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

ExtensionsExtensions 11

ExtensionsExtensions

DefinitionsDefinitions of fuzzy sets of fuzzy setsOperationsOperations with fuzzy sets with fuzzy sets


Types of fuzzy setsTypes of fuzzy sets

Interval-valued fuzzy setInterval-valued fuzzy setType two fuzzy setType two fuzzy setType m fuzzy setType m fuzzy setL-fuzzy setL-fuzzy set


Interval-value fuzzy setInterval-value fuzzy set11

A A membership functionmembership function based on the latter based on the latter approach does not assign to each element of approach does not assign to each element of the universal set one real number, but the universal set one real number, but a a closed interval of real numbers between the closed interval of real numbers between the identified lower and upper boundsidentified lower and upper bounds..

1,0:~ XxA

where ([0,1])([0,1]) denotes the family of all closed intervals of real numbers in closed intervals of real numbers in [0,1].[0,1].


Interval-value fuzzy setInterval-value fuzzy set22


Type 2 fuzzy setType 2 fuzzy set11

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


Type 2 fuzzy setType 2 fuzzy set22


Type m fuzzy setType m fuzzy set

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


L-fuzzy setL-fuzzy set

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

LXxA

:~


集合之二元關係集合之二元關係給定集合給定集合 AA 及集合及集合 BB ，，直積直積 AABB 的每個的每個子集子集 RR 都叫都叫從從 AA 到到 BB 的關係的關係，當，當 (a,b)(a,b)RR時，稱時，稱 a,ba,b 適合關係適合關係 RR ，記作，記作 aRbaRb 。。

設非空的設非空的 RR 是集合是集合 AA 上的二元關係上的二元關係每個每個 aaAA都有都有 aRaaRa，則稱其具備，則稱其具備自反性自反性。。若由若由 aRbaRb必可推出必可推出 bRabRa，則稱其具備，則稱其具備對稱性對稱性。。若由若由 aRbaRb及及 bRabRa必可推出必可推出 a=ba=b，則稱其具備，則稱其具備反對稱性反對稱性。。

若由若由 aRbaRb及及 bRcbRc必可推出必可推出 aRcaRc，則稱其具備，則稱其具備傳遞性傳遞性。。


偏序偏序 (partially ordered)(partially ordered) 集集集合上具備集合上具備自反性自反性、、反對稱性反對稱性及及傳遞性傳遞性的的關係叫做關係叫做偏序關係偏序關係。。

具具偏序關係偏序關係之集合稱為之集合稱為偏序集偏序集。。


Lattice (Lattice ( 格格 ))

若非空之若非空之偏序集偏序集 LL 的任何個元素構成的集的任何個元素構成的集合，既有合，既有上確界上確界也有也有下確界下確界，則偏序集，則偏序集 (L(L ，，≤≤ ))叫做叫做格格。。


格的主要性質格的主要性質 11

設設 (L(L ，，≤≤ ))是格如果在是格如果在 LL上規定二元運算上規定二元運算∨∨及及∧，∧， a b=sup{a∨a b=sup{a∨ ，， b} b} ，， a b=inf{a∧a b=inf{a∧ ，，b} b} ，則此二元運算滿足如下算律。，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∧


格的主要性質格的主要性質 22

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∨ ∧


Operations on fuzzy setsOperations on fuzzy sets

Fuzzy complementFuzzy complementFuzzy intersection (t-norms)Fuzzy intersection (t-norms)Fuzzy union (t-conorms)Fuzzy union (t-conorms)Aggregation operationsAggregation operations


Fuzzy complementsFuzzy complements11

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 )



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 )



ExampleExample

tafor 0tafor 1ac Satisfy c1, c2

aac cos121 Satisfy c1,c2,c3

,1

11 aa

ac Satisfy c1~c4


Equilibrium of a fuzzy Equilibrium of a fuzzy complement ccomplement c

Any value a for which c(a)=aAny value a for which c(a)=aTheorem 1Theorem 1: Every fuzzy complement has : Every fuzzy complement has aa

t most onet most one equilibrium equilibriumTheorem 2Theorem 2: Assume that a given fuzzy co: Assume that a given fuzzy co

mplement mplement cc has an equilibrium e has an equilibrium ecc, which b, which by theorem 1 is unique. Then y theorem 1 is unique. Then aa≤c(a) iff a≤c(a) iff a≤ec and aa≥c(a) iff a≥c(a) iff a≥≥ec

Theorem 3Theorem 3: If : If cc is a is a continuouscontinuous fuzzy com fuzzy complement, then plement, then cc has a has a unique unique equilibrium.equilibrium.


