chapter 5 fuzzy numbercomp.eng.ankara.edu.tr/files/2016/02/chapter5.pdf1)addition and subtraction...

1

Chapter 5�

Fuzzy Number

2

Concept of Fuzzy Number

!   Interval n  interval A = [a1, a3] a1, a3 ∈ , a1 < a3

n  Expressing the interval as membership function

ℜ

⎪⎩

⎪⎨

⎧

>

≤≤

<

=

3

31

1

,0,1,0

)(ax

axaax

xAµ

a1 a3

µA(x)

x

1

3

Fuzzy Number

!   Definition(Fuzzy number) n  convex fuzzy set

n  normalized fuzzy set

n  it’s membership function is piecewise continuous

n  It is defined in the real number

!   α-cut interval of fuzzy number Aα = [a1

(α), a3(α)]

(αʹ < α) ⇒ (a1(αʹ) ≤ a1

(α), a3(αʹ) ≥ a3

(α))

4

Fuzzy Number

a1 a3

µA(x)

x

1

a2

Fuzzy Number A = [a1, a2, a3]

a1(0)

µA(x)

x

1

α

αʹ

a1(αʹ) a1

(α) a3(α) a3

(αʹ) a3(0)

A

Aα = [a1(α), a3

(α)]

Aαʹ = [a1(αʹ), a3

(αʹ)]

α-cut of fuzzy number (α’ < α) ⇒ (Aα ⊂ Aαʹ)

5

Operation of Interval

∀a1, a3, b1, b3 ∈ ℜ, A = [a1, a3], B = [b1, b3]

n  Addition

[a1, a3] (+) [b1, b3] = [a1 + b1, a3 + b3]

n  Subtraction

[a1, a3] (–) [b1, b3] = [a1 – b3, a3 – b1]

n  Multiplication

[a1, a3] (•) [b1, b3]

= [a1 • b1 ∧ a1 • b3 ∧ a3 • b1 ∧ a3 • b3 ,

a1 • b1 ∨ a1 • b3 ∨ a3 • b1 ∨ a3 • b3]

6

Operation of Interval

n  Division

[a1, a3] (/) [b1, b3]

= [a1 / b1 ∧ a1 / b3 ∧ a3 / b1 ∧ a3 / b3 ,

a1 / b1 ∨ a1 / b3 ∨ a3 / b1 ∨ a3 / b3]

excluding the case b1 = 0 or b3 = 0

n  Inverse interval

[a1, a3]–1 = [1 / a1 ∧ 1 / a3, 1 / a1 ∨ 1 / a3]

excluding the case a1 = 0 or a3 = 0

7

Operation of Interval

!   Example n  A = [3, 5], B = [–2, 7]

]12,1[]75,23[)( =+−=+ BA]7,4[])2(5,73[)( −=−−−=− BA

]35,10[])2(3,75)2(573)2(3[)(

−=

∨−••∧−•∧•∧−•=• !BA

]7/5,5.2[])2/(3,7/5)2/(57/3)2/(3[(/)

−=

∨−∧−∧∧−= !BA

⎥⎦

⎤⎢⎣

⎡−=⎥⎦

⎤⎢⎣

⎡∨

−∧

−=−= −−

71,

21

71

)2(1,

71

)2(1]7,2[ 11B

8

Operation of Fuzzy Number

!   Operation of α-cut Interval n  α-cut interval of fuzzy number A = [a1, a3]

Aα = [a1(α), a3

(α)], ∀α ∈ [0, 1], a1, a3, a1(α), a3

(α) ∈ ℜ

n  [a1(α), a3

(α)] (+) [b1(α), b3

(α)] = [a1(α) + b1

(α), a3(α) + b3

(α)]

n  [a1(α), a3

(α)] (–) [b1(α), b3

(α)] = [a1(α) – b3

(α), a3(α) – b1

(α)]

9

Operation of Fuzzy Number

!   Addition: A (+) B

!   Subtraction: A (–) B

!   Multiplication: A (•) B

!   Division: A (/) B

))()(()()( yxz BAyxzBA µµµ ∧∨=+=

+

))()(()()( yxz BAyxzBA µµµ ∧∨=−=

−

))()(()()( yxz BAyxzBA µµµ ∧∨=•=

•

))()(()(/(/) yxz BAyxzBA µµµ ∧∨=

=

10

Operation of Fuzzy Number

!   Minimum: A (∧) B

!   Maximum: A (∨) B

!   multiply a scalar value to the interval a ∈

a[b1, b3] = [a • b1 ∧ a • b3, a • b1 ∨ a • b3]

!   multiply scalar value to α-cut interval

a[b1(α), b3

(α)] = [a • b1(α) ∧ a • b3

(α), a • b1(α) ∨ a • b3

(α)]

