Journal of Uncertain Systems
Vol.7, No.2, pp.92-107, 2013
Online at: www.jus.org.uk
929292929216
Intuitionistic Fuzzy Number and Its Arithmetic Operation
with Application on System Failure
G.S. Mahapatra1,*, T.K. Roy2
1
Department of Mathematics, Siliguri Institute of Technology, P.O.- Sukna, Siliguri-734009
Darjeeling, West Bengal, India
2
Department of Mathematics, Bengal Engineering and Science University, Shibpur
P.O.-B. Garden, Howrah–711103, India
Received 27 December 2011; Revised 23 June 2012
Abstract
We have introduced intuitionistic fuzzy number and its arithmetic operations based on extension principle of
intuitionistic fuzzy sets. Here two types of intuitionistic fuzzy sets, namely triangular intuitionistic fuzzy number and
trapezoidal intuitionistic fuzzy number is presented. We also present that the arithmetic operation of two or more
intuitionistic fuzzy number is again an intuitionistic fuzzy number. The starting failure of an automobile system is
presented by intuitionistic fuzzy system. Each components failure is represented by trapezoidal intuitionistic fuzzy
number of the system failure model to compute the imprecise failure. Finally, the presented concepts are analyzed
through suitable numerical example.
© 2013 World Academic Press, UK. All rights reserved.
Keywords: intuitionistic fuzzy number, triangular intuitionistic fuzzy number, trapezoidal intuitionistic fuzzy number,
extension principle, failure rate
1 Introduction
Intuitionistic fuzzy set (IFS) is one of the generalizations of fuzzy sets theory [22]. Out of several higher-order fuzzy
sets, IFS first introduced by Atanassov [1] have been found to be compatible to deal with vagueness. The conception
of IFS can be viewed as an appropriate/alternative approach in case where available information is not sufficient to
define the impreciseness by the conventional fuzzy set. In fuzzy sets the degree of acceptance is considered only but
IFS is characterized by a membership function and a non-membership function so that the sum of both values is less
than one [2]. Presently IFSs are being studied and used in different fields of science. Among the research works on
IFS we can mention Atanassov [2-6], Atanassov and Gargov [7], Szmidt and Kacprzyk [19], Buhaescu [9], Ban [8],
Deschrijver and Kerre [13], Stoyanova [18]. With the best of our knowledge, Burillo et al. [10] proposed definition of
intuitionistic fuzzy number (IFN) and studied perturbations of IFN and the first properties of the correlation between
these numbers. Several researchers [15, 17, 21] considered the problem of ranking a set of IFNs to define a fuzzy rank
and a characteristic vagueness factor for each IFN.
In the real world problems, the collected data or system parameters are often imprecise because of incomplete or
non-obtainable information, and the probabilistic approach to the conventional reliability analysis is inadequate to
account for such built-in uncertainties in data. Therefore concept of fuzzy reliability has been introduced and
formulated either in the context of the possibility measures or as a transition from fuzzy success state to fuzzy failure
state [11, 12]. Cheng and Mon [16] considered that components are with different membership functions, then
interval arithmetic and  -cuts were used to evaluate fuzzy system reliability. Verma [20] presented the dynamic
reliability evaluation of the deteriorating system using the concept of probist reliability as a triangular fuzzy number.
Mahapatra and Roy [14] evaluate system reliability by considering reliability of components as triangular
intuitionistic fuzzy number.
*
Corresponding author. Email: g_s_mahapatra@yahoo.com (G.S. Mahapatra); Tel. +91-9433135327; Fax: +91-353-2778003.
Journal of Uncertain Systems, Vol.7, No.2, pp.92-107, 2013
93
In this paper, we have presented IFN according to the approach of fuzzy number presentation. Triangular
intuitionistic fuzzy number (TIFN) and trapezoidal intuitionistic fuzzy number (TrIFN) are defined, and their
arithmetic operations based on intuitoinistic fuzzy extension principle and (α, β)-cut method is presented. The grade
of a membership function indicates a subjective degree of preference of a decision maker within a given tolerance and
grade of a non-membership function indicates a subjective degree of negative response of a decision maker within a
given tolerance. Here we consider failure of components of starting failure of an automobile system as TrIFN.
Intuitionitic fuzzy fault tree analysis is presented for starting failure of the automobile system. Arithmetic operations
of TrIFN are used to evaluate imprecise system failure.
2 Basic Concept of Intuitionistic Fuzzy Sets
Fuzzy set theory was first introduced by Zadeh [22] in 1965. Let X be universe of discourse defined
by X   x1 , x2 ,..., xn  . The grade of membership of an element xi  X in a fuzzy set is represented by real value in
[0,1]. It does indicate the evidence for xi  X , but does not indicate the evidence against xi  X . Atanassov [1]
~i
presented the concept of IFS, an IFS A in X is characterized by a membership function  ~i ( x) and a nonA
membership function  ~i ( x) . Here  ~i ( x) and  ~i ( x) are associated with each point in X, a real number in [0,1] with
A
A
A
~i
~i
the values of  ~i ( x) and  ~i ( x) at X representing the grade of membership and non-membership of x in A . When A
A
A
is an ordinary (crisp) set, its membership function can take only two values zero and one. An IFS becomes a fuzzy set
~i
A when  ~i ( x)  0 but  ~i ( x)   0,1 x  A .
A
A
~i
Definition 2.1 Intuitionistic Fuzzy Set: Let a set X be fixed. An IFS A in X is an object having the
~i


