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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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

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( ), ( ), ( )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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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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