Fuzzy union (t-conorms) / InterseFuzzy union (t-conorms) / Intersection (t-norms)ction (t-norms)

Union u:[0,1]Union u:[0,1][0,1][0,1]→[0,1]→[0,1] Intersection i:Intersection i:[0,1][0,1][0,1][0,1]→[0,1]→[0,1]

xxux BABA ~~~~ ,

xxix BABA ~~~~ ,


Axiomatic skeletonAxiomatic skeleton(t-conorms / t-norms)(t-conorms / t-norms)11

Axiom u1/i1Axiom u1/i1 (Boundary conditions) (Boundary conditions)u(0,0)=0; u(0,1)=u(1,0)=u(1,1)=1u(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)=0i(1,1)=1; i(0,1)=i(1,0)=i(0,0)=0

Axiom u2/i2Axiom u2/i2 (Commutative) (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)


Axiomatic skeletonAxiomatic skeleton(t-conorms / t-norms)(t-conorms / t-norms)22

Axiom u3/i3Axiom u3/i3 (monotonic) If a (monotonic) If a≤a’, b≤b’U(a,b) U(a,b) ≤u(a’,b’)i(a,b) ≤i(a’,b’)

Axiom u4/i4Axiom u4/i4 (associative) (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))


Additional requirementsAdditional requirements(t-conorms / t-norms)(t-conorms / t-norms)11

Axiom u5/i5Axiom u5/i5 (continuous) (continuous)u/i u/i is a continuous functionis a continuous function

Axiom u6/i6Axiom u6/i6 (idempotent)( (idempotent)( 冪等冪等 ))u(a,a)=i(a,a)=au(a,a)=i(a,a)=a


Additional requirementsAdditional requirements(t-conorms / t-norms)(t-conorms / t-norms)22

Axiom u7/i7Axiom u7/i7 (subidempotency) (subidempotency)u(a,a)>au(a,a)>a i(a,a)<ai(a,a)<a

Axiom u8/i8Axiom u8/i8 (strict monotonicity) (strict monotonicity)aa11<a<a22 and b and b11<b<b22 implies u(a implies u(a11,b,b11)<u(a)<u(a22,b,b22))aa11<a<a22 and b and b11<b<b22 implies i(a1,b1)<i(a2,b2) implies i(a1,b1)<i(a2,b2)

Theorem 4Theorem 4The The standard fuzzy union/intersectionstandard fuzzy union/intersection is th is th

e e onlyonly idempotent and continuous t-conoridempotent and continuous t-conorm/t-norm m/t-norm (i.e., the only function that satisfi(i.e., the only function that satisfies Axiom u1/i1~u6/i6)es Axiom u1/i1~u6/i6)


Some t-conorms/t-normsSome t-conorms/t-norms(Drastic union/intersection(Drastic union/intersection11))

