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Fuzzy Integrals in Multi-Crite-ria Decision Making
Dec. 2011 Jiliang University China
Multi-Criteria Decision Making Problem Aggregation• Requirements of aggregation operators
• Common aggregation operators Fuzzy Measure and Integrals Properties of Fuzzy Integral Importance and Interaction of Criteria Decision Making in Pattern Recognition Summary
Contents
Multi-Criteria Decision Making Problem
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)(),...,,( :Example
operator.n aggregatioan called is where
))(),...,(),((),...,,(
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).()( that relation order thebuild and
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,...,, criteriaor attributes .t.result w.ror Consquence
,...,, actsor esalternativ ofSet
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Aggregation in MCDM
pii
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eAlternativ
Criteria
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2x
3x
4x
Utility
)( 11 xu
)( 22 xu
)( 33 xu
)( 44 xu
Operatorn Aggregatio
)(H Evaluation of
Result Final
Mathematical Properties• Properties of extreme values
• Idempotency
• Continuity
• Non-decreasing w.r.t. each argument
• Stability under the same positive linear trans-form
Requirements of Aggregation Operator
1)1,...,1,1( ,0)0,...,0,0( HH
aaaaH ),...,,(
RtrtaarHtratraH nn ,0 ,),...,(),...,( 11
Behavioral Properties• Expressing the weights of unequal importance on criteria
• Expressing the behavior of decision maker from perfect toler-ance (disjunctive behavior) to total intolerance (conjunctive be-havior)• Accept when some criteria are met
• Demand all criteria have to be equally met
• Expressing compensatory effect: • Redundancy when two criteria express the same things
• Synergy of two criteria: little importance separately but impor-tant jointly
• Easy semantic interpretation of aggregation operator
Requirements of Aggregation Operator
Quasi-arithmetic Mean
Example: Mean and Generalized Mean
Common Aggregation Operator
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iinf af
nfaaM
1
11 )(
1),...,(
n
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fww afwfaaM
n1
11,..., )(),...,(
1
),....,(or 1
),....,()(1
11
1
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iiinf
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iinf awaaMa
naaMxxf
pn
i
piinf
p awaaMxxf/1
11 ),....,()(
Median: mid-ordered data after sorting Weighted minimum and maximum
• When all weights are 1, then weighted minimum becomes the min-operator
• The larger weight value represents the more degree of impor-tance in the aggregation process
• When all weights are 0, then weighted maximum becomes the max-operator
Common Aggregation Operator
ii
nin
iinin
awaa
awaa
11
11
),...,(wmax
)1(),...,(wmin
Ordered weighted averaging (OWA)• Weighted average of ordered input
Note:
Common Aggregation Operator
)()2()1(
11)(1,...,
....
1 ),...,(OWA1
n
n
ii
n
iiinww
aaa
wawaan
Median )0,0..,0,1,0,..0,0(),....(
Minimum)0,0,....,0,1(),....(
mean Trimmed )0,2
1,....,
2
1,0(),....(
Average )1
,....,1
(),....(
1
1
1
1
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ww
wwnn
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nnww
Fuzzy measure
Additivity, Super-additivity, Sub-additivity
Fuzzy Measures
)()( implies )2(
1)( ,0)( (1)
axioms. following satisfying ]1,0[)(:
BAXBA
X
XP
additivity sub :)()()(
additivitysuper :)()()(
measurey probabilitin additivity :)()()(
BABA
BABA
BABA
BA
Sugeno’s g-lamda measure
Fuzzy Measure
.or measure Sugeno called is gThen
.1 somefor g(B)g(A)g(B)g(A))(
, with )( allFor
condition. following thesatisfying measurefuzzy a is
measureg
BAg
BAXPA, B
g
)(1
)()()(-)()()( 2.
additivity-sub satisfies0
additivity-super satisfies0
additivity satisfies 0 .1
:Note
0
BAg
BgAgBAgBgAgBAg
g
g
g
Fuzzy Measure
1/1)1(
......)(
},...,,{for general,In
)(}),,({
or
)()()( ))()()()()()((
)()()()(
, , :Note
function.density fuzzy called is })({:
},...,,{For
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ggggggggggggxxxg
CgBgAgCgAgCgBgBgAg
CgBgAgCBAg
CACBBA
xgg
xxxX
Fuzzy Measure
. aconstruct
equation fromn calculatio },...,{
. ingcorrespond
aconstruct can one then given, is },...,{ If :Colloary
1/1)1()(
) (-1,in solution unique a hasequation following The :Theorem
21
21
measureg
ggg
measureg
ggg
gXg
n
n
Xx
i
i
Note: We need only n numbers of fuzzy density instead of 2n.
