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FUZZY ARITHMETIC WITHOUT USING THE METHOD OF ALPHA CUTS

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Abstract

In this article, an alternative method to evaluate the arithmetic operations on fuzzy number has been developed, on the assumption that the Dubois-Prade left and right reference functions of a fuzzy number are distribution function and complementary distribution function respectively. Using the method, the arithmetic operations of fuzzy numbers can be done in a very simple way. This alternative method has been demonstrated with the help of numerical examples.
International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) 73
Volume 1, Issue 2, December 2010
FUZZY ARITHMETIC WITHOUT USING THE
METHOD OF
- CUTS
Supahi Mahanta1, Rituparna Chutia2, Hemanta K Baruah3
1
(Corresponding author) Research Scholar, Department of Statistics, Gauhati University, E-mail: supahi_mahanta@rediffmail.com
2Research Scholar, Department of Mathematics, Gauhati University, E-mail: Rituparnachutia7@rediffmail.com
3Professor of Statistics, Gauhati University, Guwahati, E-mail: hemanta_bh@yahoo.com, hemanta@gauhati.ac.in
Abstract: In this article, an alternative method to
evaluate the arithmetic operations on fuzzy number has
been developed, on the assumption that the Dubois-
Prade left and right reference functions of a fuzzy
number are distribution function and complementary
distribution function respectively. Using the method, the
arithmetic operations of fuzzy numbers can be done in a
very simple way. This alternative method has been
demonstrated with the help of numerical examples.
Key words and phrases: Fuzzy membership function,
Dubois-Prade reference function, distribution function,
Set superimposition, Glivenko-Cantelli theorem.
1. INTRODUCTION
The standard method of
-cuts to the
membership of fuzzy number does not always yield
results. For example, the method of
-cuts fails to find
the fuzzy membership function (fmf) of even the simple
function
X
when
X
is fuzzy. Indeed, for
X
in
particular, Chou (2009) has forwarded a method of
finding the fmf for a triangular fuzzy number
X
. We
shall in this article, put forward an alternative method
for dealing with the arithmetic of fuzzy numbers which
are not necessarily triangular.
Dubois and Prade (see e.g. Kaufmann and
Gupta (1984)) have defined a fuzzy number
 
cbaX ,,
with membership function
(1.1)
 
xL
being continuous non-decreasing function in the
interval [a, b], and
 
xR
being a continuous non-
increasing function in the interval [b, c], with
 
0cRaL
and
   
1bRbL
. Dubois and
Prade named
 
xL
as left reference function and
 
xR
as right reference function of the concerned fuzzy
number. A continuous non-decreasing function of this
type is also called a distribution function with reference
to a Lebesgue-Stieltjes measure (de Barra (1987). pp-
156).
In this article, we are going to demonstrate the
easiness of applying our method in evaluating the
arithmetic of fuzzy numbers if start from the simple
assumption that the Dubois-Prade left reference function
is a distribution function, and similarly the Dubois-
Prade right reference function is a complementary
distribution function. Accordingly, the functions
 