))()(()()( yxz BAyxzBA µµµ ∧∨=∧=

∧

))()(()()( yxz BAyxzBA µµµ ∧∨=∨=

∨

ℜ

11

Examples of Fuzzy Number Operation

!   Example : Addition A(+)B A = {(2, 1), (3, 0.5)}, B = {(3, 1), (4, 0.5)}

for all x ∈ A, y ∈ B, z ∈ A(+)B

i. for z < 5, µA(+)B(z) = 0

ii. z = 5 results from x + y = 2 + 3

µA(2) ∧ µB(3) = 1 ∧ 1 = 1

iii. z = 6 results from x + y = 3 + 3 or x + y = 2 + 4

µA(3) ∧ µB(3) = 0.5 ∧ 1 = 0.5

µA(2) ∧ µB(4) = 1 ∧ 0.5 = 0.5

1)1()5(325)( =∨=+=

+ BAµ

5.0)5.0,5.0()6(426336)( =∨=+=

+=+ BAµ

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Examples of Fuzzy Number Operation

iv. z = 7

results from x + y = 3 + 4 µA(3) ∧ µB(4) = 0.5 ∧ 0.5 = 0.5

v. for z > 7 µA(+)B(z) = 0

A(+)B = {(5, 1), (6, 0.5), (7, 0.5)} µA(x)

1

3

0 .5

2

µB(x)

1

3

0.5

4

µA (+) B(x)

1

6

0 .5

5 7

(a) Fuzzy set A (b) Fuzzy number B (c) Fuzzy set A (+) B

5.0)5.0()7(437)( =∨=

+=+ BAµ

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Examples of Fuzzy Number Operation

!   Example : Subtraction A(-)B

A = {(2, 1), (3, 0.5)}, B = {(3, 1), (4, 0.5)}

A(-)B = {(-2, 0.5), (-1, 1), (0, 0.5)}

µA(x) 1

3

0 .5

2

µB(x) 1

3

0 .5

4

µA (-) B(x)

1

-1

0.5

-2 0

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Examples of Fuzzy Number Operation

!   Example : Max operation A(∨)B

A = {(2, 1), (3, 0.5)}, B = {(3, 1), (4, 0.5)}

A(∨)B = {(3, 1), (4, 0.5)}

15

Triangular fuzzy number

!   Definition of Triangular fuzzy number A = (a1, a2, a3)

membership functions

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

>

≤≤

−

−

≤≤

−

−

<

=

,0

,

,

,0

)(

3

32

23

3

21

12

1

1

)(

ax

axaaaxa

axaaaax

ax

xAµ

a1 a3

µA(x)

x

1

a2

Triangular fuzzy number A = (a1, a2, a3)

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Triangular fuzzy number

!   α-cut interval of triangular fuzzy number

interval Aa

Aα = [a1(α), a3

(α)]

= [(a2 - a1)α + a1, -(a3 - a2)α + a3]

17

Triangular fuzzy number

!   Example triangular fuzzy number A = (-5, -1, 1)

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

>

≤≤−−

−≤≤−+

−<

=

1,0

11,21

15,45

5,0

)()(

x

xx

xx

x

xAµ

-6 -5 -4 -3 -2 -1 0 1 2

1

0.5

A0.5

α = 0.5 cut of triangular fuzzy number A = (-5, -1, 1)

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Triangular fuzzy number

!   Example n  α-cut interval from this fuzzy number

Aα = [a1(α), a3

(α)] = [4α - 5, -2α + 1]

n  If α = 0.5, substituting 0.5 for α, we get A0.5

A0.5 = [a1(0.5), a3

(0.5)] = [-3, 0]

1221

5445

+−=⇒=−

−=⇒=+

αα

αα

xx

xx

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Operation of Triangular Fuzzy Number

!   Properties of operations on triangular fuzzy number

1.  The results from addition or subtraction between triangular fuzzy numbers result also triangular fuzzy numbers.

2.  The results from multiplication or division are not triangular fuzzy numbers.

3.  Max or min operation does not give triangular fuzzy number

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Operation of Triangular Fuzzy Number

!   triangular fuzzy numbers A and B are defined

A = (a1, a2, a3), B = (b1, b2, b3)

n  Addition

n  Subtraction

n  Symmetric image -(A) = (-a3, -a2, -a1)

),,(),,)()(,,()(

332211

321321

babababbbaaaBA

+++=

+=+

),,(),,)()(,,()(

132231

321321

babababbbaaaBA−−−=

−=−

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Operation of Triangular Fuzzy Number

!   Example A = (-3, 2, 4), B = (-1, 0, 6)

A (+) B = (-4, 2, 10)

