Exponentialentropyonintuitionisticfuzzysets (1)

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Exponential entropy on intuitionistic fuzzy sets
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Kybernetika
Rajkumar Verma; Bhu Dev Sharma
Exponential entropy on intuitionistic fuzzy sets
Kybernetika, Vol. 49 (2013), No. 1, 114--127
Persistent URL: http://dml.cz/dmlcz/143243
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K Y B E R N E T I K A — V O L U M E 4 9 ( 2 0 1 3 ) , N U M B E R 1 , P A G E S 1 1 4 – 1 2 7
EXPONENTIAL ENTROPY ON INTUITIONISTIC
FUZZY SETS
Rajkumar Verma and Bhu Dev Sharma
In the present paper, based on the concept of fuzzy entropy, an exponential intuitionistic
fuzzy entropy measure is proposed in the setting of Atanassov’s intuitionistic fuzzy set theory.
This measure is a generalized version of exponential fuzzy entropy proposed by Pal and Pal.
A connection between exponential fuzzy entropy and exponential intuitionistic fuzzy entropy is
also established. Some interesting properties of this measure are analyzed. Finally, a numerical
example is given to show that the proposed entropy measure for Atanassov’s intuitionistic fuzzy
set is consistent by comparing it with other existing entropies.
Keywords: fuzzy set, fuzzy entropy, Atanassov’s intuitionistic fuzzy set, intuitionistic
fuzzy entropy, exponential entropy
Classification: 94A17
1. INTRODUCTION
The theory of fuzzy sets proposed by Zadeh [16] in 1965 has gained wide applications in
many areas of science and technology e. g. clustering, image processing, decision making
etc. because of its capability to model non-statistical imprecision or vague concepts.
Fuzziness brings in a feature of uncertainty. The first attempt to quantify the fuzziness
was made in 1968 by Zadeh [17], who introduced a probabilistic framework and defined
the entropy of a fuzzy event as weighted Shannon entropy [11] (but this measure was
not found adequate for measuring the fuzziness of a fuzzy event). In 1972, De Luca
and Termini [5] formulated axioms which the fuzzy entropy measure should comply, and
they defined the entropy of a fuzzy set based on Shannon’s function. It may be regarded
as the first correct measure of fuzziness of a fuzzy set.
Atanassov [1] introduced the notion of ‘Atanassov’s intuitionistic fuzzy set’, which
is a generalization of the concept of fuzzy set. Burillo and Bustince [3] defined the en-
tropy on Atanassov’s intuitionistic fuzzy set and on interval-valued fuzzy set. Vlachos
and Sergiagis [13] proposed a measure of intuitionistic fuzzy entropy and revealed an
intuitive and mathematical connection between the notions of entropy for fuzzy set and
Atanassov’s intuitionistic fuzzy set. Zhang and Jiang [18] defined a measure of intu-
itionistic (vague) fuzzy entropy on Atanassov’s intuitionistic fuzzy sets by generalizing
of the De Luca Termini [5] logarithmic fuzzy entropy.
In this paper, we propose a new information measure for Atanassov’s intuitionistic

