Prof. Dr. Jaroslav Ram

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Fuzzy sets I

• Prof. Dr. Jaroslav Ramík

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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)

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Examples

• Feasible daily car production
• Young man age
• Number around 8

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Example1. Feasible car production per day
X 3, 4, 5, 6, 7, 8, 9 - universe
(3 0), (4 0), (5 0,1), (6 0,5), (7 1),
(8 0,8), (9 0)
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Example 2. Young man age
X 0, 100 - universe (interval)
Approximation of empirical evaluations
(points) 20 respondents have been asked to
evaluate the membership grade
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Example 3.Number around eight
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( ) x?X ?A(x)
  gt 0
• support is a set (crisp set)!
• Height of fuzzy set , hgt( ) Sup?A(x)
  x?X
• Fuzzy set is normal (normalized), if there
  exists
• x0?X with ?A(x0) 1
• Ex. Support of from Example 1 supp( )
  5, 6, 7, 8
• hgt( ) ?A(8) 1 ? is normal!

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?-cut (?- level set)
? ?0,1, - fuzzy set, A? x ?X?A(x)??
- ?-cut of
- convex FS, if A? is convex set (interval) for
all ? ?0,1 !!!
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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

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• Union and Intersection of fuzzy sets

??(X)
? ?A?B(x) Max?A(x), ?B(x) - union
? ?A?B(x) Min?A(x), ?B(x)- intersection
Properties Commutativity, Associativity,
Distributivity,
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Example 4.
(3 0), (4 0), (5 0,1), (6 0,5), (7 1),
(8 0,8), (9 0)
(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

??(X)
? ?CA(x) 1 - ?A(x) - complement of
??(Y)
??(X) ,
? ?A?B(x,y) Min?A(x), ?B(y)
- Cartesian product (CP) CP is a fuzzy set of
X?Y ! Extension to more parts possible e.g. X,
Y, Z,
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• Complementarity conditions

??(X)
1. ? ?
2. ? X
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

?(X)
Intersection ? and Union ? operations on
Realization by Min and Max operators generalized
by t-norms and t-conorms
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• t-norms

A 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-conorms

A 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

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Examples of t-norms and t-conorms 2

• 3. TP(a,b) a.b SP (a,b) ab - a.b - product
  and probabilistic sum
• 4. TL(a,b) Max0,ab - 1 SL (a,b) Min1,ab
• - Lukasiewicz t-norm and t-conorm (satisfies
  complematarity!)
• (bounded difference, bounded sum)
• Also ?b - bold intersection, ?b - bold union
• Property
• 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
• Yagers 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)

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• Extended Union and Intersection of fuzzy sets

??(X), T - t-norm, S - t-conorm
? ?A?sB(x) S(?A(x), ?B(x)) - S-union
? ?A?TB(x) T(?A(x), ?B(x)) -T-intersection
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 1 A2 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
S
Max
G
Min
T
Extension to more elements possible!
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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

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Fuzzy numbers
- 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
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Positive and negative fuzzy numbers
- fuzzy number is - positive if ?A(x) 0 for all
x ? 0 - negative if ?A(x) 0 for all x ? 0
0
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Example 5. Fuzzy number About three
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Example 6. Triangular fuzzy number About three
spread
mean value
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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, ? gt 0, ? gt 0 - real
  numbers
• - fuzzy interval of L-R-type if
• fuzzy number of L-R-type if m n, L, R -
  decreasing functions

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Example 7. L-R fuzzy number Around eight
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Example 8. L-R fuzzy number About eight
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Example 9. L-R fuzzy interval
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Example 10. Fuzzy intervals
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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.