fuzzy sets i

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Fuzzy sets I 1

Fuzzy sets I

Prof. Dr. Jaroslav Ramk

Fuzzy sets I 2

Content Basic definitions

Examples

Operations with fuzzy sets (FS)

t-norms and t-conorms

Aggregation operators Extended operations with FS

Fuzzy numbers: Convex fuzzy set, fuzzy interval,fuzzy number (FN), triangular FN, trapezoidal FN,L-R fuzzy numbers


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Basic definitions

Set - a collection well understood anddistinguishable objects of our concept orour thinking about the collection.

Fuzzy set - a collection of objects inconnection with expression of uncertainty

of the property characterizing the objectsby grades from interval between 0 and 1.

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Fuzzy set

X - universe (of discourse) = set of objects

A : X [0,1] - membership function

= {(x, A(x))| x X} - fuzzy set of X (FS)A~


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Examples

1. Feasible daily car production

2. Young man age

3. Number around 8

4. High profit

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Example1. Feasible car production per day

= {(3; 0), (4; 0), (5; 0,1), (6; 0,5), (7; 1), (8; 0,8), (9; 0)}A~

X = {3, 4, 5, 6, 7, 8, 9} - universe


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Example 2. Young man age

Approximation of empirical evaluations (points):

20 respondents have been asked to evaluate the membershipgrade

X = [0, 100] - universe (interval)

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Example 3.Number around eight

})8x(1

1)x(R))x(,x{(A

~2A

2

+==

X = ]0, +[ - universe (interval)


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Example 4.High profit

}x1

11)x(R))x(,x{(A

~A

2

+==

X = [0, +[ - universe (interval)

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1

0 5 10 15 20 25 30 35

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Crisp set

Crisp set A of X = fuzzy set with a special membership

function: A : X {0,1} - characteristic function

Crisp set can be uniquely identified with a set:

(non-fuzzy) set A is in fact a (fuzzy) crisp set


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Support, height, normal fuzzy set

Support of fuzzy set , supp( ) = {xX| A(x) > 0}support is a set (crisp set)!

Height of fuzzy set , hgt( ) = Sup{A(x) | xX }see Example 4!

Fuzzy set is normal (normalized), if there exists

x0X with A(x0) = 1

Ex.: Support of from Example 1: supp( ) = {5, 6, 7, 8}

hgt( ) = A(8) = 1 is normal!

A~

A~

A~

A~

A~

A~ A~

A~

A~

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-cut (- level set, aspiration level)

[0,1], - fuzzy set, A = {x X|A(x)} - -cut of

< A A

- convex FS, if A is convex set (interval) for all [0,1] !!!

A~

A~

A~

A~


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Operations with fuzzy sets

(X) -Fuzzy power set = set of all fuzzy sets of X(X)

A(x) = B(x) for all x X - identity A(x) B(x) for all x X - inclusionProperties:

- transitivity

B~

A~

=

B~

,A~

B~

A~

B~A~)A~B~andB~A~( =)B

~(psup)A

~(psupB

~A~

C~

A~

)C~

B~

andB~

A~

(

ABA~B~

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Union and Intersection of fuzzy sets

B~

,A~ (X)

B~

A~

AB(x) =Max{A(x), B(x)} - union

AB(x) =Min{A(x), B(x)}- intersectionB~A~

Properties:

Commutativity, Associativity, Distributivity,

Union = fuzzy OR Intersection = fuzzy AND


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Example 5.

%A

= {(3; 0), (4; 0), (5; 0,1), (6; 0,5), (7; 1), (8; 0,8), (9; 0)} - feasible productionA~

B~

= {(3; 1), (4; 1), (5; 0,9), (6; 0,8), (7; 0,4), (8; 0,1), (9; 0)}- high costs

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Complement, Cartesian product

A~ (X)

B~ (Y)

A~

C CA(x) =1 - A(x) - complement ofA~

B~

A~

AB(x,y) =Min{A(x), B(y)}- Cartesian product (CP)

CP is a fuzzy set of XY !

Extension to more parts possible e.g. X, Y, Z,

A~ (X) ,


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Cartesian product

)y(B~

)y,x(B~

A~

)x(A~

B~A~

1

Cartesian product C~B~


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Complementarity conditions

A~ (X)

A~

CA~A~

1. =

2. = XA~

C

Min andMax do not satisfy 1., 2. ! (only for crisp sets)

later on bold intersection andunion will satisfy thecomplementarity

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Examples


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Extended operations with FS

Intersection and Union = operations on(X)

Realization by Min andMax operators

generalized by t-norms andt-conorms

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t-normsA function T: [0,1] [0,1] [0,1] is called

t-norm

if it satisfies the following properties (axioms):T1: T(a,1) = a a [0,1] - 1 is a neutral element

T2: T(a,b) = T(b,a) a,b [0,1] - commutativity

T3: T(a,T(b,c)) = T(T(a,b),c) a,b,c [0,1] - associativity

T4: T(a,b) T(c,d) whenever a c , b d - monotnicity


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t-conormsA function S: [0,1] [0,1] [0,1] is called

t-conorm

if it satisfies the following axioms:

S1: S(a,0) = a a [0,1] - 0 is a neutral element

S2: S(a,b) = S(b,a) a,b [0,1] - commutativity

S3: S(a,S(b,c)) = S(S(a,b),c) a,b,c [0,1] - associativity

S4: S(a,b) S(c,d) whenever a c , b d - monotnicity

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Examples of t-norms and t-conorms #1

1. TM = Min, SM = Max - minimum and maximum

2.

