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Volume 2, Issue 1, March 2011
Fuzzy Arithmetic with and without using
α
cut method: A Comparative Study
Palash Dutta
1
, Hrishikesh Boruah
2
, Tazid Ali
3
Dept. of Mathematics, Dibrugarh University, Dibrugarh,786004, India
palash.dtt@gmail.com, h.boruah07@gmail.com
,
tazidali@yahoo.com
Abstract
:
In [2] and [6] the authors proposed a method for construction of membership function
without using α
cut. They claim that the standard
method of α
cut [1] fails in certain situations viz in determining square root of a fuzzy number. We would like to counter their argument by proving
that α
cut method is general enough to deal with different type of fuzzy arithmetic including exponentiation, extracting nth root, taking logarithm. Infact we illustrate with examples to
show that α
cut method is simpler than their proposed method
.
Keywords
:
Fuzzy membership function, fuzzy number, alphacut
1.
Introduction
:
α
cut method is a standard method for performing different arithmetic operations like addition, multiplication, division, subtraction. In [2] and [6] the authors argue that finding membership function for square root of X where X is a fuzzy number, is not possible by the standard alphacut method. They have proposed a method of finding membership function from the simple assumption that the DuboisPrade left reference function is a distribution function and similarly the DuboisPrade right reference function is a complementary distribution function. In this paper we are going to show that alphacut method can be used for finding n
th
root of fuzzy number and infact this method is simpler than that proposed by them. However we do acknowledge that the proposed method has more mathematical beauty than the existing alphacut method.
2.
Basic Concept of Fuzzy Set Theory:
In this section, some necessary backgrounds and notions of fuzzy set theory are reviewed.
Definition 2.1:
Let
X
be
a universal set. Then the fuzzy subset
A
of
X
is defined by its membership function
: [0,1]
A
X
which assign a real number
( )
A
x
in the interval [0, 1], to each element
x X
, where the value of
( )
A
x
at
x
shows the grade of membership of
x
in
A.
Definition 2.2
: Given a fuzzy set
A
in
X
and any real number
α
[0, 1], then the
α
cut or
α
level or cut worthy set of
A
, denoted by
α
A
is the crisp set
α
A
= {
x
X
:
( )
A
x
≥
α
} The strong
a
cut, denoted by
α
+
A
is the crisp set
α
+
A
= {
x
Є
X
:
( )
A
x
>
α
} For example, let A be a fuzzy set whose membership function is given as
,( ),
A
x aa x bb a xc xb x cc b
To find the α

cut of A, we first set α
ϵ
[0,1] to both left and right reference functions of A. That is,
x ab a
and
c xc b
.
Expressing x in terms of α we have
( )
x b a a
and
( )
x c c b
. w
hich gives the α
cut of A is
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Volume 2, Issue 1, March 2011
[( ) , ( ) ]
A b a a c c b
Definition 2.3
: The support of a fuzzy set
A
defined on
X
is a crisp set defined as Supp (
A
) = {
x
Є
X
:
( )
A
x
>
0
}
Definition 2.4
:
The height of a fuzzy set
A
, denoted by
h
(
A
) is the largest membership grade obtain by any element in the set.
( ) sup ( )
A x X
h A x
Definition 2.5
: A fuzzy number is a convex normalized fuzzy set of the real line
R
whose membership function is piecewise continuous.
Definition 2.6
: A triangular fuzzy number A can be defined as a triplet [
a, b, c
]. Its membership function is defined as:
,( ),
A
x aa x bb a xc xb x cc b
Definition 2.7
: A trapezoidal fuzzy number
A
can be expressed as [
a, b, c, d
] and its membership function is defined as:
,( ) 1,,
A
x aa x bb a x b x cd xc x d d c
3. Arithmetic operation of fuzzy
numbers using α
cut method:
In this section we consider arithmetic
operation on fuzzy numbers using α
cut cut method considering the same problem as done in [2] and [3] and hence make a comparative study.
3.1. Addition of fuzzy Numbers:
Let X= [a, b, c] and Y= [p, q, r] be two fuzzy numbers whose membership functions are
,( ),
X
x aa x bb a xc xb x cc b
,( ),
Y
x p p x qq p xr xq x r r q
Then
[( ) , ( ) ]
X b a a c c b
and
[( ) , ( ) ]
Y q p p r r q
are the
–
cuts of fuzzy numbers X and Y respectively. To calculate addition of fuzzy numbers X and Y we first add the
–
cuts of X and Y using interval arithmetic.
X Y
[( ) , ( ) ][( ) , ( ) ]
b a a c c bq p p r r q
[ ( ) ,( ) ]....................(3.1)
a p b a q pc r c b r q
To find the membership function
( )
X Y
x
we equate to x both the first and second component in (3.1) which gives
( )
x a p b a q p
and
( )
x c r c r b q
Now, expressing
in terms of x and setting
0
and
1
in (3.1) we get
together with the domain of x,
( ),( ) ( )( ) ( )
x a pa p x b qb q a p
and
( ),( ) ( )( ) ( )
c r xb q x c r c r b q
which gives
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( ),( ) ( )( ) ( )( )( ),( ) ( )( ) ( )
X Y
x a pa p x b qb q a p xc r xb q x c r c r b q
3.2.
Subtraction of Fuzzy Numbers:
Let X= [a, b, c] and Y= [p, q, r] be two fuzzy numbers. Then
[( ) , ( ) ]
X b a a c c b
and
[( ) , ( ) ]
Y q p p r r q
are the
–
cuts of fuzzy numbers X and Y respectively. To calculate subtraction of fuzzy numbers X and Y we first subtract the
–
cuts of X and Y using interval arithmetic.
