fuzzy arithmetic operations over set theory

fuzzy arithmetic operation s

1. Fuzzy Numbers and Fuzzy Arithmetic
March 26, 2020FLNN, Spring 2020
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OBJECTIVES
1. Define fuzzy numbers
2. To learn how to perform arithmetic operations on fuzzy
numbers

2. Fuzzy Number
March 26, 2020FLNN, Spring 2020
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To qualify as a fuzzy number, a fuzzy set A on R must
possess the following three properties:
1. A must be a normal fuzzy set;
2.

A must be closed interval for every  (0, 1];
3. The support of A,
0+
A, must be bounded.
The fuzzy set must be normal since our conception of a set of “real
numbers close to r” is fully satisfied by r itself.

3. Fuzzy Number (figure from Klir&Yuan)
March 26, 2020FLNN, Spring 2020
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4. Fuzzy Arithmetic
March 26, 2020FLNN, Spring 2020
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Fuzzy arithmetic is based on two properties of fuzzy numbers:
1)Each fuzzy set, and thus also each fuzzy number, can uniquely be
represented by it’s -cuts.
2)-cuts of each fuzzy number are closed intervals of real numbers for all
(o, 1].
These two properties enable us to define arithmetic operations on
fuzzy numbers in terms of arithmetic operations on it’s -cuts. The latter
operations are a subject of interval analysis.

5. Arithmetic Operations on intervals
March 26, 2020FLNN, Spring 2020
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Let * denote any of the four arithmetic operations on closed interval:
addition +, subtraction -, multiplication . and division /. Then,
[a, b]*[d, e]={f*g | afb, dge}
is a general property of all arithmetic operations on closed intervals,
except that [a, b]/[d, e] is not defined when 0 [d, e]. That is, the result
of an arithmetic operation on closed intervals is again a closed
interval.

6. Arithmetic Operations on intervals
March 26, 2020FLNN, Spring 2020
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The four arithmetic operations on closed intervals are defined as
follows:
[a, b]+[d, e]=[a+d, b+e],
[a, b] - [d, e]=[a - e, b - d],
[a, b] . [d, e]=[min(ad, ae, bd, be], max(ad, ae, bd, be)],
And, provided that 0[d, e],
[a, b] / [d, e]=[a, b].[1/e, 1/d]
= [min(a/d, a/e, b/d, b/e), max(a/d, a/e, b/d, b/e)].

7. Arithmetic Operations on intervals
March 26, 2020FLNN, Spring 2020
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The following are a few examples illustrating the interval-valued
arithematic operations:
[2, 5]+[1,3]=[3, 8] [0, 1]+[-6, 5]=[-6, 6]
[2, 5]-[1, 3]=[-1, 4][0, 1]-[-6, 5]=[-5, 7]
[-1, 1].[-2, -0.5]=[-2, 2][3, 4].[2, 2]=[6, 8]
[-1, 1]/[-2, 0.5] = [-2, 2][4, 10]/[1, 2]=[2, 10]

8. Arithmetic Operations on Fuzzy Numbers
March 26, 2020FLNN, Spring 2020
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1. First method: Based on interval arithmetic
Let A and B denote fuzzy numbers and let * denote any of the four
arithmetic operations. Then, we define a fuzzy set on R, A*B, by
defining its -cut,

(A*B), as

(A*B) =

A*

B for any (0, 1].
(When *=/, we have to require that 0B for all (0, 1].
A*B can be expressed as:
A*B= .

(A*B), [0, 1]

9. Arithmetic Operations on Fuzzy Numbers
March 26, 2020FLNN, Spring 2020
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Example:
consider two triangular-shape fuzzy numbers A and B defined as
follows:

10. Arithmetic Operations on Fuzzy Numbers
March 26, 2020FLNN, Spring 2020
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Their –cuts are:
So, we obtain

11. Arithmetic Operations on Fuzzy Numbers
March 26, 2020FLNN, Spring 2020
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The resulting fuzzy numbers are then:

12. Operations on Fuzzy Numbers: Addition and Subtraction (figure from Klir&Yuan)
March 26, 2020FLNN, Spring 2020
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13. Arithmetic Operations on Fuzzy Numbers
March 26, 2020FLNN, Spring 2020
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2. Second method: Based on extension principle
Let * denote any of the four basic arithmetic operations and let A, B
denote fuzzy numbers. Then, we define a fuzzy set on R, A*B, by
the equation
(A*B)(z) = sup
z=x*y
min [A(x), B(y)]for all zR.
More specifically, we define for all zR:
(A+B)(z) = sup
z=x+y min [A(x), B(y)]for all zR.
(A-B)(z) = sup
z=x-y
min [A(x), B(y)]for all zR.

14. Arithmetic Operations on Fuzzy Numbers
March 26, 2020FLNN, Spring 2020
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Example:
Add the fuzzy numbers A and B, where
Solution:

15. Arithmetic Operations on Fuzzy Numbers
March 26, 2020FLNN, Spring 2020
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