Research Journal of Applied Sciences, Engineering and Technology 4(13): 1850-1856, 2012
ISSN: 2040-7467
© Maxwell Scientific Organization, 2012
Submitted: November 21, 2011
Accepted: January 04, 2012
Published: July 01, 2012
A Method for Defuzzification Based on General Translation
R. Saneifard and S. Khodaei
Department of Applied Mathematics, Urmia Branch, Islamic Azad University, Oroumieh, Iran
Abstract: The problem of defuzzification is examined in this study from a broader perspective as a special way
of dealing with the general problem of retranslation. The study includes an overview of different formulations
of the problem of defuzzification, as well as an overview of methods that have been suggested in the literature
to deal with the problem. Our own approach to defuzzification, which is described in the study with details, is
based on relevant measures of uncertainty-based information. This study is a companion to our recent study
that addresses the general problem of retranslation in calculating concepts. Another application of this
defuzzification noticed at the end of this article.
Key words: Correlation, defuzzification, fuzzy intervals, fuzzy sets
INTRODUCTION
The emergence of computer technology in the second
half of the 20th century opened many new possibilities for
machines. These possibilities have been discussed in
various contexts quite extensively in the literature. One
aspect of considerable interest has been a comparison of
existing and prospective capabilities of machines with
those of human beings. An overall observation at this time
is that the range of machine capabilities has visibly
expanded over the past years, from numerical
computation to symbol manipulation, processing of visual
data, learning from experience, etc. Moreover, machines
have become superior to humans in some specific
capabilities, such as large-scale processing of numerical
data, massive combinatorial searches of various kinds,
complex symbol manipulation, or sophisticated graphics.
Some important new areas have emerged due to these
machine capabilities, such as fractal geometry, cellular
automata or evolutionary computing. In spite of the
impressive advances of machines, they are still not able to
match some capabilities of human beings. Perhaps the
most exemplary of them are the remarkable and very
complex perceptual abilities of the human mind, which
allow humans to use perceptions in purposeful ways to
perform complex tasks. Although current machines are
not capable of reasoning and acting on the basis of
perceptions, a feasible research program for developing
this capability was recently proposed by (Zadeh, 1999).
The crux of this program is to approximate perceptions by
statements in natural language and, then, to use fuzzy
logic to represent these statements and deal with them as
needed. This approach, to develop perception-based
machines, is referred in the literature as computing with
words, which is a name suggested also by (Zadeh, 1996)
previously. Approximating statements in natural language
by propositions in fuzzy logic may be viewed as a
translation from natural language to a formalized
language. Alternatively, it may be viewed as a linguistic
approximation of the first kind. As it is well known, this
translation (or approximation) is strongly context
dependent. Once it is accomplished in the context of a
given application, all available resources of fuzzy logic in
the broad sense can be utilized to emulate the ordinary
(commonsense) human reasoning that pertains to the
application (Yager et al., 1987; Bezdek et al., 1999). It is
quite obvious that any relevant background knowledge
should also be utilized in the reasoning, regardless of the
nature of the reasoning process; its consequents are fuzzy
propositions, of which involves one or more fuzzy sets. In
order to connect these fuzzy propositions to perceptions
(i.e., to convey appropriate perceptions), we need to
approximate them by statements in natural language. This
means, in turn, that we need to express each of involved
fuzzy sets with a linguistic expression in natural language
that has an understandable meaning in the given context.
These issues pertain to the second kind of linguistic
approximation, which may conveniently be called a
retranslation. While the problem of translation has been
extensively studied and discussed in the literature, the
problem of retranslation is far less developed. Prior to the
late 1990s, this problem had been recognized only by a
few authors, among them (Eshragh and Mamdani, 1979;
Novak, 1989). More recently, the problem has been
addressed more substantially by Dvorak (1990), Yager
(2004), Delgado et al. (2006) and Saneifard (2009a). In
our previous study (Allahviranloo et al., 2011), we
discuss the retranslation problem and explore some
approaches to dealing with it. In this study, our primary
objective is to look at the well-known problem of
Corresponding Author: R. Saneifard, Department of Applied Mathematics, Urmia Branch, Islamic Azad University, Oroumieh,
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Res. J. Appl. Sci. Eng. Technol., 4(13): 1850-1856, 2012
defuzzification as one way, perhaps the simplest one, of
dealing with retranslation.
BASIC DEFINITIONS AND NOTATIONS
In this study, we assume that the reader is familiar
with basics of fuzzy set theory and fuzzy logic in the
broad sense. For the sake of completeness, we introduce
in this section only those concepts that are relevant to our
discussion of the retranslation problem. We denote all
fuzzy sets in this study by capital letters. Classical sets are
viewed as special fuzzy sets, called crisp sets and are thus
denoted by capital letters as well. For simplicity, we
consider only numerical linguistic variables, states that are
expressed by normal and convex fuzzy sets which are
defined in some given closed interval, X = [x1, x2], of
real numbers. These fuzzy sets, usually referred as fuzzy
intervals, are viewed as concave functions from X to
[0, 1] whose maxima are 1.
Definition 1: (Novak, 1989). For any fuzzy interval
A:X÷[0, 1], its "-cut, A", for each "0[0, 1] the closed
interval is as follows:
A" = {x0X*A(x)$"}
Definition 2: (Novak, 1989). For each given fuzzy
interval A, the canonical form is as follow:
 a L ( x ), where x  [a , b)
 1,
where x  [b, c]