X Y
[( ) , ( ) ][( ) , ( ) ]
b a a c c bq p p r r q
[( ) ( ( ) ),( ) (( ) )]
b a a r r qc c b q p p
[( ) ( ) ,( ) ( ) ].............(3.2)
a r b a r qc p c b q p
To find the membership function
( )
X Y
x
we equate to x both the first and second component in (3.2) which gives
( ) ( ) ,
x a r b a r q
and
( ) ( )
x c p c b q p
Now, expressing
in terms of x and setting
0
and
1
in (3.2) we get
together with the domain of x,
( ),( ) ( )( ) ( )
x a r a r x b qb q a r
and
( ),( ) ( )( ) ( )
c p xb q x c pc p b q
which gives
( ),( ) ( )( ) ( )( )( ),( ) ( )( ) ( )
X Y
x a r a r x b qb q a r xc p xb q x c pc p b q
3.3.
Multiplication of Fuzzy Numbers:
Let X= [a, b, c] and Y= [p, q, r] be two positive fuzzy numbers. Then
[( ) , ( ) ]
X b a a c c b
and
[( ) , ( ) ]
Y q p p r r q
are the
–
cuts of fuzzy numbers X and Y respectively. To calculate multiplication of fuzzy numbers X and Y we first multiply the
–
cuts of X and Y using interval arithmetic.
*
X Y
[( ) , ( ) ]*[( ) , ( ) ]
b a a c c bq p p r r q
[(( ) )*(( ) ),( ( ) )*( ( ) )].............(3.3)
b a a q p pc c b r r q
To find the membership function
( )
XY
x
we equate to x both the first and second component in (3.3) which gives
2
( )( ) (( ) ( ) )
x b a q p b a p q p a ap
and
2
( )( ) (( ) ( ) )
x c b r q r q c c b r cr
Now, expressing
in terms of x and setting
0
and
1
in (3.3) we get
together with the domain of x,
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which gives,
3.4.
Division of Fuzzy Numbers
:
Let X= [a, b, c] and Y= [p, q, r] be two positive fuzzy numbers. Then
[( ) , ( ) ]
X b a a c c b
and
[( ) , ( ) ]
Y q p p r r q
are the
–
cuts of fuzzy numbers X and Y respectively. To calculate division of fuzzy numbers X and Y we first divide the
–
cuts of X and Y using interval arithmetic.
X Y
=
[( ) , ( ) ][( ) , ( ) ]
b a a c c bq p p r r q
( ) ( ), .....(3.4)( ) ( )
b a a c c br r q q p p
To find the membership function
/
( )
X Y
x
we equate to x both the first and second component in (3.4) which gives
( )( )
b a a xr r q
and
( )( )
c c b xq p p
. Now, expressing
in terms of x and setting
0
and
1
in (3.4) we get
together with the domain of x,
( ) ( )
xr ab a q r x
,
/ /
a r x b q
and
( ) )
c pxc b q p x
,
/ /
b q x c p
which gives
/
, / /( ) ( )( ), / /( ) )
X Y
xr aa r x b qb a q r x xc pxb q x c pc b q p x
3.5.
Inverse of fuzzy number
:
Let
X
= [a, b, c] be a positive fuzzy number. Then
[( ) , ( ) ]
X b a a c c b
is the
cut of the fuzzy numbers
X
. To calculate inverse of the fuzzy number
X
we
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first take the inverse of the
cut of
X
using interval arithmetic.
1 1[( ) , ( ) ]
X b a a c c b
1 1, ......(3.5)( ) ( )
c c b b a a
To find the membership function
1/
( )
X
x
we equate to x both the first and second component in (3.5), which gives
1( )
xc c b
and
1( )
xb a a
Now, expressing
in terms of x and setting
0
and
1
in (3.5) we get
together with the domain of x,
1 1 1,( )
cx x x c b c b
and
1 1 1,( )
ax x x b a b a
which gives,
1/
1 1 1,( )( )1 1 1,( )
X
cx x x c b c b xax x x b a b a
3.6.
Exponential of a Fuzzy number:
Let X= [a, b, c] > 0 be a fuzzy number. Then
[( ) , ( ) ]
A b a a c c b
is the
cut of the fuzzy numbers A. To calculate exponential of the fuzzy number
A
we first take the exponential of the
cut of A using interval arithmetic.
exp( ) exp([( ) , ( ) ])
A b a a c c b
[exp(( ) ),exp( ( ) )]........(3.6)
b a a c c b
To find the membership function
exp( )
( )
X
x
we equate to x both the first and second component in (3.6) which gives
exp(( ) )
x b a a
and
exp( ( ) )
x c c b
Now, expressing
in terms of x and setting
0
and
1
in (3.6) we get
together with the domain of x,
ln( ),exp( ) exp( )
x aa x xb a
and
ln( ),exp( ) exp( )
c xb x cc b
which gives
exp( )
ln( ),exp( ) exp( )( )ln( ),exp( ) exp( )
X
x aa x xb a xc xb x cc b
3.7.
Logarithm of a fuzzy number:
Let X= [a, b, c] > 0 be a fuzzy number. Then
[( ) , ( ) ]
X b a a c c b
is the
cut of the fuzzy number X. To calculate logarithm of the fuzzy number X we first take the logarithm of the
cut of X using interval arithmetic.
ln( ) ln([( ) , ( ) ])
X b a a c c b
[ln(( ) ),ln( ( ) )]........(3.7)
b a a c c b
To find the membership function
ln( )
( )
X
x
we equate to x both the first and second component in (3.7) which gives
ln(( ) )
x b a a
and
ln( ( ) )
x c c b
. Now, expressing
in terms of x and setting
0
and
1
in (3.7) we get
together with the domain of x,