@2002 Adriano Cruz NCE e IM - UFRJ No. 2
Fuzzy NumbersFuzzy Numbers
A fuzzy number is fuzzy subset of the universe of a numerical number.
– A fuzzy real number is a fuzzy subset of the domain of real numbers.
– A fuzzy integer number is a fuzzy subset of the domain of integers.
@2002 Adriano Cruz NCE e IM - UFRJ No. 3
Fuzzy Numbers - ExampleFuzzy Numbers - Example
u(x)
x5 10 15
Fuzzy real number 10
u(x)
x5 10 15
Fuzzy integer number 10
@2002 Adriano Cruz NCE e IM - UFRJ No. 4
Functions with Fuzzy ArgumentsFunctions with Fuzzy Arguments
A crisp function maps its crisp input argument to its image.
Fuzzy arguments have membership degrees.
When computing a fuzzy mapping it is necessary to compute the image and its membership value.
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Crisp MappingsCrisp Mappings
XXYYf(X)f(X)
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Functions applied to intervalsFunctions applied to intervals
Compute the image of the interval.
An interval is a crisp set.
x
y
I
y=f(I)
@2002 Adriano Cruz NCE e IM - UFRJ No. 7
MappingsMappings
XXYY
f(X)f(X)
Fuzzy argument?
@2002 Adriano Cruz NCE e IM - UFRJ No. 8
Extension PrincipleExtension Principle
Suppose that ff is a function from XX to YY and AA is a fuzzy set on XX defined as
A A = = µµAA((xx11)/)/xx1 1 + + µµAA((xx22)/)/xx2 2 + ... + + ... + µµAA((xxnn)/)/xxnn
The extension principle states that the image of fuzzy set AA under the mapping f(.)f(.) can be expressed as a fuzzy set BB.
B B = = ff((AA) = ) = µµAA((xx11)/)/yy1 1 + + µµAA((xx22)/)/yy2 2 + ... + + ... + µµAA((xxnn)/)/yynn
where yyii==ff((xxii))
@2002 Adriano Cruz NCE e IM - UFRJ No. 9
Extension PrincipleExtension Principle
If f(.) is a many-to-one mappingmany-to-one mapping, then there exist x1, x2 X, x1 x2, such that f(x1)=f(x2)=y*, y*Y.
The membership grade at y=y* is the maximum of the membership grades at x1 and x2
more generally, we have)(max)(
)(1xy A
yfxB
@2002 Adriano Cruz NCE e IM - UFRJ No. 10
Monotonic Continuous FunctionsMonotonic Continuous Functions
For each point in the interval
– Compute the image of the interval.
– The membership degrees are carried through.
I
@2002 Adriano Cruz NCE e IM - UFRJ No. 11
Monotonic Continuous FunctionsMonotonic Continuous Functions
x
y
x
y
u(x)u(y)
@2002 Adriano Cruz NCE e IM - UFRJ No. 12
Monotonic Continuous Ex.Monotonic Continuous Ex.
Function: y=f(x)=0.6*x+4
Input: Fuzzy number - around-5
– Around-5 = 0.3 / 3 + 1.0 / 5 + 0.3 / 7
f(around-5) = 0.3/f(3) + 1/f(5) + 0.3/f(7) f(around-5) = 0.3/0.6*3+4 + 1/ 0.6*5+4 + 0.3/
0.6*7+4
f(around-5) = 0.3/5.8 + 1.0/7 + 0.3/8.2I
@2002 Adriano Cruz NCE e IM - UFRJ No. 13
Monotonic Continuous Ex. Monotonic Continuous Ex.
f(x)
x5 10u(x)
x753
10
4
1 0.3
1
0.3
8.2
5.8
@2002 Adriano Cruz NCE e IM - UFRJ No. 14
Nonmonotonic Continuous Nonmonotonic Continuous FunctionsFunctions
For each point in the interval
– Compute the image of the interval.
– The membership degrees are carried through.
– When different inputs map to the same value, combine the membership degrees.
@2002 Adriano Cruz NCE e IM - UFRJ No. 15
Nonmonotonic Continuous Nonmonotonic Continuous FunctionsFunctions
x
y
x
y
u(x)u(y)
@2002 Adriano Cruz NCE e IM - UFRJ No. 16
Nonmonotonic Continuous Ex.Nonmonotonic Continuous Ex.
