fuzzy arithmetic operations over set theory

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

Fuzzy Number
March 26, 2020
FLNN, Spring 2020
2
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.

Fuzzy Number (figure from Klir&Yuan)
March 26, 2020
FLNN, Spring 2020
3

Fuzzy Arithmetic
March 26, 2020
FLNN, Spring 2020
4
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.

Arithmetic Operations on intervals
March 26, 2020
FLNN, Spring 2020
5
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.

March 26, 2020
FLNN, Spring 2020
6
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)].

March 26, 2020
FLNN, Spring 2020
7
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]

Arithmetic Operations on Fuzzy Numbers
March 26, 2020
FLNN, Spring 2020
8
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]

March 26, 2020
FLNN, Spring 2020
9
Example:
consider two triangular-shape fuzzy numbers A and B defined as
follows:

March 26, 2020
FLNN, Spring 2020
10
Their –cuts are:
So, we obtain

March 26, 2020
FLNN, Spring 2020
11
The resulting fuzzy numbers are then:

Operations on Fuzzy Numbers: Addition and Subtraction (figure from Klir&Yuan)
March 26, 2020
FLNN, Spring 2020
12

March 26, 2020
FLNN, Spring 2020
13
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) = supz=x*y min [A(x), B(y)] for all zR.
More specifically, we define for all zR:
(A+B)(z) = supz=x+y min [A(x), B(y)] for all zR.
(A-B)(z) = supz=x-y min [A(x), B(y)] for all zR.

March 26, 2020
FLNN, Spring 2020
14
Example:
Add the fuzzy numbers A and B, where
Solution:

March 26, 2020
FLNN, Spring 2020
15

fuzzy arithmetic operations over set theory

More Related Content

Similar to fuzzy arithmetic operations over set theory (20)

Recently uploaded (20)

fuzzy arithmetic operations over set theory