Exponential entropy on intuitionistic fuzzy sets 115
fuzzy sets. We call it exponential intuitionistic fuzzy entropy. It is based on the concept
of exponential fuzzy entropy defined by Pal and Pal [9]. To define this entropy function
fuzzy set theoretic approach has been used. Such an approach is found particularly
useful in situations where data is available in terms of intuitionistic fuzzy set values
but implementation requirements are only fuzzy. So far the practice has been to simply
ignore the hesitation part. A better result has been obtained by not ignoring but by
merging the hesitation part suitably. We suggest a mathematical method for it. This
may help application of IFS data in industry, where the tools used are of fuzzy set theory.
The paper is organized as follows: In Section 2 some basic definitions related to
probability, fuzzy set theory and Atanassov’s intuitionistic fuzzy set theory are briefly
discussed. In Section 3 a new information measure called, ‘exponential intuitionistic
fuzzy entropy’ is proposed, which satisfies the axiomatic requirements [12]. Some math-
ematical properties of the proposed measure are then studied in this section. In Section 4
a numerical example is given comparing our measure with other entropies proposed in
[14] and [18].
2. PRELIMINARIES
In this section we present some basic concepts related to probability theory, fuzzy sets
and Atanassov’s intuitionistic fuzzy sets, which will be needed in the following analysis.
First, let us cover probabilistic part of the preliminaries.
Let ∆n = {P = (p1, . . . , pn) : pi ≥ 0,
Pn
i=1 pi = 1}, n ≥ 2 be a set of n-complete
probability distributions.
For any probability distribution P = (p1, . . . , pn) ∈ ∆n, Shannon’s entropy [11], is
defined as
H(P) = −
n
X
i=1
pi log pi. (1)
It is to be noted from the logarithmic entropic measure (1) that as pi → 0, it’s
corresponding self information of this event, I(pi) = −log(pi) → ∞ but I(pi = 1) =
−log(1) = 0. Thus we see that self information of an event has conceptual problem, as
in practice, the self information of an event, whether highly probable or highly unlikely,
is expected to lie between two finite limits.
Some advantages for considering exponential entropy: In Shannon’s theory,
which is widely acclaimed, we find that the measure of self information of an event with
probability pi is taken as log(1/pi), a decreasing function of pi. The same decreasing
character alternatively may be maintained by considering it as a function of (1 − pi)
rather than of (1/pi).
The additive property, which is considered crucial in Shannon’s approach, of the self
information function for independent events may not have a strong relevance (impact) in
practice in some situations. Alternatively, as in the case of probability law, the joint self
information may be product rather than sum of the self informations in two independent
cases.
The above considerations suggest the self information as an exponential function of
(1 − pi).

116 R. K. VERMA AND B. D. SHARMA
Based on the these considerations, Pal and Pal [9] proposed another measure called
exponential entropy given by
eH(P) =
n
X
i=1
pie(1−pi)
− 1. (2)
These authors point out that the exponential entropy has an advantage over Shannon’s
entropy. For example, for the uniform probability distribution P = ( 1
n , 1
n , . . . , 1
n ), expo-
nential entropy has a fixed upper bound
lim
n→∞
H
1
n
,
1
n
, . . . ,
1
n

= e − 1 (3)
which is not the case for Shannon’s entropy.
Definition 2.1. Fuzzy Set: A fuzzy set Ã defined in a finite universe of discourse
X = (x1, . . . , xn) is given by (Zadeh [16]):
Ã = {hx, µÃ(x)i |x ∈ X}, (4)
where µÃ : X → [0, 1] is the membership function of Ã. The number µÃ(x) describes
the degree of membership of x ∈ X to Ã.
De Luca and Termini [5] defined fuzzy entropy for a fuzzy set Ã corresponding (1) as
H(Ã) = −
1
n
n
X
i=1
h
µÃ(xi) log µÃ(xi) + 1 − µÃ(xi)

log 1 − µÃ(xi)
i
. (5)
Fuzzy exponential entropy for fuzzy set Ã corresponding to (2) has also been introduced
by Pal and Pal [9] as
eH(Ã) =
1
n(
√
e − 1)
n
X
i=1
h
µÃ(xi)e1−µÃ(xi)
+ (1 − µÃ(xi))eµÃ(xi)
− 1
i
. (6)
Further, Atanassov [1] generalized the idea of fuzzy sets, by what is called Atanassov’s
intuitionistic fuzzy sets, defined as follows:
Definition 2.2. Atanassov’s Intuitionistic Fuzzy Set: An Atanassov’s intuitionistic
fuzzy set A in a finite universe of discourse X = (x1, . . . , xn) is given by:
A =

hx, µA(x), νA(x)i |x ∈ X

, (7)
where
µA : X → [0, 1] and νA : X → [0, 1] (8)
with the condition
0 ≤ µA(x) + νA(x) ≤ 1, ∀ x ∈ X. (9)
The numbers µA(x) and νA(x) denote the degree of membership and degree of non-
membership of x ∈ X to A, respectively.

