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Ranking of Intuitionistic Fuzzy Number by
Centroid Point
Satyajit Das and Debashree Guha Department of Mathematics, Indian Institute of Technology, Patna, India
Email: {satyajit, debashree}@iitp.ac.in
Abstract—The notion of Intuitionistic Fuzzy Numbers (IFNs)
has been improved in many decision making problems.
Ranking of IFNs is one of the techniques that conceptualize
IFNs to illustrate order or preference in decision making.
Ranking of IFNs plays a very important role in multi-
criteria decision making, optimization and in many different
fields but ranking of IFNs is not a very easy process. As far
as our knowledge is concerned, the number of existing
methods for ranking of IFNs in the literature is very few. In
this paper a new method has been proposed for ranking of
IFNs by determining centroid point of IFNs. Examples have
been given to compare the proposed method with the
existing ranking results. The results show that the new
method can overcome the drawbacks of the existing
methods.
Index Terms—intuitionistic fuzzy number, ranking, centroid
point.
I. INTRODUCTION
The idea of intuitionistic fuzzy set (IFS) introduced by
Atanassov [1] is the generalization of Zadeh’s [2] fuzzy
set. An IFS is characterized by membership degree as
well as non-membership degree. Since its introduction,
the IFS theory has been studied and applied in different
areas including decision making. Now in modeling a
decision problem, ranking is a very important issue. In
this regard, many authors have paid considerable
attention to investigate the ranking methods of IFSs. In
1994 Chen and Tan [3] defined a score function of
intuitionistic fuzzy values (IFVs) for ranking IFVs. Li
and Rao [4] defined different types of score function to
compare IFVs. Some time two IFVs may have same
score value. In this situation ranking is not possible by
using score function. To overcome this case, Hong and
Choi [5] defined a new function known as accuracy
function of IFVs. Xu and Yager [6] used both the score
function and accuracy function for ranking IFVs.
However, it has been observed that the research
concentrated on finite universe of discourse only. In view
of this, recently the research on the concept of
intuitionistic fuzzy numbers (IFNs), with the universe of
discourse as the real line, has received attention and
definitions of IFNs [7]-[9] have been proposed. Further,
several ranking methods have also been proposed to solve
the ranking problems of IFNs. Chen and Hwang [10]
Manuscript received April 15, 2013; revised May 7, 2013.
introduced a crisp score function to rank IFNs. In 2008
Nayagam et al. [11] introduced a new score function for
ranking triangular intuitionistic fuzzy numbers (TIFNs)
and further they modified it in [12]. Jianqiang and Zhong
[13] used both the score function and accuracy function
to ranking TrIFNs. In case of inter-valued intuitionistic
fuzzy numbers (IVIFNs) Lee [14] proposed a novel
method for ranking of IVIFNs by utilizing score function
and deviation function. In 2008 Xu and Yager [15]
proposed a new method for ranking IFNs by determining
the distance from the IFNs to the positive and negative
ideal points. By calculating normalized Hamming
distance from IFNs to positive and negative ideal solution
a ranking method has been given by Wu and Cao [16]
and they applied it in multi attribute group decision
making problem. A method for comparing IFNs based on
metrics in the space of IFNs was proposed by
Grzegorzewski Wei and Tang [17] proposed a possibility
degree method for ranking IFNs. A new ranking method
was developed by Li [18] on the basis of the concept of a
ratio of the value index and ambiguity index of IFNs.
Rezvani [19] also proposed a ranking process of TrIFNs
by determining value and ambiguity of TrIFNs.
However, after analyzing the aforementioned ranking
procedures it has been observed that, for some cases, they
fail to calculate the ranking results correctly. Furthermore,
many of them produce different ranking outcomes for the
same problem. Under these circumstances, the decision
maker may not be able to carry out the comparison and
recognition properly. This creates problem in practical
applications. In order to overcome these problems of the
existing methods, a new method for ranking IFNs has
been proposed in this paper which is based on centroid
point of IFNs.
This paper has been organized as follows: In section-II
some basic concepts of IFNs have been reviewed.
