FUZZY ARITHMETIC WITHOUT USING THE METHOD OF α - CUTS

International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) Volume 1, Issue 2, December 2010

73

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: [email protected] 2 Research Scholar, Department of Mathematics, Gauhati University, E-mail: [email protected] 3 Professor of Statistics, Gauhati University, Guwahati, E-mail: [email protected], [email protected]

Lx , a  x  b   X x   Rx , b  x  c 0, otherwise 

Abstract: In this article, an alternative method to evaluate the arithmetic operations on fuzzy number has been developed, on the assumption that the DuboisPrade left and right reference functions of a fuzzy

Lx  being continuous non-decreasing function in the

number are distribution function and complementary distribution function respectively. Using the method, the arithmetic operations of fuzzy numbers can be done in a

(1.1)

interval [a, b], and

Rx  being a continuous non-

increasing function in the interval [b, c], with

very simple way. This alternative method has been

La   Rc   0 and Lb  Rb  1 . Dubois and

demonstrated with the help of numerical examples.

Prade named

Lx  as left reference function and Rx 

as right reference function of the concerned fuzzy Key words and phrases: Fuzzy membership function, Dubois-Prade reference function, distribution function, Set superimposition, Glivenko-Cantelli theorem.

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). pp156).

1. INTRODUCTION

In this article, we are going to demonstrate the

 -cuts to the

easiness of applying our method in evaluating the

membership of fuzzy number does not always yield

arithmetic of fuzzy numbers if start from the simple

results. For example, the method of  -cuts fails to find

assumption that the Dubois-Prade left reference function

the fuzzy membership function (fmf) of even the simple

is a distribution function, and similarly the Dubois-

The standard method of

function

X when X is fuzzy. Indeed, for

X in

Prade right reference function is a complementary

particular, Chou (2009) has forwarded a method of

distribution function. Accordingly, the functions

finding the fmf for a triangular fuzzy number X . We

and

1  Rx

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

X  a, b, c with membership function

densities

L x 

would have to be associated with

d Lx  and d 1  Rx  in [a,b] and dx dx

[b,c] respectively (Baruah (2010 a, b)).

number

2. SUPERIMPOSITION OF SETS

International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) Volume 1, Issue 2, December 2010

The superimposition of sets defined by Baruah

74

((n-1) /n) in the entire interval, a

(1),

a

(2),………,

a

(n)

(1999), and later used successfully in recognizing

being values of a1, a2, ………, an arranged in increasing

periodic patterns (Mahanta et al. (2008)), the operation

order of magnitude, and b

of set superimposition is defined as follows: if the set A

values of b1, b2, ………, bn arranged in increasing order

is superimposed over the set B, we get

of magnitude.

A(S) B= (A-B) ∪ (A ∩ B) (2) ∪ (B-A)

(1),

b

(2),………,

b

(n)

being

We now define a random vector X = (X1, X2,

(2.1)

……,Xn) as a family of Xk, k = 1, 2,……, n, with every 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 [a 1, b1] and [a2, b2] superimposed gives

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

(2),

…, x(n) are

magnitude. Further let the sub-σ fields induced by Xkbe independent and identical. Define now

= [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

Фn(x)

(b1, b2), and b (2) = max (b1, b2).

= 0, if x < x (1), = (r-1)/n, if x (r-1)≤ x ≤ x (r), r= 2, 3, …, n,

Indeed, in the same way if [a1, b1] (1/2) and [a2, (1/2)

x

ordered values of x1, x2, ……, xnin increasing order of

[a1, b1] (S) [a2, b2]

b2]

(1),

= 1, if x ≥ x (n) ;

(2.4)

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

Фn(x) here is an empirical distribution function of a theoretical distribution function Ф(x).

give rise to As there is a one to one correspondence [a1, b1]

(1/2)

(S) [a2, b2]

(1/2)

between

(1/n)

, [a2, b2]

… [an, bn](1/n) all with membership value equal to 1/n

everywhere,

and

the

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.

