Fuzzy sets I Prof. Dr. Jaroslav Ramík Fuzzy sets I

Content Basic definitions Examples Operations with fuzzy sets (FS) t-norms and t-conorms Aggregation operators Extended operations with FS Fuzzy numbers: Convex fuzzy set, fuzzy interval, fuzzy number (FN), triangular FN, trapezoidal FN, L-R fuzzy numbers Fuzzy sets I

Basic definitions Set - a collection well understood and distinguishable objects of our concept or our thinking about the collection. Fuzzy set - a collection of objects in connection with expression of uncertainty of the property characterizing the objects by grades from interval between 0 and 1. Fuzzy sets I

Fuzzy set X - universe (of discourse) = set of objects A : X  [0,1] - membership function = {(x, A(x))| x X} - fuzzy set of X (FS) Fuzzy sets I

Examples Feasible daily car production Young man age Number around 8 Fuzzy sets I

Example1. “Feasible car production per day” X = {3, 4, 5, 6, 7, 8, 9} - universe = {(3; 0), (4; 0), (5; 0,1), (6; 0,5), (7; 1), (8; 0,8), (9; 0)} Fuzzy sets I

Example 2. “Young man age” X = [0, 100] - universe (interval) Approximation of empirical evaluations (points): 20 respondents have been asked to evaluate the membership grade Fuzzy sets I

Example 3.“Number around eight” X = (0, +) - universe (interval) Fuzzy sets I

Crisp set Crisp set A of X = fuzzy set with a special membership function: A : X  {0,1} - characteristic function Crisp set can be uniquely identified with a set: (non-fuzzy) set A is in fact a (fuzzy) crisp set Fuzzy sets I

Support, height, normal fuzzy set Support of fuzzy set , supp( ) = {xX| A(x) > 0} support is a set (crisp set)! Height of fuzzy set , hgt( ) = Sup{A(x) | xX } Fuzzy set is normal (normalized), if there exists x0X with A(x0) = 1 Ex.: Support of from Example 1: supp( ) = {5, 6, 7, 8} hgt( ) = A(8) = 1  is normal! Fuzzy sets I

-cut (- level set)  [0,1], - fuzzy set, A = {x X|A(x)} - -cut of - convex FS, if A is convex set (interval) for all  [0,1] !!! Fuzzy sets I

Operations with fuzzy sets (X) -Fuzzy power set = set of all fuzzy sets of X (X)  A(x) = B(x) for all x X - identity  A(x)  B(x) for all x X - inclusion - transitivity Fuzzy sets I

Union and Intersection of fuzzy sets (X)  AB(x) =Max{A(x), B(x)} - union  AB(x) =Min{A(x), B(x)}- intersection Properties: Commutativity, Associativity, Distributivity,… Fuzzy sets I

Example 4. = {(3; 0), (4; 0), (5; 0,1), (6; 0,5), (7; 1), (8; 0,8), (9; 0)} = {(3; 1), (4; 1), (5; 0,9), (6; 0,8), (7; 0,4), (8; 0,1), (9; 0)} Fuzzy sets I

Complement, Cartesian product (X)  CA(x) =1 - A(x) - complement of (X) , (Y)  AB(x,y) =Min{A(x), B(y)} - Cartesian product (CP) CP is a fuzzy set of XY ! Extension to more parts possible e.g. X, Y, Z,… Fuzzy sets I

Complementarity conditions (X) 1.  =  2.  = X Min and Max do not satisfy 1., 2. ! (only for crisp sets) …later on …”bold” intersection and union will satisfy the complementarity… Fuzzy sets I

Examples Fuzzy sets I

Extended operations with FS Intersection  and Union  = operations on (X) Realization by Min and Max operators generalized by t-norms and t-conorms Fuzzy sets I

t-norms A function T: [0,1]  [0,1]  [0,1] is called t-norm if it satisfies the following properties (axioms): T1: T(a,1) = a a  [0,1] - “1” is a neutral element T2: T(a,b) = T(b,a) a,b  [0,1] - commutativity T3: T(a,T(b,c)) = T(T(a,b),c) a,b,c  [0,1] - associativity T4: T(a,b)  T(c,d) whenever a  c , b  d - monotnicity Fuzzy sets I

t-conorms A function S: [0,1]  [0,1]  [0,1] is called t-conorm if it satisfies the following axioms: S1: S(a,0) = a a  [0,1] - “0” is a neutral element S2: S(a,b) = S(b,a) a,b  [0,1] - commutativity S3: S(a,S(b,c)) = S(S(a,b),c) a,b,c  [0,1] - associativity S4: S(a,b)  S(c,d) whenever a  c , b  d - monotnicity Fuzzy sets I