Section-III represents the centroid formula for trapezoidal
intuitionistic fuzzy numbers (TrIFNs). This section also
describes the proposed ranking process of normal TrIFNs.
A set of examples have also been provided in section-IV,
to compare the proposed ranking method with the
existing methods. Some conclusions have been made in
section -V.
II. PRELIMINARIES
This section describes basic definition and some
arithmetic operations related to IFN.
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Journal of Industrial and Intelligent Information Vol. 1, No. 2, June 2013
©2013 Engineering and Technology Publishingdoi: 10.12720/jiii.1.2.107-110
Definition 1: [20] Let A is a TrIFN and its membership
and non-membership functions are defined as follows:
( ), ;
( )
, ;( )
( ), ;
( )
0 , .
A
x aw a x b
b a
w b x cx
d xw c x d
d c
x a or x d
…. (1)
( ) ( '), ' ;
( ')
, ;( )
( ) ( ' ), ';
( ' )
0 , ' '.
A
b x x a ua x b
b a
u b x cx
x c d x uc x d
d c
x a or x d
…. (2)
where 0 1; 0 1; 1; , , , , ', 'w u w u a b c d a d R
For sake of simplicity, throughout this paper we have
considered 'a a and '.d d Symbolically, then TrIFN
has been represented as ([ , , , ]; , ).A a b c d w u In
particular, if b c then TrIFN transform to TIFN.
Definition 2: [16] Let 1 1 1 1 1 1([ , , , ]; , )A a b c d w u and
2 2 2 2 2 2([ , , , ]; , )B a b c d w u be two TrIFNs and 0 be a
scalar, then
11 2 2 1 2 1 2 1 2 1 2 1 2) ([ , , , ]; , )i A B a a b b c c d d w w w w u u
1 2 1 2 1 2 1 2 1 2 1 2 1 2( ) ([ , , , ]; , )ii A B a a b b c c d d w w u u u u
1 1 1 1 1 1( ) ([ , , , , ];1 , )(1 )iii A a b c d w u
1 1 1 1 1 1( ) ([ , , , ]; , )(1 )iv a b c d w uA
III. NEW RANKING METHOD
Let ([ , , , ]; , )A a b c d w u be a TrIFN, which has been
shown in Fig-1. In order to find out the centroid of TrIFN,
the area under the membership and non membership
function has been considered together. First of all the
whole TrIFN has been split into five rectangles: ARUP,
REBU, EFCB, FSVC and SDQV where coordinates of
the corner points of rectangles have been given below:
: ( ,0), : ( , ), : ( , ), : ( ,0),A a B b w C c w D d
: ( ,0), : ( ,0), : ( ,1), : ( ,1),
: ( ) /( 1),0 ,
: ( ) /( 1),0 ,
: ( ) /( 1), /( 1) ,
: ( ) /( 1), /( 1) .
E b F c P a Q d
R aw au b w u
S dw du c w u
U aw au b w u w w u
V dw du c w u w w u
Now, the centroid point has been determined by using
the formulae ( )
( )
xf x dxX
f x dx
and
( )
( )
yg y dyY
g y dy
where the
specific region bounded by continuous function ( )f x and
( )g y respectively. The required centroid point ( , )A AX Y of
TrIFN A has been given below: 1
2A
x
xX , where
1
1
1
aw au bb c
w u L Law au ba b
w u
x xg dx xf dx xwdx
1
1
R
dw du cd
w u R dw du cc
w u
xf dx xg dx
(3)
and
1
1
1
1
2
R
aw au bbw u
L Law au ba
w u
c
b
dw du cdw u
R dw du cc
w u
x g dx f dx
wdx
f dx g dx
(4)
1,
2A
y
yY
where,
0
1 110 0
1
1 110 0
1
1 ( )
[ ]
[ ]L
wR L
w
w uR Rw
w u
w
w uLw
w u
y y h h dy
yd dy yh dy yk dy
yh dy yk dy ay dy
(5)
and
0
1 110 0
1
1 110 0
1
2 ( )
[ ]
[ ]L
wR L
w
w uR Rw
w u
w
w uLw
w u
y h h dy
d dy h dy k dy
h dy k dy a dy
(6)