form A   x,  ~i ( x), ~i ( x)  : x  X , where the  ~i ( x) : X  [0,1] and  ~i ( x) :X[0,1] define the degree of
A
A
A
A
membership and degree of non-membership respectively, of the element xX to the set A i , which is a subset of X,
for every element of xX, 0   ~i ( x)   ~i ( x)  1 .
A
A
~i
Definition 2.2  ,   -level Intervals or  ,   -cuts: A set of  ,   -cut, generated by an IFS A , where
 ,    0,1 are fixed numbers such that     1 is defined as
A ,  
 x,  ( x), ( x) : x  X ,  (x)   , ( x)   ,  ,   0,1 .
~i
~i
~i
~i
A
A
A
A
~i
We define  ,   -level interval or  ,   -cut, denoted by A ,  , as the crisp set of elements x which belong to A at
~i
least to the degree α and which belong to A at most to the degree β.
3 Presentation of Intuitionistic Fuzzy Numbers and Its Properties
~i
Definition 3.1 Intuitionistic Fuzzy Number: An IFN A is defined as follows:
i)
ii)
an intuitionistic fuzzy subset of the real line
normal, i.e., there is any x0  R such that  ~i  x0   1 (so  ~i  x0   0 )
iii)
a convex set for the membership function  ~i  x  , i.e.,
A
A

iv)
A
~i
A
  x  1    x   min    x  ,   x  
1
2
~i
A
1
~i
A
2
a concave set for the non-membership function  ~i  x  , i.e.,
A
x1 , x2  R,    0,1
94
G.S. Mahapatra and T.K. Roy: Intuitionistic Fuzzy Number and Its Arithmetic Operation

~i
A
  x  1    x   max   x  ,  x  
1
2
~i
A
1
~i
A
2
x1 , x2  R,    0,1 .
~i
Definition 3.2 Triangular Intuitionistic Fuzzy Number: A TIFN A is a subset of IFS in R with following
membership function and non-membership function as follows:
 x  a1
 a2  x
 a  a for a1  x  a2
 a  a  for a1  x  a2
 2 1
 2 1
 a3  x
 x  a2
 ~i  x   
for a2  x  a3 and  ~i  x   
for a2  x  a3
A
A
 a3  a2
 a3  a2
 0
 1 otherwise
otherwise




~i
where a1  a1  a2  a3  a3 and TIFN (Fig. 1) is denoted by A TIFN   a1 , a2 , a3 ; a1, a2 , a3  .
Figuer 1: Membership and non-membership functions of TIFN
Note 1 Here  ~i  x  increases with constant rate for x   a1 , a2  and decreases with constant rate for x   a2 , a3  but

~i
A
 x
A
decreases with constant rate for x   a1, a2  and increases with constant rate for x   a2 , a3  .
~i
Definition 3.3 Trapezoidal Intuitionistic Fuzzy Number: A TrIFN (Fig. 2) A is a subset of IFS in R with membership
function and non-membership function as follows
 x  a1
 a  a for a1  x  a2
 2 1
 1
for a2  x  a3
 ~i  x   
A
 a4  x for a  x  a
3
4
 a4  a3

otherwise
 0
 a2  x
 a  a  for a1  x  a2
 2 1
for a2  x  a3
 0
and  ~i  x   
A
 x  a3 for a  x  a 
3
4
 a4  a3

otherwise
 1
~i
where a1  a1  a2  a3  a4  a4 and TrIFN is denoted by A TrIFN   a1 , a2 , a3 , a4 ; a1, a2 , a3 , a4 
Figure 2: Membership and non-membership function of TrIFN
Note 2 Here  ~i  x  increases with constant rate for x   a1 , a2  and decreases with constant rate for x   a3 , a4  but

 x
A
~i
A
decreases with constant rate for x   a1, a2  and increases with constant rate for x   a3 , a4  .
Journal of Uncertain Systems, Vol.7, No.2, pp.92-107, 2013
95
Here we have presented a chart of IFN with transformation rule to fuzzy numbers and interval and real number.
Figure 3: Chart of transformation rule on IFN
4
Extension Principle for Intuitionistic Fuzzy Sets
~i
Let f : X  Y be a mapping from a set X to a set Y. then the extension principle allows us to define the IFS B in Y
~i
induced by the IFS A in X through f as follows
~i

B   y,  ~i ( y ), ~i ( y )  : y  f  x  , x  X
B
B

with
 sup ~i ( x) : f 1  y   

 ~i ( y )   y  f  x  A
B

0
: f 1  y   
 inf  ~i ( x) : f 1  y   

and  ~i ( y )   y  f  x  A
B
0
: f 1  y   

where f 1  y  is the inverse image of y .
4.1 Cartesian Product of Intuitionistic Fuzzy Sets
~i
~i
Let A1 ,..., A n be IFSs in X 1 ,..., X n with the corresponding membership functions  ~i ( y ),...,  ~i ( y ) and nonA1
An
~i
~i
membership function  ~i ( y ),..., ~i ( y ) respectively. Then the Cartesian product of the IFSs A1 ,..., A n denoted by
A1
~i
~i
An
A1   A n is defined as IFS in X 1    X n whose membership functions and non-membership functions are
expressed by
 ~i ~i ( x1 ,..., xn )  min   ~i ( x1 ),...,  ~i ( x1 )  and ~i ~i ( x1 ,..., xn )  max  ~i ( x1 ),..., ~i ( x1 )  .
A1  A1
An
A1  A1
An
 An

 An

96
G.S. Mahapatra and T.K. Roy: Intuitionistic Fuzzy Number and Its Arithmetic Operation
4.2 Extension Principle in Cartesian Space
Let f : X 1    X n  Y be a mapping from X 1    X n to a set Y such that y  f  x1 ,..., xn  . Then the extension
~i
~i
principle allows us to define the IFS B in Y induced by the IFS A1   A n in X 1    X n through f as follows

~i

B   y,  ~i ( y ), ~i ( y )  : y  f  x1 ,..., xn  ,  x1 ,..., xn    x1 ,..., xn 
B
B
with

sup

x ,..., xn  f 1  y   
 x1 ,..., xn X1 ,..., X n A1  An  1
 ~i  y   
B

0
f 1  y   
and

inf

x ,..., xn  f 1  y   
 x1 ,..., xn X1 ,..., X n A1  An  1
 ~i  y   
B

0
f 1  y   
where f 1  y  is the inverse image of y .
5 Arithmetic Operations on Intuitionistic Fuzzy Numbers
In this section, we have presented arithmetic operations of IFNs based on intuitionistic fuzzy extension principle and
approximation ((α, β)-cuts) method.
5.1 Arithmetic Operations of Intuitionistic Fuzzy Numbers based on Extension Principle
The arithmetic operation (*) of two IFNs is a mapping of an input vector X   x1 , x2  define in the Cartesian product
T
~i
~i
space R  R onto an output y define in the real space R. If A1 and A2 are IFN then their outcome of arithmetic
operation is also a IFN determined with the formula





