extension principle

Extension Principle 1

Extension Principle Extension Principle — Concepts— Concepts

To generalize crisp To generalize crisp mathematical mathematical conceptsconcepts to to fuzzy setsfuzzy sets..


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


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


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~


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


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))


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


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


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~


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

~~


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


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


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 ~~


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 ~~


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~ ~:\~


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 ~ ~ ~ ~


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


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 ~ ~~ ~~~ ~


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


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~


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


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.


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]


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


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~


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


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 ,,,~


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~


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~


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~


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~

extension principle

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