xL
and
 
xR1
would have to be associated with
densities
 
xL
dx
d
and
 
xR
dx
d1
in [a,b] and
[b,c] respectively (Baruah (2010 a, b)).
2. SUPERIMPOSITION OF SETS
International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) 74
Volume 1, Issue 2, December 2010
The superimposition of sets defined by Baruah
(1999), and later used successfully in recognizing
periodic patterns (Mahanta et al. (2008)), the operation
of set superimposition is defined as follows: if the set A
is superimposed over the set B, we get
A(S) B= (A-B) (A B) (2) (B-A) (2.1)
where S represents the operation of superimposition,
and (A B) (2) represents the elements of (A B)
occurring twice. It can be seen that for two intervals [a1,
b1] and [a2, b2] superimposed gives
[a1, b1] (S) [a2, b2]
= [a (1), a (2)] [a (2), b (1)] (2) [b (1), b (2)]
where a (1) = min (a1, a2), a (2) = max (a1, a2), b (1) = min
(b1, b2), and b (2) = max (b1, b2).
Indeed, in the same way if [a1, b1] (1/2) and [a2,
b2] (1/2) represent two uniformly fuzzy intervals both
with membership value equal to half everywhere,
superimposition of [a1, b1] (1/2) and [a2, b2] (1/2) would
give rise to
[a1, b1](1/2) (S) [a2, b2] (1/2)
= [a (1), a (2)] (1/2) [a (2), b (1)] (1) [b (1), b (2)] (1/2). (2.2)
So for n fuzzy intervals [a1, b1] (1/n), [a2, b2]
(1/n) [an, bn](1/n) all with membership value equal to 1/n
everywhere,
[a1, b1] (1/n)(S) [a2, b2] (1/n)(S) ……… (S) [an, bn] (1/n)
= [a (1), a (2)] (1/n) [a (2), a (3)] (2/n)……… [a (n-1), a (n)]
((n-1) /n) [a (n), b (1)] (1) [b (1), b (2)] ((n-1) /n) ……… [b (n-
2), b (n-1)] (2/n) [b (n-1), b (n)] (1/n), (2.3)
where, for example, [b (1), b (2)]((n-1) /n) represents the
uniformly fuzzy interval [b (1), b (2)] with membership
((n-1) /n) in the entire interval, a (1), a (2),………, a (n)
being values of a1, a2, ………, an arranged in increasing
order of magnitude, and b (1), b (2),………, b (n) being
values of b1, b2, ………, bn arranged in increasing order
of magnitude.
We now define a random vector X = (X1, X2,
……,Xn) as a family of Xk, k = 1, 2,……, n, with every
Xkinducing a sub-σ field so that X is measurable. Let
(x1, x2xn) be a particular realization of X, and let X(k)
realize the value x(k) where x (1), x (2), …, x(n) are
ordered values of x1, x2, ……, xnin increasing order of
magnitude. Further let the sub-σ fields induced by Xkbe
independent and identical. Define now
Фn(x) = 0, if x < x (1),
= (r-1)/n, if x (r-1) x x (r), r= 2, 3, …, n,
= 1, if x ≥ x (n) ; (2.4)
Фn(x) here is an empirical distribution function of a
theoretical distribution function Ф(x).
As there is a one to one correspondence
between a Lebesgue-Stieltjes measure and the
distribution function, we would have
Π (a, b) = Ф (b) Ф (a) (2.5)
whereΠ is a measure in (Ω, A,
Π
), A being the σ- field
common to every xk.
Now the Glivenko-Cantelli theorem (see e.g.
Loeve (1977), pp-20) states that
Фn(x) converges to Ф(x) uniformly in x. This means,
sup │Фn(x) - Ф(x) │ → 0 (2.6)
International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) 75
Volume 1, Issue 2, December 2010
Observe that (r-1)/n in (2.4), for x (r-1) x x
(r), are membership values of [a (r -1), a (r)] ((r -1) /n) and
[b (n r + 1), b (n - r)] ((r -1) /n) in (2.3), for r = 2, 3, …, n.
Indeed this fact found from superimposition of
uniformly fuzzy sets has led us to look into the
possibility that there could possibly be a link between
distribution functions and fuzzy membership.
In the sections 3, 4, 5 and 6 we are going to
discuss the arithmetic of fuzzy numbers.
3. ADDITION OF FUZZY NUMBERS
Consider
 
