1. Extensions1 Extensions Definitions of fuzzy sets Definitions of fuzzy sets Operations with fuzzy sets Operations with fuzzy sets

2. Extensions2 Types of fuzzy sets Interval-valued fuzzy set Interval-valued fuzzy set Type two fuzzy set Type two fuzzy set Type m fuzzy set Type m fuzzy set L-fuzzy set L-fuzzy set

3. Extensions3 Interval-value fuzzy set 1 A membership function based on the latter approach does not assign to each element of the universal set one real number, but a closed interval of real numbers between the identified lower and upper bounds. A membership function based on the latter approach does not assign to each element of the universal set one real number, but a closed interval of real numbers between the identified lower and upper bounds.  ([0,1]) closed intervals of real numbers in [0,1]. where  ([0,1]) denotes the family of all closed intervals of real numbers in [0,1].

4. Extensions4 Interval-value fuzzy set 2

5. Extensions5 Type 2 fuzzy set 1 A fuzzy set whose membership values are type 1 fuzzy set on [0,1]. (fuzzy set whose membership function itself is a fuzzy set) A fuzzy set whose membership values are type 1 fuzzy set on [0,1]. (fuzzy set whose membership function itself is a fuzzy set)

6. Extensions6 Type 2 fuzzy set 2

7. Extensions7 Type m fuzzy set A fuzzy set in X whose membership values are type m-1 (m>1) fuzzy sets on [0,1]. A fuzzy set in X whose membership values are type m-1 (m>1) fuzzy sets on [0,1].

8. Extensions8 L-fuzzy set The membership function of an L-fuzzy set maps into a partially ordered set, L. The membership function of an L-fuzzy set maps into a partially ordered set, L.

9. Extensions9 集合之二元關係 給定集合 A 及集合 B ，直積 A  B 的每個子集 R 都叫從 A 到 B 的關係，當 (a,b)  R 時，稱 a,b 適合關係 R ，記作 aRb 。 給定集合 A 及集合 B ，直積 A  B 的每個子集 R 都叫從 A 到 B 的關係，當 (a,b)  R 時，稱 a,b 適合關係 R ，記作 aRb 。 設非空的 R 是集合 A 上的二元關係 設非空的 R 是集合 A 上的二元關係 每個 a  A 都有 aRa ，則稱其具備自反性。 每個 a  A 都有 aRa ，則稱其具備自反性。 若由 aRb 必可推出 bRa ，則稱其具備對稱性。 若由 aRb 必可推出 bRa ，則稱其具備對稱性。 若由 aRb 及 bRa 必可推出 a=b ，則稱其具備反 對稱性。 若由 aRb 及 bRa 必可推出 a=b ，則稱其具備反 對稱性。 若由 aRb 及 bRc 必可推出 aRc ，則稱其具備傳 遞性。 若由 aRb 及 bRc 必可推出 aRc ，則稱其具備傳 遞性。

10. Extensions10 偏序 (partially ordered) 集 集合上具備自反性、反對稱性及傳遞性的 關係叫做偏序關係。 集合上具備自反性、反對稱性及傳遞性的 關係叫做偏序關係。 具偏序關係之集合稱為偏序集。 具偏序關係之集合稱為偏序集。

11. Extensions11 Lattice ( 格 ) 若非空之偏序集 L 的任何個元素構成的集合， 既有上確界也有下確界，則偏序集 (L ， ≤) 叫 做格。 若非空之偏序集 L 的任何個元素構成的集合， 既有上確界也有下確界，則偏序集 (L ， ≤) 叫 做格。

12. Extensions12 格的主要性質 1 設 (L ， ≤) 是格如果在 L 上規定二元運算∨及∧， a ∨ b=sup{a ， b} ， a ∧ b=inf{a ， b} ，則此 二元運算滿足如下算律。 設 (L ， ≤) 是格如果在 L 上規定二元運算∨及∧， a ∨ b=sup{a ， b} ， a ∧ b=inf{a ， b} ，則此 二元運算滿足如下算律。 冪等律 冪等律 a ∨ a=a a ∨ a=a a ∧ a=a a ∧ a=a

13. Extensions13 格的主要性質 2 交換律 交換律 a ∨ b=b ∨ a a ∨ b=b ∨ a a ∧ b=b ∧ a a ∧ b=b ∧ a 結合律 結合律 (a ∨ b) ∨ c=a ∨ (b ∨ c) (a ∨ b) ∨ c=a ∨ (b ∨ c) (a ∧ b) ∧ c=a ∧ (b ∧ c) (a ∧ b) ∧ c=a ∧ (b ∧ c) 吸收律 吸收律 a ∧ (a ∨ b)=a a ∧ (a ∨ b)=a a ∨ (a ∧ b)=a a ∨ (a ∧ b)=a

