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Fuzzy sets II1 Prof. Dr. Jaroslav Ramík. Fuzzy sets II2 Content Extension principle Extended binary operations with fuzzy numbers Extended operations.

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Presentation on theme: "Fuzzy sets II1 Prof. Dr. Jaroslav Ramík. Fuzzy sets II2 Content Extension principle Extended binary operations with fuzzy numbers Extended operations."— Presentation transcript:

1 Fuzzy sets II1 Prof. Dr. Jaroslav Ramík

2 Fuzzy sets II2 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 2 nd type Fuzzy relations

3 Fuzzy sets II3 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.)

4 Fuzzy sets II4 Example 1. Addition of fuzzy numbers EP:

5 Fuzzy sets II5 Theorem 1. Let the operation  denotes + or · (add or multiply) - fuzzy numbers,  [0,1] -  -cuts Then is defined by its  -cuts as follows  [0,1]

6 Fuzzy sets II6 Extension principle for functions X 1, X 2,…,X n, Y - sets n - fuzzy sets on X i, i = 1,2,…,n g : X 1  X 2  …  X n  Y - function of n variables i.e. (x 1,x 2,…,x n )  y = g (x 1,x 2,…,x n ) Then the extended function is defined by

7 Fuzzy sets II7 Remarks g -1 (y) = {(x 1,x 2,…,x n ) | y = g (x 1,x 2,…,x n )} - co-image of y Special form of EP: g (x 1,x 2 ) = x 1 +x 2 or g (x 1,x 2 ) = x 1 *x 2 Instead of Min any t-norm T can be used - more general for of EP

8 Fuzzy sets II8 Example 2. Fuzzy Min and Max

9 Fuzzy sets II9 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 Left spread Right spread

10 Fuzzy sets II10 Example 3. L-R fuzzy number “About eight”

11 Fuzzy sets II11 Example 4. L(u) = Max(0,1 ‑ u) R(u) =

12 Fuzzy sets II12 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 Addition

13 Fuzzy sets II13 Opposite FN = (m, ,  ) LR - FN of L-R-type = (m, ,  ) LR - opposite FN of L-R-type to “Fuzzy minus”

14 Fuzzy sets II14 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 Subtraction

15 Fuzzy sets II15 Example 5. Subtraction

16 Fuzzy sets II16 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 Multiplication 1. 2.

17 Fuzzy sets II17 Example 6. Multiplication = (2,1,2) LR, = (4,2,2) LR  (8,8,12) LR  formula formula 2. ……. exact function

18 Fuzzy sets II18 Inverse FN = (m, ,  ) LR > 0 - FN of L-R-type - approximate formula 1 - approximate formula 2 We define inverse FN only for positive (or negative) FN !

19 Fuzzy sets II19 Example 7. Inverse FN = (2,1,2) LR  formula formula 2. ……. exact function f.1: f.2:

20 Fuzzy sets II20 = (m, ,  ) LR, = (n, ,  ) LR > 0 where L, R are shape functions Define Combinations of approximate formulae, e.g. Division

21 Fuzzy sets II21 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 A i  F, i = 1, 2,... then  i A i  F  - elementary space (space of outcomes - elementary events) F -  -Algebra of events of 

22 Fuzzy sets II22 Probability measure F -  -Algebra of events of  p : F  [0,1] - probability measure on F satisfying: (W1) if A  F then p(A)  0 (W2) p(  ) = 1 (W3)if A i  F, i = 1, 2,..., A i  A j = , i  j then p(  i A i ) =  i p(A i ) -  -additivity (W3*)if A,B  F, A  B= , then p(A  B ) = p(A ) + p(B) - additivity

23 Fuzzy sets II23 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, A  B then p(A)  p(B) - monotonicity (FM4) if A 1, A 2,...  F, A 1  A 2 ... then g(A i ) = g( A i ) - continuity

24 Fuzzy sets II24 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

25 Fuzzy sets II25 Possibility measure P(  ) - Power set of  (st of all subsets of  )  : P(  )  [0,1] - possibility measure on  satisfying: (P1)  (  ) = 0 (P2)  (  ) = 1 (P3) if A i  P(  ), i = 1, 2,... then  (  i A i ) = Sup i {p(A i )} (P3*) if A,B  P(  ), then  (A  B ) = Max{  (A ),  (B)}

26 Fuzzy sets II26 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

27 Fuzzy sets II27 Example 8.  = A  B  C F = { , A, B, C, A  B, B  C, A  C, A  B  C}

28 Fuzzy sets II28 Possibility distribution  - possibility measure on P(  ) Function  :   [0,1] defined by  (x) =  ({x}) for  x  is 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 

29 Fuzzy sets II29 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

30 Fuzzy sets II30 Probability of fuzzy event Finite universe  ={x 1, …,x n } - 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

31 Fuzzy sets II31 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

32 Fuzzy sets II32 Example 9. = (4, 1, 2) LR L(u) = R(u) = e -u - density function of random value = 0,036 - “around 4”

33 Fuzzy sets II33 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

34 Fuzzy sets II34 Fuzzy sets of the 2 nd type The function value of the membership function is again a fuzzy set (FN) of [0,1]

35 Fuzzy sets II35 Example 10.

36 Fuzzy sets II36 Example 11. Linguistic variable “Stature”- Height of the body

37 Fuzzy sets II37 Fuzzy relations X - universe - (binary) fuzzy (valued) relation on X = fuzzy set on X  X is given by the membership function  R : X  X  [0,1] FR is: Reflexive:  R (x,x) = 1  x  X Symmetric:  R (x,y) =  R (y,x)  x,y  X Transitive:Sup z [Min{  R (x,z),  R (z,y)}]   R (x,y) Equivalence: reflexive & symmetric & transitive

38 Fuzzy sets II38 Example 12. Binary fuzzy relation : “x is much greater than y” e.g.  R (8,1) = 7/9 = 0,77… - is antisymmetric: If  R (x,y) > 0 then  R (y,x) = 0  x,y  X

39 Fuzzy sets II39 Example 13. Binary fuzzy relation : “x is similar to y” x/y ,00,50,30,20 20,51,00,60,50,2 30,30,61,00,70,4 40,20,50,71,00,8 500,20,4 1,0 is equivalence ! X = {1,2,3,4,5}

40 Fuzzy sets II40 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 2 nd type Fuzzy relations

41 Fuzzy sets II41 References [1]J. Ramík, M. Vlach: Generalized concavity in fuzzy optimization and decision analysis. Kluwer Academic Publ. Boston, Dordrecht, London, [2] H.-J. Zimmermann: Fuzzy set theory and its applications. Kluwer Academic Publ. Boston, Dordrecht, London, [3] H. Rommelfanger: Fuzzy Decision Support - Systeme. Springer - Verlag, Berlin Heidelberg, New York, [4] H. Rommelfanger, S. Eickemeier: Entscheidungstheorie - Klassische Konzepte und Fuzzy - Erweiterungen, Springer - Verlag, Berlin Heidelberg, New York, 2002.


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