Fuzzy sets I 1

Fuzzy sets I

Prof. Dr. Jaroslav Ramík

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 and distinguishable objects of our concept or our thinking about the collection.

• Fuzzy set - a collection of objects in connection with expression of uncertainty of the property characterizing the objects by 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

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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 membership grade

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

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Example 3.“Number around eight”

})8x(

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

~2A

2

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

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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 }

• 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)

[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 - inclusion

- 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~ ]1, 0[

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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,…

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Example 4.

A

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

B~

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

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Complement, Cartesian product

A~ (X)

B~ (Y)

A~

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

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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Complementarity conditions

A~ (X)

A~

CA~A~

1. = 2. = XA

~C

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

…later on …”bold” intersection and union will satisfy the complementarity…

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Examples

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Extended operations with FS

Intersection and Union = operations on (X)

Realization by Min and Max operators

generalized by t-norms and t-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 sum

Property:

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, and a,b [0,1]

otherwise0

1aforb

1bfora

)b,a(Tw

otherwise1

0aforb

0bfora

)b,a(Sw

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Examples of t-norms and t-conorms #2

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

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

- Lukasiewicz t-norm and t-conorm (satisfies complematarity!)

(bounded difference, bounded sum)

Also: b - bold intersection, b - bold unionProperty: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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Examples of t-norms and t-conorms #3

5. q [1,+)

a,b [0,1] Yager’s t-norm and t-conorm

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

Property:

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 operatorsA 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 (CO) #1

CO = Aggregation operator G satisfying

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

Example 1. 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 by

PTS(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 numbers

A~

- fuzzy set of R (real numbers)

- convex

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

- A is closed interval for all [0,1]

Then is called fuzzy interval

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

then is called fuzzy number

A~

A~

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Positive and negative fuzzy numbers

A~

- fuzzy number is

- positive if A(x) = 0 for all x 0

- negative if A(x) = 0 for all x 0

0

0A~

0B~

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Example 5. Fuzzy number “About three”

otherwise0

6,3xfor3

x6

3,1xfor2

1x

)x(A

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Example 6. Triangular fuzzy number “About three”

otherwise0

6,3xfor3

x6

3,1xfor2

1x

)x(A

mean value spread

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L-R fuzzy intervals

• L, R : [0,+) [0,1] - non-increasing, non-constant functions - shape functions

• L(0) = R(0) = 1, m, n, > 0, > 0 - real numbers• - fuzzy interval of L-R-type if

• fuzzy number of L-R-type if m = n, L, R - decreasing functions

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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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 fuzzy set, -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 and t-

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. 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.