Chapter 3. Fuzzy Arithmetic 3- Fuzzy arithmetic: ~Addition(+) and subtraction (-): Let A = [a and B = [b, b in R If x [a and y [b, b than x+y [a +b +b Symbolically,we write A(+)B = [a (+)[b, b = [a +b +b subtraction: A(-)B = [a (-) [b, b = [a-b, a -b ~Image Ā If x [a,then its image x [-a, -a That is, if A=[a then its image is Ā=[-a, -a Note that: A(+)Ā = [a (+)[-a, -a = [a -a, a -a 0 A(-)B = A(+) B=[a,a (+)[-b, -b = [a -b, a -b ~Multiplication( ) and division ( : ): Consider A = [a and B = [b, b in R + (nonnegative real line) A( )B = [a,a ( )[b, b = [a b b If A, B R, then there are nine possible combinations 3 If k R + and A R, then k A = [k, k( )[a = [ka,ka 4For the division of two intervals of confidence in R + 0, we have A( : )B = [a ( : )[b, b = [a /b, a /b Inverse A - : If x [a,a R 0+, where R 0+ is the positive real line, then its inverse is /x [/a, /a and - A = [a,a = [, a a a a A (:)B A ( i) B -=[a,a ( i)[, = [, b b b b for k>0 a a A (:)k A ( i )[, =[, k k k k 3-
Max ( ) and min ( )operations: A( )B = [a ( )[b, b = [a b b A( )B = [a ( )[b, b = [a b b Algebraic properties of addition and multiplication on intervals Property Addition(+) Multiplication( ) 3-3 Intervals For all A, B, C R For all A,B,C R + Commutativity A(+)B = B(+)A A( )B = B( )A Associativity (A(+)B)(+)C = A(+)(B(+)C) (A( )B)( )C = A( )(B( )C) Neutral number A(+)0 = 0(+)A = A A( ) = ( )A = A Image and inverse A(+)Ā = Ā(+)A 0 A( )A - = A - ( )A Given two fuzzy numbers A and B in R, for a specific [0,, we will obtain two closed intervals, A [ a () () from fuzzy number A, and B [b (),b () from fuzzy number B. The interval arithmetic can be applied to these two closed intervals. Let( ) denote an arithmetic operation on fuzzy numbers such as addition(+), subtraction (-), multiplication( )or division( : ). Using the extension principle, the result A ( ) B, where A and B are two fuzzy numbers, can be obtained: A( )B (z)= sup min { A (x), B (y)} z=x y Using the -cuts, we have (A ( )B) = A ( )B, for all [0, where A = [a, and B = [b, b For two fuzzy numbers A and B in R, we have A (+) B = [a +b, a +b A (-) B = [a -b, a -b For two fuzzy numbers A and B in R + =[0, ) we have 3-4 A ( ) B = [a b,a b A (:) B = a b a b ( ) ( ) [, ( ) ( )
The multiplication of a fuzzy number A R by an ordinary number k R + : (k A) = k A = [ka,ka or,equivalently, k A (x) = A (x/k), for all x R Ex 5.6: Compute A(+)B and A(-)B,where 3-5 0 x -6 0 x - (x+6)/4-6 <x -, (x+)/5 - < x 4 A(x)= B(x)= (-x+3)/5 - <x 3 (-x+0)/6 4 < x 0 0 x >3 0 x >0 A B A(+)B -0-5 0 5 0 5 Find -cuts [ a, a and [b, b ( ) ( ) a + 6 -a + 3 =, and = 4 5 Solving a and a, we obtain A = [ a = [ 4-6,-5 +3 Similarly, B = [ b, b = [ 5 -,-6 +0 A (+) B = [9-7,- +3 [c, c = (A(+)B) 3 x = 9-7, = x= - +3, = x + 7 9 -x+3 3-6 A( + ) B (x)= 0 x -7 (x+7)/9-7<x (-x+3)/ <x 3 0 x>3