Definition 2.3. Hesitation Margin: For each Atanassov’s intuitionistic fuzzy set A in
X, if
πA(x) = 1 − µA(x) − νA(x), (10)
then πA(x) is called the Atanassov’s intuitionistic index (or a hesitation degree) of the
element x ∈ X to A.
For studying sets, there is need to consider set relations and operations, which in the
study of Atanassov’s intuitionistic fuzzy sets are defined as follows.
Definition 2.4. Set Operations on Atanassov’s Intuitionistic Fuzzy Set[2]: Let AIFS(X)
denote the family of all Atanassov’s intuitionistic fuzzy sets in the universe X, and let
A, B ∈ AIFS(X) given by
A = {hx, µA(x), νA(x)i |x ∈ X},
B = {hx, µB(x), νB(x)i |x ∈ X},
then some set operations can be defined as follows:
(i) A ⊆ B iff µA(x) ≤ µB(x) and νA(x) ≥ νB(x) ∀ x ∈ X;
(ii) A = B iff A ⊆ B and B ⊆ A;
(iii) AC
= {hx, νA(x), µA(x)i |x ∈ X};
(iv) A ∪ B = {hx, (µA(x) ∨ µB(x)), (νA(x) ∧ νB(x))i |x ∈ X};
(v) A ∩ B = {hx, (µA(x) ∧ µB(x)), (νA(x) ∨ νB(x))i |x ∈ X};
(vi) A = {hx, µA(x), 1 − µA(x)i |x ∈ X};
(vii) ♦A = {hx, 1 − νA(x), νA(x)i |x ∈ X};
(viii) A@B =
n D
x,
µA + µB
2 , νA + νB
2
E
|x ∈ X
o
.
Method for Transforming AIFSs into FSs: Li, Lu and Cai [8], as briefly outlined
below, proposed a method for transforming ‘Atanassov’s intuitionistic fuzzy sets’ (vague
sets) into ‘fuzzy sets’ by distributing hesitation degree equally with membership and
non-membership.
Definition 2.5. Let A = {hx, µA(x), νA(x)i |x ∈ X} be an Atanassov’s intuitionistic
fuzzy set defined in a finite universe of discourse X. Then the fuzzy membership function
µ ˜
A∗ (x) to ˜
A∗ ( ˜
A∗ be the fuzzy set corresponding to Atanassov’s intuitionistic fuzzy set
A) is defined as:
µ ˜
A∗ (x) = µA(x) +
πA(x)
2
=
µA(x) + 1 − νA(x)
2
. (11)

This area of study has attracted quite some attention for applications in decision-making.
Finally we may as well mention some other related measures with which we compare
our study.
Zhang and Jiang [18] presented a measure of intuitionistic (vague) fuzzy entropy
based on a generalization of measure (5) as
EZJ (A) = −
1
n
n
X
i=1

µA(xi) + 1 − νA(xi)
2

log

2

+

νA(xi) + 1 − µA(xi)
2

log

2

. (12)
Ye [15] introduced two effective measures of intuitionistic fuzzy entropy based on a
generalization of the fuzzy entropy defined by Prakash et al. [10] given by
EJY 1(A) =
1
n
n
X
i=1

sin π

4

+ sin π

4

− 1

×
1
√
2 − 1

, (13)
EJY 2(A) =
1
n
n
X
i=1

cos π

4

+ cos π

4

− 1

×
1
√
2 − 1

. (14)
Later, Wei et al. [14] have shown that the two entropy functions (13) and (14) proposed
by Ye [15] are mathematically the same and gave a simplified version as
EW GG(A) =
1
n
n
X
i=1

√
2 cos π

µA(xi) − νA(xi)
4

− 1

×
1
√
2 − 1

. (15)
Throughout this paper, we denote the set of all Atanassov’s intuitionistic fuzzy sets in
X by AIFS(X). Similarly, FS(X) is the set of all fuzzy sets defined in X.
In the next section we introduce an entropy measure on Atanassov’s intuitionistic
fuzzy sets called “exponential intuitionistic fuzzy entropy” corresponding to (6) and
verify axiomatic basis of the same.
3. EXPONENTIAL INTUITIONISTIC FUZZY ENTROPY
Let A be an Atanassov’s intuitionistic fuzzy set defined in the finite universe of discourse,
X = (x1, . . . , xn). Then, according to the Definition 2.5, an Atanassov’s intuitionistic
fuzzy set can be transformed into a fuzzy set to structure an entropy measure of the
intuitionistic fuzzy set by means of
µ ˜
A∗ (xi) = µA(xi) +
πA(xi)
2
=
2
.