 ~i ~i 


max  ~i  x1  , ~i  x2     x1 , x2 , y  R 
 A1 * A2   y    y, sup  min  A~i  x1  , A~i  x2    , y inf

x
x
*
A
A
1
2
2
2
 1

 1
 



 y  x1 * x2 

to calculate the arithmetic operation of IFNs it is sufficient to determine the membership function and nonmembership function as follows




 ~i ~i  y   sup  min   ~i  x1  ,  ~i  x2    and  ~i ~i  y   inf  max  ~i  x1  , ~i  x2    .
y  x1 * x2
A2
A2
A1 * A2
A1 * A2
y  x1 * x2 
 A1

 A1


5.2 Arithmetic Operations of Intuitionistic Fuzzy Numbers based on (, β)-cuts Method
~i
If A is an IFN, then  ,   -level interval or  ,   -cut is given by
 A1   , A2    for degree of acceptance    0,1
A ,   
with     1.
  A1    , A2     for degree of rejection    0,1
dA1  
dA  
dA1   
dA   
 0, 2
 0    0,1 , A1 1  A2 1 and
(ii)
Here (i)
 0, 2
 0    0,1 ,
d
d
d
d
A1  0   A2  0  .


It is expressed as A ,    A1   , A2    ;  A1    , A2     ,     1,  ,    0,1 .
~i
For instance, if A   a1 , a2 , a3 , a4 ; a1, a2 , a3 , a4  is a TrIFN, then  ,   -level intervals or  ,   -cuts is
Journal of Uncertain Systems, Vol.7, No.2, pp.92-107, 2013
97


A ,    A1   , A2    ;  A1    , A2     ,     1,  ,    0,1
where A1    a1    a2  a1  , A2    a4    a4  a3  ; A1     a2    a2  a1  , A2     a3    a4  a3  .
~i
~i
~i
Property 5.1 (a) If TrIFN A   a1 , a2 , a3 , a4 ; a1, a2 , a3 , a4  and y  ka (with k>0), then Y  k A is a TrIFN
 ka1 , ka2 , ka3 , ka4 ; ka1, ka2 , ka3 , ka4  .
~i
~i
(b) If y  ka (with k<0), then Y  k A is a TrIFN  ka4 , ka3 , ka2 , ka1 ; ka4 , ka3 , ka2 , ka1  .
Proof: (a) When k>0, with the transformation y  ka , we can find the membership function for membership
~i
~i
(acceptance) function of TrIFN Y  k A by -cut method.
~i
Left-hand and right-hand -cut of A is  ~i  x      a1    a2  a1  , a4    a4  a3   for any    0,1 , i.e.,
A
x   a1    a2  a1  , a4    a4  a3   . So, y   ka    ka1    ka2  ka1  , ka4    ka4  ka3   .
~i
~i
Thus, we get the membership function of Y  k A as
 y  ka1
 ka  ka
1
 2
 1
 ~i  y   
y
 ka4  y
 ka4  ka3

0

Hence the rule is proved for membership function.
for ka1  y  ka2
for ka2  y  ka3
(5.1)
for ka3  y  ka4
otherwise.
~i
For non-membership function, -cut of A is  ~i  x      a2    a2  a1  , a3    a4  a3   for any    0,1 ,
A
i.e., x   a2    a2  a1  , a3    a4  a3   . So, y   ka    ka2    ka2  ka1  , ka3    ka4  ka3   .
~i
~i
Thus, we get the non-membership function of Y  k A as
 ka2  y
 ka  ka  for ka1  y  ka2
1
 2
for ka2  y  ka3
 0
 ~i  y   
A
y
ka

3

for ka3  y  ka4
 ka4  ka3

otherwise.
 1
Hence rule is proved for non-membership function.
~i
(5.2)
~i
Thus we have Y  k A   ka1 , ka2 , ka3 , ka4 ; ka1, ka2 , ka3 , ka4  is a TrIFN.
(b) Similarly we can proof that, if y  ka and k<0 , then
 y  ka4
 ka  ka
4
 3
 1
 ~i ( y )  
y
 ka1  y
 ka1  ka2

0

~i
for ka4  y  ka3
for ka3  y  ka2
for ka2  y  ka1
otherwise.
 ka3  y
 ka  ka 
4
 3

0
and  ~i ( y )  
y
y
ka2


 ka1  ka2

1

~i
for ka4  y  ka3
for ka3  y  ka2
(5.3)
for ka2  y  ka1
otherwise.
~i
~i
~i
Property 5.2 If A   a1 , a2 , a3 , a4 ; a1, a2 , a3 , a4  and B   b1 , b2 , b3 , b4 ; b1, b2 , b3 , b4  are two TrIFNs, then C  A B is
~i
~i
also TrIFN A  B   a1  b1 , a2  b2 , a3  b3 , a4  b4 ; a1  b1, a2  b2 , a3  b3 , a4  b4  .
98
G.S. Mahapatra and T.K. Roy: Intuitionistic Fuzzy Number and Its Arithmetic Operation
Proof: With the transformation z=x+y, we can find the membership function of acceptance (membership) IFS
~i
~i
~i
C  A B by -cut method.
~i
-cut for membership function of A is  a1    a2  a1  , a4    a4  a3      0,1 , i.e.,
x   a1    a2  a1  , a4    a4  a3   .
~i
-cut for membership function of B is b1    b2  b1  , b4    b4  b3      0,1 , i.e. ,
y  b1    b2  b1  , b4    b4  b3   .
So,
z (=x+y)   a1  b1     a2  a1    b2  b1   , a4  b4     a4  a3    b4  b3    .
~i
~i
~i
So, the membership (acceptance) function of C  A B is
z  a1  b1

  a  a    b  b  for a1  b1  z  a2  b2
2
1
 2 1

1
for a2  b2  z  a3  b3
 ~i  z   
C
a
b
z


4
4

for a3  b3  z  a4  b4
  a4  a3    b4  b3 

0
otherwise.

