PART 4 Fuzzy Arithmetic 1. Fuzzy numbers 2. Linguistic variables 3. Operations on intervals 4....

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PART 4 Fuzzy Arithmetic 1. Fuzzy numbers 2. Linguistic variables 3. Operations on intervals 4. Operations on fuzzy numbers 5. Lattice of fuzzy numbers 6. Fuzzy equations FUZZY SETS AND FUZZY LOGIC Theory and Applications
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Transcript of PART 4 Fuzzy Arithmetic 1. Fuzzy numbers 2. Linguistic variables 3. Operations on intervals 4....

PART 4Fuzzy Arithmetic

1. Fuzzy numbers2. Linguistic variables3. Operations on intervals4. Operations on fuzzy numbers5. Lattice of fuzzy numbers6. Fuzzy equations

FUZZY SETS AND

FUZZY LOGICTheory and Applications

Fuzzy numbers

• Three properties1) A must be a normal fuzzy set;

2) αA must be a closed interval for every

3) the support of A, 0+A, must be bounded.

A is a fuzzy set on R.

];1 ,0(

Fuzzy numbers

Fuzzy numbers

• Theorem 4.1

Let Then, A is a fuzzy number if and only if there exists a closed interval

such that

).(RFA

] ,[ ba

), ,(for

) ,(for

] ,[for

)(

)(

1

)(

bx

ax

bax

xr

xlxA

Fuzzy numbers

• Theorem 4.1 (cont.)

where is a function from that is

monotonic increasing, continuous from the right,

and such that ; is a

function from that is monotonic decreasing, continuous from the left, and such

that

l 1] [0, to) ,( a

) ,(for 0)( 1 xxl r1] [0, to) ,( b

) ,(for 0)( 2 xxr

Fuzzy numbers

Fuzzy numbers

Fuzzy numbers

• Fuzzy cardinality

Given a fuzzy set A defined on a finite universal set X, its fuzzy cardinality, , is a fuzzy number defined on N by the formula

for all

|~

| A

|)(||~

| AA

).(A

Linguistic variables

• The concept of a fuzzy number plays a fundamental role in formulating quantitative fuzzy variables.

• The fuzzy numbers represent linguistic concepts, such as very small, small, medium, and so on, as interpreted in a particular context, the resulting constructs are usually called linguistic variables.

Linguistic variables

• base variable

Each linguistic variable the states of which are expressed by linguistic terms interpreted as specific fuzzy numbers is defined in terms of a base variable, the values of which are real numbers within a specific range.

A base variable is a variable in the classical sense, exemplified by any physical variable (e.g., temperature, etc.) as well as any other numerical variable, (e.g., age, probability, etc.).

Linguistic variables

• Each linguistic variable is fully characterized by a quintuple (v, T, X, g, m).– v : the name of the variable.– T : the set of linguistic terms of v that refer to

a base variable whose values range over a universal set X.

– g : a syntactic rule (a grammar) for generating linguistic terms.

– m : a semantic rule that assigns to each linguistic term t T.

Linguistic variables

Operations on intervals

• Let * denote any of the four arithmetic operations on closed intervals: addition + , subtraction —, multiplication • , and division /. Then,

)].e/ ,/ ,/ ,/max(

),e/ ,/ ,/ ,/[min(

]d1 ,1[] ,[] ,/[] ,[

)],e , , ,max( ),e , , ,[min(] ,[] ,[

], ,[] ,[] ,[

], ,[] ,[] ,[

}, ,|{] ,[] ,[

bdbeada

bdbeada

ebaedba

bbdaeadbbdaeadedba

dbeaedba

ebdaedba

egdbfagfedba

Operations on intervals

• Properties

Let ].1 ,1[ ],0 ,0[ ], ,[ ], ,[ ], ,[ 212121 10ccCbbBaaA

).( )( .4

).(

, .3

).( )()(

),()( .2

).(

, .1

utivitysubdistribCABACBA

identityAAA

AAA

ityassociativCBACBA

CBACBA

itycommutativABBA

ABBA

11

00

Operations on intervals

). ( //

,

,

,

: then, and If .7

.1 and 0 .6

.)( then ], ,[ if e,Furthermor ).(

)( then , and every for 0 If .5

tymonotoniciinclusionFEBA

FEBA

FEBA

FEBA

FBEA

A/AA-A

CaBaCBaaaAvitydistributi

CABACBACcBbcb

Operations on fuzzy numbers

• First method

Let A and B denote fuzzy numbers. * denote any of the four basic arithmetic operations.

for any

Since is a closed interval for each

and A, B are fuzzy numbers, is

also a fuzzy number.