Then, in analogy with the definition of exponential fuzzy entropy given in (6), we propose
the exponential intuitionistic fuzzy entropy measure for Atanassov’s intuitionistic fuzzy
set A as follows:
eE(A) =
1
n(
√
e − 1)
n
X
i=1

2

e1−
µA(xi)+1−νA(xi)
2

+

1 −
2

e
µA(xi)+1−νA(xi)
2

− 1

, (16)
which can also be written as
eE(A) =
1
n(
√
e − 1)
n
X
i=1

2

e
νA(xi)+1−µA(xi)
2

+

2

e
µA(xi)+1−νA(xi)
2

− 1

. (17)
In the next theorem, we establish properties that according to Szmidt and Kacprzyk
[12], justify our proposed measure to be a bonafide/valid ‘intuitionistic fuzzy entropy’:
Theorem 3.1. The eE(A) measure in (17) of the exponential intuitionistic fuzzy en-
tropy satisfies the following propositions:
P1. eE(A) = 0 iff A is a crisp set, i. e., µA(xi) = 0, νA(xi) = 1 or µA(xi) = 1, νA(xi) = 0
for all xi ∈ X.
P2. eE(A) = 1 iff µA(xi) = νA(xi) for all xi ∈ X.
P3. eE(A) = eE(A) iff A ≤ B, i. e., µA(xi) ≤ µB(xi) and νA(xi) ≥ νB(xi), for
µB(xi) ≤ νB(xi) or µA(xi) ≥ µB(xi) and νA(xi) ≤ νB(xi), for µB(xi) ≥ νB(xi)
for any xi ∈ X.
P4. eE(A) = eE(AC
).
P r o o f . P1. Let A be a crisp set with membership values being either 0 or 1 for all
xi ∈ X. Then from (17) we simply obtain that
eE(A) = 0. (18)
Now, let
2
= zA(xi). (19)
In view of (19), expression in (17) can be written as
eE(A) =
1
n(
√
e − 1)
n
X
i=1
h
zA(xi)e1−zA(xi)
+ (1 − zA(xi))ezA(xi)
− 1
i
. (20)

From Pal and Pal [9], we know that (20) becomes zero if and only if zA(xi) = 0 or 1,
∀ xi ∈ X i. e.,
(µA(xi) + 1 − νA(xi))
2
= 0 i. e., νA(xi) − µA(xi) = 1, ∀ xi ∈ X (21)
or
(µA(xi) + 1 − νA(xi))
2
= 1 i. e., µA(xi) − νA(xi) = 1, ∀ xi ∈ X. (22)
And from Definition 2.2,we have
µA(xi) + νA(xi) ≤ 1, ∀ xi ∈ X. (23)
Now solving equation (21) with (23), we get
µA(xi) = 0, νA(xi) = 1, ∀ xi ∈ X.
Next solving equation (22) with (23), we get
µA(xi) = 1, νA(xi) = 0, ∀ xi ∈ X.
Therefore eE(A) reduces to zero only if either µA(xi) = 0, νA(xi) = 1 or µA(xi) = 1,
νA(xi) = 0 for all xi ∈ X, proving the result.
P2. Let µA(xi) = νA(xi) for all xi ∈ X. From (17) we obtain eE(A) = 1.
From equation (20), we have
eE(A) =
1
n
n
X
i=1
f(zA(xi)),
where
f(zA(xi)) =
hzA(xi)e1−zA(xi)
− 1
(
√
e − 1)
i
∀ xi ∈ X. (24)
Now, let us suppose that eE(A) = 1, i. e.
1
n
n
X
i=1
f(zA(xi)) = 1
or
f(zA(xi)) = 1 ∀ xi ∈ X. (25)
Differentiating (25) with respect to zA(xi) and equating to zero, we get
∂f
∂(zA(xi))
= e1−zA(xi)
− zA(xi)e1−zA(xi)
− ezA(xi)
= 0
or
(1 − zA(xi))e1−zA(xi)
= zA(xi)ezA(xi)
∀ xi ∈ X. (26)