Hence additions rule is proved for membership function.
(5.4)
~i
For non-membership function, β-cut of A is  a2    a2  a1  , a3    a4  a3      0,1 , i.e.,
x   a2    a2  a1  , a3    a4  a3   .
~i
β-cut for non-membership function of B is b2    b2  b1  , b3    b4  b3      0,1 , i.e.,
y  b2    b2  b1  , b3    b4  b3   .
So, z (=x+y)   a2  b2     a2  a1    b2  b1   , a3  b3     a4  a3    b4  b3    .
~i
~i
~i
So, the non-membership (rejection) function of C  A B is
a2  b2  z

  a  a     b  b 
2
1
 2 1

0
 ~i  z   
C
z  a3  b3

  a4  a3    b4  b3 

1

Hence additions rule is proved for non-membership function.
~i
for a1  b1  z  a2  b2
for a2  b2  z  a3  b3
(5.5)
for a3  b3  z  a4  b4
otherwise.
~i
Thus we have A B   a1  b1 , a2  b2 , a3  b3 , a4  b4 ; a1  b1, a2  b2 , a3  b3 , a4  b4  is a TrIFN.
~i
~i
~i
Note 3 If we have the transformation C  k1 A k2 B (k1, k2 are (not all zero) real numbers), then the IFS
~i
~i
~i
C  k1 A k2 B is the following TrIFN:
 k 2 b1 , k1 a2  k 2 b2 , k1 a3  k2 b3 , k1 a4  k 2 b4 ; k1 a1  k 2 b1, k1 a2  k 2 b2 , k1 a3  k 2 b3 , k1 a4  k 2 a4  if k1  0, k2  0
or k1  0, k2  0 ,
(ii)  k1 a1  k 2 b4 , k1 a2  k2 b3 , k1 a3  k 2 b2 , k1 a4  k 2 b1 ; k1 a1  k 2 b4 , k1 a2  k2 b3 , k1 a3  k 2 b2 , k1 a4  k 2 b1  if k1  0, k2  0
or k1  0, k2  0 ,
(iii)  k1 a4  k 2 b1 , k1 a3  k 2 b2 , k1 a2  k 2 b3 , k1 a1  k 2 b4 ; k1 a4  k 2 b1, k1 a3  k 2 b2 , k1 a2  k 2 b3 , k1 a1  k 2 b4  if k1  0, k2  0 ,
(i)
k a
1 1
or k1  0, k2  0
Journal of Uncertain Systems, Vol.7, No.2, pp.92-107, 2013
(iv)
k a
99
 k 2 b4 , k1 a3  k 2 b3 , k1 a2  k 2 b2 , k1 a1  k 2 b1 ; k1 a4  k 2 b4 , k1 a3  k 2 b3 , k1 a2  k 2 b2 , k1 a1  k 2 b1  if k1  0, k2  0
or k1  0, k2  0 .
1
4
~i
~i
~i
~i
~i
Property 5.3 If A   a1 , a2 , a3 , a4 ; a1, a2 , a3 , a4  and B   b1 , b2 , b3 , b4 ; b1, b2 , b3 , b4  are two TrIFN, then P  A  B is
~i
~i
approximated TrIFN A  B   a1b1 , a2 b2 , a3b3 , a4 b4 ; a1b1, a2 b2 , a3b3 , a4 b4  .
Proof: With the transformation z=x×y, we can find the membership function of acceptance (membership) IFS
~i
~i
~i
P  A  B by -cut method.
~i
-cut for membership function of A is  ~i  x      a1    a2  a1  , a4    a4  a3      0,1 , i.e.,
A
x   a1    a2  a1  , a4    a4  a3   .
~i
-cut for membership function of B is ~i  x     b1    b2  b1  , b4    b4  b3      0,1 , i.e.,
B
y  b1    b2  b1  , b4    b4  b3   .
So, z (=x×y)   a1    a2  a1    b1    b2  b1   ,  a4    a4  a3    b4    b4  b3    .
~i
~i
~i
So, the membership (acceptance) function of P  A  B is
 B  B2  4 A  a b  z 
1
1
1 1
 1
for a1b1  z  a2 b2
2 A1


1
for a2 b2  z  a3b3

(5.6)
 ~i  z   
P
 B2  B22  4 A2  a4 b4  z 
for a3b3  z  a4 b4

2 A2


0
otherwise

where A1   a2  a1  b2  b1  , B1  b1  a2  a1   a1  b2  b1  , A2   a4  a3  b4  b3  and B2    b4  a4  a3   a4  b4  b3   .
~i
For non-membership function, β-cut of A is  ~i  x      a2    a2  a1  , a3    a4  a3      0,1 , i.e.,
A
~i
x   a2    a2  a1  , a2    a3  a2   , β-cut of B is  ~i  x     b2    b2  b1  , b3    b4  b3      0,1 , i.e.,
B
y  b2    b2  b1  , b2    b3  b2   .
So, z(=x×y)   a2    a2  a1    b2    b2  b1   ,  a3    a4  a3    b3    b4  b3    . So, the non-membership
~i
~i
~i
(rejection) function of P  A  B is
  B   B 2  4 A  a b  z 
1
1
1 1
1  1
for a1b1  z  a2 b2
2 A1


