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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
otherwise
cxbxR
bxaxL
x
X,0
,
,
(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
0 cRaL
and
1 bRbL
. 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, x2… xn) 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
1 xxy
and
28
2
x
xy
. Let
yxz
, so we shall
have
13
1 xxxz
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 2 yty
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
1 xxy
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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[2]. Baruah, Hemanta K.; (2010 a), “Construction of
The Membership Function of a Fuzzy Number”, ICIC
Express Letters (Accepted for Publication: to appear in
February, 2011).
[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.
[5]. de Barra, G.; (1987), “Measure Theory and
Integration”, Wiley Eastern Limited, New Delhi.
[6]. Kaufmann A., and M. M. Gupta; (1984),
“Introduction to Fuzzy Arithmetic, Theory and
Applications”, Van Nostrand Reinhold Co. Inc.,
Wokingham, Berkshire.
[7]. Loeve M., (1977), “Probability Theory”, Vol.I,
Springer Verlag, New York.
[8]. Mahanta, Anjana K., Fokrul A. Mazarbhuiya and
Hemanta K. Baruah; (2008), “Finding Calendar Based
Periodic Patterns”, Pattern Recognition Letters, 29 (9),
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