Examples of t-norms and t-conorms #1 1. TM = Min, SM = Max - minimum and maximum 2. - drastic product, drastic sum Property: TW(a,b)  T(a,b)  TM(a,b) , SM(a,b)  S(a,b)  SW(a,b) for every t-norm T, resp. t-conorm S, and a,b  [0,1] Fuzzy sets I

Examples of t-norms and t-conorms #2 3. TP(a,b) = a.b SP (a,b) = a+b - a.b - product and probabilistic sum 4. TL(a,b) = Max{0,a+b - 1} SL (a,b) = Min{1,a+b} - Lukasiewicz t-norm and t-conorm (satisfies complematarity!) (bounded difference, bounded sum) Also: b - bold intersection, b - bold union Property: T*(a,b) = 1 - T(1-a,1-b) , S*(a,b) = 1 - S(1-a,1-b) If T is a t-norm then T* is a t-conorm ( T and T* are dual ) If S is a t-conorm then S* is a t-norm ( S and S* are dual ) Fuzzy sets I

Examples of t-norms and t-conorms #3 5. q [1,+) a,b  [0,1] Yager’s t-norm and t-conorm 6. Einstein, Hamacher, Dubois-Prade product and sum etc. Property: If q =1, then Tq, (Sq) is Lukasiewicz t-norm (t-conorm) If q = +, then Tq, (Sq) is Min (Max) Fuzzy sets I

Extended Union and Intersection of fuzzy sets (X), T - t-norm, S - t-conorm  AsB(x) =S(A(x), B(x)) - S-union  ATB(x) =T(A(x), B(x)) -T-intersection Properties: Commutativity, Associativity?,… Fuzzy sets I

Aggregation operators A function G: [0,1]  [0,1]  [0,1] is called aggregation operator if it satisfies the following properties (axioms): A1: G(0,0) = 0 - boundary condition 1 A2: G(1,1) = 1 - boundary condition 2 A3: G(a,b)  G(c,d) whenever a  c , b  d - monotnicity NO commutativity or associativity conditions! All t-norms and t-conorms are aggregation operators! May be extended to more parts, e.g. a,b,c,… Fuzzy sets I

Compensative operators (CO) #1 CO = Aggregation operator G satisfying Min(a,b)  G(a,b)  Max(a,b) Example 1. Averages: 1: G(a,b) = (a +b)/2 - arithmetic mean (average) 2: G(a,b) = - geometric mean 3: G(a,b) = - harmonic mean S Max G Min T Extension to more elements possible! Fuzzy sets I

Compensative operators #2 Examples. Compensatory operators: 1: TW(a,b) = .Min(a,b) + (1- ) - fuzzy „and“ SW(a,b) = .Max(a,b) + (1- ) - fuzzy „or“ (by Werners) 2: ATS(a,b) = .T(a,b) + (1 - ).S(a,b) - COs by PTS(a,b) =T(a,b) . S(a,b)1- Zimmermann and Zysno T - t-norm, S - t-conorm,   [0,1] - compensative parameter CO compensate trade-offs between conflicting evaluations extension to more elements possible Fuzzy sets I

Fuzzy numbers - fuzzy set of R (real numbers) - convex - normal (there exists x0  R with A(x0) = 1) - A is closed interval for all  [0,1] Then is called fuzzy interval Moreover if there exists only one x0  R with A(x0) = 1 then is called fuzzy number Fuzzy sets I

Positive and negative fuzzy numbers - fuzzy number is - positive if A(x) = 0 for all x  0 - negative if A(x) = 0 for all x  0 Fuzzy sets I

Example 5. Fuzzy number “About three” Fuzzy sets I

Example 6. Triangular fuzzy number “About three” spread mean value Fuzzy sets I

L-R fuzzy intervals L, R : [0,+)  [0,1] - non-increasing, non-constant functions - shape functions L(0) = R(0) = 1, m, n,  > 0,  > 0 - real numbers - fuzzy interval of L-R-type if fuzzy number of L-R-type if m = n, L, R - decreasing functions Fuzzy sets I

Example 7. L-R fuzzy number “Around eight” Fuzzy sets I

Example 8. L-R fuzzy number “About eight” Fuzzy sets I

Example 9. L-R fuzzy interval Fuzzy sets I

Example 10. Fuzzy intervals Fuzzy sets I

Summary Basic definitions: set, fuzzy set, membership function, crisp set, support, height, normal fuzzy set, -level set Examples: daily production, young man age, around 8 Operations with fuzzy sets: fuzzy power set, union, intersection, complement, cartesian product Extended operations with fuzzy sets: t-norms and t-conorms, compensative operators Fuzzy numbers: Convex fuzzy set, fuzzy interval, fuzzy number (FN), triangular FN, trapezoidal FN, L-R fuzzy numbers Fuzzy sets I

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 I