cbaX ,,
and
 
rqpY ,,
be
two triangular fuzzy numbers.
Suppose
YXZ
 
rcqbpa ,,
be the
fuzzy number of
YX
. Let the fmf of
X
and
Y
be
 
x
X
and
 
y
Y
as mentioned below
   
 
otherwise
cxbxR
bxaxL
x
X,0
,
,
(3.1)
and
   
 
otherwise
cybyR
byayL
y
Y,0
,
,
(3.2)
where
 
xL
and
 
yL
are the left reference functions
and
 
xR
and
 
yR
are the right reference functions
respectively. We assume that
 
xL
and
 
yL
are
distribution function and
 
xR
and
 
yR
are
complementary distribution function. Accordingly, there
would exist some density functions for the distribution
functions
 
xL
and
 
xR1
. Say,
   
bxaxL
dx
d
xf ,
and
   
cxbxR
dx
d
xg ,1
We start with equating
 
xL
with
 
yL
, and
 
xR
with
 
yR
. And so, we obtain
 
xy 1
and
 
xy 2
respectively. Let
yxz
, so we have
 
xxz 1
and
 
xxz 2
, so that
 
zx 1
and
 
zx 2
, say. Replacing
x
by
 
z
1
in
 
xf
,
and by
 
z
2
in
 
xg
, we obtain
   
zxf 1
and
   
zxg 2
say.
Now let,
 
 
 
11
dx d z m z
dz dz

and
 
 
 
22
dx d z m z
dz dz

The distribution function for
YX
, would now be given
by
   
11 ,
x
ap
z m z dz a p x b q
 
and the complementary distribution function would be
given by
   
22
1,
x
bq
z m z dz b q x c r
 
We claim that this distribution function and the
complementary distribution function constitute the fuzzy
membership function of
YX
as,
 
   
   
11
22
,
1,
0,
x
ap
x
XY bq
z m z dz a p x b q
x z m z dz b q x c r
otherwise

 
 
4. SUBTRACTION OF FUZZY NUMBERS
International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) 76
Volume 1, Issue 2, December 2010
Let
 
cbaX ,,
and
 
rqpY ,,
be two
fuzzy numbers with fuzzy membership function as in
(3.1) and (3.2). Suppose
YXZ
. Then the fuzzy
membership function of
YXZ
would be given
by
 
YXZ
.
Suppose
 
 
pqrY ,,
be the fuzzy
number of
 
Y
. We assume that the Dubois-Prade
reference functions
 
yL
and
 
yR
as distribution and
complementary distribution function respectively.
Accordingly, there would exist some density functions
for the distribution functions
 
yL
and
 
yR1
.
Say,
   
qypyL
dy
d
yf ,
and
   
ryqyR
dy
d
yg ,1
Let
yt
so that
 
tm
dt
dy 1
, say. Replacing
ty
in
 
yf
and
 
yg
, we obtain
 
tyf 1
and
 
tyg 2
, say. Then the fmf of
 
Y
would be given by
 
   
   
otherwise
pyqdttmt
qyrdttmt
yy
q
y
r
Y
,0
,1
,
1
2
Then we can easily find the fmf of
YX
by addition
of fuzzy numbers
X
and
 
Y
as described in the
earlier section.
5. MULTIPLICATION OF FUZZY
NUMBERS
Let
 
cbaX ,,
,
 
0,, cba
and
 
rqpY ,,
,
 
0,, rqp
be two triangular
fuzzy numbers with fuzzy membership function as in
(3.1) and (3.2). Suppose
 
rcqbpaYXZ .,.,..
be
the fuzzy number of
YX.
.
 
xL
and
 
yL
are the left
reference functions and
 
xR
and
 
yR
are the right
reference functions respectively. We assume that
 
xL
and
 
yL
are distribution function and
 
xR
and
 
yR
are complementary distribution function. Accordingly,
there would exist some density functions for the
distribution functions
 
xL
and
 
xR1
. Say,
   
bxaxL
dx
d
xf ,
and
   
cxbxR
dx
d
xg ,1
We again start with equating
 
xL
with
 
yL
,
and
 
xR
with
 
yR
. And so, we obtain
 
xy 1
and
 
xy 2
respectively. Let
yxz .
, so we have
 
xxz 1
.
and
 
xxz 2
.
, so that
 
zx 1
and
 
zx 2
, say. Replacing x by
 
z
1
in
 
xf
,
and by
 
z
2
in
 
xg
, we obtain
   
zxf 1
and
   
zxg 2
say.
Now let,
 
 
 