].1 ,0(BABA )(

].1 ,0(

.)(1] [0,αα

BABA

)( BA

BA

Operations on fuzzy numbers

• Second method

)].( ),(min[sup)B)(/(A

)],( ),(min[sup)B)((A

)],( ),(min[sup)B)((A

)],( ),(min[sup)B)((A

)],( ),(min[sup)B)((A

allfor

/yBxAz

yBxAz

yBxAz

yBxAz

yBxAz

z

yxz

yxz

yxz

yxz

yxz

R

Operations on fuzzy numbers

Operations on fuzzy numbers

Operations on fuzzy numbers

• Theorem 4.2

Let * { + , - , •, / }, and let A, B denote continuous fuzzy numbers. Then, the fuzzy set

A * B defined by

is a continuous fuzzy number.

)]( ),(min[sup)B)((A yBxAzyxz

Lattice of fuzzy numbers

• MIN and MAX

)].( ),(min[sup))( ,(

)],( ),(min[sup))( ,(

) ,max(

) ,min(

yBxAzBA

yBxAzBA

yxz

yxz

MAX

MIN

Lattice of fuzzy numbers

Lattice of fuzzy numbers

Lattice of fuzzy numbers

• Theorem 4.3

Let MIN and MAX be binary operations on R.

Then, for any , the following properties hold:

RCBA , ,

Lattice of fuzzy numbers

Lattice of fuzzy numbers

• Lattice

It also can be expressed as the pair , where is a partial ordering defined as:

MAXMIN , ,R ,R

intervals. closed are where

,)ax( iff

,)in( iff

:cuts-relevant theof in terms ordering partial

thedefine alsocan we],10( all and any for

)( iff

ely,alternativ or, )( iff

BA,

BBA, BA

ABA, BA

, αRA, B

BA, BBA

AA, BBA

m

m

MAX

MIN

Lattice of fuzzy numbers

].10( allfor

iff

have we,any for then

, and iff ][][

is, that way,

usual in the intervals closed of ordering partial thedefine weIf

)].(max ),(max[)(max

)],(min ),(min[)(min

Then,

22112121

2211

2211

,

BA BA

A, B

baba, bb, aa

, ba, baBA,

, ba, baBA, αα

αα

R

Fuzzy equations

• A + X = B

The difficulty of solving this fuzzy equation is caused by the fact that X = B - A is not the solution.

Let A = [a1, a2] and B = [b1, b2] be two closed intervals, which may be viewed as special fuzzy numbers. B - A = [b1- a2 , b2 - a1], then

Fuzzy equations

Let X = [x1, x2].

]. ,[

. iffsolution a hasequation the

. that required sit' interval,an bemust

.

.

,

,

]. ,[] ,[ Then,

2211

2211

21

222

111

222

111

212211

ababX

abab

xxX

abx

abx

bxa

bxa

bbxaxa

Fuzzy equations

Let αA = [αa1, αa2], αB = [αb1, αb2], and

αX = [αx1, αx2] for any . ]1 ,0(

]10(

22221111

2211

.

bygiven isequation fuzzy theof solution the

. implies (ii)

and ],10(every for (i)

:iffsolution a has

, α

ααββββαα

XX

X

ababababβα

, αabab

BXA

Fuzzy equations

• A . X = B

A, B are fuzzy numbers on R+. It’s easy to show that X = B / A is not a solution of the equation.

]10(

22221111

2211

.

bygiven isequation fuzzy theof solution the

.//// implies (ii)

and ],10(every for // (i)

:iffsolution a has

, α

ααββββαα

XX

X

ababababβα

, αabab

BXA

Exercise 4

• 4.1

• 4.2

• 4.5

• 4.6

• 4.9