extensions1 extensions definitions of fuzzy sets definitions of fuzzy sets operations with fuzzy...
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ExtensionsExtensions 11
ExtensionsExtensions
DefinitionsDefinitions of fuzzy sets of fuzzy setsOperationsOperations with fuzzy sets with fuzzy sets
ExtensionsExtensions 22
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
ExtensionsExtensions 33
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].
ExtensionsExtensions 44
Interval-value fuzzy setInterval-value fuzzy set22
ExtensionsExtensions 55
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)
ExtensionsExtensions 66
Type 2 fuzzy setType 2 fuzzy set22
ExtensionsExtensions 77
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].
ExtensionsExtensions 88
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
:~
ExtensionsExtensions 99
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。。
ExtensionsExtensions 1010
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。。
ExtensionsExtensions 1111
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。。
ExtensionsExtensions 1212
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∧
ExtensionsExtensions 1313
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∨ ∧
ExtensionsExtensions 1414
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
ExtensionsExtensions 1515
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 )
ExtensionsExtensions 1616
Fuzzy complementsFuzzy complements22
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 )
ExtensionsExtensions 1717
Fuzzy complementsFuzzy complements33
ExampleExample
tafor 0tafor 1ac Satisfy c1, c2
aac cos121 Satisfy c1,c2,c3
,1
11 aa
ac Satisfy c1~c4
ExtensionsExtensions 1818
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.
ExtensionsExtensions 1919
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 ~~~~ ,
ExtensionsExtensions 2020
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)
ExtensionsExtensions 2121
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))
ExtensionsExtensions 2222
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
ExtensionsExtensions 2323
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)
ExtensionsExtensions 2424
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
ExtensionsExtensions 2525
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 ,~~
ExtensionsExtensions 2626
Some t-conorms/t-normsSome t-conorms/t-norms33
nAA ~...~1
Algebraic sumAlgebraic sum
XxxxCBA
|,~~~
BAC ~~~
where
xxxxx BABABA ~~~~~~
ExtensionsExtensions 2727
Some t-conorms/t-normsSome t-conorms/t-norms44
nAA ~...~1
Bounded sumBounded sum
XxxxC BA |,~~~
BAC ~~~
where
xxx BABA ~~~~ ,1min
ExtensionsExtensions 2828
Some t-conorms/t-normsSome t-conorms/t-norms55
nAA ~...~1
Bounded differenceBounded difference
XxxxC BA |,~~~
BAC ~~~
where
1,0max ~~~~ xxx BABA
ExtensionsExtensions 2929
Some t-conorms/t-normsSome t-conorms/t-norms66
nAA ~...~1
Algebraic productAlgebraic product
XxxxCBA
|,~~~
BAC ~~~
where
xxx BABA ~~~~
ExtensionsExtensions 3030
ExampleExample
Let Let
6.0.7,1,5,5.0,3~ xA
6.0,5,1,3~ xB
ExtensionsExtensions 3131
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)
ExtensionsExtensions 3232
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
ExtensionsExtensions 3333
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.
ExtensionsExtensions 3434
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
ExtensionsExtensions 3535
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..
ExtensionsExtensions 3636
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)
ExtensionsExtensions 3737
The full scope of fuzzy aggregation The full scope of fuzzy aggregation operationsoperations
ExtensionsExtensions 3838
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
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