7 Transformations of Fuzzy Sets Fuzzy Systems Engineering Toward - - PowerPoint PPT Presentation

7 Transformations of Fuzzy Sets

Fuzzy Systems Engineering Toward Human-Centric Computing

7.1 The extension principle 7.2 Composition of fuzzy relations 7.3 Fuzzy relational equations 7.4 Associative memories 7.5 Fuzzy numbers and fuzzy arithmetic

Contents

Pedrycz and Gomide, FSE 2007

7.1 The extension principle

Pedrycz and Gomide, FSE 2007

Extension principle

• Extends point transformations to operations involving

– sets – fuzzy sets

• Given a function f: X → Y and a set (or fuzzy set) A on X

the extension principle allows to map A into a set (or fuzzy set) on Y through f

Pedrycz and Gomide, FSE 2007

Pointwise transformation

f : X → Y yo = f (xo)

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

x y f xo yo

f is a function

Pedrycz and Gomide, FSE 2007

Set transformation

5 10 2 4 6 8 10

x y f

2 4 6 8 10 1

x A(x) A

2 4 6 8 10 0.5 1

B(y) B

f : X → Y, A ∈ P(X)

B= f (A) = { y ∈ Y | y = f (x), ∀x ∈ X}

) ( sup ) (

) ( /

x A y B

x f y x =

=

B ∈ P(Y)

Pedrycz and Gomide, FSE 2007

5 10 2 4 6 8 10

x y f

5 10 0.2 0.4 0.6 0.8 1

x A(x) A

2 4 6 8 10 0.5 1 B(y)

B

Fuzzy set transformation

f : X → Y, A ∈ F(X)

B= f (A), B ∈ F(Y)

) ( sup ) (

) ( /

x A y B

x f y x =

=

   ≤ < + − ≤ ≤ + − − = 10 5 if 5 ) 5 ( 2 . 5 if 5 ) 5 ( 2 . ) (

2 2

x x x x x f

A = A(x, 3, 5, 8)

Pedrycz and Gomide, FSE 2007

Example

• 4 -3 -2 -1 0 1 2 3 4

2 4 6 8 10 x

y f

• 4 -3 -2 -1 0 1 2 3 4

0.2 0.4 0.6 0.8 1

x A(x) A

2 4 6 8 10 0.5 1 B(y)

B

y= f (x) = x2 A = A(x, –2, 2, 3)

Pedrycz and Gomide, FSE 2007

Example

y= f (x) = x2 X = {–3, –2, – 1, 0, 1, 2, 3} Y = {0, 1, 4, 9}

• 4 -3 -2 -1 0 1 2 3 4

2 4 6 8 10

x f

• 4 -3 -2 -1 0 1 2 3 4

0.2 0.4 0.6 0.8 1

x A(x) A

2 4 6 8 10 0.5 1 B(y)

B y

B = {1/0, max(0.2,0.3)/1, max(0, 0.1)/4, 0/9} = {1/0, 0.3/1, 0.1/4, 0/9}

Pedrycz and Gomide, FSE 2007

Generalization

X = X1× X2 × ... × Xn Ai ∈ F(Xi), i = 1,…,n y = f (x), x = [x1, x2, …, xn]

)]} ( , ), ( ), ( [ {min sup ) (

2 2 1 1 ) ( | n n x f y

x A x A x A y B

• =

=

x

B ∈ F(Y)

Pedrycz and Gomide, FSE 2007

Properties

) ( . 6 ) ( . 5 ) ( ) ( . 4 ) ( ) ( . 3 . 2 . 1

1 1 1 1 1 1 2 1 2 1 + + = = = = = =

= ⊇ = ⊆ = = ⊆ ⇒ ⊆ ∅ = ∅ =

α α α α

A f B A f B B A f A f B A f A f B B A A A iff B

n i i n i i n i i n i i n i i n i i i i

• Pedrycz and Gomide, FSE 2007

B+

α = {y∈Y| B(y) > α }

strong α –cut

7.2 Compositions of fuzzy relations

Pedrycz and Gomide, FSE 2007

Given the fuzzy relations G : X×Z → [0,1] W : Z×Y → [0,1] R = G ° W sup-t composition

