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
• 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 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.
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
• 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
• 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
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
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