A( x )  
a
(
x
),
where x  (c, d ]
 R

0, otherewise
C
f ( 21 )  0
C
f (0)  f (1)  1
1
C
 f ( )d 
0
T" = [a+(b!a)", d-(d-c)"]
where a"L and a"R are the inverse function of aL and aR,
respectively. The crisp sets:
(3)
Every trapezoidal fuzzy interval T is thus uniquely
characterized via the quadruple T = (a, b, c, d). A special
case in which b = c, that is called a triangular fuzzy
interval, is also employed in this study. An important
concept for dealing with the problem of retranslation is
the degree of subsection, s(AfB) of fuzzy set A in fuzzy
set B (both defined on the same interval X), which is
expressed by the formula (Zadeh, 1999):
s( A  B ) 
(2)
1
2
In most examples in this study, we use trapezoidal
fuzzy intervals, T, in which "L and "R are linear
functions. That is, aL(x) = x-a/b-a and aR(x) = d-x/d-c.
Then, for each a0(0, 1] there is:
(1)
where x0X = [x1, x2] and ", b, c, d are real numbers in X
such that a#b#c#d, aL is a continuous and increasing
function from aL(a) = 0 to aL(b) = 1 and aR is a continuous
decreasing function from aR(c) = 1 to aR(d) = 0. For each
value of "0[0, 1], the a-cut of A, A", is a closed interval
of real numbers defined by the formula:
A" = [a"L, a"R]
") = f(1/2+") for all "0[0, 1/2], which reaches its
minimum in 1/2, is called the bi-symmetrical weighted
function. Moreover, the bi-symmetrical weighted function
is called regular if:
 min{ A( x), B( x)}dx
 A( x)dx
(4)
The minimum operator in this formula represents the
standard intersection of fuzzy sets. As it is well known,
this is the only intersection of fuzzy sets that is cut worthy
in the sense that(A1B)" = A"1B", hold for all "0[0, 1].
To deal with the retranslation problem, we also need
to measure the non-specificity and fuzziness of the fuzzy
sets involved.
Definition 4: For any given normal and convex fuzzy set
A, we define a well-justified measure of non-specificity,
NS, as follow:
Supp(A) = {x0 X|A(x)>0}
Core(A) = {x0 X|A(x) = 1}
1
NS ( A) 
are called, respectively, a support of A and a core of A.
Clearly, Supp(A) = (a, d) and Core (A) = (b, c).
Definition 3: (Grzegorzewski and Mrowka, 2005) A
function f:[0, 1]÷[0, 1] symmetric around 1/2 i.e., f(1/2-
 f ( ) Log [1  L
2
m
( A )]2 d
(5)
0
where f:[0, 1]÷[0, 1] is a bi-symmetrical (regular)
weighted function (Grzegorzewski and Mrowka, 2005).
One can, of course, propose many regular bi-symmetrical
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Res. J. Appl. Sci. Eng. Technol., 4(13): 1850-1856, 2012
weighted functions and hence obtain different bisymmetrical weighted distances. Further on we will
consider mainly a following function:
1

1  2 , where   [0, 2 ]
f ( )  
 2  1, where   [ 1 , 1]

2
(6)
Example 1: Let G be a fuzzy number with membership
function as follow: X = [-10, 10]:
x
when x  [0,2)
2,

