extension principle
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Extension Principle 1
Extension Principle Extension Principle — Concepts— Concepts
To generalize crisp To generalize crisp mathematical mathematical conceptsconcepts to to fuzzy setsfuzzy sets..
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Extension Principle 2
Extension PrincipleExtension Principle
Let Let XX be a cartesian product of universes be a cartesian product of universes X=XX=X11……XXrr, and be r fuzzy sets in , and be r fuzzy sets in XX11,…,X,…,Xrr, respec, respectively. tively. ff is a mapping from is a mapping from XX to a universe to a universe YY, , y=fy=f(x(x11,…,x,…,xrr)), Then the extension principle allows us t, Then the extension principle allows us to define a fuzzy set in o define a fuzzy set in YY by by
rAA ~,...,~1
B~
XxxxxfyyyB rrB ,..., ,,...,,~11~
wherewhere
otherwise 0
0 ,...,minsup 1~1~
,...~11
1
yfifxxy
rAAyfxx
Br
r
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Extension Principle 3
Example 1Example 1
4.0,2,1,1,8.0,0,5.0,1~ A
f(x)=x2
4.0,4,1,1,8.0,0~~ AfB
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Extension Principle 4
Fuzzy NumbersFuzzy Numbers
To qualify as a fuzzy number, a fuzzy set oTo qualify as a fuzzy number, a fuzzy set on n RR must possess at least the following must possess at least the following three prthree propertiesoperties::– must be a must be a normal fuzzy setnormal fuzzy set
– must be a closed interval for every must be a closed interval for every αα(0,1](0,1] (con(convex)vex)
– thethe support support of , must be of , must be boundedbounded
A~
A~
A~
A~
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Extension Principle 5
Positive (negative) fuzzy numberPositive (negative) fuzzy number
A fuzzy number is called A fuzzy number is called positive (negative)positive (negative) if its membership function is such that if its membership function is such that
A~
0 0 ,0~ xxxA
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Extension Principle 6
Increasing (Decreasing) Increasing (Decreasing) OperationOperationA binary operation A binary operation in in RR is called is called
increasing (decreasing)increasing (decreasing) if if
forfor x x11>y>y11 and and xx22>y>y22
xx11xx22>y>y11yy22 (x(x11xx22<y<y11yy22))
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Extension Principle 7
Example 2Example 2
f(x,y)=x+yf(x,y)=x+y is an increasing operation is an increasing operation f(x,y)=xf(x,y)=x••yy is an increasing operation on is an increasing operation on RR++
f(x,y)=-(x+y)f(x,y)=-(x+y) is an decreasing operation is an decreasing operation
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Extension Principle 8
Notation of fuzzy numbers’ Notation of fuzzy numbers’ algebraic operationsalgebraic operations If the normal algebraic operations If the normal algebraic operations +,-,*,/+,-,*,/ are are
extended to operations on fuzzy numbers they extended to operations on fuzzy numbers they shall be denoted by shall be denoted by
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Extension Principle 9
Theorem 1Theorem 1
If and are fuzzy numbers whose memberIf and are fuzzy numbers whose membership functions are ship functions are continuous continuous and and surjectivesurjective frfromom R to [0,1]R to [0,1] and and is a is a continuous increasing continuous increasing (decreasing) binary operation(decreasing) binary operation, then is , then is a fuzzy number whose membership function is a fuzzy number whose membership function is continuous and surjective from continuous and surjective from RR to to [0,1][0,1]..