Sup-t composition

)} , ( ) , ( [ {min sup ) , ( y z W t z x G y x R

z Z ∈

=

∀(x,y) ∈ X×Y R : X×Y → [0,1]

Pedrycz and Gomide, FSE 2007

Example

] 2 / ) ( exp[ ) , ( } { max } { sup ) , ( ] ) ( exp[ ) , ( ] ) ( exp[ ) , (

2 ) ( ) ( ) ( ) ( 2 2

2 2 2 2

y x y x R e e e e y x R y z y z W z x z x G

x z z x z x z z x z

− − = = = − − = − − =

− − − − ∈ − − − − ∈ Z Z

t = product sup-product composition

Pedrycz and Gomide, FSE 2007

G : X×Z R = G ° W

] ) ( exp[ ) , (

2

z x z x G − − = ] 2 / ) ( exp[ ) , (

2

y x y x R − − =

Pedrycz and Gomide, FSE 2007

procedure SUP-T-COMPOSITION (G,W) returns composition of fuzzy relations static: fuzzy relations: G = [gik], W=[wkj] 0nm: n×m matrix with all entries equal to zero t: a t-norm R = 0nm for i = 1:n do for j = 1:m do for k = 1:p do tope ← gik t wkj rij ← max(rij, tope) return R

Sup-t composition for matrix relations

Pedrycz and Gomide, FSE 2007

Example

            =           = 6 . 3 . 8 . 7 . 7 . 5 . 1 . 6 . 3 . 4 . 3 . 8 . 2 . . 1 8 . 6 . 5 . 5 . 6 . . 1 W G

r11=max(1.0∧0.6, 0.6∧0.5, 0.5∧0.7, 0.5∧0.3) = max (0.6, 0.5, 0.5, 0.3) = 0.6 ……………….…… r32=max(0.8∧0.1, 0.3∧0.7, 0.4∧0.8, 0.3∧0.6) = max (0.1, 0.3, 0.4, 0.3) = 0.4

          = = 4 . 6 . 8 . 7 . 6 . 6 . W G R

• t = min = ∧

Pedrycz and Gomide, FSE 2007

Properties

S P Q P S Q R P Q P R Q P R P Q P R Q P R Q P R Q P

• ⊆

⊆ ∩ ⊆ ∩ ∪ = ∪ = then If . 4 ) ( ) ( ) ( . 3 ) ( ) ( ) ( . 2 ) ( ) ( . 1

∪ ∪ ∪ ∪ , ∩ ∩ ∩ ∩ are standard operations

associativity distributivity over union weak distributivity over intersection monotonicity

Pedrycz and Gomide, FSE 2007

Interpretations

) , ( sup )] , ( 1 [ sup )] , ( ) ( [ sup ) ( . 3 )] ( ) ( | [ truth ) ( . 2 )] ( ) ( [ sup ) ( . 1 y x R y x R t y x tR x y B x R and x A x y B x tR x A y B

x x x y y x X X X X

X

∈ ∈ ∈ ∈

= = = ∃ = =

possibility existential quantifier projection

Pedrycz and Gomide, FSE 2007

Given the fuzzy relations G : X×Z → [0,1] W : Z×Y → [0,1] R = G • W inf-s composition

Inf-s composition

)} , ( ) , ( [ {min inf ) , ( y z W s z x G y x R

z Z ∈

=

∀(x,y) ∈ X×Y R : X×Y → [0,1]

Pedrycz and Gomide, FSE 2007

procedure INF-S-COMPOSITION(G,W) returns composition of fuzzy relations static: fuzzy relations: G = [gik], W = [wkj] 1nm: n×m matrix with all entries equal to unity s: a s-norm R = 1nm for i = 1:n do for j = 1:m do for k = 1:p do sope ← gik s wkj rij ← min(rij, sope) return R

