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.