 2.4  0.7 x , when x  [2,3)

G ( x )   0.3,
when x  [3,4)
15
.  0.3x , when x  [4,5)

otherwise
 0,

15  10

], when   (0, 0.3]
[2 ,
3
G 
[2 , 24  10 ], when   (0.3, 1]

3
a
(7)
2
0.3
Definition 6: The degree of validity of choosing a
standard fuzzy interval A that has a linguistic
interpretation for representing a given convex fuzzy set G,
v(F|G), as the degree to which G is contained in F, define
as follow:
 X min{G( x), F ( x)}dx
 G( x)dx
(11)
There is:
NS (G ) 
Clearly, f(A) = 0 if and only if A is a crisp (classical)
set and the maximum degree of fuzziness is obtained for
the unique fuzzy set in which A(x) = 1-A(x) = 0.5 for all
x0 X. Two criteria that are considered in this study as
essential are valid and informative.
v( F | G) 
(10)
Clearly,
Definition 5: Given a convex fuzzy set A, its fuzziness,
F(A), can be measured by the overlap of A and its
complement. Using standard operations of
complementation and intersection of fuzzy sets, we have:
 X min{ A( x), 1  A( x)}dx
(9)
Clearly, if i(F1)$i(F2) then F1 is preferable to F2
according to informativeness.
In Eq. (5), Lm(A") denotes for each "0[0, 1] the
Lebesgue measure of A". In this case, Lm(A") is the
length of the interval A" for each "0[0, 1]. The reason for
choosing logarithm base 2 in this formula is to measure
ambiguity in a convenient measurement unit: NS(A) = 1
when Lm(A") = 1. Calculating ambiguity is more
complicated for fuzzy sets defined on Rn when n>1, but
this is beyond the scope of this study. Function NS is a
special case of a more general measurement of nonspecificity (applicable to convex subsets of the ndimensional Euclidean space), which is called a Hartleylike measure (Yager, 2004).
f ( A) 
NS ( F )
Log 2 [1  Lm ( X )]
i( F )  1 
(8)
X
For any pair of standard fuzzy intervals, F1 and F2,
that compete for representing a given fuzzy set G, clearly,
if v(F1|G)$v(F2|G) then F1 is preferable to F2 according to
validity.
Definition 7: The degree of informative of F, i(F), is
concerned and is define as the normalized reduction of
non-specificity with respect to the non-specificity of X as
follow:

0
 15  16 
Log2 1 
 d 
3


1
 Log21  (24  24 ) d  2.6317
2
0.3
and
i (G)  1 
NS (G )
 0.4008
Log2 (21)
DEFUZZIFICATION (AN OVERVIEW OF
PROPOSED METHODS)
The term “defuzzification”, as we use it in this study
(i.e., as a replacement of a given fuzzy set by a
representative crisp set) firstly employed by Recasens
et al. (1999). Although this study corrected the
terminology in some sense and attracted attention to the
problem of representing fuzzy sets by crisp sets, this
problem was already addressed in a special way more
than ten years earlier by Bezdek et al. (1999) in their
research about the interval-valued mean of a fuzzy
interval. Given a fuzzy interval A with its "-cut A" =
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a
 
L , aR
 for all "0[0, 1], the interval-valued mean of
Res. J. Appl. Sci. Eng. Technol., 4(13): 1850-1856, 2012
A is, according to Bezdek et al. (1999), the crisp interval
N(A) = [nL, nR]. where, n L 
a