Pedrycz and Gomide, FSE 2007

Example

            =           = 6 . 3 . 8 . 7 . 7 . 5 . 1 . 6 . 3 . 4 . 3 . 8 . 2 . . 1 8 . 6 . 5 . 5 . 6 . . 1 W G

r11= min (1.0+0.6-0.6, 0.6+0.5-0.3, 0.5+0.7-0.35, 0.5+0.3-0.15) = min (1.0, 0.8, 0.85, 0.65) = 0.65 ……………….…… r32= min (0.8+0.1-0.08, 0.3+0.7-0.21, 0.4+0.8-0.32, 0.3+0.6-01.8) = min (0.82, 0.79, 0.88, 0.72) = 0.72

          =

• =

72 . 51 . 064 44 . 80 . 65 . W G R

s = probabilistic sum

Pedrycz and Gomide, FSE 2007

Example

] ) ( exp[ ) , ( , ] ) ( exp[ ) , (

2 2

y z y z W z x z x G − − = − − =

G : X×Z R = G • W

Pedrycz and Gomide, FSE 2007

Properties

S P Q P S Q I R P Q P R Q P R P Q P R Q P R Q P R Q P

• ⊇
• ⊆
• ∩
• =

∩

• ∪
• ⊇

∪

• =
• then

f . 4 ) ( ) ( ) ( . 3 ) ( ) ( ) ( . 2 ) ( ) ( . 1

∪ ∪ ∪ ∪ , ∩ ∩ ∩ ∩ are standard operations

associativity weak distributivity over union distributivity over intersection monotonicity

Pedrycz and Gomide, FSE 2007

Interpretations

)] ( ) ( | [ truth ) ( . 2 )] ( ) ( [ inf )] ( ) ( [ inf )] ( ) ( [ inf ) ( . 1 x R

• r

x A x y B x A s x R x sA x R x sR x A y B

y y x y x y x

∀ = = = =

∈ ∈ ∈ X X X

necessity universal quantifier

Pedrycz and Gomide, FSE 2007

Given the fuzzy relations G : X×Z → [0,1] W : Z×Y → [0,1] R = G ϕ W inf-ϕ composition

Inf-ϕ ϕ ϕ ϕ composition

)} , ( ) , ( { inf ) , ( y z W z x G y x R

z

ϕ

Z ∈

=

∀(x,y) ∈ X×Y

] 1 , [ , }, | ] 1 , [ { ∈ ∀ ≤ ∈ = b a b atc c b aϕ

ϕ : [0,1] → [0,1

Pedrycz and Gomide, FSE 2007

Example

            =           = 6 . 3 . 8 . 7 . 7 . 5 . 1 . 6 . 3 . 4 . 3 . 8 . 2 . . 1 8 . 6 . 5 . 5 . 6 . . 1 W G

If t is the bounded difference: a t b = max (0, a + b – 1) then a ϕ b = min (1, 1 – a + b) Lukasiewicz implication

          = = 3 . 8 . 06 7 . 1 . 6 . W G R ϕ

Pedrycz and Gomide, FSE 2007

Properties

S P Q P S Q R P Q P R Q P R P Q P R Q P R Q P R Q P ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ⊆ ⊆ ∩ = ∩ ∪ ⊇ ∪ = then If . 4 ) ( ) ( ) ( . 3 ) ( ) ( ) ( . 2 ) ( ) ( . 1

• ∪

∪ ∪ ∪ , ∩ ∩ ∩ ∩ are standard operations

associative weak distributivity over union distributivity over intersection monotonicity

Pedrycz and Gomide, FSE 2007

Interpretation

)] ( ) ( [ ) ( )] ( ) ( [ inf )] ( ) ( [ inf )] ( ) ( [ inf )] ( ) ( [ inf ) ( x R x A x y B x R x A x R x A x R x A x R x A y B

y y x y x y x y x

⇒ ∀ = ⊂ = ⇒ = ⇒ = =

∈ ∈ ∈ ∈ X X X X

ϕ

⇓

Pedrycz and Gomide, FSE 2007

7.3 Fuzzy relational equations

Pedrycz and Gomide, FSE 2007

R on X×Y

R U V Single–input, single–output fuzzy system

U on X V on Y

Fundamental problems – given U and V, determine R estimation – given V and R, determine U inverse