otherwise. 1

0,a when 0,b when

,max ba

bau

otherwise. 0

,1a when ,1b when

,min ba

bai


Some t-conorms/t-normsSome t-conorms/t-norms22

nAA ~...~1

Cartesian Product of fuzzy setCartesian Product of fuzzy set

i i1 2 n

1 n i iA xA A ... Ai

x | x x ...x , x Xmin

nAA ~...~1

mth power of a fuzzy set A~

Xxxx m

AAm ,~~



nAA ~...~1

Algebraic sumAlgebraic sum

XxxxCBA

|,~~~

BAC ~~~

where

xxxxx BABABA ~~~~~~



nAA ~...~1

Bounded sumBounded sum

XxxxC BA |,~~~

BAC ~~~

where

xxx BABA ~~~~ ,1min



nAA ~...~1

Bounded differenceBounded difference

XxxxC BA |,~~~

BAC ~~~

where

1,0max ~~~~ xxx BABA



nAA ~...~1

Algebraic productAlgebraic product

XxxxCBA

|,~~~

BAC ~~~

where

xxx BABA ~~~~


ExampleExample

Let Let

6.0.7,1,5,5.0,3~ xA

6.0,5,1,3~ xB


Some theorem of t-conorms/t-noSome theorem of t-conorms/t-normsrms

Theorem 5Theorem 5For all a,b For all a,b [0,1], u(a,b) ≥max(a,b)≥max(a,b)

Theorem 6Theorem 6For all a,b For all a,b [0,1], u(a,b) ≤umaxmax(a,b)(a,b)

Theorem 7Theorem 7For all a,b For all a,b [0,1], i(a,b) ≤min(a,b)min(a,b)

Theorem 8Theorem 8For all a,b For all a,b [0,1], i(a,b) ≥i≥iminmin(a,b)(a,b)


Aggregation operationsAggregation operations

DefinitionDefinitionOperations by which Operations by which several fuzzy sets several fuzzy sets

are combined to produce a single setare combined to produce a single set..

h:[0,1]n→[0,1] for n≥2

xxxhxnAAAA ~~~~ ...,,

21


Axiomatic requirementsAxiomatic requirements

Axiom h1Axiom h1h(0,0,…,0)=0 and h(1,1,…,1)=1 (boundary ch(0,0,…,0)=0 and h(1,1,…,1)=1 (boundary c

onditions)onditions)Axiom h2Axiom h2

For any pair (aFor any pair (aii,b,bii), a), aii,b,bii [0,1] if a[0,1] if aii≥b≥bii i, thei, the

n h(an h(aii) ) ≥h(b≥h(bii) (monotonic increasing)) (monotonic increasing)

Axiom h3Axiom h3h is a continuous function.h is a continuous function.


Additional axiomaticAdditional axiomatic requirementsrequirements

Axiom h4Axiom h4h is a symmetric function in all its argumeh is a symmetric function in all its argume

nts; that is, h(ants; that is, h(a11,a,a22,…,a,…,ann)=h(a)=h(ap(1)p(1),a,ap(2)p(2),…,a,…,ap(n)p(n)) )

for any permutation p on Nfor any permutation p on Nnn..

Axiom h5Axiom h5H is an idempotent function; that is, h(a,a,H is an idempotent function; that is, h(a,a,

…,a)=a…,a)=a


Averaging operationsAveraging operations

Any aggregation operation Any aggregation operation hh that that satisfies satisfies Axioms h2Axioms h2 and and h5h5 satisfies also satisfies also the inequalities:the inequalities: min(a1,a2,…,an)min(a1,a2,…,an)≤h(a1,a2,…,an) ≤h(a1,a2,…,an) ≤max(a1,a2,…,an)≤max(a1,a2,…,an)

All aggregations between the standard All aggregations between the standard fuzzy intersection and the standard fuzzy fuzzy intersection and the standard fuzzy union are idempotent.union are idempotent.

These aggregation operations are These aggregation operations are usually called usually called averaging operationsaveraging operations..


Example for averaging operationExample for averaging operation

Generalized means (satisfies h1 through h4)Generalized means (satisfies h1 through h4)

1

2121

...,...,,

naaa

aaah nn

α:parameter, αR (R (αα≠≠0)0)


The full scope of fuzzy aggregation The full scope of fuzzy aggregation operationsoperations


Criteria for selecting appropriate Criteria for selecting appropriate aggregation operatorsaggregation operators

Axiomatic strengthAxiomatic strengthEmpirical fitEmpirical fitAdaptabilityAdaptabilityNumerical efficiencyNumerical efficiencyCompensationCompensationRange of compensationRange of compensationAggregating behaviorAggregating behaviorRequired scale level of membership Required scale level of membership

functionsfunctions

Documents

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