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Fuzzy Sets and Systems
Lecture 3(Fuzzy Arithmatic)
Bu- Ali Sina UniversityComputer Engineering Dep.
Spring 2010
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Fuzzy Arithmetic
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Outline
Fuzzy numbersArithmetic operations on intervalsArithmetic operations on fuzzy numbers
– Addition– Subtraction– Multiplication– Division
Fuzzy equations
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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
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Fuzzy NumbersThe membership function of a fuzzy number is
of the form A: ℜ→[0,1]
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Fuzzy number, Fuzzy interval
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Examples of fuzzy numbers
Fuzzy number
Fuzzy interval
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Fuzzy numbers of Around 3
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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)
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Interval addition(+)[a, b] + [c, d] = [a+c, b+d]
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Interval subtraction(-)[a, b] - [c, d] = [a-d, b-c]
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Interval multiplication(.)
[a, b] · [c, d] = [min(ac, ad, bc, bd), max(ac, ad, bc, bd)]
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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)]
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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)⊆
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Arithmetic operations on fuzzy numbersAn example
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Arithmetic operations on fuzzy numbers
A * B = ∪ α∈[0,1]α(A*B)
Where * represents any of +, - , · and /operations
α(A*B) = αA*αB
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Fuzzy numbers for illustrating arithmetic operations
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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]
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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]
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Fuzzy additionαA =[2α-1, 3-2α]αB = [2α+1, 5-2α]
α(A+B) =[4α, 8-4α]
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Fuzzy Addition
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Fuzzy Addition Examples
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Fuzzy Addition Examples
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Fuzzy Addition Examples
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Fuzzy Addition Examples
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Fuzzy subtractionαA =[2α-1, 3-2α]αB = [2α+1, 5-2α]
α(A-B) =[4α-6, 2-4α]
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Fuzzy multiplication(2α-1)(5-2α) (3-2α)(5-2α)
(2α-1) (2α+1)
x = -4α2+12α-5
x = 4α2-16α+15x = 4α2-1
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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
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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
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Fuzzy division
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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
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In the form of α- cut (general form): and
Example
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Assignment4.1,4.5,4.6,
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