Pedrycz and Gomide, FSE 2007

Solution to the estimation problem Sup-t composition

X={x1, x2,…,xn} Y={y1, y2,…,ym} U: X → [0,1] U = [u1, u2,…, ui,…, un] = [ui] (1×n) V: Y → [0,1] V = [v1, v2,…, vj,…, vm] = [vj] (1×m)

) ( ] [ ] 1 , [ :

1 1 11

m n r r r r r r R R

ij nm n ij m

× =           = → ×

• Y

X

Pedrycz and Gomide, FSE 2007

Se = {R∈F(X)×F(Y) | V = U ° R} a ϕ b = sup {c∈[0,1] | a t c ≤ b

solution set

ϕ operator

Proposition if Se ≠ ∅, then the unique maximal solution R of the sup-t relational equation V = U ° R is

∧

V U R

Tϕ

= ˆ

R is maximal (in the sense that, if R ∈ Se , then R ⊆ R)

∧ ∧

Pedrycz and Gomide, FSE 2007

R

^

maximal solution minimal solutions

Se

Pedrycz and Gomide, FSE 2007

procedure ESTIMATION-SOLUTION (U,V) returns fuzzy relation static: fuzzy unary relations U = [ui], V = [vj] t: a t-norm define ϕ operator for i = 1:n do for j = 1:m do rij ← ui ϕ vj return R

^

^ Pedrycz and Gomide, FSE 2007

Example

U = [0.8, 0.5, 0.3] V = [0.4, 0.2, 0.0, 0.7]

   > ≤ = ⇒ = b a b b a b a t if if 1 min ϕ

[ ]

7 . . 2 . 4 . 3 . 5 . 8 . ˆ ϕ           = R

Pedrycz and Gomide, FSE 2007

          = 7 . 3 . . 3 . 2 . 3 . 4 . 3 . 7 . 5 . . 5 . 2 . 5 . 4 . 5 . 7 . 8 . . 8 . 2 . 8 . 4 . 8 . ˆ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ R           = . 1 . 2 . . 1 . 1 . 2 . 4 . 7 . . 2 . 4 . ˆ R

⇓

maximal solution

Pedrycz and Gomide, FSE 2007

          = 5 . . . 3 . 5 . . . . 7 . . 2 . 4 .

1

R           = . 1 . 2 . 6 . 2 . . . 4 . 7 . . 2 . .

2

R

R1 ∈ Se and R2 ∈ Se R1 ⊂ R and R2 ⊂ R

^ ^ Pedrycz and Gomide, FSE 2007

Fuzzy relational system

X={x1, x2,…,xn}, Y={y1, y2,…,ym} Uk: X → [0,1] Uk = [u1k, u2k,…, uik,…, unk] = [uik] (1×n) Vk: Y → [0,1] Vk = [v1k, v2k,…, vjk,…, vmk] = [vjk] (1×m) k = 1,….,N

) ( ] [ ] 1 , [ :

1 1 11

m n r r r r r r R R

ij nm n ij m

× =           = → ×

• Y

X

Pedrycz and Gomide, FSE 2007

∅ ≠ = ∅ ≠ = × ∈ =

=

• N

k k e N e k k k e

S S R U V F F R S

1

} | ) ( ) ( { Y X

k T k k N k k

V U R R R ϕ = =

=

ˆ ˆ ˆ

1

• N

k R U V

k k

, , 1 ,

• =

=

⇓

maximal solution

Pedrycz and Gomide, FSE 2007

Relation-relation fuzzy equations

X={x1, x2,…,xn}, Y={y1, y2,…,ym}, Z={z1, z2,…,zp} U: Z×X → [0,1] U = [uki] (p×n) V: Z×Y → [0,1] V = [vkj] (p×m)

) ( ] [ ] 1 , [ :

1 1 11

m n r r r r r r R R R U V

ij nm n ij m

× =           = → × =

• Y

X

Pedrycz and Gomide, FSE 2007

Uk = [uk1, uk2,…, uki,…, ukn] (1×n) k-th row of U Vk = [vk1, vk2,…, vki,…, vkm] (1×n) k-th row of V Rj = [r1j, r2j,…, rij,…, rnj]T (n×1) j-th column of R Let