M~ N~
M~ N~
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Extension Principle 10
Theorem 2Theorem 2 If , If , F(R)F(R) (set of real fuzzy (set of real fuzzy
number) with and continuous number) with and continuous membership functions, then by membership functions, then by application of the extension principle for application of the extension principle for the binary operation the binary operation : R : R R R→R the →R the membership function of the fuzzy membership function of the fuzzy number is given by number is given by
M~ N~
xN~ xM~
M~ N~
xx NMzyxz
NM ~~ ,min sup
~~
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Extension Principle 11
Special Extended OperationsSpecial Extended Operations
If If f:Xf:X→Y→Y, , X=XX=X11 the extension principle the extension principle reduces for all reduces for all F(R)F(R) to to M~
~~1
supzfx
MMf xz
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Extension Principle 12
Example 3Example 311
For For f(x)=-xf(x)=-x the opposite of a fuzzy number the opposite of a fuzzy number is given with , where is given with , where
If If f(x)=1/xf(x)=1/x, then the inverse of a fuzzy num, then the inverse of a fuzzy number is given with ber is given with
M~
XxxxM M ~,~
xx MM ~~
M~
XxxxMM
1~1 ,~ , where
1 MM
1x
x
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Extension Principle 13
Example 3Example 322
For For λλR\{0}R\{0} and and f(x)=f(x)=λλxx then the scalar then the scalar multiplication o a fuzzy number is given multiplication o a fuzzy number is given by , whereby , where XxxxM M ~,~
xx MM ~~
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Extension Principle 14
Extended Addition Extended Addition
Since addition is an Since addition is an increasing operationincreasing operation → → extended addition extended addition of fuzzy of fuzzy numbers that numbers that RFNMNMNMf ~,~ ,~~~,~
is a fuzzy number is a fuzzy number — that is — that is
RFNM ~~
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Extension Principle 15
Properties of Properties of
( )( )( )( ) is commutativeis commutative is associativeis associative00RRF(R)F(R) is the is the neutral elementneutral element for for , ,
that is , that is , 0=0= , , F(R)F(R)For For there there does not exist an inverse eledoes not exist an inverse ele
mentment, that is,, that is,
NM ~~M~ N~
M~ M~ M~
RMMRRFM 0~ ~:\~
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Extension Principle 16
Extended Product Extended Product
Since multiplication is an Since multiplication is an increasing increasing operationoperation on on RR++ and a decreasing and a decreasing operation on operation on RR--, the product of positive , the product of positive fuzzy numbers or of negative fuzzy fuzzy numbers or of negative fuzzy numbers results in a positive fuzzy numbers results in a positive fuzzy number.number.
Let be a positive and a negative Let be a positive and a negative fuzzy number then is also negative fuzzy number then is also negative and results in a and results in a negative fuzzy number.negative fuzzy number.
M~ N~
M~
NMNM ~ ~ ~ ~
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Extension Principle 17
Properties of Properties of
is commutativeis commutative is associativeis associative , , 11RRF(R)F(R) is the is the neutral neutral
elementelement for , that is , , for , that is , , F(R)F(R)
For there For there does not exist an inverse does not exist an inverse elementelement, that is,, that is,
M~
M~ N~M~(( )) N~ == (( ))
M~ 1=1= M~
M~ 1=1= M~
1M~ ~:\~ 1- MRRFM
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Extension Principle 18
Theorem 3Theorem 3
If is If is eithereither a a positive or a negativepositive or a negative fuz fuzzy number, and and are zy number, and and are bothboth eithe either r positive or negativepositive or negative fuzzy numbers the fuzzy numbers then n
M~
N~ P~
PMNMPNM ~ ~~ ~~~ ~
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Extension Principle 19
Extended SubtractionExtended Subtraction
Since subtraction is Since subtraction is neither an neither an increasing nor a decreasing operationincreasing nor a decreasing operation,,
is written as is written as ( )( )M~ N~ M~ N~
~~~ ~ ,min supyxz
NMNM yxz
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Extension Principle 20
Extended Division Extended Division
Division is also Division is also neither an increasing nor neither an increasing nor a decreasing operationa decreasing operation. If and are . If and are strictly positive fuzzy numbers thenstrictly positive fuzzy numbers then
M~ N~
/
~~~ ~ ,min supyxz
NMNM yxz
The same is true if and are strictly The same is true if and are strictly negative.negative.
M~ N~
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Extension Principle 21
NoteNoteExtended operations on the basis of Extended operations on the basis of
min-max min-max can’t directly applied to “fuzzy can’t directly applied to “fuzzy numbers” with numbers” with discrete supportsdiscrete supports..
ExampleExample– Let ={(1,0.3),(2,1),(3,0.4)}, ={(2,0.7),(3,1),Let ={(1,0.3),(2,1),(3,0.4)}, ={(2,0.7),(3,1),
(4,0.2)} then (4,0.2)} then M~
N~
M~ N~ ={(2,0.3),(3,0.3),(4,0.7),(6,1),(8,0.2),={(2,0.3),(3,0.3),(4,0.7),(6,1),(8,0.2),(9,0.4),(12,0.2)}(9,0.4),(12,0.2)}
No longer be convex No longer be convex → not fuzzy number→ not fuzzy number
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Extension Principle 22
Extended Operations for LR-Extended Operations for LR-Representation of Fuzzy SetsRepresentation of Fuzzy SetsExtended operations with fuzzy Extended operations with fuzzy
numbers involve rather extensive numbers involve rather extensive computations as long as no restrictions computations as long as no restrictions are put on the type of membership are put on the type of membership functions allowed.functions allowed.
LR-representationLR-representation of fuzzy sets of fuzzy sets increases computational efficiencyincreases computational efficiency without limiting the generality beyond without limiting the generality beyond acceptable limits.acceptable limits.
