Fuzzy sets II 1

Fuzzy sets II

Prof. Dr. Jaroslav Ramík

Fuzzy sets II 2

Content

• Extension principle• Extended binary operations with fuzzy numbers• Extended operations with L-R fuzzy numbers• Extended operations with t-norms• Probability, possibility and fuzzy measure• Probability and possibility of fuzzy event• Fuzzy sets of the 2nd type• Fuzzy relations

Fuzzy sets II 3

Extension principle (EP)by L. Zadeh, 1965

• EP makes possible to extend algebraical operations with NUMBERS to FUZZY SETS

• Even more: EP makes possible to extend REAL FUNCTIONS of real variables to FUZZY FUNCTIONS with fuzzy variables

• Even more: EP makes possible to extend CRISP CONCEPTS to FUZZY CONCEPTS

(e.g. relations, convergence, derivative, integral, etc.)

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Example 1. Addition of fuzzy numbers N~

M~

S~

N~

M~

S~

EP:

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Theorem 1.

Let

• the operation denotes + or · (add or multiply)

• - fuzzy numbers, [0,1]

• - -cuts

Then is defined by its -cuts as follows

N~

,M~

]d,c[N~

],b,a[M~

]d*c,b*a[S~

N~~M

~S~

[0,1]

Fuzzy sets II 6

Extension principle for functions• X1, X2,…,Xn, Y - sets

• n - fuzzy sets on Xi , i = 1,2,…,n

• g : X1X2 …Xn Y - function of n variables

i.e. (x1,x2 ,…,xn ) y = g (x1,x2 ,…,xn )

Then the extended function

is defined by

iA~

otherwise0

)y(gif)x(),...,x(MinSup)y(

1nA1A

)y(g)x,...,(xBn11-

n1

)A~

,...,A~

,A~

(g~B~

)A~

,...,A~

,A~

( n21n21

Fuzzy sets II 7

Remarks

• g-1(y) = {(x1,x2 ,…,xn ) | y = g (x1,x2 ,…,xn )} - co-image of y

• Special form of EP: g (x1,x2) = x1+x2 or g (x1,x2) = x1*x2

• Instead of Min any t-norm T can be used - more general for of EP

)y(),x(MinSup2R)y,x(

)z( NMNM

yxz

Fuzzy sets II 8

Example 2. Fuzzy Min and Max

))y(),x((MinSup)z( NM)y,x(Maxz

Max

))y(),x((MinSup)z( NM)y,x(Minz

Min

)N~

,M~

(ni~

M)N~

,M~

(xa~M

Fuzzy sets II 9

Extended operations with L-R fuzzy numbers

• L, R : [0,+) [0,1] - decreasing functions - shape functions

• L(0) = R(0) = 1, m - main value, > 0, > 0

• = (m, , )LR - fuzzy number of L-R-type if

•

.mxifmx

R

,mxifxm

L

)x(A~

A~

Left spread Right spread

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Example 3. L-R fuzzy number “About eight”

28xA e)x( 1,8nm,e)x(R)x(L

2mx

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Example 4.

L(u) = Max(0,1 ‑ u) R(u) = 2u1

1

LRLR )1,2,3(),,m(M~

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Theorem 2.Let

= (m,,)LR , = (n,,)LR

where L, R are shape functions

Then is defined as

Example: (2,3,4)LR (1,2,3)LR = (3,5,7)LR

N~

M~

N~~M

~S~

LR),,nm(S~

~

Addition

Fuzzy sets II 13

Opposite FN

= (m,,)LR - FN of L-R-type

= (m,, )LR - opposite FN of L-R-type to

M~

M~~ M

~

“Fuzzy minus” M~

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Theorem 3.Let

= (m,,)LR , = (n,,)LR

where L, R are shape functions

Then is defined as

Example: (2,3,4)LR (1,2,3)LR = (1,6,6)LR

N~

M~

N~~M

~S~

LR),,nm(S~

~

Subtraction

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Example 5. Subtraction

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Theorem 4.Let

= (m,,)LR , = (n,,)LR

where L, R are shape functions

Then is defined by approximate formulae:

Example by 1.: (2,3,4)LR (1,2,3)LR (2,7,10)LR

N~

M~

N~~M

~S~

LR).m.n,.m.n,n.m(S~

~

Multiplication

LR)..m.n,..mn,n.m(S~

1.