14. Extensions14 Operations on fuzzy sets Fuzzy complement Fuzzy complement Fuzzy intersection (t-norms) Fuzzy intersection (t-norms) Fuzzy union (t-conorms) Fuzzy union (t-conorms) Aggregation operations Aggregation operations

15. Extensions15 Fuzzy complements 1 C:[0,1]→[0,1] To produce meaningful fuzzy complements, function c must satisfy at least the following two axiomatic requirements: (axiomatic skeleton for fuzzy complements) Axiom c1.  c(0)=1 and c(1)=0 (boundary conditions)( 邊界條件 ) Axiom c2.  For all a,b  [0.1], if a≤b, then c(a)≥c(b) (monotonicity)( 單調 )

16. Extensions16 Fuzzy complements 2 Two of the most desirable requirements: Axiom c3.  c is a continuous function.( 連續 ) Axiom c4.  c is involutive, which means that c(c(a))=a for each a  [0,1]( 復原 )

17. Extensions17 Fuzzy complements 3 Example Satisfy c1, c2 Satisfy c1,c2,c3 Satisfy c1~c4

18. Extensions18 Equilibrium of a fuzzy complement c Any value a for which c(a)=a Any value a for which c(a)=a Theorem 1: Every fuzzy complement has at most one equilibrium Theorem 1: Every fuzzy complement has at most one equilibrium Theorem 2: Assume that a given fuzzy complement c has an equilibrium e c, which by theorem 1 is unique. Then a≤c(a) iff aa≥c(a) iff a≥ Theorem 2: Assume that a given fuzzy complement c has an equilibrium e c, which by theorem 1 is unique. Then a≤c(a) iff a≤e c and a≥c(a) iff a≥e c Theorem 3: If c is a continuous fuzzy complement, then c has a unique equilibrium. Theorem 3: If c is a continuous fuzzy complement, then c has a unique equilibrium.

19. Extensions19 Fuzzy union (t-conorms) / Intersection (t-norms) Union u:[0,1]  [0,1]→[0,1] Union u:[0,1]  [0,1]→[0,1] Intersection i:[0,1]  [0,1]→[0,1] Intersection i:[0,1]  [0,1]→[0,1]

20. Extensions20 Axiomatic skeleton (t-conorms / t-norms) 1 Axiom u1/i1 (Boundary conditions) Axiom u1/i1 (Boundary conditions) u(0,0)=0; u(0,1)=u(1,0)=u(1,1)=1 u(0,0)=0; u(0,1)=u(1,0)=u(1,1)=1 i(1,1)=1; i(0,1)=i(1,0)=i(0,0)=0 i(1,1)=1; i(0,1)=i(1,0)=i(0,0)=0 Axiom u2/i2 (Commutative) Axiom u2/i2 (Commutative) u(a,b)=u(b,a) u(a,b)=u(b,a) i(a,b)=i(b,a) i(a,b)=i(b,a)

21. Extensions21 Axiomatic skeleton (t-conorms / t-norms) 2 Axiom u3/i3 (monotonic) If a Axiom u3/i3 (monotonic) If a≤a’, b≤b’ U(a,b) U(a,b) ≤u(a’,b’) i(a,b) ≤i(a’,b’) Axiom u4/i4 (associative) Axiom u4/i4 (associative) u(u(a,b),c)=u(a,u(b,c)) u(u(a,b),c)=u(a,u(b,c)) i(i(a,b),c)=i(a,i(b,c)) i(i(a,b),c)=i(a,i(b,c))

22. Extensions22 Additional requirements (t-conorms / t-norms) 1 Axiom u5/i5 (continuous) Axiom u5/i5 (continuous) u/i is a continuous function u/i is a continuous function Axiom u6/i6 (idempotent)( 冪等 ) Axiom u6/i6 (idempotent)( 冪等 ) u(a,a)=i(a,a)=a u(a,a)=i(a,a)=a