[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)]

∪………∪ [a (n-1), a (n)]

(2/n)

∪ [a (n), b (1)] (1) ∪ [b (1), b (2)] ((n-1) /n)∪ ………∪ [b (n-

(2/n) ∪ [b (n-1), b (n)] (1/n), 2), b (n-1)]

where, for example, [b

measure

(2.2)

(1/n)

((n-1) /n)

Lebesgue-Stieltjes

distribution function, we would have

= [a (1), a (2)] (1/2)∪ [a (2), b (1)] (1)∪ [b (1), b (2)] (1/2). So for n fuzzy intervals [a1, b1]

a

(1),

uniformly fuzzy interval [b

b (1),

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,

(2.3) (2)]

b

((n-1) /n) (2)]

sup │Фn(x) - Ф(x) │ → 0 represents the

with membership

(2.6)

International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) Volume 1, Issue 2, December 2010

Observe that (r-1)/n in (2.4), for x (r), are

membership values of ((r -1) /n)

[b (n – r + 1), b (n - r)]

(r-1)≤

[a (r -1), a (r)]

x≤x

((r -1) /n)

and

in (2.3), for r = 2, 3, …, n.

Indeed this fact found from superimposition of

75

We start with equating

Rx  with R y  . And so, we obtain y  1 x  and y   2 x  respectively. Let z

uniformly fuzzy sets has led us to look into the

z  x  1 x 

possibility that there could possibly be a link between

and x   2 z  , say. Replacing

distribution functions and fuzzy membership.

and by  2 z  in g

In the sections 3, 4, 5 and 6 we are going to

two

triangular

Suppose Z

fuzzy

numbers.

 X  Y  a  p, b  q, c  r  be the

and

and

dx d   2  z    m2  z  dz dz

The distribution function for X  Y , would now be given by x

   z  m  z  dz, a  p  x  b  q 1

and the complementary distribution function would be (3.1)

1 (3.2)

would exist some density functions for the distribution

f x  

and

1  Rx . Say,

d Lx , a  x  b dx

d 1  Rx , b  x  c g x   dx

   z  m  z  dz, b  q  x  c  r 2

2

bq

complementary distribution function. Accordingly, there

and

given by x

x  and L y  are the left reference functions and Rx  and R y  are the right reference functions respectively. We assume that L x  and L y  are distribution function and Rx  and R y  are x 

1

a p

where L

functions L

we obtain f x   1 z  and

dx d   1  z    m1  z  dz dz

 X x  and  Y  y  as mentioned below

L y , a  y  b   Y  y   R y , b  y  c 0, otherwise 

x by  1 z  in f x  ,

Now let,

fuzzy number of X  Y . Let the fmf of X and Y be

Lx , a  x  b   X x   Rx , b  x  c 0, otherwise 

x  ,

x   1 z 

g x    2 z  say.

3. ADDITION OF FUZZY NUMBERS

X  a, b, c and Y   p, q, r  be

 x  y , so we have

and z  x   2 x  , so that

discuss the arithmetic of fuzzy numbers.

Consider

Lx  with L y  , and

We claim that this distribution function and the complementary distribution function constitute the fuzzy membership function of X  Y as,

 x   1  z  m1  z  dz , a  p  x  b  q a p  x   X Y  x   1   2  z  m2  z  dz , b  q  x  c  r  bq 0 , otherwise  

4. SUBTRACTION OF FUZZY NUMBERS

International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) Volume 1, Issue 2, December 2010

Let

X  a, b, c and Y   p, q, r  be two

5.

MULTIPLICATION

fuzzy numbers with fuzzy membership function as in

NUMBERS

(3.1) and (3.2). Suppose Z  X  Y . Then the fuzzy

Let X

membership function of Z  X  Y would be given by Z

 X   Y  .

 Y    r,q, p

Suppose number of

 Y  .

reference functions complementary

and Y

OF

 a, b, c ,

  p, q, r  ,

FUZZY

a, b, c  0

 p, q, r  0

be two triangular

fuzzy numbers with fuzzy membership function as in be the fuzzy

(3.1) and (3.2). Suppose Z

 X .Y  a. p, b.q, c.r  be

We assume that the Dubois-Prade

the fuzzy number of X .Y .

Lx  and L y  are the left

L y  and R y  as distribution and

distribution

function

respectively.

reference functions and

for the distribution functions

L y  and 1  R y  .