L
d and
nR 
[ 0,1]
a

R
d .
[ 0 ,1]
Since N(A) is based on viewing A as a random set, it is
sometimes referred to as expected interval of A (Hong
et al., 1995) or a probabilistic mean interval (Saneifard
et al., 2007). It is significant that the interval N(A) was
also obtained by (Hung et al., 2002) Saneifard (2009b) by
computing the average level of A via the Novak (1989)
interval and by Grzegorzewski and Mrowka (2005) as the
best crisp approximation of A in terms of the minimum
Euclidean distance between N(A) and A. Recognizing the
defuzzification via the Aumann integral is a kind of
averaging procedure, Delgado et al. (2006) try to identify
reasonable requirements for averaging operations in this
context for prospective defuzzification procedures on this
basis. Carlsson et al. (2005) further discuss the
requirements and argue that a continuity-type requirement
is also needed. Zadeh (1999) shows the interval N(A) that
nR-nL = IX A(x)dx and asks the question: is N(A) well
placed in relation to A among other crisp intervals with
the same length is N(A)?. He then calculates the crisp
interval H(A) = [hL, hR] such that hR-hL = nR-nL and
minimized the Hamming distance between A and H(A).
An alternative crisp interval, B(A), for representing a
given fuzzy interval A was introduced by (Carlsson et al.,
2005) and further investigated by Yager (2004). This
interval, B(A) = [bL, bR], where bL = 2I[0, 1] a"L"d" and bR
= 2I[0,1] a"R"d", is based on possibilistic interpretation of
A and it is called a possibilistic mean interval of A. Its left
and right endpoints are called, respectively, the
possibility-weighted averages of the minima and maxima
of the "-cuts of A.
In a study that well surveys the literature on crisp
interval and point representations of fuzzy intervals,
Saneifard (2007) introduces, in addition to the intervals
N(A) and B(A), the concept of a median interval of A,
M(A) = [mL, mR], where,
mL

b
A( x )dx 
a

mL
mR
d
c
mR
A( x )dx ,  A( x)dx   A( x)dx
and a, b, c, d are the real numbers employed in the
canonical form of A expressed by Eq. (1). In addition, she
defines the concept of a central of A, C(A)=[cL, cR], as:
C(A) = N (A)1B(A)1M(A)
investigates percentile characterizations of "-cuts in
connection with three properties of each given fuzzy set,
its height, width and cardinality. One study which its title
contains the term “defuzzification” (Bezdek et al., 1999)
is actually dealing with neither defuzzification nor
disambiguation, but with the problem of standardization
discussed in our previous study (Saneifard, 2009a, b). A
method is introduced in this study for converting a general
fuzzy set on X to the symmetric triangular fuzzy number
that is nearest to the given fuzzy set according to a
specific metric distance.
Ambiguity-preserving defuzzification: In the context of
the retranslation problem, defuzzification (as viewed in
this study) is a very special way of standardization in the
sense introduced in our previous study (Saneifard et al.,
2007). The aim of defuzzification in this context is to
eliminate linguistic uncertainty (fuzziness) while
preserving information-based uncertainty (ambiguity).
That is, given a fuzzy set G , we want to find a crisp set F
(a defuzzification of G) such that the ambiguities of G
and F are equal. That is, we require that NS(F) = NS(G),
where NS is defined by Eq. (5). When G is fuzzy interval,
this equation has the form:
f ( ) Log2 [1  Lm ( F )]2 
1
 f ( )Log [1  (G )] d
2
0
where the right-hand side is a constant. While the length
of the sought crisp interval F is determined by solving this
equation for Lm(F), the location remains undetermined. To
determine by it, we need to employ some additional
criteria. The most important among these criteria is the
criterion of validity. We define the degree of validity of F
with respect to G, as the degree to which G is contained
in F Formally, (8). The crisp interval F can be viewed as
a fully valid approximation of G if only if F contains G
and, hence, V(F|G) = 1. However, requiring full validity
of F with respect to G would need that F be the support
for G and this implies that Lm(F)>Lm(G). In order to
satisfy Eq. (12), clearly, we need to sacrifice some
validity, but it is desirable to sacrifice as little of it as
possible. This results in a two-stage defuzzification
procedure:
C
C
or, alternatively,
CL(A) = max [nL, bL, mL], CR(A) = min [nL, bL, mL]
In her more recent study, Saneifard (2007) focuses an
approximation of fuzzy sets by their specific "-cuts. She
(12)
2
Determine Lm(F) by solving Eq. (12).
Determine the location of crisp interval F whose
length is Lm(F) for which v(F|G) is maximized.
This procedure can be implemented in slightly
different ways depending on G. Let us examine some of
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Res. J. Appl. Sci. Eng. Technol., 4(13): 1850-1856, 2012
them. The proposed defuzzification procedure is
particularly easy to implement when the canonical
representation of the given fuzzy interval, expressed by
Eq. (1), is such that function ""L is strictly increasing and
function a"R is strictly decreasing. In this case, there
exists an "-cut of G, G"*, such that, Lm(G"*) = Lm(F),
where Lm(F) is obtained by solving Eq. (12). Moreover,
among all crisp intervals whose length is Lm(F), G"* is the
only one with maximal validity. In the discussed case
(when ""L and "dR are strictly monotone) A"* can be
determined more directly by solving the " equation:
when functions a"L and a"R in the canonical representation
of a given fuzzy interval A are not strictly monotone, no
"-cut of A may exist that satisfies Eq. (13). This is due to
discontinuities in the "-cut representation. In this case, the
defuzzified interval F with the length Lm(F) determined by
Eq. (12) and maximum validity can be found at one of the
levels of discontinuity and may not be unique. These
issues are further discussed in the context of the following
example.
Example 3: Consider a fairly complex fuzzy interval A
defined by the following "-cut representation:
f ( ) Log2 [1  Lm ( Aa* )]2
1