23. Extensions23 Additional requirements (t-conorms / t-norms) 2 Axiom u7/i7 (subidempotency) Axiom u7/i7 (subidempotency) u(a,a)>a u(a,a)>a i(a,a)<a i(a,a)<a Axiom u8/i8 (strict monotonicity) Axiom u8/i8 (strict monotonicity) a 1 <a 2 and b 1 <b 2 implies u(a 1,b 1 )<u(a 2,b 2 ) a 1 <a 2 and b 1 <b 2 implies u(a 1,b 1 )<u(a 2,b 2 ) a 1 <a 2 and b 1 <b 2 implies i(a1,b1)<i(a2,b2) a 1 <a 2 and b 1 <b 2 implies i(a1,b1)<i(a2,b2) Theorem 4 Theorem 4 The standard fuzzy union/intersection is the only idempotent and continuous t- conorm/t-norm (i.e., the only function that satisfies Axiom u1/i1~u6/i6) The standard fuzzy union/intersection is the only idempotent and continuous t- conorm/t-norm (i.e., the only function that satisfies Axiom u1/i1~u6/i6)

24. Extensions24 Some t-conorms/t-norms (Drastic union/intersection 1 )

25. Extensions25 Some t-conorms/t-norms 2 Cartesian Product of fuzzy set Cartesian Product of fuzzy set mth power of a fuzzy set

26. Extensions26 Some t-conorms/t-norms 3 Algebraic sum Algebraic sum where

27. Extensions27 Some t-conorms/t-norms 4 Bounded sum Bounded sum where

28. Extensions28 Some t-conorms/t-norms 5 Bounded difference Bounded difference where

29. Extensions29 Some t-conorms/t-norms 6 Algebraic product Algebraic product where

30. Extensions30 Example Let Let

31. Extensions31 Some theorem of t-conorms/t- norms Theorem 5 Theorem 5 For all a,b  ≥max(a,b) For all a,b  [0,1], u(a,b) ≥max(a,b) Theorem 6 Theorem 6 For all a,b  max (a,b) For all a,b  [0,1], u(a,b) ≤u max (a,b) Theorem 7 Theorem 7 For all a,b  min(a,b) For all a,b  [0,1], i(a,b) ≤min(a,b) Theorem 8 Theorem 8 For all a,b  ≥i min (a,b) For all a,b  [0,1], i(a,b) ≥i min (a,b)

32. Extensions32 Aggregation operations Definition Definition Operations by which several fuzzy sets are combined to produce a single set. Operations by which several fuzzy sets are combined to produce a single set. h:[0,1] n →[0,1] for n≥2

33. Extensions33 Axiomatic requirements Axiom h1 Axiom h1 h(0,0,…,0)=0 and h(1,1,…,1)=1 (boundary conditions) h(0,0,…,0)=0 and h(1,1,…,1)=1 (boundary conditions) Axiom h2 Axiom h2 For any pair (a i,b i ), a i,b i  [0,1] if a i ≥b i  i, then h(a i ) ≥h(b i ) (monotonic increasing) For any pair (a i,b i ), a i,b i  [0,1] if a i ≥b i  i, then h(a i ) ≥h(b i ) (monotonic increasing) Axiom h3 Axiom h3 h is a continuous function. h is a continuous function.

34. Extensions34 Additional axiomatic requirements Axiom h4 Axiom h4 h is a symmetric function in all its arguments; that is, h(a 1,a 2,…,a n )=h(a p(1),a p(2),…,a p(n) ) for any permutation p on N n. h is a symmetric function in all its arguments; that is, h(a 1,a 2,…,a n )=h(a p(1),a p(2),…,a p(n) ) for any permutation p on N n. Axiom h5 Axiom h5 H is an idempotent function; that is, h(a,a,…,a)=a H is an idempotent function; that is, h(a,a,…,a)=a

35. Extensions35 Averaging operations Any aggregation operation h that satisfies Axioms h2 and h5 satisfies also the inequalities: Any aggregation operation h that satisfies Axioms h2 and h5 satisfies also the inequalities: min(a1,a2,…,an)≤h(a1,a2,…,an) ≤max(a1,a2,…,an) min(a1,a2,…,an)≤h(a1,a2,…,an) ≤max(a1,a2,…,an) All aggregations between the standard fuzzy intersection and the standard fuzzy union are idempotent. All aggregations between the standard fuzzy intersection and the standard fuzzy union are idempotent. These aggregation operations are usually called averaging operations. These aggregation operations are usually called averaging operations.

36. Extensions36 Example for averaging operation Generalized means (satisfies h1 through h4) Generalized means (satisfies h1 through h4)  R (α≠0) α:parameter, α  R (α≠0)

37. Extensions37 The full scope of fuzzy aggregation operations

38. Extensions38 Criteria for selecting appropriate aggregation operators Axiomatic strength Axiomatic strength Empirical fit Empirical fit Adaptability Adaptability Numerical efficiency Numerical efficiency Compensation Compensation Range of compensation Range of compensation Aggregating behavior Aggregating behavior Required scale level of membership functions Required scale level of membership functions