A (-) B = (-9, 2, 5)

-3

-1

0 2 4

1

0.5

6

A

B

-4

0 2

1

0.5

10

A (+) B

-9 0 2

1

0.5

5

A (-) B

(a) Triangular fuzzy number A, B (b) A (+) B (c) A (-) B

22

Operation of Triangular Fuzzy Number

!   Example

triangular fuzzy numbers A and B :

A = (-3, 2, 4), B = (-1, 0, 6)

α-level intervals from α-cut operation

]66,1[])(,)[(],[

]42,35[])(,)[(],[

323112)(

3)(

1

323112)(

3)(

1

+−−=

+−−+−==

+−−=

+−−+−==

αααα

αααα

ααα

ααα

bbbbbbbbB

aaaaaaaaA

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Operation of Triangular Fuzzy Number

!   Example n  Aα (+) Bα = [6α - 4, -8α + 10]

α = 0 and α = 1,

A0 (+) B0 = [-4, 10] A1 (+) B1 = [2, 2] = 2

n  Aα (-) Bα = [11α - 9, -3α + 5]

α = 0 and α = 1

A0 (-) B0 = [-9, 5] A1 (-) B1 = [2, 2] = 2

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Approximation of Triangular Fuzzy Number

!   Example

A = (1, 2, 4), B=(2, 4, 6)

When α = 0, A0(•)B0 = [2, 24]

When α = 1, A0(•)B1 = [2+4+2, 4-20+24] = [8, 8] = 8

A (•) B ≅ (2 , 8, 24)

]62,22[]6)46(,2)24([]42,1[]4)24(,1)12([

+−+=+−−+−=

+−+=+−−+−=

αααα

αααα

α

α

BA

]24204,242[])62)(42(),22)(1([]62,22)[](42,1[)(

22 +−++=

+−+−++=

+−+•+−+=•

αααααααααααααα BA

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Approximation of Triangular Fuzzy Number

!   Example

When α = 0

When α = 1

approximated value of A (/) B

])22/()62(),62/()1([(/) ++−+−+= αααααα BA

]2,17.0[]2/4,6/1[(/) 00

=

=BA

5.0]4/2,4/2[

])22/()42(),62/()11([(/) 11

=

=

++−+−+=BA

)2,5.0,17.0((/) =BA

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Other Types of Fuzzy Number

!   Trapezoidal Fuzzy Number

Definition(Trapezoidal fuzzy number)

A = (a1, a2, a3, a4)

membership function

⎪⎪⎪⎪

⎩

⎪⎪⎪⎪

⎨

⎧

>

≤≤−

−≤≤

≤≤−

−<

=

4

4334

4

32

2112

1

1

,0

,

,1

,

,0

)(

ax

axaaaxa

axa

axaaaax

ax

xAµ

a1 a4

µA(x)

x

1

a2 a3

Trapezoidal fuzzy number A = (a1, a2, a3, a4)

27

1)Addition and subtraction between fuzzy numbers become trapezoidal fuzzy number.

2)Multiplication, division, and inverse need not be trapezoidal fuzzy number.

3)Max and Min of fuzzy number is not always in the

form of trapezoidal fuzzy number.

Operations of Trapezoidal Fuzzy Number

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!   Addition

!   Subtraction

),,,(),,,)()(,,,()(

44332211

43214321

bababababbbbaaaaBA++++=

+=+

),,,()( 14233241 babababaBA −−−−=−

Operations of Trapezoidal Fuzzy Number

29

!   Example Multiplication A = (1, 5, 6, 9), B = (2, 3, 5, 8)

Aα = [4α + 1, –3α + 9], Bα = [α + 2, –3α + 8]

When α = 0

When α = 1

approximated value

]72519,294[])83)(93(),2)(14([)(

22 +−++=

+−+−++=•

αααααααααα BA

]72,2[)( 00 =• BA

]30,15[]72519,294[)( 11

=

+−++=• BA

]72,30,15,2[)( ≅• BA

Operations of Trapezoidal Fuzzy Number

30

Operations of Trapezoidal Fuzzy Number

!   Example

10 20 30 40 50 60 70

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

(1,5, 6, 9) = A

(2,3, 5, 8) = B

( 1 , 5 , 6 , 9 ) ( • ) (2,3,5,8) = A (•) B

(2,15, 30, 72)

x

Multiplication of trapezoidal fuzzy number A (•) B

31

Operations of Trapezoidal Fuzzy Number

!   flat fuzzy number

∃m1, m2 ∈ ℜ, m1 < m2

µA(x) = 1, m1 ≤ x ≤ m2 In this case, membership function in x < m1 and x < m2 need not be a lin

e

1

m1 m2

Flat fuzzy number