0
for a2 b2  z  a3b3

 ~i  z   
(5.7)
P
 B2  B22  4 A2  a4 b4  z 
for a3b3  z  a4 b4
1 
2 A2


1
otherwise

where A1   a2  a1  b2  b1  , B1  b1  a2  a1   a1  b2  b1  , A2   a4  a3  b4  b3  and B2    b4  a4  a3   a4  b4  b3   .
~i
~i
~i
So P  A  B represented by (5.6) and (5.7) is a trapezoidal shaped IFN. It can be approximated to a TrIFN
~i
~i
A  B   a1b1 , a2 b2 , a3b3 , a4 b4 ; a1b1, a2 b2 , a3b3 , a4 b4  (shown in Fig.3 with - - - - line).
100
G.S. Mahapatra and T.K. Roy: Intuitionistic Fuzzy Number and Its Arithmetic Operation
Figure 4: Membership and non-membership functions for product of two TrIFN
Theorem 1 The divergences due to approximation of trapezoidal shaped IFN to TrIFN for multiplication of two
TrIFN are as follows:
(a) Maximum left divergence for membership (non-membership) function = 1 4 product of left spread of TrIFN
~i
~i
A and B for membership (non-membership) function.
(b) Maximum right divergence for membership (non-membership) function = 1 4 product of right spread of TrIFN
~i
~i
A and B for membership (non-membership) function.
~i
~i
Proof: A  B represented by (5.6) and (5.7) is a trapezoidal shaped IFN. It can be approximated to a TrIFN
~i
P   a1b1 , a2 b2 , a3b3 , a4 b4 ; a1b1, a2 b2 , a3b3 , a4 b4  .
 ,    cut of the above approximated TrIFN, which is given by P ,  
 P1   , P2   ,  P1   , P2    where P1    a1b1    a2b2  a1b1  , P2    a4b4    a4b4  a3b3  , P1   
a2 b2    a2 b2  a1b1  , P2     a3b3    a4 b4  a3b3  .
The corresponding left divergent for membership function   lm    is
 lm     a1    a2  a1    b1    b2  b1     a1b1    a2 b2  a1b1   .
d  lm  
 0 gives  *  0.5   0,1 . The maximum left divergence for
To find out the optimum divergence
Now
consider
membership is  lm 
*
  a
2
d
 a1  b2  b1  / 4.
Again the corresponding right divergent for membership function   rm    is
 rm     a4    a4  a3    b4    b4  b3     a4 b4    a4 b4  a3b3   .
To find out the optimum divergence
d  rm  
d
 0 gives  *  0.5   0,1 . The maximum right divergence for
membership is  lm  *     a4  a3  b4  b3  / 4.
The corresponding left divergent for non-membership function  ln    is
 ln      a2    a2  a1    b2    b2  b1     a2 b2    a2 b2  a1b1   .
To find out the optimum divergence
d  ln   
d
 0 gives  *  0.5   0,1 . The maximum left divergence for non-
membership is  ln   *     a2  a1  b2  b1  / 4.
Again the corresponding right divergent for non-membership function  rn    is
 rn      a3    a4  a3    b3    b4  b3     a3b3    a4 b4  a3b3   .
Journal of Uncertain Systems, Vol.7, No.2, pp.92-107, 2013
To find out the optimum divergence
d  rn   
d
101
 0 gives  *  0.5   0,1 . The maximum right divergence for non-
membership is  rn   *     a4  a3  b4  b3  / 4.
Note 4 It may conclude that when spreads are increasing, divergences due to approximation for product of two
TrIFNs are also increasing. Divergences are insignificant for very small spreads. In such a situation product of two
TrIFNs can directly be written as approximated TrIFN.
~i
~i
~i
~i
~i
Property 5.4 If A   a1 , a2 , a3 , a4 ; a1, a2 , a3 , a4  and B   b1 , b2 , b3 , b4 ; b1, b2 , b3 , b4  are two TrIFN, then D  A  B is
approximated TrIFN
~i
a a a a a a a a
A  B   b1 , b2 , b3 , b4 ; b1 , b2 , b3 , b4  .
 4 3 2 1 4 3 2 1 
Proof: With the transformation z=x÷y, we can find the membership function of acceptance (membeship) IFS
~i
~i
~i
~i
D  A  B by -cut method.
~i
-cut for membership function of A is  ~i  x      a1    a2  a1  , a4    a4  a3      0,1 , i.e.,
A
x   a1    a2  a1  , a4    a4  a3   .
-cut
for
membership
function
of
~i
B
is

~i
B
 x   
b1    b2  b1  , b4    b4  b3      0,1 , i.e., y  b1    b2  b1  , b4    b4  b3   . So,
 a1   a2  a1  a4   a4  a3  
z (=x÷y)  
,
.
b   b b  b   b b 
 4  4 3  1  2 1  
~i
~i
~i
So, we have the membership (acceptance) function of D  A  B as
b4 z  a1

  a  a   z b  b 
4
3
 2 1

1

 ~i  z   
D

a1  b1 z

  a4  a3   z  b2  b1 

0
a
for b1  z 
1
3
a3
b2
a3
b2
a4
b1
a
for b2  z 
for
a2
b3
z
(5.8)
otherwise.
~i
For non-membership function, β-cut of A is  ~i  x      a2    a2  a1  , a3    a4  a3      0,1 , i.e.,
A
~i
x   a2    a2  a1  , a2    a3  a2   . β-cut of B is  ~i  x     b2    b2  b1  , b3    b4  b3      0,1 , i.e.,
B
y  b2    b2  b1  , b2    b3  b2   . So,
 a2    a2 a1  a3    a4  a3  
,
z (=x÷y)  
.
b    b b  b    b b 
 3  4 3  2  2 1  
~i
~i
~i
So, we have the non-membership (rejection) function of D  A  B as follows
a2  b3 z

  a  a    z  b  b 
4
3
 2 1

0

 ~i  z   
D

b2 z  a3
 
a
a


 4 3   z  b2  b1 

1
a
for b1  z 
4
for
a2
b3
a
for b3
2
a2
b3
z
a3
b2
z
a4
b1
otherwise.
(5.9)
102
G.S. Mahapatra and T.K. Roy: Intuitionistic Fuzzy Number and Its Arithmetic Operation
~i
~i
~i
So D  A  B represented by (5.8) and (5.9) is a trapezoidal shaped IFN. It can be approximated to TrIFN
~i

