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

n

iiin

nnn

tn

p

xuxxxu

H

xuxuxuHxxxu

yuxuyx

XuΩ

Xxxxx

Ω

121

221121

21

21

)(),...,,( :Example

operator.n aggregatioan called is where

))(),...,(),((),...,,(

:functionsutility component ofn Aggregatio

).()( that relation order thebuild and

:function utility of in terms in onebest theFind

,...,, criteriaor attributes .t.result w.ror Consquence

,...,, actsor esalternativ ofSet

R

Aggregation in MCDM

pii

,...1

eAlternativ

Criteria

1C

2C

3C

4C

1x

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

n

iinf af

nfaaM

1

11 )(

1),...,(

n

iiin

fww afwfaaM

n1

11,..., )(),...,(

1

),....,(or 1

),....,()(1

11

1

n

iiinf

n

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

n

n

n

n

ww

wwnn

ww

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

21121

1 11

21

3212313221321321

2

21

Xx

i

nnn

j

n

jk

kjn

j

j

n

ii

n

i

g

ggggggXg

xxxX

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

lkyzyzzCE kk

l

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