Rx  and R y  are the right

reference functions respectively. We assume that

Accordingly, there would exist some density functions and

L x 

L y  are distribution function and Rx  and R y 

are complementary distribution function. Accordingly,

Say,

there would exist some density functions for the

d L y , p  y  q f y  dy and

76

gy 

distribution functions

f x  

d 1  R y , q  y  r dy

dy Let t   y so that  1  mt  , say. Replacing dt

y  t in

f  y  and g  y  ,

we

g x  

and

Lx  and 1  Rx  . Say,

d Lx , a  x  b dx

d 1  Rx , b  x  c dx

We again start with equating

obtain

Lx  with L y  ,

f  y   1 t  and g  y   2 t  , say. Then the fmf of

and

Rx  with R y  . And so, we obtain y  1 x 

 Y  would be given by

and

y  2 x  respectively. Let z  x. y , so we have

y  r  y  q   2 t mt dt , r  y   Y  y   1  1 t mt dt ,  q  y   p  q 0 , otherwise  

z  x.1 x  and z  x.2 x  , so that x   1 z  and x

and by

earlier section.

 Y 

 2 z  in g x  ,

we obtain

f x   1 z  and

g x    2 z  say. dx d   1  z    m1  z  dz dz

Now let,

Then we can easily find the fmf of X  Y by addition of fuzzy numbers X and

  2 z  , say. Replacing x by  1 z  in f x  ,

and

as described in the

dx d   2  z    m2  z  dz dz

The distribution function for X .Y , would now be given by

International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) Volume 1, Issue 2, December 2010 x

L y  and 1  R y  .Say,

for the distribution functions

 z m z dz, ap  x  bq 1

77

1

ap

f y 

and the complementary distribution function would be given by x

1   2 z m2 z dz, bq  x  cr

and

gy 

d 1  R y , q  y  r dy

bq

We are claiming that this distribution function

d L y , p  y  q dy

t  y 1 so that

Let

and the complementary distribution function constitute the fuzzy membership function of

Replacing

X .Y as,

Y  would be given by 1

6. DIVISION OF FUZZY NUMBERS

 a, b, c ,

and Y

  p, q, r  ,

a, b, c  0

 p, q, r  0

f  y  and g  y  , we obtain

f  y   1 t  and g  y   2 t  , say. Then the fmf of

x   1 z m1 z dz , ap  x  bq ap  x   XY x   1    2 z m2 z dz, bq  x  cr  bq 0 , otherwise  

Let X

y  t 1 in

dy 1   2  mt  , say. dt t

be two triangular

fuzzy numbers with fuzzy membership function as in

y r 1  y  q 1   2 t mt dt , r 1  y  Y 1  y   1  1 t mt dt , q 1  y  p 1  q1 0 , otherwise   Next, we can easily find the fmf of multiplication

of

fuzzy

numbers

X

X Y

and Y 1

(3.1) and (3.2). Suppose Z 

X . Then the fuzzy Y

described in the earlier section.

membership function of Z 

X Y

numerical examples for the above discussed methods.

by Z  X .Y

1

of Y

1





1

.

Y 1  r 1, q 1, p 1 be the fuzzy number

distribution

Example 1: Let X

 1,2,4

and

Y  3,5,6 be two

triangular fuzzy numbers with fmf

L y  and R y  as distribution and

complementary

In the next section we are going to cite some

7. NUMERICAL EXAMPLES

.

. We assume that the Dubois-Prade reference

functions

as

would be given

At first, we have to find the fmf of Y Suppose

by

function

respectively.

Accordingly, there would exist some density functions

 x  1, 1  x  2 4  x   X x    , 2 x4  2 , otherwise 0

(7.1)

International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) Volume 1, Issue 2, December 2010

y 3  2 , 3 y 5   Y  y   6  y, 5  y  6 0 , otherwise  

And

Here X

 Y  4,7,10 .

in

we

(7.2)

and y  2  x  

Now, of (Y ) .

Equating

the

y  1 x   2 x  1

obtain

x8 . Let z  x  y , so we shall 2

density functions have

Now

3x  8 , 2

and

1  2 z  . 2

d  1 z   1 dz 3

m2  z  

d  2 z   2 . dz 3

Example 2: Let X

 1,2,4

and

Y  3,5,6 be two

Let

t  y

so

that

y  t , which

density

function

gy 

d 1  6  y   1  2 t , 5  y  6 . dy

Then the fmf of

 Y  would be given by

6  y  6  5 ,  6  y  5  3  y  Y  y    ,  5  y  3 3  5  , otherwise 0   Then by addition of fuzzy numbers

X  1,2,4 and

 Y    6,5,3 the fmf of X  Y is given by, x  5  2 ,  5  x  3  1  x  X   Y   x    , 3  x 1 4  , otherwise 0 