- drastic product, drastic sumProperty:

TW(a,b) T(a,b) TM(a,b) , SM(a,b) S(a,b) SW(a,b)

for every t-norm T, resp. t-conorm S, anda,b [0,1]

=

=

= otherwise01aforb

1bfora

)b,a(Tw

=

=

=otherwise1

0aforb

0bfora

)b,a(Sw


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3. TP(a,b) = a.b SP (a,b) = a+b - a.b -product andprobabilistic sum

4. TL(a,b) = Max{0,a+b - 1} SL (a,b) = Min{1,a+b}

- Lukasiewicz t-norm and t-conorm (satisfies complementarity!)(bounded difference, bounded sum)

Also: b - bold intersection, b - bold union

Properties:Let T*(a,b) = 1 - T(1-a,1-b) , S*(a,b) = 1 - S(1-a,1-b)

If T is a t-norm then T* is a t-conorm ( T and T* are dual )

If S is a t-conorm then S* is a t-norm ( S and S* are dual )

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5. q [1,+)

a,b [0,1]

Yagers t-norm and t-conorm

6. Einstein, Hamacher, Dubois-Prade product and sum etc.

7. Average (a+b)/2 is not a t-norm !!

Properties:

If q =1, then Tq, (Sq) is Lukasiewicz t-norm (t-conorm)

If q = +, then Tq, (Sq) is Min (Max)

( )

+= q

1

qqq ba,1Min)b,a(S

( )

+= q

1qq

q )b1()a1(1,0Max)b,a(T


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Extended Union and Intersection of fuzzy sets

B~

,A~ (X), T = t-norm, S = t-conorm

B~

A~

S AsB(x) =S(A(x), B(x)) - S-union

ATB(x) =T(A(x), B(x)) -T-intersectionB~

A~

T

Properties:

Commutativity, Associativity?,

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Aggregation operators

A function G: [0,1] [0,1] [0,1] is called

aggregation operator

if it satisfies the following properties (axioms):A1: G(0,0) = 0 -boundary condition 1A2: G(1,1) = 1 -boundary condition 2

A3: G(a,b) G(c,d) whenever a c , b d - monotnicity

NO commutativity or associativity conditions!

All t-norms and t-conorms are aggregation operators!

May be extended to more parts, e.g. a,b,c,


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Compensative operators (COs) #1

CO = Aggregation operator G satisfying

Min(a,b) G(a,b) Max(a,b)

Examples. Averages:

1: G(a,b) = (a +b)/2 - arithmetic mean (average)

2: G(a,b) = - geometric mean

3: G(a,b) = - harmonic mean

b.a

b1

a1 1+

Extension to more elements possible!

Max

Min

S

T

G

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Compensative operators #2

Examples. Compensatory operators:

1: TW(a,b) = .Min(a,b) + (1- ) - fuzzy and

SW(a,b) = .Max(a,b) + (1- ) - fuzzy or (by Werners)

2: ATS(a,b) = .T(a,b) + (1 - ).S(a,b) - COs byPTS(a,b) =T(a,b)

. S(a,b)1- Zimmermann and Zysno

T - t-norm, S - t-conorm, [0,1] - compensative parameter

CO compensate trade-offs between conflicting evaluations

extension to more elements possible

2

ba +

2

ba +


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Fuzzy numbersA~

- fuzzy set ofR (real numbers)

- A is convex (i.e. interval) for all [0,1]

- normal (there exists x0 R with A(x0) = 1)

- A is closed interval (with the end points) for all [0,1]

Then is called fuzzy interval

Moreover if there existsonly one x0 R with A(x0) = 1

then is called fuzzy number

A~

A~

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Fuzzy numbersgraphs of membership functions in R

1

1

Normal, convex, compact fuzzy sets

Not normal, non-convex fuzzy sets

triangular bell shaped

a b c


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Positive and negative fuzzy numbers

A~

- fuzzy number is

-positive ifA(x) = 0 for all x 0

- negative ifA(x) = 0 for all x 0

0

0A~

>0B~

0, > 0 - real numbers - fuzzy interval of L-R-type, or bell-shaped f. i. if

fuzzy number of L-R-type if m = n, L, R - decreasingfunctions

A~

.nxifnx

R

,nxmif1

,mxifxm

L

)x(A~


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Example 7. L-R fuzzy number Around eight

2A )8x(1

1)x(

+= 1,8nm,

x1

1)x(R)x(L

2====

+==

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Example 8. L-R fuzzy number About eight

( )28xA e)x(

= 1,8nm,e)x(R)x(L

2mx

======


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L-R fuzzy numbers Around eightdiffrence

-0,2

0

0,2

0,4

0,6

0,8

1

1,2

0 2 4 6 8 10 12 14 16

f1(x)

f2(x)

21

1)(

xxL

+=

2

)( xexL =

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Example 9. L-R fuzzy interval

( ) 1,2,5n,4m,e)x(R,e)x(L2

2

5x2

4x

======

0

1

0 1 2 3 4 5 6 7 8 9 10 11 12


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Example 10. Fuzzy intervals N~andM~

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Summary Basic definitions: set, fuzzy set, membership

function, crisp set, support, height, normal fuzzyset, -level set

Examples: daily production, young man age,around 8

Operations with fuzzy sets: fuzzy power set,union, intersection, complement, cartesian product

Extended operations with fuzzy sets: t-norms andt-conorms, compensative operators

Fuzzy numbers: Convex fuzzy set, fuzzy interval,fuzzy number (FN), triangular FN, trapezoidal FN,L-R fuzzy numbers


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References

[1] J. Ramk, M. Vlach: Generalized concavity in fuzzyoptimization 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.

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