[ ]

              =               =               = =

m p p p m m m p p

R U R U R U R U R U R U R U R U R U R R R U U U V V V R U V

• 2

1 2 2 2 1 2 1 2 1 1 1 2 1 2 1 2 1

then

Pedrycz and Gomide, FSE 2007

[ ] [ ] [ ]

R U R U R U R U V R U R U R U R U V R U R U R U R U V

p m p p p p m m

• =

= = = = =

2 1 2 2 2 2 1 2 2 1 1 2 1 1 1 1

Therefore, using the previous result we get

T k kT k kT k p k k

U U V U R R R ) ( ˆ ˆ ˆ

1

= = =

=

ϕ

• Pedrycz and Gomide, FSE 2007

Multi–input, single–output fuzzy equations R

U2 V Up U1 Ui ∈ F(Xi) , i = 1,…, p V ∈ F(Y) R ∈ F(X1 × X2 ×…. × Xp × Y)

R U V tU t tU U U R U U U V

p p

• =

= = then If

2 1 2 1

→

V U R

Tϕ

= ˆ

→

Pedrycz and Gomide, FSE 2007

Solution to the estimation problem Inf-s composition

X={x1, x2,…,xn}, Y={y1, y2,…,ym} U: X → [0,1] U = [ui] (1×n) V: Y → [0,1] V = [vj] (1×m) R: X×Y → [0,1] R = [rjj] (n×m)

R U V

• =

Pedrycz and Gomide, FSE 2007

Se

s = {R∈F(X)×F(Y) | V = U • R}

a β b = inf {c∈[0,1] | a s c ≥ b

solution set

β operator

Proposition if Se

s ≠ ∅, then the unique minimal solution R of the sup-t relational

equation V = U • R is

∧

V U R

Tβ

= ˆ

R is minimal (in the sense that, if R ∈ Se

s , then R ⊆ R)

∧ ∧

Pedrycz and Gomide, FSE 2007

Solution to the inverse problem Sup-t composition

X={x1, x2,…,xn}, Y={y1, y2,…,ym} U: X → [0,1] U = [ui] (1×n) V: Y → [0,1] V = [vj] (1×m) R: X×Y → [0,1] R = [rjj] (n×m)

R U V

• =

Pedrycz and Gomide, FSE 2007

Si = {U∈F(X) | V = U ° R} vj θ sji = min (vj ϕ sji, j =1,….,m), i = 1,…,n

solution set

θ operator

Proposition if Si ≠ ∅, then the unique maximal solution U of the sup-t relational equation V = U ° R is

∧ T

R V U θ = ˆ

U is maximal (in the sense that, if U ∈ Si , then U ⊆ U)

∧ ∧

Pedrycz and Gomide, FSE 2007

procedure INVERSE-SOLUTION (R,V) returns fuzzy unary relation static: fuzzy relations: R=[rij], V=[vj] M: large number t: a t-norm define: ϕ operator for i = 1:n do u ← M for j = 1:m do u ← min(u, vj ϕ rij) ui ← u return U

∧ ∧

Pedrycz and Gomide, FSE 2007

          = . 1 . 2 . . 1 . 1 . 2 . 4 . 7 . . 2 . 4 . ˆ R

Example

V = [0.4, 0.2, 0.0, 0.7]

   > ≤ = ⇒ = b a b b a b a t if if 1 min ϕ

Pedrycz and Gomide, FSE 2007

[ ] [ ] [ ]

4 . 7 . . 1 7 . . 01 . . , 2 . 2 . , 4 . . 1 min( 7 . . 1 , . . , 2 . 2 . , 4 . 4 . min( 7 . 7 . , . . , 2 . 2 . , 4 . 4 . min( . 1 . 1 7 . . . . 2 . 2 . 2 . . 1 4 . 4 . 7 . . 2 . 4 . min . 1 . 1 7 . . . . 2 . 2 . 2 . . 1 4 . 4 . 7 . . 2 . 4 . ˆ =           =                           =             =