2.

Fuzzy sets II 17

Example 6. Multiplication

M~

N~

= (2,1,2)LR , = (4,2,2)LR

N~~M

~S~

(8,8,12)LR LR)16,6,8(S~

formula 1. - - - - formula 2. ……. exact function

Fuzzy sets II 18

Inverse FN

= (m,,)LR > 0 - FN of L-R-type

- approximate formula 1

- approximate formula 2

M~

RL221 )

m,

m,

m

1(M

~

RL1 )

)m(m,

)m(m,

m

1(M

~

11 M~~)M

~~(

We define inverse FN only for positive (or negative) FN !

Fuzzy sets II 19

Example 7. Inverse FN

1M

~= (2,1,2)LR

RL1

1 )4

1,

2

1,

2

1(M

~ RL

11 )

2

1,

4

1,

2

1(M

~

formula 1. - - - - formula 2. ……. exact function

f.1: f.2:

Fuzzy sets II 20

= (m,,)LR , = (n,,)LR > 0

where L, R are shape functions

Define

Combinations of approximate formulae, e.g.

N~M

~

1N~~M

~N~

/~

M~

S~

LR))n(n

nm,

)n(n

nm,

n

m(S

~

Division

Fuzzy sets II 21

Probability, possibility and fuzzy measure

Sigma Algebra (-Algebra) on :

F - collection of classical subsets of the set satisfying:

(A1) F

(A2) if A F then CA F

(A3) if Ai F, i = 1, 2, ... then i Ai F

- elementary space (space of outcomes - elementary events)

F - -Algebra of events of

Fuzzy sets II 22

Probability measure

F - -Algebra of events of p : F [0,1] - probability measure on Fsatisfying:(W1) if A F then p(A) 0 (W2) p() = 1

(W3) if Ai F , i = 1, 2, ..., Ai Aj = , ij

then p(i Ai ) = i p(Ai ) - -additivity

(W3*) if A,B F , AB= ,

then p(AB ) = p(A ) + p(B) - additivity

Fuzzy sets II 23

Fuzzy measure

F - -Algebra of events of g : F [0,1] - fuzzy measure on F

satisfying:

(FM1) p() = 0

(FM2) p() = 1

(FM3) if A,B F , AB then p(A) p(B) - monotonicity

(FM4) if A1, A2,... F , A1 A2 ...

then g(Ai ) = g( Ai ) - continuityi

limi

lim

Fuzzy sets II 24

Properties

• Additivity condition (W3) is stronger than monotonicity (MP3) & continuity (MP4) i.e.

• (W3) (MP3) & (MP4)

• Consequence: Any probability measure is a fuzzy measure but not contrary

Fuzzy sets II 25

Possibility measure

P() - Power set of (st of all subsets of ) : P() [0,1] - possibility measure on satisfying:(P1) () = 0 (P2) () = 1

(P3) if Ai P() , i = 1, 2, ...

then (i Ai ) = Supi {p(Ai )}

(P3*) if A,B P() ,

then (AB ) = Max{(A ), (B)}

Fuzzy sets II 26

Properties

• Condition (P3) is stronger than monotonicity (MP3) & continuity (MP4) i.e.

• (P3) (MP3) & (MP4)

• Consequence: Any possibility measure is a fuzzy measure but not contrary

Fuzzy sets II 27

Example 8.

A B C A B A C B C

P 0 0,3 0,5 0,2 0,8 0,5 0,7 1

g 0 0,4 0,6 0,3 0,8 0,6 0,7 1 0 0,7 1 0,3 1 0,7 1 1

= ABC

F = {, A, B, C, AB, BC, AC, ABC}

Fuzzy sets II 28

Possibility distribution - possibility measure on P()

• Function : [0,1] defined by

(x) = ({x}) for xis called a possibility distribution on

Interpretation: is a membership function of a fuzzy set , i.e. (x) = A(x) x ,

A(x) is the possibility that x belongs to

A~

Fuzzy sets II 29

Probability and possibility of fuzzy event

Example 1: What is the possibility (probability) that tomorrow will be a nice weather ?

