropy [14]:
(48)ESK(A)=1n∑i=1n(μA(xi)∧νA(xi)+πA(xi)μA(xi)∨νA(xi)+πA(xi)),
Zhang and Jiang’s entropy [18]:
(49)EZJ(A)=1n∑i=1n(μA(xi)∧νA(xi)μA(xi)∨νA(xi)),
and Hung and Yang’ entropies [16]:
(50)Ehc2(A)=1n∑i=1n(1-(μA2(xi)+νA2(xi)+πA2(xi))),Er1/2(A)=2n∑i=1nlog⁡(μA1/2(xi)+νA1/2(xi)+πA1/2(xi)).
From the viewpoint of mathematical operations, the entropy values of the above defined IFSs, A1/2, A, A2, A3, A4, have the following requirement:
(51)E(A1/2)>E(A)>E(A2)>E(A3)>E(A4).
The values of different entropy measures under A1/2, A, A2, A3, and A4 are shown in Table 1.Table 1
Values of the different entropy measures underA1/2,A,A2,A3,A4.
A
1
/
2
A
A
2
A
3
A
4
E
BB
0.0923
0.1200
0.1320
0.1344
0.1360
E
ZL
0.3788
0.3600
0.3160
0.2320
0.1911
E
SK
0.3194
0.3073
0.3010
0.2121
0.1758
E
VS
0.5067
0.4931
0.3746
0.2969
0.2476
E
ZJ
0.2486
0.2117
0.2261
0.0949
0.0457
E
hc
2
0.3276
0.3400
0.2903
0.2525
0.2258
E
r
1
/
2
0.6609
0.6809
0.6045
0.5216
0.4567
E
0.3
0.5944
0.5885
0.5090
0.4159
0.3428
E
0.8
0.5633
0.5373
0.3973
0.2647
0.1988
E
2
0.5039
0.4721
0.3185
0.1618
0.1144
E
5
0.4272
0.4186
0.2719
0.0991
0.0646
E
10
0.3373
0.2996
0.2529
0.1514
0.0921
E
15
0.3216
0.2838
0.2468
0.1389
0.0354
E
50
0.2999
0.2616
0.2389
0.1217
0.0657
E
100
0.2941
0.2571
0.2373
0.1182
0.0627Based on Table1, we see that the entropy measures EZL, ESK, EVS, E0.3, E0.8, E2, E5, E10, E15, and E100 satisfy the requirement given in (51), but EBB, EZJ, Ehc2, and Er1/2 do not satisfy the requirement.Therefore, the behavior of intuitionistic fuzzy entropy of order-α, Eα(A), is good for the viewpoint of structured linguistic variables.


## 5. Conclusions

This work introduces a new entropy measure called intuitionistic fuzzy entropy of order-α in the setting of intuitionistic fuzzy set theory. Some properties of this measure have been also studied. This measure generalizes Bhandari and Pal [8] fuzzy entropy of order-α and Vlachos and Sergiadis [17] logarithmic intuitionistic fuzzy entropy. Introduction of parameter provides new flexibility and wider application of intuitionistic fuzzy entropy to different situations.



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*Source: 789890-2014-09-03.xml*