T

U ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ ϕ θ θ

Pedrycz and Gomide, FSE 2007

Relation-relation fuzzy equations

X={x1, x2,…,xn}, Y={y1, y2,…,ym}, Z={z1, z2,…,zp} U: Z×X → [0,1] U = [uki] (p×n) V: Z×Y → [0,1] V = [vkj] (p×m)

) ( ] [ ] 1 , [ :

1 1 11

m n r r r r r r R R R U V

ij nm n ij m

× =           = → × =

• Y

X

Pedrycz and Gomide, FSE 2007

Uk = [uk1, uk2,…, uki,…, ukn] (1×n) k-th row of U Vk = [vk1, vk2,…, vki,…, vkm] (1×n) k-th row of V Rj = [r1j, r2j,…, rij,…, rnj]T (n×1) j-th column of R As before, let

[ ]

              =               =               = =

m p p p m m m p p

R U R U R U R U R U R U R U R U R U R R R U U U V V V R U V

• 2

1 2 2 2 1 2 1 2 1 1 1 2 1 2 1 2 1

thus

Pedrycz and Gomide, FSE 2007

[ ] [ ] [ ]

R U R U R U R U V R U R U R U R U V R U R U R U R U V

p m p p p p m m

• =

= = = = =

2 1 2 2 2 2 1 2 2 1 1 2 1 1 1 1

Using the previous result we get

p i R V U

T i i

, , 1 , ˆ

• =

= θ

Pedrycz and Gomide, FSE 2007

Multi–input, single–output fuzzy equations

Ui ∈ F(Xi) , i = 1,…, p V ∈ F(Y) R ∈ F(X1 × X2 ×…. × Xp × Y)

)] , , , , ( ) ( ) ( ) ( [ sup ) (

2 1 2 2 1 1 2 1

y x x x R t x U t t x U t x U y V R U U U V

n p p x p

• X

∈

= = R U U U U R R V U

p i i i T i i

• 1

1 1

ˆ

+ −

= = θ

Pedrycz and Gomide, FSE 2007

Solvability conditions for maximal solutions

• hgt (U) ≥ hgt (V)

estimation problem

• maxi rij ≥ vj

necessary condition for inverse problem

• concise and practically relevant solvability is difficult (in general)
• if system is not solvable, then look for approximate solution

Pedrycz and Gomide, FSE 2007

7.4 Associative memories

Pedrycz and Gomide, FSE 2007

X={x1, x2,…,xn}, Y={y1, y2,…,ym} Uk: X → [0,1] Uk = [u1k, u2k,…, uik,…, unk] = [uik] (1×n) Vk: Y → [0,1] Vk = [v1k, v2k,…, vjk,…, vmk] = [vjk] (1×m) k = 1,….,N Uk and Vk are patterns to be encoded into memory R

Sup-t fuzzy associative memories

Pedrycz and Gomide, FSE 2007

k T k k N k k

V U R R R ϕ = =

=

,

1

• Encoding

R U V

k k

• =
• Decoding

Sup-t fuzzy associative memories

Pedrycz and Gomide, FSE 2007

• U1, U2,…,UN

form a partition

• adjacent and overlap at ½
• hgt(Uk ∩ Uk–1) = 0.5 and ∑kUk(x) = 1 ∀x∈X
• x = [x1, x2,…,xn]

Semioverlapping fuzzy sets

Pedrycz and Gomide, FSE 2007

Proposition if fuzzy patterns Uk are semioverlapped, then the pairwise encoding of Uk and Vk , k = 1,…, N using produces perfect recall realized as

R U V

k k

• =

k T k k N k k

V U R R R ϕ = =

=

,

1

• Pedrycz and Gomide, FSE 2007

X={x1, x2,…,xn}, Y={y1, y2,…,ym} Uk: X → [0,1] Uk = [u1k, u2k,…, uik,…, unk] = [uik] (1×n) Vk: Y → [0,1] Vk = [v1k, v2k,…, vjk,…, vmk] = [vjk] (1×m) k = 1,….,N Uk and Vk are patterns to be encoded into memory R

