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

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**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

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**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

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**Example 1. Addition of fuzzy numbers**

EP: Fuzzy sets II

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**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

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**Extension principle for functions**

X1, X2,…,Xn, Y - sets n - fuzzy sets on Xi , i = 1,2,…,n g : X1X2 …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

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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

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**Example 2. Fuzzy Min and Max**

Fuzzy sets II

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**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

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**Example 3. L-R fuzzy number “About eight”**

Fuzzy sets II

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

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**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

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**Opposite FN = (m,,)LR - FN of L-R-type**

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

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**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

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Example 5. Subtraction Fuzzy sets II

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**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

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**Example 6. Multiplication**

= (2,1,2)LR , = (4,2,2)LR (8,8,12)LR formula formula ……. exact function Fuzzy sets II

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**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

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**Example 7. Inverse FN = (2,1,2)LR f.2: f.1:**

formula formula ……. exact function Fuzzy sets II

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**Division = (m,,)LR , = (n,,)LR > 0**

where L, R are shape functions Define Combinations of approximate formulae, e.g. Fuzzy sets II

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**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

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**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 = , ij then p(i Ai ) = i p(Ai ) - -additivity (W3*) if A,B F , AB= , then p(AB ) = p(A ) + p(B) - additivity Fuzzy sets II

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**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 , AB then p(A) p(B) - monotonicity (FM4) if A1, A2,... F , A1 A2 ... then g(Ai ) = g( Ai ) continuity Fuzzy sets II

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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

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**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 (AB ) = Max{(A ), (B)} Fuzzy sets II

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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

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**F = {, A, B, C, AB, BC, AC, ABC}**

Example 8. = ABC F = {, A, B, C, AB, BC, AC, ABC} Fuzzy sets II

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**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

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**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

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**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

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**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

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**Example 9. = (4, 1, 2)LR L(u) = R(u) = e-u - “around 4”**

- density function of random value = 0,036 Fuzzy sets II

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**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

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**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

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Example 10. Fuzzy sets II

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**Linguistic variable “Stature”- Height of the body**

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

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**Fuzzy relations X - universe**

- (binary) fuzzy (valued) relation on X = fuzzy set on XX is given by the membership function R : XX [0,1] FR is: Reflexive: R (x,x) = 1 xX Symmetric: R (x,y) = R (y,x) x,yX Transitive: Supz[Min{R (x,z), R (z,y)}] R (x,y) Equivalence: reflexive & symmetric & transitive Fuzzy sets II

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**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,yX Fuzzy sets II

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**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

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**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

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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. Fuzzy sets II

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