~i
A B 
a1 a2 a3 a4 a1 a2 a3 a4
, , , ; , , ,
b4 b3 b2 b1 b4 b3 b2 b1
.
Theorem 2 Divergences due to approximation of trapezoidal shaped IFN to TrIFN for division of two TrIFNs
~i
~i
A   a1 , a2 , a3 , a4 ; a1, a2 , a3 , a4  and B   b1 , b2 , b3 , b4 ; b1, b2 , b3 , b4  are as follows:
(a) Maximum left and right divergences for membership function are

b4  b3
b4  b3
a2
b3

b2  b1
 b41 and
a
b2  b1


 b23
a
a4
b1
respectively.
(b) Maximum left and right divergences for non-membership function are

b4  b3
b4  b3
a2
b3
a


b2  b1
 b41 and
b2  b1
a4
b1
 b23
a

respectively.
~i
~i
~i
Proof: Let D  A  B represented by (5.8) and (5.9) is a trapezoidal shaped IFN. It can be approximated to a TrIFN
~i
~i
A B 

a1 a2 a3 a4 a1 a2 a3 a4
, , , ; , , ,
b4 b3 b2 b1 b4 b3 b2 b1
.
Now consider  ,    cut of the above approximated TrIFN, which is given by

where D1   
a1
b4


a2
b3

 b41 , D2   
a

D ,    D1   , D2    ,  D1    , D2    

a4
b1

a4
b1

 b23 , D1    
a

a2
b3

a2
b3

 b41 , D2    
a
The corresponding left divergent for membership function   lm    is  lm   
d  lm  
find out the optimum divergence
membership is  lm  *   
b4  b3
b4  b3

d
a2
b3
 0 gives  * 
b4
b4  b3
a3
b2
a1   a2  a1 
b4   b4  b3 




a4
b1
a1
b4

 b23 .
a


a2
b3
 b41
a
 . To
  0,1 . The maximum left divergence for

 b41 .
a
Again the corresponding right divergent for membership function   rm    is
 rm   
To find out the optimum divergence
 
membership is  lm  *  
b2  b1
b2  b1

a4
b1
d  rm  
d
a4   a4  a3 
b1   b2  b1 



a4
b1
 0 gives  * 

b1
b2  b1
a4
b1
 b23
a
 .
  0,1 . The maximum right divergence for

 b23 .
a
The corresponding left divergent for non-membership function  ln    is  ln    
To find out the optimum divergence
 
membership is  ln  *  
Again
 rn    
the
a3    a4  a3 
b2    b2  b1 
b4  b3
b4  b3

a2
b3
corresponding


a3
b2


a4
b1
 b23
a
d  ln   
d
 0 gives  * 
b3
b4  b3
a2    a2  a1 
b3    b4  b3 


a2
b3


a2
b3
a
 b41
 .
  0,1 . The maximum left divergence for non-

 b41 .
a
right
divergent
for
non-membership
 . To find out the optimum divergence
The maximum right divergence for non-membership is  rn   *   
b2  b1
b2  b1

a4
b1
d  rn   
d
function
 0 gives  * 
 rn   
b2
b2  b1
is
  0,1 .

 b23 .
a
Note 5 When spreads increases, divergences due to approximation for division of two TrIFNs also increases.
Divergences are insignificant for very small spreads. In such a situation division of two TrIFNs can directly be
written as approximated TrIFN.
Journal of Uncertain Systems, Vol.7, No.2, pp.92-107, 2013
103
6 Numerical Exposure of Arithmetic Operation on Intuitionistic Fuzzy
Number
Here we have presented numerical example of arithmetic operations of IFNs based on intuitionistic fuzzy extension
principle and on approximation ((α, β)-cuts) method.
6.1 Addition of Two TrIFN by Intuitionistic Fuzzy Extension Principle
~i
~i
~i
~i
~i
Let A  1.5, 2,3.5, 4;0.5, 2,3.5, 4.5  , B   2,3, 4,5.5;1.5,3, 4, 6  and C  A  B (Fig.5). Then

~i
C
 z   sup
    x  ,   y  : x  y  z and 
min
~i
~i
~i
A
B
C
 z   inf
 
max
~i
A
 x  , B  y 
~i
 : x  y  z .
Let us exhibit the computational procedure involve in above equation for membership function, first pick a value


for z, then evaluate min  ~i  x  ,  ~i  y  for x and y which add up to z=4.5. We have done this for certain values of x
A
B
and y as shown in Table 1. It appear that the max occurs for x=2 and y=2.5, therefore   4.5   0.5 . Now do this for
~i


C
other values of z. Similarly for non-membership function, evaluate max  ~i  x  , ~i  y  for x and y which add up to
A
B
z=4.5. We have done this for certain values of x and y as shown in Table 2. The min occurs for x=1.75 and y=2.75 so
~i
that  ~i  4.5   0.16666 . Now do this for other values of z. Finally, we get C   3.5,5, 7.5,9.5; 2,5, 7.5,10.5  a TrIFN.
C
Figure 5: Addition of two TrIFN based on IF Extension principle
Table 1: Finding membership function of sum of two TrIFN
x
1.5
1.75
2
2.25
2.5

~i
 x
A
0
0.5
1
1
1
y

~i
 y
B
3
2.75
2.5
2.25
2
min(col#2, col#4)
1
0.75
0.5
0.25
0
0
0.25
0.5
0.25
0
6.2 Addition of Two TrIFN by (α, β)-cut Method
~i
~i
Let us consider two TrIFN P  1.5, 2,3.5, 4;0.5, 2,3.5, 4.5  and Q   2,3, 4,5.5;1.5,3, 4, 6  . Using Eq.(5.4) and
~i
~i
Eq.(5.5), the addition of these two TrIFN is defined by P  Q   3.5,5, 7.5,9.5; 2,5, 7.5,10.5  with membership and
non-membership function as follows
104
G.S. Mahapatra and T.K. Roy: Intuitionistic Fuzzy Number and Its Arithmetic Operation
 x  3.5
 1.5