 f ( )Log2[1  ( A )]2 d
.  ], when 
[2.5 , 13.3  17
[0.8 , 13.3  17
.  ], when 

Aa  
125
2
5
6
[
.

.
,

  ], when 

[  35
. , 6   ],
when 
(13)
0
 [0.0, 0.2]
 [0.2, 0.6]
 [0.6, 0.8]
. ]
 [0.8, 10
for ". This is illustrated in the following example.
Example 2: Let us consider a fuzzy interval A discussed
in Ralescu (2000, 2002), where,
when x  [0, 0.5]
 2 x,

2
A( x )   4( x  x ), when x  [0.5,1]
 0,
otherwise

and
A  [ 0.5 , 0.5(1 
1   )  0.5 ]
To obtain the "-cut that has the same ambiguity, we
first calculate the integral on the right-hand side of Eq.
(13) to obtain Lm(A):
Using Eq. (5), we determine that NS(A) = 2.15621.
Solving Eq. (13) for each of the four range of the "-cuts
of A gives a solution that is outside the respective range
and, hence, is not valid. Solving now Eq. (12), we find
that Lm(F) = 3.45742. This value must fit into one of the
three "-cut discontinuities, at " = 0.2, " = 0.6 or " = 0.8.
It fits only in the discontinuity at " = 0.6. At this level, the
left-hand plateau is the smaller one and has the range [3,
3.25]. In order to maximize validity, the left-end point of
F, fR, is then equal to fL+Lm(F). That is, fR = fL+3.45742.
Hence all crisp intervals in the range from [3, 6.45742]
to[3.25, 6.70742] are acceptable defuzzifications: they all
preserve the ambiguity NS (A) and maximize the validity
v(F|A). A choice of a unique defuzzification from this
range may be based on additional criteria. If no additional
criteria are employed, it is reasonable to consider as a
defuzzification average of the whole range, F"vg. In this
example:
Favg 
1

 Log 2 [1  0.5(1  1   )  0.5 ]2 d  235833
.
0
Applications: In this section, the researchers introduce
some of applications of the central interval approximation
of fuzzy numbers. These indices can be applied for
comparison of fuzzy numbers namely fuzzy correlations
in fuzzy environments and expert’s systems.
we solve the equation:
 Log 2 [1  0.5(1  1   )  0.5 ]2  235833
.
for ". The solution (obtained by Mathematica) is " =
0.559462. The "-cut of A for this value of A, Ad*, is then
taken as the defuzzification of A:
F(A)A"* = [0.274781, 0.835573]
([3,6.4574]  [3.25,6.7074])
. , 6.58]
 [315
2
Correlation between fuzzy numbers: In many
applications, the correlation between fuzzy numbers is
interested. Several authors have proposed different
measures of correlation between membership functions,
intuitionistic fuzzy sets and correlation (Ezzati and
Saneifard, 2010; Bustince and Burillo, 1995; Hung and
Wu, 2002) defined a correlation by means of expected
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Res. J. Appl. Sci. Eng. Technol., 4(13): 1850-1856, 2012
interval. They defined the correlation coefficient between
fuzzy numbers A and B as follow:
 ( A, B) 
E* ( A) E* ( B)  E *( A) E *( B)
E*2 ( A)  E *2 ( A) E*2 ( B)  E *2 ( B)
numbers such that for large values of k this article has:
lim  D ( A, B) 
k
(14)
a  a 32 b22  b32
Moreover if A and B be two triangular fuzzy numbers,
whenever k÷4, we have:
where,
1