Then the fmf of X  Y would be given by,

x  4  3 , 4 x7  10  x  X Y  x    , 7  x  10 3  , otherwise 0  

 Y   6,5,3 be the fuzzy number

d  y  3 1     1 t , 3  y  5 and dy  2  2

f x  and g x  respectively, we

m1 z  

Suppose,

f y 

x by  1 z  and  2 z  in the

f x   1  1 z  and g x  

(7.2).

t   1 . Then the f  y  and g  y  would be, say,

z 1 2z  8 so that x   1 z   and x   2 z   , 3 3 respectively. Replacing

and

implies m

have

z  x  1 x   3x  1 and z  x  2 x  

(7.1)

Z  X  Y or Z  X  (Y ) .

distribution function and complementary distribution function,

78

Example 3: Let X

 1,2,4

and

Y  3,5,6 be two

triangular fuzzy numbers with membership functions as





in (7.1) and (7.2). Suppose, X .Y  3,10,24 be the fuzzy number of X .Y . Equating the distribution function and complementary distribution function, we

triangular fuzzy numbers with membership functions as obtain

y  1 x   2 x  1 and y  2 x  

x8 . 2

International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) Volume 1, Issue 2, December 2010

Let z

 x. y ,

so

we

f x  and

1  6  y 1 1 ,  y  5 65 6 1  3 1 1 y  1 y   ,  y 3 Y 53 5 , otherwise 0    

f x   1  1 z 

Then by multiplication of fuzzy numbers

shall

have

z  x.1 x   2 x  x and 2

8x  x 2 z  x. 2  x   2 1

x   1 z  

,

so

and

 2 z 

g x  respectively, and g x  

Now

in the density functions we

that

1  8z 4

and x   2 z   4  16  2 z . Replacing

 1 z 

have

x by

1  2 z  . 2

m1 z  

Then the fmf of

and

d  1 z  and m2 z   d  2 z  . dz dz

 1,2,4

and

Y  3,5,6 be two

Then the fmf of Y

1



the

fuzzy

membership

X would be given by, Y

Example 5: Let X

 2,4,5 be a triangular fuzzy number

Y  4,16,25 which is a non-triangular fuzzy

number with membership functions respectively as,

x  2  2 , 2 x4   X x   5  x, 4  x  5 0 , otherwise  

triangular fuzzy numbers with membership functions as in (7.1) and (7.2). Suppose, Z 



X  1,2,4

2 6x  1 1  x 1 , 6  x  5  4  4  3x 2  X x    , x 3 Y  2 x  1 5 0 , otherwise  

and

Example 4:



Y 1  61 ,51 ,31

function of

X .Y would be given by

 1  8x  5 , 3  x  10  4   8  16  2 x  X .Y x    , 10  x  24 2  , otherwise 0   Let X

79

X 1 or Z  X .Y . Y



 61 ,51 ,31 is given as

and

 y 2 , 4  y  16   2  Y  y   5  y , 16  y  25 0 , otherwise  

We can find the fmf of

X  Y which is given by,

International Journal of Latest Trends in Computing (E-ISSN: 2045-5364) Volume 1, Issue 2, December 2010

 1  4x  2 , 6  x  20  4  11  1  4 x  X Y x    , 20  x  30 2  , otherwise 0   All four demonstrations above can be verified to be true, using the method of  -cuts.

80

[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

9. CONCLUSION

to

Fuzzy

Arithmetic,

Theory

and

Applications”, Van Nostrand Reinhold Co. Inc.,

The standard method of

 -cuts to the

Wokingham, Berkshire.

membership of a fuzzy number does not always yield

[7]. Loeve M., (1977), “Probability Theory”, Vol.I,

results. We have demonstrated that an assumption that

Springer Verlag, New York.

the Dubois-Prade left reference function is a distribution

[8]. Mahanta, Anjana K., Fokrul A. Mazarbhuiya and

function and that the right reference function is a

Hemanta K. Baruah; (2008), “Finding Calendar Based

complementary distribution function leads to a very

Periodic Patterns”, Pattern Recognition Letters, 29 (9),

simple way of dealing with fuzzy arithmetic. Further,

1274-1284.

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.

REFERENCES [1]. Baruah, Hemanta K.; (1999), “Set Superimposition and Its Applications to the Theory of Fuzzy Sets”, Journal of the Assam Science Society, Vol. 40, Nos. 1 & 2, 25-31. [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).