where :[ , ] [0, ]Lf a b w and :[ , ] [0, ]Rf c d w are the left
and right part of the membership function of TrIFN .A
Figure 1. Trapezoidal intuitionistic fuzzy number
:[ , ] [0, ]Lg a b u and :[ , ] [0, ]Rg c d u are the left and
right part of the non-membership function of TrIFN
A which have been shown in Fig. 1. :[0, ] [ , ]Lh w a b and
:[0, ] [ , ]Rh w c d are the inverse functions of Lf and Rf respectively; :[0, ] [ , ]Lk u a b and :[0, ] [ , ]Rk u c d
are
the inverse functions of Lg and Rg respectively which
have been shown in Fig. 2. In case of TrIFN, functions
108
Journal of Industrial and Intelligent Information Vol. 1, No. 2, June 2013
©2013 Engineering and Technology Publishing
( ), ( ), ( )L R Lf x f x g x and ( )Rg x and their inverse functions
( ), ( ), ( )L R Lh y h y k y and ( )Rk y can be analytically expressed
as follows:
( )( ) , ;
( )
( )( ) , ;
( )
L
R
w x af x a x b
b a
w x df x c x d
c d
( ) ( )( ) , ;
( )
( ) ( )( ) , ;
( )
L
R
x b u a xg x a x b
a b
x c u d xg x c x d
d c
( )( ) , 0 ;
( )( ) , 0 ;
( ) ( )( ) , 1;
1
( ) ( )( ) , 1.
1
L
R
L
R
b a yh y a y w
w
d c yh y d y w
w
a b y b auk y u y
u
d c y c duk y u y
u
In particular if 1w and 0u then
2 2 2 2(3 3 ),
2(3 3 )
7( ) 5( )
18( ) 6( )
A
A
a b c d
a b c d
d a c b
d a c b
X
Y
(7)
It is known that XA denotes the representative location
of IFN A on the real line and YA presents the average
height of the IFN. In order to rank IFNs, the importance
of the degree of representative location is higher than the
average height. Therefore, the ranking may be done in the
following way [21]:
For any two different IFNs A and B , we have
(a)If A BX X , then A B ;
(b)If A BX X , then A B ;
(c)If A BX X , then
if A BY Y , then A B ;
else if A BY Y , then A B ;
else A BY Y , then A B .
We rank A and B based on their X’s values if they are
different. If their X’s values are equal then the attention
has been given to the Y’s values.
Figure 2. Inverse function of TrIFN
IV. COMPARISON WITH THE EXISTING METHOD
In this section some examples of IFNs have been
presented (see Table I) to compare the proposed ranking
process with the existing methods [16], [18], [20], [21]. A
comparison between the results of the proposed process
and the result of the existing methods has been illustrated
in Table I.
TABLE I. A COMPARISON OF THE PROPOSED RANKING PROCESS
WITH THE EXISTIN METHOD
The expressions of existing
ranking process
Examples The
proposed
method
Wu and Cao [18]
1 1 2 2
1 1 2 2
1 1 2 2
1 1 2 2
( , )
1[ (1 ) (1 ).1
8
(1 ) (1 ).1
(1 ) (1 ).1
(1 ) (1 ).1 ]
d A r
a
b
b
b
Where [( , , , ); , ]A a b c d and
[(1,1,1,1);1,0]r
If ( , ) ( , )i jd A r d A r then i jA A
Example-1
([0.57,0.73,0.83];
0.73,0.20),
([0.58,0.74,0.819];
0.72,0.20).
A
B
( , ) 0.45
( , ) 0.45
d A r
d B r
A B
0.6973
0.3610
0.6957
0.3600
A
A
B
B
X
Y
X
Y
A B
Jianqiang and Zhong [20]
1( ) [( ) (1 )]
8
( ) ( ) ( )
( ) ( ) ( )
I A a b c d
S A I A
H A I A
Where [( , , , ); , ]A a b c d
If ( ) ( )i jS A S A then i jA A ;
If ( ) ( )i jS A S A then
i jA A if ( ) ( )i jH A H A
Example-2
([0.56,0.74,0.80,
0.90];0.50,0.50),
([0.50,0.70,0.85,
0.95];0.50,0.50).