Using the fact that f(x) = xex
is a bijection function, we can write
(1 − zA(xi)) = zA(xi) ∀ xi ∈ X (27)
or
zA(xi) = 0.5 ∀ xi ∈ X (28)
and find
h ∂2
f
∂(zA(xi))2
i
zA(xi)=0.5
0 ∀ xi ∈ X. (29)
Hence f(zA(xi)) is a concave function and has a global maximum at zA(xi) = 0.5. Since
eE(A) = 1
n
Pn
i=1 f(zA(xi)), So eE(A) attains the maximum value when zA(xi) = 0.5 or
µA(xi) = νA(xi) for all xi ∈ X.
P3. In order to show that (17) fulfills P3, it suffices to prove that the function
g(x, y) =
hx + 1 − y
2

e( y+1−x
2 )
+
y + 1 − x
2

e( x+1−y
2 )
− 1
i
(30)
where x, y ∈ [0, 1], is increasing with respect to x and decreasing for y. Taking the
partial derivatives of g with respect to x and y, respectively, yields
∂g
∂x
=
1
2
y + 1 − x
2

e
y+1−x
2

−

x + 1 − y
2

e
x+1−y
2

(31)
∂g
∂y
=
1
2
x + 1 − y
2

e
x+1−y
2

−

y + 1 − x
2

e
y+1−x
2

(32)
In order to find critical point of g, we set ∂g
∂x = 0 and ∂g
∂y = 0. This gives
x = y. (33)
From (31) and (33), we have
∂g
∂x
≥ 0, when x ≤ y (34)
and
∂g
∂x
≤ 0, when x ≥ y (35)
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 obtain that
∂g
∂y
≤ 0, when x ≤ y (36)
and
∂g
∂y
≥ 0, when x ≥ y. (37)

Let us now consider two sets A, B ∈ IFS(X) with A ⊆ B. Assume that the finite
universe of discourse X = (x1, . . . , xn) is partitioned into two disjoint sets X1 and X1
with X1 ∪ X2.
Let us further suppose that all xi ∈ X1 are dominated by the condition
µA(xi) ≤ µB(xi) ≤ νB(xi) ≤ νA(xi),
while for all xi ∈ X2
µA(xi) ≥ µB(xi) ≥ νB(xi) ≥ νA(xi).
Then from the monotonicity of g(x, y) and (17), we obtain that eE(A) ≤ eE(B) when
A ⊆ B.
P4. It is clear that AC
= {hx, νA(xi), µA(xi)i |x ∈ X} for all xi ∈ X, i. e.,
µAC (xi) = νA(xi) and νAC (xi) = µA(xi)
then, from (17) we have
eE(A) = eE(AC
).
Hence eE(A) is a valid measure of Atanassov’s intuitionistic fuzzy entropy.
This proves the theorem.
Particular case: It is interesting to notice that if an Atanassov’s intuitionistic fuzzy
set is an ordinary fuzzy set, i. e., for all xi ∈ X, νA(xi) = 1−µA(xi), then the exponential
intuitionistic fuzzy entropy reduces to exponential fuzzy entropy as proposed in [9].
We now turn to study of properties of eE(A). The proposed exponential intuitionistic
fuzzy entropy eE(A), just like fuzzy entropy measure, satisfies the following interesting
properties.
Theorem 3.2. Let A and B two Atanassov’s intuitionistic fuzzy sets in a finite universe
of discourse X = (x1, . . . , xn), where A(xi) = hµA(xi), νA(xi)i , B(xi) = hµB(xi), νB(xi)i
such that they satisfy for any xi ∈ X either A ⊆ B or A ⊇ B, then we have
eE(A ∪ B) + eE(A ∩ B) = eE(A) + eE(B).
P r o o f . Let us separate X into two parts X1 and X2, where
X1 = {xi ∈ X : A(xi) ⊆ B(xi)} and X2 = {xi ∈ X : A(xi) ⊇ B(xi)}.
That is, for all xi ∈ X1
µA(xi) ≤ µB(xi) and νA(xi) ≥ νB(xi) (38)
and for all xi ∈ X2
µA(xi) ≥ µB(xi) and νA(xi) ≤ νB(xi). (39)