11
dx d z m z
dz dz

and
 
 
 
22
dx d z m z
dz dz

The distribution function for
YX.
, would now be given
by
International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) 77
Volume 1, Issue 2, December 2010
   
x
ap
bqxapdzzmz ,
11
and the complementary distribution function would be
given by
   
x
bq
crxbqdzzmz ,1 22
We are claiming that this distribution function
and the complementary distribution function constitute the
fuzzy membership function of
YX.
as,
 
   
   
otherwise
crxbqdzzmz
bqxapdzzmz
xx
bq
x
ap
XY
,0
,1
,
22
11
6. DIVISION OF FUZZY NUMBERS
Let
 
cbaX ,,
,
 
0,, cba
and
 
rqpY ,,
,
 
0,, rqp
be two triangular
fuzzy numbers with fuzzy membership function as in
(3.1) and (3.2). Suppose
Y
X
Z
. Then the fuzzy
membership function of
Y
X
Z
would be given
by
1
.
YXZ
.
At first, we have to find the fmf of
1
Y
.
Suppose
 
1111 ,, pqrY
be the fuzzy number
of
1
Y
. We assume that the Dubois-Prade reference
functions
 
yL
and
 
yR
as distribution and
complementary distribution function respectively.
Accordingly, there would exist some density functions
for the distribution functions
 
yL
and
 
yR1
.Say,
   
qypyL
dy
d
yf ,
and
   
ryqyR
dy
d
yg ,1
Let
1
yt
so that
 
tm
tdt
dy 2
1
, say.
Replacing
1
ty
in
 
yf
and
 
yg
, we obtain
 
tyf 1
and
 
tyg 2
, say. Then the fmf of
 
1
Y
would be given by
 
   
   
otherwise
pyqdttmt
qyrdttmt
yy
q
y
r
Y
,0
,1
,
1
1
111
1
11
2
Next, we can easily find the fmf of
Y
X
by
multiplication of fuzzy numbers
X
and
1
Y
as
described in the earlier section.
In the next section we are going to cite some
numerical examples for the above discussed methods.
7. NUMERICAL EXAMPLES
Example 1:
Let
 
4,2,1X
and
 
6,5,3Y
be two
triangular fuzzy numbers with fmf
 
otherwise
x
x
xx
x
X
,0
42,
2
4
21,1
(7.1)
International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) 78
Volume 1, Issue 2, December 2010
And
 
otherwise
yy
y
y
y
Y,0
65,6
53,
23
(7.2)
Here
 
10,7,4YX
. Equating the
distribution function and complementary distribution
function, we obtain
 
12
1xxy
and
 
28
2
x
xy
. Let
yxz
, so we shall
have
 
13
1xxxz
and
 
283
2
x
xxz
,
so that
 
31
1
z
zx
and
 
382
2
z
zx
,
respectively. Replacing
x
by
 
z
1
and
 
z
2
in the
density functions
 
xf
and
 
xg
respectively, we
have
   
zxf 1
1
and
   
zxg 2
2
1
.
Now
   
3
1
11 z
dz
d
zm
and
   
3
2
22 z
dz
d
zm
.
Then the fmf of
YX
would be given by,
 
otherwise
x
x
x
x
x
YX
,0
107,
3
10
74,
34
Example 2:
Let
 
4,2,1X
and
 
6,5,3Y
be two
triangular fuzzy numbers with membership functions as
in (7.1) and (7.2). Suppose,
YXZ
or
)( YXZ
.
Now,
 