Fuzzy Measures and Integrals
)()1()()()()2()1(
)()(11
,...,, and )(...)()(0 where
)()(),...,(
w.r.t.]1,0[:function a of integral Sugeno
niiin
iinin
xxxAxfxfxf
AxfxfxfS
Xf
)()1()()(
)()2()1()0(
1)()1()(1
,...,,
and )(...)()()(0 where
))()()(),...,(
w.r.t.]1,0[:function a of integralChoquet
niii
n
n
iiiin
xxxA
xfxfxfxf
AxfxfxfxfC
Xf
Fuzzy Measures and Integrals
82.03.09.082.06.013.0)(),...,(
3.0,6.0,9.0
54.0,,43.0,,82.0,
1.0,4.0,3.0
: w.r.t.function a of integral Sugeno
1
nxfxfS
cfbfaf
cbcaba
cba
f
636.03.03.082.03.013.0)(),...,(
: w.r.t.function a of integralChiquet
1 nxfxfC
f
Sugeno and Choquet integral are idempotent, contin-uous, and monotonically non-decreasing operators.
Choquet integral with additive measure coincide with a weighted arithmetic mean.
Choquet integral is stable under positive linear trans-forms.
Choquet integral is suitable for cardinal aggregation where numbers have a real meaning.
Sugeno integral is suitable for ordinal aggregation where only order makes sense.
Properties of Fuzzy Integrals
Any OWA operator is a Choquet integral. Sugeno and Choquet integral contains all order sta-
tistics, thus in particular, min, max, and the median. Weighted minimum and weighted maximum are
special case of Sugeno integral
Properties of Fuzzy Integrals
Example:
Importance of Criteria and Interaction
2 3 literphysicsmath www
25.15 8
102163 183_
AStudent
Rank Order: A > C > B
Importance of Criteria and Interaction
1,, 0
synergy of because 3.045.09.0,
synergy of because 3.045.09.0,
redundancy of because 45.045.05.0,
3.0 45.0
literaturephysicsmath
literaturephysics
literaturemath
physicsmath
eliterarturphysicsmath
Rank Order: C > A > B
Index for Importance
1
!
!!1 with
indexShapley : of Importance Global
1
n
jj
XxXA
jXj
j
xΛ
X
AAXAAxAAxΛ
x
j
Multiplied by n
Index of Interaction
, |,
and !
!!2 with
[-1,1] |,,
: and between Index n Interactio Average
,
AxAxAxxAAxxI
X
AAXA
AxxIAxxI
xx
jijiji
X
xxXAjiXji
ji
ji
Note: Redundancy and synergy
Identification Based on Semantics• Importance of criteria
• Interaction between criteria
• Symmetric criteria {math, physics}
• Veto effect
• Pass effect
Identification of Fuzzy Measure
}{ allfor 0)(
),...,,,...,(),...,,...,( 1111
j
njjjnj
xXAA
aaaaGaaaaH
j
njjjnj
xAA
aaaaGaaaaH
allfor 1)(
),...,,,...,(),...,,...,( 1111
Identification Based on Learning Data
Identification of Fuzzy Measure
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kkknk ,...,1 ),,( data learningfor ),...,(
error theminimize that measurefuzzy heIdentify t
1
21
2
M. Grabish, H. T. Nguyen, and E. A. Walker, Fundamentals of Uncertainty Calculi, with Applications to Fuzzy Inference, Kluwer Academic, 1995
Decision Making in Pattern recognition
x
1C
2C
3C
4C
1x
2x
3x
4x
)(1 H
Decision
Class
)(2 H
Feature level simple classi-fier
Aggregation of class member-ships
Input pattern Class
label
Decision Making in Pattern recognition
1C
2C
3C
4C
)(HDecision
Class
x
Input pattern
Complexclassi-fiers
Aggregation of class memberships
Classlabel
Multi-Criteria Decision Making Problem and Aggregation Op-erators
Fuzzy Integrals have useful properties required for aggrega-tion operator in multi-criteria decision making• Not only degree of importance foe a separate criterion but also redun-
dancy and synergy effects between criteria Identification of Fuzzy measure based on• Semantic involved in the decision making problem
• Learning data
• Semantics and learning data Application are diverse • Pattern Recognition
• Multi-sensor Fusion
Summary