Example 2: What is the possibility (probability) that the profit of the firm A in 2003 will be high ?

• nice weather, high profit - fuzzy events

Fuzzy sets II 30

Probability of fuzzy event Finite universe

={x1, …,xn} - finite set of elementary outcomes

F - -Algebra on P - probability measure on F

- fuzzy set of , with the membership

function A(x) - fuzzy event,

A F for [0,1]

P( ) = - probability of fuzzy event

A~

A~

n

1iiiA })x({P).x(

A~

Fuzzy sets II 31

Probability of fuzzy event Real universe

= R - real numbers - set of elementary outcomes

F - -Algebra on R

P - probability measure on F given by density fction g

- fuzzy set of R, with the membership

function A(x) - fuzzy event

A F for [0,1]

P( ) = - probability of fuzzy event

A~

A~ dx)x(g).x(A

A~

Fuzzy sets II 32

Example 9.

= (4, 1, 2)LR L(u) = R(u) = e-u

A~

.otherwise0

0xfore2)x(g

x2

- density function of random value

dxe2edxe2e)A

~(P x2

4

x24

0

)x4( 24x

225

25

e5

2e2)

e

1e(e2e

5

2e2limee2 2

044

z

5

x2

A

4

0x4

= 0,036

- “around 4”

Fuzzy sets II 33

Possibility of fuzzy event

- set of elementary outcomes

: [0,1] - possibility distribution

- fuzzy set of , with the membership

function A(x) - fuzzy event

A F for [0,1]

P( ) = - possibility of fuzzy event

A~

A~ )}x(),x({MinSup A

x

A~

Fuzzy sets II 34

Fuzzy sets of the 2nd type

• The function value of the membership function is again a fuzzy set (FN) of [0,1]

Fuzzy sets II 35

Example 10.

Fuzzy sets II 36

Example 11.Linguistic variable “Stature”- Height of the body

Stature

Tall Short Very tall Middle Very short

Fuzzy sets II 37

Fuzzy relations

• X - universe• - (binary) fuzzy (valued) relation on X = fuzzy set on XX

is given by the membership function R : XX [0,1]

FR is:

• Reflexive: R (x,x) = 1 xX

• Symmetric: R (x,y) = R (y,x) x,yX

• Transitive: Supz[Min{R (x,z), R (z,y)}] R (x,y)

• Equivalence: reflexive & symmetric & transitive

R~

R~

Fuzzy sets II 38

Example 12.Binary fuzzy relation : “x is much greater than y”

xy10for1

y10xyfory9

yxyxfor0

)y,x(R

e.g. R(8,1) = 7/9 = 0,77…

- is antisymmetric: If R (x,y) > 0 then R (y,x) = 0 x,yX

R~

R~

Fuzzy sets II 39

Example 13.Binary fuzzy relation : “x is similar to y”R

~

x/y 1 2 3 4 51 1,0 0,5 0,3 0,2 02 0,5 1,0 0,6 0,5 0,23 0,3 0,6 1,0 0,7 0,44 0,2 0,5 0,7 1,0 0,85 0 0,2 0,4 0,4 1,0

R~

is equivalence !

X = {1,2,3,4,5}

Fuzzy sets II 40

Summary• Extension principle• Extended binary operations with fuzzy numbers• Extended operations with L-R fuzzy numbers• Extended operations with t-norms• Probability, possibility and fuzzy measure• Probability and possibility of fuzzy event• Fuzzy sets of the 2nd type• Fuzzy relations

Fuzzy sets II 41

References

[1] J. Ramík, M. Vlach: Generalized concavity in fuzzy optimization and decision analysis. Kluwer Academic Publ. Boston, Dordrecht, London, 2001.

[2] H.-J. Zimmermann: Fuzzy set theory and its applications. Kluwer Academic Publ. Boston, Dordrecht, London, 1996.

[3] H. Rommelfanger: Fuzzy Decision Support - Systeme. Springer - Verlag, Berlin Heidelberg, New York, 1994.

[4] H. Rommelfanger, S. Eickemeier: Entscheidungstheorie - Klassische Konzepte und Fuzzy - Erweiterungen, Springer - Verlag, Berlin Heidelberg, New York, 2002.