 1
 ~ i ~i  x   
P Q
 9.5  x
 2
 0

5  x
 3

if 5  x  7.5
 0
and  ~i ~i  x   
P Q
 x  7.5
if 7.5  x  9.5
 3
 1
otherwise

if 3.5  x  5
if 2  x  5
if 5  x  7.5
if 7.5  x  10.5
otherwise.
Table 2: Finding non-membership function of sum of two TrIFN

x
~i
 x
0.5
0.75
1
1.25
1.5
1.75
2
2.25
2.5
2.75
3

y
A
1
0.83333
0.66666
0.5
0.33333
0.16666
0
0
0
0
0
4
3.75
3.5
3.25
3
2.75
2.5
2.25
2
1.75
1.5
~i
 y
max(col#2, col#4)
B
0
0
0
0
0
0.16666
0.33333
0.5
0.66666
0.83333
1
1
0.83333
0.66666
0.5
0.33333
0.16666
0.33333
0.5
0.66666
0.83333
1
6.3 Multiplication of Two TrIFN by (α, β)-cut Method
~i
~i
Let us consider two TrIFN P  1.5, 2,3.5, 4;0.5, 2,3.5, 4.5  and Q   2,3, 4,5.5;1.5,3, 4, 6  . Using Eq.(5.6) and
Eq.(5.7),
the
multiplication
 3, 6,14, 22;0.75, 6,14, 27 
of
these
two
TrIFN
is
defined
by
the
approximate
TrIFN
~i
~i
PQ 
with membership and non-membership function as follows
x 3
6  x
if 3  x  6
if 0.75  x  6
 3
 5.25


if 6  x  14
if 6  x  14
 1
 0
 ~ i ~i  x   
and  ~i ~i  x   
P Q
P

Q
 22  x if14  x  22
 x  14 if 14  x  27
 13
 8
 1
 0
otherwise
otherwise.


The corresponding left, right divergence for membership and non-membership functions are 0.25, 0.1875 and
0.5625, 0.5 respectively.
6.4 Division of Two TrIFN by (α, β)-cut Method
~i
~i
Let us consider two TrIFN P   2,5, 6,8;1.5,5, 6,9  and Q  1,3, 4,5;0.5,3, 4, 6  . Using Eq.(5.8) and Eq.(5.9), the
~i
~i
division of TrIFN is defined by P  Q   0.4,1.25, 2,8;0.25,1.25, 2,18  with membership and non-membership
function as follows
Journal of Uncertain Systems, Vol.7, No.2, pp.92-107, 2013
105
 x  0.4
1.25  x
if 0.4  x  1.25
if 0.25  x  1.25
 0.85
 1


