Fuzzy Sets and Systems

Lecture 3(Fuzzy Arithmatic)

Bu- Ali Sina UniversityComputer Engineering Dep.

Spring 2010

Fuzzy Arithmetic

Outline

Fuzzy numbersArithmetic operations on intervalsArithmetic operations on fuzzy numbers

– Addition– Subtraction– Multiplication– Division

Fuzzy equations

Fuzzy numbersTo represent an inaccurate number we use fuzzy numbers

Example– About two o’clock– Around six-thirty– Approximately six

A number word and a linguistic modifier

Fuzzy number– A fuzzy set defined in the set of real number– Degree 1 of central value– Membership degree decrease from 1 to 0 on both side

In the other word

4 86

1

– Normal fuzzy sets– The α-cuts of fuzzy number are closed intervals– The support of every fuzzy number is the open interval (a,d)– Convex fuzzy sets

Fuzzy NumbersThe membership function of a fuzzy number is

of the form A: ℜ→[0,1]

Fuzzy number, Fuzzy interval

Examples of fuzzy numbers

Fuzzy number

Fuzzy interval

Fuzzy numbers of Around 3

Arithmetic operations on intervals

Fuzzy arithmetic is based on two properties ofFuzzy numbers– Each fuzzy set and also each fuzzy number

can fully be represented by its α-cuts– α-cuts of each fuzzy number are closed

intervals of real numbers for all α (0,1]

Arithmetic operations on fuzzy numbersare defined in terms of arithmeticoperations on their α-cuts (on closed interval)

Interval addition(+)[a, b] + [c, d] = [a+c, b+d]

Interval subtraction(-)[a, b] - [c, d] = [a-d, b-c]

Interval multiplication(.)

[a, b] · [c, d] = [min(ac, ad, bc, bd), max(ac, ad, bc, bd)]

Interval division(/)

[a, b] / [c, d] = [a, b] · [1/d, 1/c]=[min(a/c, a/d, b/c, b/d), max(a/c, a/d, b/c, b/d)]

Closed intervals arithmetic operations properties

We consider:A=[a1,a2],B=[b1,b2],C=[c1,c2],0=[0,0], 1=[1,1]

� A+B=B+A, A.B=B.A� (A+B)+C=A+(B+C), (A.B).C=A.(B.C)� A=0+A=A+0, A=1.A=A.1� A.(B+C) (A.B)+(A.C)⊆

Arithmetic operations on fuzzy numbersAn example

Arithmetic operations on fuzzy numbers

A * B = ∪ α∈[0,1]α(A*B)

Where * represents any of +, - , · and /operations

α(A*B) = αA*αB

Fuzzy numbers for illustrating arithmetic operations

Construction of α-cuts of A(x)

A(αa1 )=(αa1 +1)/2= α

A(αa2 )=(3- αa2)/2= α

αA = [αa1, αa2]= [2α-1, 3-2α]

Construction of α-cuts of B(x)

B(αb1 )=(αb1 -1)/2= αB(αb2 )=(5- αb2)/2= α

αB = [αb1, αb2]= [2α+1, 5-2α]

Fuzzy additionαA =[2α-1, 3-2α]αB = [2α+1, 5-2α]

α(A+B) =[4α, 8-4α]

Fuzzy Addition

Fuzzy Addition Examples

Fuzzy Addition Examples

Fuzzy Addition Examples

Fuzzy Addition Examples

Fuzzy subtractionαA =[2α-1, 3-2α]αB = [2α+1, 5-2α]

α(A-B) =[4α-6, 2-4α]

Fuzzy multiplication(2α-1)(5-2α) (3-2α)(5-2α)

(2α-1) (2α+1)

x = -4α2+12α-5

x = 4α2-16α+15x = 4α2-1

Fuzzy Multiplication Example

}38.0

21

16.0{2"2

~++=== elyApproximatA

}21

7.018

8.015

8.014

7.0121

108.0

76.0

66.0

56.0{

}21

)7.0,8.0min(18

)1,8.0min(...6

)1,6.0min(5

)8.0,6.0min({

)77.0

61

58.0()

38.0

21

16.0(12"12

~

++++++++=

++++=

++×++===× elyApproximatBA

}77.0

61

58.0{6"6

~++=== elyApproximatB

Fuzzy Multi. Examples

}87.0

41

25.0

12.0{

}8

)1,7.0min(4

)]1,1min(),5.0,7.0max[min(2

)]1,5.0min(),1,2.0max[min(1

)5.0,2.0min({

)21

15.0()

37.0

21

12.0()(),(

+++=

++

+=

+×++=×= productarithmeticBABAf

}47.0

21

12.0{ ++=A }

21

15.0{ +=B

If the mapping is not one-to-one, we get:)]}(),({min[max),( 21),(21

21

xxxx BAxxfyBA µµµ=× =

4

Fuzzy division

Fuzzy equationsHere we study two form of fuzzy equations:

– A+x=B– A.x=B where x is unknown fuzzy number

X=B- Ais not a correct answer becauseA+(B- A)=[a1,a2]+[b1- a2,,b2- a1]=[a1+b1- a2,a2+b2- a1]

Which is not a correct answer

Let X=[x1,x2] then x1=b1- a1, x2=b2- a2and x1<=x2 thus b1-a1<=b2- a2

In the form of α- cut (general form): and

Example

Assignment4.1,4.5,4.6,