1
[ E* ( A), E * ( B)]   L A ( )d , R A ( )d 


0

0


(15)
D ( A, B) 
This correlation coefficient shows not only the degree of
relationship between the fuzzy numbers, but also whether
these fuzzy numbers are positively or negatively related.
The researchers extend (14) by interval H(A) = [H*(A),
H*(A)] instead of (15), then:
 D ( A, B) 
H* ( A) H* ( B)  H * ( A) H * ( B)
H*2 ( A) H *2 ( A) H*2 ( B)  H *2 ( B)
(16)
Proof: The proof is obvious.
Example 4: Let A" = ["1+( "2- "1) ", "4 ( "4- "3) "] and
B" = [b1+(b2-b1) ", b4(b4-b3) "] be two trapezoidal fuzzy
numbers and f(r) = (k+1) "k, k = 1, 2, … is the weighted
function. Then:
PD ( A, B ) 
 ( k  1)a2  a1 )(( k  1)b2  b1 )  


 (( k  1)a3  a4 )(( k  1)b3  b4 ) 
a 2 b2
, a 2 , b2  0
 5x ,
0  x  0.2
1  5x ,0  x  0.2,

G6  
G7   2  5x , 0.2  x  0.4
 0, 0.2  x  1.
 0,
0.4  x  1

0  x  0.2,
0  x  0.4
 0,
 0,
 5x  2, 0.4  x  0.6
 5x  1, 0.2  x  0.4,


G8  
G9  
 4  5x , 0.6  x  0.8
 3  5x , 0.4  x  0.6,
 0,
 0,
0.6  x  1.
0.8  x  1
 0,
0  x  0.6,
0  x  0.8
 0,

G10   5x , 3 0.6  x  0.8, G11  
x
,
5
4
0.8  x  1


 5  5x 0.8  x  1.

Proposition 1: For any fuzzy numbers A and B0F we
have:
DD(A, B) = DD(B.A)
If A = B then DD(A, B) = 1
If A = cB for some c>0 DD(A, B) = 1
|DD(A, B)|#1
a 2 b2
Example 5: Yen (2002) used to six grade levels (6thgrade to 11th-grade) as student’s mathematical learning
progress. He assigned the linguistic values G6, G7, G8,
G9, G10 and G11, respectively to these grade levels,
respectively and transferred these linguistic values to
corresponding reasonable normal fuzzy numbers G6, G7,
G8, G9, G10 and G11, respectively with triangular
membership functions as follows:
DD (A, B) is called the weighted correlation coefficient
between two fuzzy numbers A and B . This correlation
coefficient lies in [-1, 1] and gives us more information
compared to correlation coefficient in (Bustince and
Burillo, 1995; Yu, 1993) and some others, which lie
within[0, 1]. It has all mentioned properties from
correlation coefficient that introduced in Yu (1993), the
researchers review properties DD(A, B) as follow:
C
C
C
C
a 2 b2  a 3b3
2
2
The researchers use the weighted correlation
coefficient to compute DD(G6, Gi),i = 7, 8, 9, 10, 11. The
result of comparison is summarized in Table 1. This
example shows that DD(G6, Gi), is decreasing in i,
intuitively. But the methods of (Hong and Hwang, 1995;
Murty et al., 1985; Sahnoun and Dieesare, 1991) have not
decreasing behavior. For our method weighted function f
interact on correlation value between two fuzzy numbers.
Hence, there is reasonable advantage in using the
proposed formula (16).
CONCLUSION

2
2
  (( k  1)a2  a1   (( k  1)a3  a4 ) 


 (( k  1)b  b )2  (( k  1)b  b )2 


2
1
3
4
The above example shows the parametric function
interacts on the correlation coefficient between two fuzzy
The correlation coefficients computed in this study,
which lie in [-1, 1], give us more information than the
correlation coefficients computed by Hong and Hwang
(1995), Murty et al. (1985) and Sahnoun and Dieesare
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Res. J. Appl. Sci. Eng. Technol., 4(13): 1850-1856, 2012
Table 1: Comparative results of example 5
DD(G6, G7)
Method
Hong and Hwang (1995)
0.852
Murty et al. (1985)
0.667
Sahnoun and Dieesare (1991)
0.224
Proposed method with k = 1
0.949
Proposed method with k = 2
0.894
DD(G6, G8)
0.778
0.500
0.000
0.857
0.814
(1991). The correlation coefficients computed by our
method which show us not only the degree of the
relationship between the intuitionist fuzzy sets, but also
the fact that these two sets are positive or negative related,
which are better than the correlation coefficients from
other methods, they review only the strength of the
relation.
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