Inf-s fuzzy associative memories

Pedrycz and Gomide, FSE 2007

k T k k N k k

V U R R R β = =

=

,

1

• Encoding

R U V

k k

• =
• Decoding

Inf-s fuzzy associative memories

Pedrycz and Gomide, FSE 2007

7.5 Fuzzy numbers and fuzzy arithmetic

Pedrycz and Gomide, FSE 2007

Algebraic operations on fuzzy numbers

Fuzzy interval Fuzzy number

       ∈ ∈ ∈ =

• therwise

] , ( if ) ( ] , [ if 1 ) , [ if ) ( ) ( d c x x g c b x b a x x f x A

A A

1

A(x) R

a b c d 1

A(x) R

a m b

(a) (b)

fA gA fA gA

fA right semicontinuous gA left semicontinuous

Pedrycz and Gomide, FSE 2007

1 1 2.5 2.5 1 1 2.2 2.2 3.0 3.0 2.5 2.2 3.0 real number 2.5 fuzzy number about 2.5 real interval [2.2, 3.0] fuzzy interval around [2.2, 3.0]

R R R R

Examples

Pedrycz and Gomide, FSE 2007

Computing with fuzzy numbers

• Consider a 2 h travel at a speed of about 110 km/h.

What was the distance you traveled?

• In a given manufacturing process, there are five operations completed

in series. Each manufacturing task has durations of about T1, T2,…, Tn time units. What is the completion time of the process?

• Two fundamental methods to perform algebraic operations

– based on interval arithmetic and α-cuts – extension principle

Pedrycz and Gomide, FSE 2007

Interval arithmetic and α-cuts

)] / , / , / , / max( ), / , / , / , / [min( ] , /[ ] , [ )] , , , max( ), , , , [min( ] , ].[ , [ ] , [ ] , [ ] , [ ] , [ ] , [ ] , [ d b c b d a c a d b c b d a c a d c b a bd bc ad ac bd bc ad ac d c b a c b d a d c b a d c b a d c b a = = − − = − + + = +

Pedrycz and Gomide, FSE 2007

If * is any of the four basic algebraic operations and A and B are fuzzy sets on R and α∈[0,1], then (A*B)α = Aα*Bα

)] )( ( [ sup ) )( ( ) (

] 1 , [ ] 1 , [

x B A x B A B A B A ∗ = ∗ ∗ = ∗

∈ ∈

α

α α α

• Pedrycz and Gomide, FSE 2007

Example

A(x,a,m,b), B(x,c,n,d) triangular fuzzy numbers Aα = [(m – a)α + a, (m – b) α + b], Bα = [(n – c)α + c, (n – d) α + d] A = A(x,1,2,3), B = B(x,2,3,5) Aα = [α + 1, – α + 3], Bα = [α + 2, – 2α + 5] (A+B)α = [2α + 3, – 3α + 3] (A – B)α = [3α – 4, – 2α + 1] (AB)α = [(α + 1)(α + 2), (– α + 3) (– 2α + 5)] (A/B)α = [(α + 1)/(–2α + 5), (– α + 3)/(α + 2)]

Pedrycz and Gomide, FSE 2007

• 5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x (a) Addition A+B A B

• 5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x (b) Subtraction A-B A B

• 5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x (c) Multiplication AB A B

• 5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 9 10

11 12 13 14 15 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 x (d) Division B/A A B

Pedrycz and Gomide, FSE 2007

Fuzzy arithmetic and the extension principle

R ∈ ∀ = ∗

=

z y B x A z B A

y x z

, )] ( ), ( min[ sup ) )( (

*

Extension principle and standard operations on real numbers

∗ ∈ {+, −, ⋅ , / }

In general, if t is a t-norm and ∗: R2→ R then

R ∈ ∀ = ∗

=

z y B t x A z B A

y x z

, )] ( ) ( [ sup ) )( (

*

Pedrycz and Gomide, FSE 2007

Example

A(x,a,m,b), B(x,c,n,d) triangular fuzzy numbers

tm(A+B) = (A+B)

using minimum t-norm

td(A+B) = (A+B)