3-7 A (-) B = [ 0-6,-0 + 4 =(A(-)B) x+6 0-6 = x, = 0 4 x -0 + 4 = x, = 0 A ( ) B (x ) = Ex 5.7: Compute A( )B, A(:)B 0 x 6 x + 6-6 < x 6 0 -x + 4-6 < x 4 0 0 x > 4 0 x 0 x x- x- < x 3 < x 4 A (x )= B (x ) = -x+6 -x+7 3 < x 6 4 < x 7 3 3 0 x > 6 0 x > 7 A B A( )B 3-8 0 5 0 5 0 5 30 35 40 compute -cuts ( ) ( ) = a a = + a ( ) + 6 ( ) = = 3 + 6 3 A = [ a = [ +, -3 +6 similarly, from μ B (x), we have B = [ +, -3 +7 A ( )B = [( + ) ( + ), (-3 +6)(-3 +7) = [ 4 + 6 +, 9-39 + 4= (A( )B) Solve 4 + 6 + = x, = (-3 ± + 4x )/4 9-39 + 4=x, = (3 ± + 4x )/6 a
[0, 0 x ( 3 + + 4 x ) / 4 <x A( i ) B(x)= (3 + 4 x ) / 6 <x 4 0 x>4 For A (:)B = [ + -3 +6, 3 + 7 + 0 x 7 7x- 3 < x 3x+ 7 4 -x+6 3 x+3 4 < x 3 0 x > 3 A (:) B (x )= ~k( )A, k=5 0 x 5 x-5 x 0 5<x 5 k ( i ) A(x)= A( )= 5 -x+30 5<x 30 5 0 x>30 3-9 3-0 Properties of operations on fuzzy numbers:.if A and B are fuzzy numbers in R, then A(+)B and A(-)B are also fuzzy numbers.if A and B are fuzzy numbers in R +, then A( )B and A(:)B are also fuzzy numbers 3.There are no image and inverse fuzzy numbers Ā and A -, respectively, such that A(+)Ā=0 and A( ) A - = 4. (A(-)B)(+)B A and (A(:)B) ( )B A To prove the above properties, the concept of -cuts is usually used Thm 5-: Let A,B,and C be fuzzy numbers in R + ; then the following distributives holds (A(+)B)( ) C = (A ( ) C)(+)(B ( )C) pf: ((A(+)B)( ) C ) =(A (+)B )(.)C = ( [a (+) [b, b )( ) [c, c = [ a +b +b ( ) [c, c = [ a c +b c c + b c
and ((A( )C)(+)(B ( ) C)) =([ a ( )[c,c )(+) ([b,b ( ) [c, c ) =[a c + b c,a c + b c Remark: (A( )B)(+)C (A(+)C)( )( B(+)C) Fuzzy minimum and fuzzy maximum: Two fuzzy numbers A and B in R are comparable, if a b and a b and we can write A B for all [0. The fuzzy minimum and fuzzy maximum of two fuzzy numbers A and B in R are denoted as A( )B and A( )B and are defined, respectively, by 3- A ( )B [min(a,b ),min(a,b ) =[a b,a b A ( )B [max(a,b ),max(a,b ) =[a b,a b Ex 5.8: Consider two fuzzy numbers A and B with 0 x -3 0 x -4 μ A (x) = x+3-3 <x - and μ B (x)= (x+4)/5-4<x (-x+5)/7 -<x 5 -x+ <x 0 x>5 0 x> = a + 3 a = -3 A = [ 3, 7 + 5 = (- a +5 ) / 7 a = -7 + 5 B =[ 5-4, - + A ( ) B = [( -3) (5-4), (-7 +5) (-+) [ c, c there may be four combinations [a, [a, b, [b,[b, b -3 = 5-4, =0.5 ; -7 +5 = - +, =0.5 [5-4, - + 0 0.5 A ( )B = [ -3, - + 0.5< 0.5 [ -3, -7 +5 0.5< 3-
x = 5 4, = x + 4, = 0, x = 4, = 0.5, x =.75 5 x= +, = x+, = 0, x=, = 0.5, x=.5 0 x 4 x + 4.0-4 < x.75 5 x 3 -.75 x ( ) ( x ) A = B 0.5 -x + 5 - < x.5 7 - x +.5 < x 0 x > Similarly, A ( )B =[( -3) (5-4), (-7 +5) (- +) [c, c [ -3, -7 +5 0 0.5 A ( )B = [5-4, -7 +5 0.5< 0.5 [5-4, - + 0.5< A( )B 0.5 3-3 A ( ) B (x )= 0 x 3 x + 3-3 < x.7 5 x+4 -.7 5 < x 5 -x + < x.5 -x + 5.5 < x 5 7 0 x > 5 3-4 L-R fuzzy number: ~ to increase computational efficiency ~parametric function ~ In general, a L-R reference (or shape) function, φ(x) [0, L(x) - x <0 φ(x)= x = 0 x R R(x) 0<x< L(x) and R(x) map R + [0, and are monotonic (nonincreasing)