From definition in (17), we have
eE(A ∪ B) =
1
n(
√
e − 1)
n
X
i=1

µA∪B(xi) + 1 − νA∩B(xi)
2

e
νA∩B(xi)+1−µA∪B(xi)
2

+

νA∩B(xi) + 1 − µA∪B(xi)
2

e
µA∪B(xi)+1−νA∩B(xi)
2

− 1

=
1
n(
√
e − 1)
X
xi∈X1

µB(xi) + 1 − νB(xi)
2

e
νB(xi)+1−µB(xi)
2

+

νB(xi) + 1 − µB(xi)
2

e
µB(xi)+1−νB(xi)
2

− 1

+
X
xi∈X2

2

e
νA(xi)+1−µA(xi)
2

+

2

e
µA(xi)+1−νA(xi)
2

− 1
#
. (40)
Again from definition in (17), we have
eE(A ∩ B) =
1
n(
√
e − 1)
n
X
i=1

µA∩B(xi) + 1 − νA∪B(xi)
2

e
νA∪B(xi)+1−µA∩B(xi)
2

+

νA∪B(xi) + 1 − µA∩B(xi)
2

e
µA∩B(xi)+1−νA∪B(xi)
2

− 1

=
1
n(
√
e − 1)
X
xi∈X1

2

e
νA(xi)+1−µA(xi)
2

+

2

e
µA(xi)+1−νA(xi)
2

− 1

+
X
xi∈X2

µB(xi) + 1 − νB(xi)
2

e
νB(xi)+1−µB(xi)
2

+

νB(xi) + 1 − µB(xi)
2

e
µB(xi)+1−νB(xi)
2

− 1
#
. (41)
Now adding (40) and (41), we get
eE(A ∪ B) + eE(A ∩ B) = eE(A) + eE(B).

Theorem 3.3. For every A ∈ AIFS(X),
(i) A@♦A is a fuzzy set;
(ii) (A@♦A) = (♦A@A);
(iii) (A@♦A) = Ã∗
;
where Ã∗
is the fuzzy set corresponding to Atanassov’s intuitionistic fuzzy set A.
P r o o f . (i) From Definition 2.4, we have
A = {hx, µA(x), 1 − µA(x)i |x ∈ X}; (42)
♦A = {hx, 1 − νA(x), νA(x)i |x ∈ X}. (43)
Now taking @ with (42) and (43), we get
A@♦A =
n
x,
µA + 1 − νA
2
,
νA + 1 − µA
2

|x ∈ X
o
. (44)
It can be easily observed that,
µA + 1 − νA
2
+
νA + 1 − µA
2
= 1 x ∈ X.
(ii) It obviously follows from equation (44).
(iii) From equations (44) and (11), we have
A@♦A =

x,
µA + 1 − νA
2
,
νA + 1 − µA
2

|x ∈ X

;
Ã∗
=

x,
µA + 1 − νA
2
,
νA + 1 − µA
2

|x ∈ X

.
eE(A) = eE(A@♦A).
P r o o f . From equation (17), we have
eE(A) =
1
n(
√
e − 1)
n
X
i=1