if 1.25  x  2
if 1.25  x  2
 1
 0
and  ~i ~i  x   
 ~ i ~i  x   
P Q
P Q
x

2

8
x


if 2  x  18
if 2  x  8
 6
 16
 0
 1
otherwise
otherwise


The corresponding left and right divergence for membership are 0.047368 for α=0.527864 and 1.607695 for
α=0.633974 respectively. Left and right divergence for non-membership are 0.10102 for β=0.55051 and 6.723266 for
β=0.710102 respectively.
7 Application of System Failure using Intuitionistic Fuzzy Number
Starting failure of an automobile depends on different facts. The facts are battery low charge, ignition failure and fuel
supply failure. There are two sub-factors of each of the facts. The fault-tree of failure to start of the automobile is
shown in the Fig.6.
~i
represents the system failure to start of automobile.
Ffs
~i
represents the failure to start of automobile due to Battery Low Charge.
Fblc
~i
represents the failure to start of automobile due to Ignition Failure.
Fif
~i
represents the failure to start of automobile due to Fuel Supply Failure.
Ffsf
~i
represents the failure to start of automobile due to Low Battery Fluid.
Flbf
~i
represents the failure to start of automobile due to Battery Internal Short.
Fbis
~i
represents the failure to start of automobile due to Wire Harness Failure.
Fwhf
~i
represents the failure to start of automobile due to Spark Plug Failure.
Fspf
~i
represents the failure to start of automobile due to Fuel Injector Failure.
Ffif
~i
represents the failure to start of automobile due to Fuel Pump Failure.
Ffpf
The intuitonistic fuzzy failure to start of an automobile can be calculated when the failures of the occurrence of
basic fault events are known. Failure to start of an automobile can be evaluated by using the following steps:
Step 1.
~i
~i
~i
~i
Fblc  1 Ө(1Ө Flbf
Fif  1 Ө(1Ө Fwhf
~i
~i
Ffsf  1 Ө(1Ө Ffif
~i
)(1 Ө F )
(7.1)
bis
~i
)(1Ө F )
spf
~i
)(1Ө F )
fpf
Step 2.
~i
~i
~i
~i
Ffs  1 Ө(1Ө Fblc )(1Ө Fif )(1Ө Ffsf )
(7.2)
Here we present numerical explanation of starting failure of the automobile using fault tree analysis with
intuitionistic fuzzy failure rate. The components failure rates as TrIFN are given by
~i
~i
Flbf   0.02, 0.05, 0.06, 0.07;0.01, 0.05, 0.06, 0.08  , Fbis   0.02, 0.03, 0.05, 0.06;0.01, 0.03, 0.05, 0.07  ,
~i
~i
Fwhf   0.03,0.04,0.05,0.07;0.01,0.04,0.05,0.09  , Fspf   0.02,0.04,0.06,0.08;0.01,0.04,0.06,0.09  ,
~i
Ffif   0.03, 0.05, 0.07, 0.08;0.02, 0.05, 0.07, 0.09  and Ffpf   0.04,0.06,0.07,0.08;0.03,0.06,0.07,0.09  .
~i
106
G.S. Mahapatra and T.K. Roy: Intuitionistic Fuzzy Number and Its Arithmetic Operation
Figure 6: Fault-tree of failure to start of an automobile
So in the step1 (using Eq.7.1) the results are as follows
~i
Fblc   0.0396, 0.0785, 0.107, 0.1258; 0.0199, 0.0785, 0.107, 0.1444  ,
~i
Fif   0.0494, 0.0784, 0.107, 0.144;0.0199, 0.0784, 0.107, 0.1719  ,
~i
Ffsf   0.0688, 0.107, 0.1351, 0.1536;0.0494, 0.107, 0.1351, 0.1719  .
In the step 2 (using Eq.7.2), we obtain the failure to start of the automobile. The fuzzy failure to start of an
automobile (Fig.6) is represented by the following TrIFN
~i
Ffs   0.149855, 0.241615, 0.310286, 0.366922;0.086857, 0.241615, 0.310286, 0.413272  .
Membership/non-membership value
So the failure to start of the automobile is about an interval [0.241615, 0.310286] with tolerance level of
acceptance is [0.149855, 0.366922] and tolerance level of rejection is [0.086857, 0.413272].
Here the left and right divergences are not significant since elements of these TrIFN are very small. Therefore,
we consider TrIFN instead of trapezoidal shaped IFN to compute system failure.
1
0.8
0.6
0.4
0.2
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
System failure to start
0.4
0.45
Figure 7: TrIFN representing the system failure to start of an automobile
8 Conclusion
In this paper, we have proposed a definition of IFN according to the fuzzy number presentation approach. Also
arithmetic operations of proposed TrIFN are evaluated based on intuitionistic fuzzy extension principle and (α, β)cuts method. The major advantage of using IFSs instead of fuzzy sets is that IFSs separate the positive and the
negative evidence for the membership of an element in a set. We discuss the fault-tree of failure to start of an
automobile with components having failure rates as trapezoidal intuitionistic fuzzy numbers. Finally, our approaches
Journal of Uncertain Systems, Vol.7, No.2, pp.92-107, 2013
107
and computational procedures are efficient and simple to implement for calculation in intuitionistic fuzzy
environment for all field of engineering and sciences where vagueness is occur.
Acknowledgement
This research is supported by the CSIR, India via research scheme under Grant No (25(0151)/06/EMR-II). The
authors are grateful to editor-in-chief and learned referees for their constructive and critical comments which have
enhanced the quality of the manuscript.
References
[1] Atanassov, K.T., Intuitionistic fuzzy sets, VII ITKR’s Session, Sofia, Bulgarian, 1983.
[2] Atanassov, K.T., Intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol.20, pp.87–96, 1986.
[3] Atanassov, K.T., and G. Gargov, Interval-valued intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol.31, no.3, pp.343–349,
1989.
[4] Atanassov, K.T., More on intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol.33, no.1, pp.37–46, 1989.
[5] Atanassov, K.T., Intuitionistic Fuzzy Sets, Physica-Verlag, Heidelberg, New York, 1999.
[6] Atanassov, K.T., Two theorems for Intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol.110, pp.267–269, 2000.
[7] Atanassov, K.T., and G. Gargov, Elements of intuitionistic fuzzy logic, Part I, Fuzzy Sets and Systems, vol.95, no.1, pp.39–52,
1998.
[8] Ban, A.I., Nearest interval approximation of an intuitionistic fuzzy number, Computational Intelligence, Theory and
Applications, Springer-Verlag, Berlin, Heidelberg, pp.229–240, 2006.
[9] Buhaesku, T., Some observations on intuitionistic fuzzy relations, Itinerant Seminar of Functional Equations, Approximation
and Convexity, pp.111–118, 1989.
[10] Burillo, P., H. Bustince, and V. Mohedano, Some definition of intuitionistic fuzzy number, Fuzzy based Expert Systems,
Fuzzy Bulgarian Enthusiasts, 1994.
[11] Cai, K.Y., C.Y. Wen, and M.L. Zhang, Fuzzy reliability modeling of gracefully degradable computing systems, Reliability
Engineering and System Safety, vol.33, pp.141–157, 1991.
[12] Cheng, C.H., and D.L. Mon, Fuzzy system reliability analysis by interval of confidence, Fuzzy Sets and Systems, vol.56,
pp.9–35, 1993.
[13] Deschrijver, G., and E.E. Kerre, On the relationship between intuitionistic fuzzy sets and some other extensions of fuzzy set
theory, Journal of Fuzzy Mathematics, vol.10, no.3, pp.711–724, 2002.
[14] Mahapatra, G.S., and T.K. Roy, Reliability evaluation using triangular intuitionistic fuzzy numbers arithmetic operations,
International Journal of Computational and Mathematical Sciences, vol.3, no.5, pp.225–232, 2009.
[15] Mitchell, H.B., Ranking-intuitionistic fuzzy numbers, International Journal of Uncertainty, Fuzziness and Knowledge-Based
Systems, vol.12, no.3, pp.377–386, 2004.
[16] Mon, D.L., and C.H. Cheng, Fuzzy system reliability analysis for components with different membership functions, Fuzzy
Sets and Systems, vol.64, pp.145–157, 1994.
[17] Nayagam, V.L.G., G. Venkateshwari, and G. Sivaraman, Modified ranking of intuitionistic fuzzy numbers, Notes on
Intuitionistic Fuzzy Sets, vol.17, no.1, pp.5–22, 2011.
[18] Stoyanova, D., More on Cartesian product over intuitionistic fuzzy sets, BUSEFAL, vol.54, pp.9–13, 1993.
[19] Szmidt, E., and J. Kacprzyk, Distances between intuitionistic fuzzy sets, Fuzzy Sets and Systems, vol.114, no.3, pp.505–518,
2000.
[20] Verma, A.K., A. Srividya, and R.P. Gaonkar, Fuzzy dynamic reliability evaluation of a deteriorating system under imperfect
repair, International Journal of Reliability, Quality and Safety Engineering, vol.11, no.4, pp.387–398, 2004.
[21] Wei, C.P., and X. Tang, A new method for ranking intuitionistic fuzzy numbers, International Journal of Knowledge and
Systems Science, vol.2, no.1, pp.43–49, 2011.
[22] Zadeh, L.A., Fuzzy sets, Information and Control, vol.8, pp.338–353, 1965.