3,5,6 Y
be the fuzzy number
of
)( Y
. Let
yt
so that
ty
, which
implies
 
1tm
. Then the density function
 
yf
and
 
yg
would be, say,
 
53,
2
1
231
yt
y
dy
d
yf
and
 
65,161 2yty
dy
d
yg
.
Then the fmf of
 
Y
would be given by
 
otherwise
y
y
y
y
y
Y
,0
35,
53
3
56,
56
6
Then by addition of fuzzy numbers
 
4,2,1X
and
 
 
3,5,6 Y
the fmf of
YX
is given by,
 
 
otherwise
x
x
x
x
x
YX
,0
13,
4
1
35,
25
Example 3:
Let
 
4,2,1X
and
 
6,5,3Y
be two
triangular fuzzy numbers with membership functions as
in (7.1) and (7.2). Suppose,
 
24,10,3. YX
be the
fuzzy number of
YX.
. Equating the distribution
function and complementary distribution function, we
obtain
 
12
1xxy
and
 
28
2
x
xy
.
International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) 79
Volume 1, Issue 2, December 2010
Let
yxz .
, so we shall have
 
xxxxz 2
12.
and
 
2
8
.2
2xx
xxz
, so that
 
4811
1z
zx
and
 
zzx 2164
2
. Replacing
x
by
 
z
1
and
 
z
2
in the density functions
 
xf
and
 
xg
respectively, we have
   
zxf 1
1
and
   
zxg 2
2
1
.
Now
   
z
dz
d
zm 11
and
   
z
dz
d
zm 22
.
Then the fmf of
YX.
would be given by
 
otherwise
x
x
x
x
x
YX
,0
2410,
22168
103,
4581
.
Example 4:
Let
 
4,2,1X
and
 
6,5,3Y
be two
triangular fuzzy numbers with membership functions as
in (7.1) and (7.2). Suppose,
Y
X
Z
or
1
.
YXZ
.
Then the fmf of
 
1111 3,5,6 Y
is given as
 
otherwise
y
y
y
y
y
Y,0 3
1
5
1
,
35
3
15
1
6
1
,
56
1
6
1
Then by multiplication of fuzzy numbers
 
4,2,1X
and
 
1111 3,5,6 Y
the fuzzy membership
function of
Y
X
would be given by,
   
otherwise
x
xx
x
x
x
x
Y
X
,0
3
4
5
2
,
12 34 5
2
6
1
,
1
16
Example 5:
Let
 
5,4,2X
be a triangular fuzzy number
and
 
25,16,4Y
which is a non-triangular fuzzy
number with membership functions respectively as,
 
otherwise
xx
x
x
x
X,0
54,5
42,
22
and
 
otherwise
yy
y
y
y
Y
,0
2516,5
164,
22
We can find the fmf of
YX
which is given by,
International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) 80
Volume 1, Issue 2, December 2010
 
otherwise
x
x
x
x
x
YX
,0
3020,
24111
206,
4241
All four demonstrations above can be verified to be true,
using the method of
-cuts.
9. CONCLUSION
The standard method of
-cuts to the
membership of a fuzzy number does not always yield
results. We have demonstrated that an assumption that
the Dubois-Prade left reference function is a distribution
function and that the right reference function is a
complementary distribution function leads to a very
simple way of dealing with fuzzy arithmetic. Further,
this alternative method can be utilized in the cases
where the method of
-cuts fails, e.g. in finding the
fmf of
X
.
10. ACKNOWLEDGEMENT
This work was funded by a BRNS Research
Project, Department of Atomic Energy,
Government of India.
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[3]. Baruah, Hemanta K.; (2010 b), " The Mathematics of Fuzziness: Myths and Realities ", Lambert Academic Publishing, Saarbrucken, Germany. [4]. Chou, Chien-Chang; (2009), " The Square Roots of Triangular Fuzzy Number ", ICIC Express Letters. Vol. 3, Nos. 2, pp. 207-212.