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Fuzzy sets II

Fuzzy sets II. Prof. Dr. Jaroslav Ramík. 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

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Fuzzy sets II

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  1. Fuzzy sets II Prof. Dr. Jaroslav Ramík Fuzzy sets II

  2. 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 2nd type • Fuzzy relations Fuzzy sets II

  3. 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.) Fuzzy sets II

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

  5. 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] Fuzzy sets II

  6. Extension principle for functions • X1, X2,…,Xn, Y - sets • n - fuzzy sets on Xi , i = 1,2,…,n • g : X1X2…Xn Y - function of n variables i.e. (x1,x2,…,xn )  y = g (x1,x2,…,xn ) Then the extended function is defined by Fuzzy sets II

  7. Remarks • g-1(y) = {(x1,x2,…,xn ) | y = g (x1,x2,…,xn )} - co-image of y • Special form of EP: g (x1,x2) = x1+x2 or g (x1,x2) = x1*x2 • Instead of Min any t-norm T can be used - more general for of EP Fuzzy sets II

  8. Example 2. Fuzzy Min and Max Fuzzy sets II

  9. 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 Fuzzy sets II

  10. Example 3. L-R fuzzy number “Abouteight” Fuzzy sets II

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

  12. Theorem 2. Addition 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 Fuzzy sets II

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

  14. Theorem 3. Subtraction 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 Fuzzy sets II

  15. Example 5. Subtraction Fuzzy sets II

  16. Theorem 4. Multiplication 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 1. 2. Fuzzy sets II

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

  18. 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 ! Fuzzy sets II

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

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

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

  22. 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 Ai  F , i = 1, 2, ..., Ai Aj = , ij then p(iAi ) = i p(Ai ) - -additivity (W3*) if A,B F , AB= , then p(AB) = p(A) + p(B) - additivity Fuzzy sets II

  23. 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Bthen p(A)  p(B) - monotonicity (FM4) if A1, A2,...  F , A1 A2  ... then g(Ai ) = g( Ai ) - continuity Fuzzy sets II

  24. 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 Fuzzy sets II

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

  26. 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 Fuzzy sets II

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

  28. 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  Fuzzy sets II

  29. 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 Fuzzy sets II

  30. Probability of fuzzy eventFinite universe  ={x1, …,xn} - 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 Fuzzy sets II

  31. Probability of fuzzy eventReal 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 Fuzzy sets II

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

  33. 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 Fuzzy sets II

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

  35. Example 10. Fuzzy sets II

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

  37. 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: Supz[Min{R (x,z), R (z,y)}]  R (x,y) • Equivalence: reflexive & symmetric & transitive Fuzzy sets II

  38. 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 Fuzzy sets II

  39. x/y 1 2 3 4 5 1 1,0 0,5 0,3 0,2 0 2 0,5 1,0 0,6 0,5 0,2 3 0,3 0,6 1,0 0,7 0,4 4 0,2 0,5 0,7 1,0 0,8 5 0 0,2 0,4 0,4 1,0 Example 13. Binary fuzzy relation : “x is similar to y” X = {1,2,3,4,5} is equivalence ! Fuzzy sets II

  40. 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 2nd type • Fuzzy relations Fuzzy sets II

  41. 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. Fuzzy sets II

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