Function: y=f(x)=x2-6x+11 Input: Fuzzy number - around-4
Around-4 = 0.3/2+0.6/3+1/4+0.6/5+0.3/6
y = 0.3/f(2)+0.6/f(3)+1/f(4)+0.6/f(5)+0.3/f(6)
y = 0.3/3+0.6/2+1/3+0.6/6+0.3/11
y = 0.6/2+(0.3 v 1)/3+0.6/6+0.3/11
y = 0.6/2 + 1/3 + 0.6/6 + 0.3/11
I
@2002 Adriano Cruz NCE e IM - UFRJ No. 17
Nonmonotonic Continuous Nonmonotonic Continuous FunctionsFunctions
x
y
x
y
u(x)u(y)
2 3 4 5 6
0.30.61
0.30.61
1 0.3v
@2002 Adriano Cruz NCE e IM - UFRJ No. 18
Function Example 1Function Example 1
Consider
Consider fuzzy set
Result
41)(
2xxfy
22|/||21~
xxxA
Y
B yyAfB /)()(~~
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Function Example 2Function Example 2
Result according to the principle
Y
A
Y
B xfxyyAfB )(/)(/)()(~~
|1|)(
||21)(
12
2
2
yx
xx
yx
A
A
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Function Example 3Function Example 3
@2002 Adriano Cruz NCE e IM - UFRJ No. 21
Extension PrincipleExtension Principle
Let f be a function with n arguments that maps a point in X1xX2x...xXn to a point in Y such that y=f(x1,…,xn).
Let A1x…xAn be fuzzy subsets of X1xX2x...xXn
The image of A under f is a subset of Y defined by
0)(0
0)()]([)(
1
1
)(),,(),,( 111
yfif
yfifxy
iAiyfxxxxB
i
nn
@2002 Adriano Cruz NCE e IM - UFRJ No. 22
Arithmetic OperationsArithmetic Operations
Applying the extension principle to arithmetic operations it is possible to define fuzzy arithmetic operations
Let x and y be the operands, z the result.
Let A and B denote the fuzzy sets that represent the operands x and y respectively.
@2002 Adriano Cruz NCE e IM - UFRJ No. 23
Fuzzy additionFuzzy addition
Using the extension principle fuzzy addition is defined as
zyx
yxBABA yxz
,
))()(()(
@2002 Adriano Cruz NCE e IM - UFRJ No. 24
Fuzzy addition - ExamplesFuzzy addition - Examples
A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5
B =(11~)= 0.5/10 + 1/11 + 0.5/12
A+B=(0.3^0.5)/(1+10) + (0.6^0.5)/(2+10) + (1^0.5)/(3+10) + (0.6^0.5)/(4+10) + (0.3^0.5)/(5+10) + (0.3^1)/(1+11) + (0.6^1)/(2+11) + (1^1)/(3+11) + (0.6^1)/(4+11) + (0.3^1)/(5+11) +( 0.3^0.5)/(1+12) + (0.6^0.5)/(2+12) + (1^0.5)/(3+12) + (0.6^0.5)/(4+12) + (0.3^0.5)/(5+12)
@2002 Adriano Cruz NCE e IM - UFRJ No. 25
Fuzzy addition - ExamplesFuzzy addition - Examples
A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5 B =(11~)= 0.5/10 + 1/11 + 0.5/12 Getting the minimum of the
membership values A+B=0.3/11 + 0.5/12 + 0.5/13 + 0.5/14 + 0.3/15 +
0.3/12 + 0.6/13 + 1/14 + 0.6/15 + 0.3/16 + 0.3/13 + 0.5/14 + 0.5/15 + 0.5/16 + 0.3/17
@2002 Adriano Cruz NCE e IM - UFRJ No. 26
Fuzzy addition - ExamplesFuzzy addition - Examples
A = (3~) = 0.3/1+0.6/2+1/3+0.6/4+0.3/5
B =(11~)= 0.5/10 + 1/11 + 0.5/12
Getting the maximum of the duplicated values
A+B=0.3/11 + (0.5 V 0.3)/12 + (0.5 V 0.6 V 0.3)/13 + (0.5 V 1 V 0.5)/14 + (0.3 V 0.6 V 0.5)/15 + (0.3 V 0.5)/16 + 0.3/17
A+B=0.3 / 11 + 0.5 / 12 + 0.6 / 13 + 1 / 14 + 0.6 / 15 + 0.5 / 16 + 0.3 / 17
@2002 Adriano Cruz NCE e IM - UFRJ No. 27
Fuzzy additionFuzzy addition
A, x=3
B, y=11
0.3
0.60.5
C, x=14
@2002 Adriano Cruz NCE e IM - UFRJ No. 28
Fuzzy ArithmeticFuzzy Arithmetic
Using the extension principle the remaining fuzzy arithmetic fuzzy operations are defined as:
zyx
yxBAA
zyx
yxBABA
zyx
yxBABA
yxz
yxz
yxz
/
,/
*
,*
,
)()()(
)()()(
)()()(