~L(x) (or R(x)) is a reference function iff () L(x) = L(-x) () L(0) = (3) L(x) is nonincreasing on [0,+ ~φ(x) is a normal convex function fuzzy number (LR) Thus, L-R fuzzy number is represented by a pair of functions L&R with nonincreasing monotonicity Ex: L(x) = max(0,- x p ), x>=0, p>0, L(x)=L(-x) L(x) = e -x, x>=0, L(x)=L(-x) L(x) = e -x Def: A fuzzy number M is called L-R fuzzy number iff m -x L ( ) x m M (x )= x = m x-m R ( ) x m β where m: mean of the fuzzy number ;, β: left and right spreads, respectively 3-5 Remarks: () when spreads are zero nonfuzzy number () When spreads increase M becomes more fuzzy (3) m < 0 we have a left translation m > 0 we have a right translation (4) If < & β<, then we have a contraction > & β>, then we have a dilation Hence, a L-R fuzzy number can be represented by 3 parameters m,, β M=(m,, β) LR Ex:Consider L(x)=, R(x)=, m=5, =, β =3 + x + x m-x 5-x L( )=L( )= x<5 5-x + ( ) M ( x ) = x=5 x-m x-5 R( )=R( )= x>5 β 3 x-5 + 3 3-6
~Consider A=(m,, β) LR, B= (n, r,δ) LR in R with nonincreasing L-R fuzzy number 3-7. Addition: A(+)B = (m,, β) LR (+) (n,r,δ) LR = (m+n, +r,β+δ) LR.Image: Ā = - (m,, β) LR = (-m, β,) RL.Subtraction: A=(m,, β) LR,, B= (n, r,δ) RL in R A(-)B = (m,, β) LR (-) (n, r,δ) RL =(m-n, +δ, β+r) LR Multiplication: k R, k( )A = k( )(m,,β) LR = (km, k, kβ) LR k>0 (km,-kβ,-k ) RL k<0 0 x<- - 0 x x Ex: L(x)= R(x)= +x - x 0, 0 x>.0 φ( x) A = (m,,β) LR = (3,,3) LR B = (n,r,δ) LR = (6,,4) LR L(x)= +x R(x)=-x 0.5 C = (p,ζ,λ) RL = (8,4,5) RL - x 0.0 x 0 x Lx ( ) = 0 x 0 x 8 x ( ) 4 x 8 -(3-x) x 3, 4 x=8 ( x) = x=3 ( x) = A C x 3 x 8 ( ) 8 x 3 - ( ) 3 x 6 5 3 0 otherwise 0 6 x 3-8 A+B = (m+n, +r, β+δ) LR = (3+6, +, 3+4) LR = (9, 3,7) LR A(-)C = (m,, β) LR -(p,ζ, λ) RL = (m-p,+λ, β+ζ) LR =(3-8, +5, 3+4) LR =(-5, 6, 7) LR M=(m,, β) LR a point ~L-R fuzzy interval: M=(m, m,, β) LR
m -x x = M x m R β L and R: nonincreasing functions L( ) x < m, > 0 ( ) x [ m, m β ( ) x > m, > 0 Consider two L-R fuzzy intervals M= (m, m,, β) LR and N=(n, n.,r,δ) LR in R with nonincreasing L and R reference functions ~Addition: m m x 3-9 M(+)N = (m, m,, β) LR (+) (n, n., r,δ) LR = (m +n, m +n, +r, β+δ) LR ~Image: M = - (m,m,, β ) LR=(-m,-m, β, ) ~Subtraction: M = (m, m,, β) LR and N = (n, n.,r,δ) RL M(-)N = (m, m,, β) LR (-) (n, n.,r,δ) RL =(m -n,m n, + δ, β+ r) LR RL 3-0 ~Multiplication: k R, k( )M = k( ) (m, m,, β) LR (km,km,k,k β ) LR k>0 = (km,km,k,k ) RL β k<0 Ex: Consider setting up a budget from three sources: source A: A certain and precise amount of $60k source B: may range from $60k to $0k. But it is reasonable to except an amount of $70k to 00k source C: may amount to $30k but not be more than $50k 0 60 K$ 0 60 70 00 0 K$ 0 30 50 K$ 50 90 60 30 K$ source A source B Source C total S
3- L(x) = R(x) = max(0, -x), x R + A = (60, 60, 0, 0) LR, B = (70, 00, 0, 0) LR, C = (30, 30, 0, 0) LR S = A (+) B (+) C = (60, 60, 0, 0) LR (+) (70, 00, 0, 0) LR (+) (30, 30, 0, 0) LR = (60, 90, 0, 40) LR the possible amount is $50,000 to $30,000 and the expected amount is $60,000 to $90,000