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Fuzzy sets II1 Prof. Dr. Jaroslav Ramík

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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

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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.)

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

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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]

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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

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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

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Fuzzy sets II8 Example 2. Fuzzy Min and Max

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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

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

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

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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

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

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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

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Fuzzy sets II15 Example 5. Subtraction

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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.

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

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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 !

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

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

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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

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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

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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

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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

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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)}

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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

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Fuzzy sets II27 Example 8. = A B C F = { , A, B, C, A B, B C, A C, A B C}

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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

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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

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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

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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

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

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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

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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]

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Fuzzy sets II35 Example 10.

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

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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

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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

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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}

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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

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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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