A
B
( ) 0
( ) 0
( ) 0.3750
( ) 0.3750
S A
S B
H A
H B
A B
0.7236
0.3715
0.7146
0.3542
A
A
B
B
X
Y
X
Y
A B
Rezvani [21]
( 2 2 )( )
6
a b c dV A
Where [( , , , ); , ]A a b c d
If ( ) ( )i jV A V A then i jA A
Example-3
([0.55,0.60,0.70,
0.75];1,0)
([0.45,0.65,0.70,
0.75];1,0).
A
B
( ) 0
( ) 0
V A
V B
A B
0.6500
0.4524
0.6039
0.4123
A
A
B
B
X
Y
X
Y
A B Li [16]
( , )( , )
1 ( , )
V aR a
A a
Where 1 2 3[( , , ); , ]a w ua a a
( , ) ( ) ( ( ) ( ))V a a a aV V V
( , ) ( ) ( ( ) ( ))A a a a aA A A
1 1 2 3( )
( )6
4a
w a a aV
1 2 31)( )
( )6
(1 4a
u a a aV
1 3 1( )
( )3
aw a a
A
3 11)( )
( )3
(1a
u a aA
Example-4
([ 6,1,2];0.6,0.5)
([ 6,1,2];0.7,0.4).
A
B
A B
2.3143
0.3899
2.2888
0.3797
A
A
B
B
X
Y
X
Y
B A
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Journal of Industrial and Intelligent Information Vol. 1, No. 2, June 2013
©2013 Engineering and Technology Publishing
From Table I we can see some drawbacks of the
existing methods and some advantages of the proposed
method, which have been elaborate below:
1) In Example 1, two different TIFNs have been
considered. By Wu and Cao’s [18] method these
two different TIFNs are not comparable. But by
the proposed method the ranking result is .A B
2) From Example 2, it is observe that for two
different TrIFNs, the ranking indices [20] give the
same value and thus they are not comparable.
However, by utilizing the proposed ranking
method we may get the ranking result as .A B
3) Similarly, in Example 3, Rezvani’s [21] approach
the ranking result is same for two different TrIFNs.
But by the proposed method ranking result is
.A B 4) In Example 4, by the ratio ranking method [16] it
is clear that the given two numbers (see Table I)
are not comparable because their ratio ranking
result is ( , ) ( , ) 0R a R b , although they have
different membership and non-membership values.
But by utilizing proposed method we can compare
these two TIFNs and ranking result is .B A
Therefore, from Table I it is clear that in all the above
cases the proposed method finds the ranking result
correctly and overcomes the drawbacks of the existing
methods.
V. CONCLUSION
In this paper, a new method for ranking IFNs has been
introduced by utilizing centroid point of IFNs. For this
purpose, the centroid point of IFNs has also been
computed. Examples have been given to compare the
proposed ranking method with the existing methods. This
ranking approach may be applicable to multi-criteria
decision making problem, which will be topic of our
future research work.
ACKNOWLEGEMENT
The authors would like to express their great thanks for
the anonymous referees for their careful reading and
valuable comments.
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Satyajit Das received his M.Sc. degree in mathematics with
ComputerApplications from National Institute of Technology, Durgapur, India in 2012. Presently, he is Research Fellow in the department of
mathemati-cs in Indian Institute of Technology, Patna, India. His current research area: Fuzzy Logic, Intuitionistic Fuzzy Set theory.
Debashree Guha received her B.Sc. and M.Sc. degree in mathematics
from Jadavpur University, Calcutta, India in 2003 and 2005, and
resepectively. She received her Ph.D degree in Mathematics from Indian Institute of Technology, Kharagpur, India in 2011.
Presently, she is the assistant professor of department of
mathematics in Indian Institute of Technology, Patna, India. Her current research interest includes: multi-attribute decision making, fuzzy
mathematical programming, aggregation operators and fuzzy logic.
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Journal of Industrial and Intelligent Information Vol. 1, No. 2, June 2013
©2013 Engineering and Technology Publishing