using drastic product t-norm

R R ∈ ∀ ∗ ≤ ∗ ≤ ∗ ∈ ∀ ≤ ≤ ∈ ∀ ≤ ⇒ ≤

= = =

z z B A z B A z B A z y B t x A y B t x A y B t x A b a b at b t a t t

d d

t t t m y x z y x z d y x z

), )( ( ) )( ( ) )( ( )], ( ) ( [ sup )] ( ) ( [ sup )] ( ) ( [ sup ] 1 , [ , ,

* * * 2 1 2 1

Pedrycz and Gomide, FSE 2007

1 1

3.0

1 1

2.0 4.0

A

1.5 2.5 4.0 1.0

B

4.0 7.0 1.0

A+B tm A+B td

6.0

x x x x

Different choices of t-norms, different results

Pedrycz and Gomide, FSE 2007

Proposition For any fuzzy numbers A and B and a continuous monotone binary operation ∗ on R, the following equality holds for all α-cuts with α∈[0,1]: (A∗B)α = Aα ∗ Bα

(Nguyen and Walker, 1999)

Pedrycz and Gomide, FSE 2007

Important consequences of the proposition:

• 1. Aα and Bα closed and bounded ∀α ⇒ (A∗B)α closed and bounded
• 2. A and B normal ⇒ (A∗B) normal
• 3. Computation of (A∗B) can be done combining the increasing and

decreasing parts of the membership functions of A and B.

Pedrycz and Gomide, FSE 2007

1 A B A∗B x y y z = x∗y 1 A B A∗B x y y z = x∗y (a) y (b) y

Computation of (A∗B) combining the increasing and decreasing parts of the membership functions

Pedrycz and Gomide, FSE 2007

Computing with triangular fuzzy numbers

         ∈ − − ∈ − −          = ∈ − − ∈ − − =

• therwise

] , [ if ) , [ if ) (

• therwise

] , [ if ) , [ f ) ( d n x n d x d n c x c n c x x B b m x m b x b m a x i a m a x x A

• A(x,a,m,b) and B(x,c,n,d) →

triangular fuzzy numbers

• membership functions

Pedrycz and Gomide, FSE 2007

Addition

) ( ) ( ) ( from and ) ( ) ( ) , [ ), , [ and ) ( ) ( and . 2 for 1 ) ( . 1 . 1 )], ( ), ( min[ sup ) ( c a n m c a z y x z c c n y a a m x n c y m a x c n c y a m a x y B x A n y m x n m z n m z z C z y B x A z C

y x z

+ − + + − = + = + − = + − = ∈ ∈ = − − = − − = = < < + < + = = ∈ ∀ =

+ =

α α α α α α R

Pedrycz and Gomide, FSE 2007

       + > + − + − + + = + < + − + + − = + − + + = + = + − = + − = ∈ ∈ = − − = − − = = > > + > n m z n m d b z d b n m z n m z c a n M c a z x C n m d b d b y x z d d n y b b m x d n y b m x n d y d m b x b y B x A n y m x n m z if ) ( ) ( ) ( if 1 if ) ( ) ( ) ( ) ( . 4 ) ( ) ( ) ( from and ) ( ) ( ] , [ ], , [ and ) ( ) ( and . 3 α α α α α α

C(x) =C(x,a+c,m+n,b+d)

C = A + B

⇓

Pedrycz and Gomide, FSE 2007

Multiplication

• Looking at the increasing part of the membership function

x = (m – a)α + a y = (n – c)α + c z = xy = [(m – a)α + a][(n – c)α + c] z = (m – a)(n – c)α2 + (m – a)αc + a(n – c)α + ac = f1(α) if ac ≤ z ≤ mn then the membership function of D =A.B is D(z) = f1

–1(z)

Pedrycz and Gomide, FSE 2007

• Looking at the decreasing part of the membership function

x = (m – b)α + b y = (n – d)α + d z = xy = [(m – b)α + b][(n – d)α + d] z = (m – b)(n – d)α2 + (m – b)αd + b(n – d)α + bd = f2(α) if mn ≤ z ≤ bd then the membership function of D =A.B is D(z) = f2

–1(z)

Pedrycz and Gomide, FSE 2007