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Extension Principle 23
Definition of L (and R) typeDefinition of L (and R) type
Map Map RR++→[0,1]→[0,1], , decreasingdecreasing, shape functi, shape functions if ons if
L(0)=1L(0)=1L(x)<1L(x)<1, for , for x>0x>0L(x)>0L(x)>0 for for x<1x<1L(1)=0L(1)=0 or [L(x)>0, or [L(x)>0, xx and and L(+∞)=0]L(+∞)=0]
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Extension Principle 24
Definition of LR-type fuzzy Definition of LR-type fuzzy numbernumber11
A fuzzy number is of A fuzzy number is of LR-typeLR-type if there if there exist reference functions exist reference functions LL(for left). (for left). RR(fo(for right), and scalars r right), and scalars αα>0>0, , ββ>0>0 with with
M~
R
~
mxfor
mx
mxforxm
LxM
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Extension Principle 25
Definition of LR-type fuzzy Definition of LR-type fuzzy numbernumber22
mm; called the ; called the mean valuemean value of , is a real of , is a real numbernumber
αα,,ββ called the called the left and right spreadsleft and right spreads, res, respectively.pectively.
is denoted by (m,is denoted by (m,αα,,ββ))LRLR
M~
M~
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Extension Principle 26
Example 4Example 4
Let Let L(x)=1/(1+xL(x)=1/(1+x22)), , R(x)=1/(1+2|x|)R(x)=1/(1+2|x|), , αα=2=2, , ββ=3=3, , m=5m=5 then then
5
352
1
13
5
5
25
1
12
52
5
xforx
xR
xforx
xL
x
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Extension Principle 27
Fuzzy IntervalFuzzy IntervalA A fuzzy intervalfuzzy interval is of LR-type if there is of LR-type if there
exist shape functions L and R and four exist shape functions L and R and four parameters , parameters , αα, , ββ and the membership function of is and the membership function of is
M~
,, 2RmmM~
mxfor
mxmfor
mxforxm
L
xM
m-x
R
1
~
The fuzzy interval is denoted byThe fuzzy interval is denoted by
LRmmM ,,,~
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Extension Principle 28
Different type of fuzzy intervalDifferent type of fuzzy interval is a is a real crisp numberreal crisp number for for mmRR →→
=(m,m,0,0)=(m,m,0,0)LRLR L, L, RR If is a If is a crisp intervalcrisp interval, , →→
=(a,b,0,0)=(a,b,0,0)LRLR L, L, RR If is a “If is a “trapezoidal fuzzy numbertrapezoidal fuzzy number” → ” →
L(x)=R(x)=max(0,1-x)L(x)=R(x)=max(0,1-x)
M~
M~
M~
M~
M~
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Extension Principle 29
Theorem 4Theorem 4
Let , be two fuzzy numbers of Let , be two fuzzy numbers of LR-tLR-typeype: : =(m,=(m,αα,,ββ))LRLR, , =(n,=(n,γγ,,δδ))LRLR Then Then– (m,(m, αα, , ββ))LRLR(n, (n, γγ,,δδ))LRLR=(m+n, =(m+n, αα++γγ, , ββ++δδ))LRLR
– -(m, -(m, αα, , ββ))LRLR=(-m, =(-m, ββ, , αα))LRLR
– (m, (m, αα, , ββ))LRLR (n, (n, γγ, , δδ))LRLR=(m-n, =(m-n, αα++δδ, , ββ++γγ))LRLR
M~ N~
M~ N~
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Extension Principle 30
Example 5Example 5
L(x)=R(x)=1/(1+xL(x)=R(x)=1/(1+x22)) =(1,0.5,0.8)=(1,0.5,0.8)LRLR
=(2,0.6,0.2)=(2,0.6,0.2)LRLR
=(3,1.1,1)=(3,1.1,1)LRLR
=(-1,0.7,1.4)=(-1,0.7,1.4)LRLR
M~
N~
N~N~M~
M~
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Extension Principle 31
Theorem 5Theorem 5Let , be fuzzy numbers → Let , be fuzzy numbers →
(m, (m, αα, , ββ))LRLR (n, (n, γγ, , δδ))LRLR ≈(mn,m≈(mn,mγγ+n+nαα,m,mδδ+n+nββ))LRLR for , positive for , positive
(m, (m, αα, , ββ))LRLR (n, (n, γγ, , δδ))LRLR ≈(mn,≈(mn,nnαα-m-mδδ,n,nββ--mmγγ))LRLR for positive, negative for positive, negative
(m, (m, αα, , ββ))LRLR (n, (n, γγ, , δδ))LRLR ≈(mn,-n ≈(mn,-nββ-m-mδδ,-n,-nαα--mmγγ))LRLR for , negative for , negative
M~ N~
N~M~
M~N~
M~ N~