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

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

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

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

the operation  denotes + or · (add or multiply) 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

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

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

Example 2. Fuzzy Min and Max Fuzzy sets II

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

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

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

Addition 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 Fuzzy sets II

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

Subtraction 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 Fuzzy sets II

Example 5. Subtraction Fuzzy sets II

Multiplication 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 1. 2. Fuzzy sets II

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

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

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

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

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 i Ai  F  - elementary space (space of outcomes - elementary events) F - -Algebra of events of  Fuzzy sets II

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(i Ai ) = i p(Ai ) - -additivity (W3*) if A,B  F , AB= , then p(AB ) = p(A ) + p(B) - additivity Fuzzy sets II

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 A1, A2,...  F , A1 A2  ... then g(Ai ) = g( Ai ) - continuity Fuzzy sets II

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

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 (i Ai ) = Supi {p(Ai )} (P3*) if A,B  P() , then (AB ) = Max{(A ), (B)} Fuzzy sets II

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

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

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

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

Probability of fuzzy event Finite 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

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 Fuzzy sets II

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

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

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

Example 10. Fuzzy sets II

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

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

Binary fuzzy relation : “x is much greater than y” 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

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

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

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