2

e
νA(xi)+1−µA(xi)
2

+

2

e
µA(xi)+1−νA(xi)
2

− 1

and
eE(A@♦A) =
1
n(
√
e − 1)
n
X
i=1

2

e
νA(xi)+1−µA(xi)
2

+

2

e
µA(xi)+1−νA(xi)
2

− 1

. (45)
eE(A@♦A) = eE(♦A@A).
P r o o f . It readily follows from Theorem 3.3(ii)and equation (45).
eE(A@♦A) = eE(((A)C
@(♦A)C
)C
).
P r o o f . From equation (45), we have
eE(A@♦A) =
1
n(
√
e − 1)
n
X
i=1

2

e
νA(xi)+1−µA(xi)
2

+

2

e
µA(xi)+1−νA(xi)
2

− 1

and
eE(((A)C
@(♦A)C
)C
) =
1
n(
√
e − 1)
n
X
i=1

2

e
νA(xi)+1−µA(xi)
2

+

2

e
µA(xi)+1−νA(xi)
2

− 1

.
In the next section we consider an example to compare our proposed entropy measure
on Atanassov’s intuitionistic fuzzy set, with others in (12) and (15).
4. NUMERICAL EXAMPLE
Example: Let A =

hxi, µA(xi), νA(xi)i |xi ∈ X

be an AIFS in X = (x1, . . . , xn).
For any positive real number n, De et al. [6] defined the AIFS An
as follows:
An
=

hxi, [µA(xi)]n
, 1 − [1 − νA(xi)]n
i |xi ∈ X

.
We consider the AIFS A on X = (x1, . . . , xn) defined as:
A = {h6, 0.1, 0.8i , h7, 0.3, 0.5i , h8, 0.5, 0.4i , h9, 0.9, 0.0i , h10, 1.0, 0.0i}.
By taking into consideration the characterization of linguistic variables, De et al. [6]
regarded A as “LARGE” on X. Using the above operations, we have

A1/2
for may be treated as “More or less LARGE”
A2
for may be treated as “Very LARGE”
A3
for may be treated as “Quite very LARGE”
A4
for may be treated as “Very very LARGE”
Now we consider these AIFSs to compare the above entropy measures. It may be
mentioned that from logical consideration, the entropies of these AIFSs are required to
follow the following order pattern:
E(A1/2
) E(A) E(A2
) E(A3
) E(A4
). (46)
Calculated numerical values of the three entropy functions for these cases are given in
the table below:
A1/2
A A2
A3
A4
EZJ 0.5819 0.5720 0.4333 0.3321 0.2698
EW GG 0.4545 0.4377 0.3029 0.2159 0.1709
Ee 0.5531 0.5343 0.3772 0.2734 0.2169
Table: Values of the different entropy measures under A1/2
, A, A2
, A3
, A4
.
Based on the Table, we see that the entropy measures EZJ and EW GG satisfy (46),
and our proposed entropy measure conforms to the same, i. e.
eE(A1/2
) eE(A) eE(A2
) eE(A3
) eE(A4
).
Therefore, the behavior of exponential intuitionistic fuzzy entropy eE(A) is also consis-
tent for the viewpoint of structured linguistic variables.
5. CONCLUSIONS
In this work, we have proposed a new entropy measure called exponential intuitionistic
fuzzy entropy in the setting of Atanassov’s intuitionistic fuzzy set theory. This measure
can be considered as a generalized version of exponential fuzzy entropy proposed by Pal
and Pal [10]. This measure is imbued with several properties. A numerical example is
given to illustrate the effectiveness of proposed entropy measure. Parametric studies that
introduce other flexibility criteria for the same membership functions, of this measure
are also under study and will be reported separately.
ACKNOWLEDGEMENTS
The authors are grateful to the associate editor and anonymous referees for their valuable
suggestions which helped in improving the presentation of the paper.
(Received June 9, 2011)

R E F E R E N C E S
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Inform. Sci. 178 (2008), 21, 4184–4191.
Rajkumar Verma, Department of Mathematics Jaypee Institute of Information Technology Uni-
versity Noida-201307, (U.P.). India.
e-mail: rkver83@gmail.com
Bhu Dev Sharma, Department of Mathematics Jaypee Institute of Information Technology Uni-
versity Noida-201307, (U.P.). India.
e-mail: bhudev.sharma@jiit.ac.in, bhudev sharma@yahoo.com
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