Fuzzy Measures and Integrals

Fuzzy Measures and Integrals

Linett Monta~noGuzman

UniversidadCatoli aBoliviana

DepartamentodeCien iasExa tas

Co habamba, Bolivia

e-mail:lmontanou b ba.edu.bo

Abstra t

Inthispaper,wewillstudytheaggregation problemwithintera ting riteria,

and we will introdu e the on epts of fuzzymeasures and integrals. Thefuzzy

integrals are powerful aggregation operators whi h are able to take into

onsi-deration the intera tionamong riteriaand the fuzzymeasures anbe usedfor

modelingtheimportan eofallsubsetsof riteria.

Wewill on entratein\theChoquetintegral",thisfuzzyintegralisaweighted

aggregationoperator,whi htakesintoa ountnotonlytheimportan eofthe

ri-teria, butalsotheimportan eofall subsetsofthem. Theinformationaboutthe

importan eandtheintera tionbetween riteriaisusedintheaggregationpro ess

ofthepartialevaluations. Thispaperanalysesome hara teristi sandproperties

ofthisoperator.

keywords: Multiple riteriaanalysis;Fuzzymeasures;aggregationoperator.

1 Introdu tion

Multi riteriaDe ision Aid(MCDA)is an importantbran h of OperationalResear h.

MCDA dis iplinedevelopsand implements de ision support toolsand methodologies

to dealwith omplex de isionproblems that involvemultiple riteria,goals or

obje -tivesof on i tingnature. ThetoolsandmethodologiesproviedbyMCDAarenotjust

mathemati almodelsaggregating riteria,pointsofvieworattributesbutfurthermore

theyarede isionsupportoriented. MostMulti riteriaDe isionAid(MCDA)methods

needsomewhere in their pro edure afundamental operation: \the aggregation". The

most ommonaggregationtoolusedtoday,isstilltheweightedarithmeti mean. This

operatorhasane essary onditionfortherepresentationofpreferen es: \the

indepen-den e". But,inmanysituations,however,the riteria onsideredarenotindependent,

theyintera t.

In fa t,in ade isionproblem there are links between riteria. Forinstan e, there

ofrelation should be taken into a ountin aggregationmethods and espe ially in the

pro ess ofdetermining the weights. Thus, thefuzzy measures and integralstakeinto

a ountthisinformation.

In the following se tions, we will introdu e someimportant notionsof thetheory

offuzzy measuresand integration. Formoredetails andproofsaboutthistheory, the

readeris referredtothefollowingpapersandbooks[1, 2,3,6,7,8,10℄.

2 Intera tion among riteria

In ade ision problem the evaluation of thealternativesunder various points of view

( riteria) often showsintera tions. These links express the orrelation or the

depen-den e with respe t to the preferen es among riteria. This does not mean that the

orrespondingpointsofviewareredundantandshouldbeeliminated. Ontheopposite,

thismeans, that theattributesthat areused tore e t thesepointsofviewarelinked

by logi alorfa tual interdependen iesand this information should beused. We will

analyse two types of intera tion among riteria: the orrelation and the preferential

dependen e.

2.1 Correlation

Theword orrelationisusedin everydaylifetodenotesomeformofasso iation.

Thestatisti aldenition ofthenotionof orrelationisthefollowing:

Denition2.1 The orrelation measures the inter-relationship between two variables

XandY. Theoutputofthismeasurementis alledthe\ orrelation oeÆ ient",denoted

by r. Itissometimes alled Pearson's orrelation oeÆ ientafter itsoriginatorand is

ameasure oflinearasso iation.

r= n n X i=1 X i Y i ( n X i=1 X i n X i=1 Y i )℄ v u u t [n n X i=1 X 2 i ( n X i=1 X i ) 2 ℄ v u u t [n n X i=1 Y 2 i ( n X i=1 Y i ) 2 ℄ (1)

The orrelation oeÆ ient is measured on a s ale that varies from 1 through 0

to +1. Complete orrelation between two variables is expressed by either +1 or 1.

When one variable in reases as the other in reases the orrelation is positive; when

onede reasesasthe otherin reasesit isnegative. Complete absen eof orrelationis

representedby0.

Instatisti altermsthenotionof orrelationisusedtodenoteanasso iationbetween

twoquantitativevariables. Itisalsoassumedthattheasso iationislinear,thatis,that

onevariablein reasesorde reasesaxedamountforaunitin reaseorde reaseofthe

The orrelationamong riteriaisprobablythebestknownandmostintuitivetypeof

dependen e. Forinstan e,inthesele tionofpersonnelproblem,theageofthe andidate

andthenumberofyearsofprofessionalexperien e, riteriausedin theevaluation,are

likelytobepositively orrelated.

2.2 Preferentialdependen e

Anotherwaytohavedependen eorintera tionamong riteriaiswhentheevaluationof

thealternativesfollowdierentpointsofview,donotrespe tthepropertyofpreferential

independen e.

A simple test, to prove the independen e in the sense of preferen es, onsists in

askingthede isionmakerabouthispreferen es,onpairsofalternatives,thatsharethe

sameprole, forasubset ofattributes;varying the ommonproleshould notreverse

thepreferen eswhenthepointsofviewareindependent.

Let us onsiderer a set of alternatives A = fa;b;:::;mg and a set of riteria

F = fg 1 ;g 2 ;:::;g n

g. Ea h alternative a 2 A is asso iated with a prole g(a) =

(g 1 (a);g 2 (a);:::;g n (a)) 2 IR n where g j

(a) represents the utility of a related to the

riterionj.

Supposethatthepreferen esoverAofthede isionmakerareknownandexpressed

byabinaryrelation.

Denition2.2 (Vin ke1992 [14 ℄)LetF bethefamilyof riteria, S asubsetofF and

S the omplementary subset in F. S ispreferentially independent in F if, given four

a tionsa;b; ;dsu hthat:

8 > > < > > : g j (a)=g j (b) 8j2S g j ( )=g j (d) 8j2S g j (a)=g j ( ) 8j2S g j (b)=g j (d) 8j2S (2) weget aPb, Pd;

whereP isthe global preferen erelationtaking intoa ount allthe riteria.

In other words, S is preferentially independent in F if the preferen es between

a tions, whi h dier only by their evaluations a ording to the riteria in S, do not

depend uponthevaluesyieldedbythe riteriaofS.

Example1 Considerthe sele tion of personnelproblem: A ompany ishiring and

hasashortlistof4 andidates: Anne,Bernard,Celine andDavid. TheHRdepartment

has a list of three riteria: age of the andidate (g

1

), number of years of professional

experien ein arelevant relatedjob (g

2

) andnumber of years of higher edu ation (g

3 ).

Candidate g 1 g 2 g 3 Anne(a) 28 4 6 Bernard(b) 28 6 4 Celine( ) 35 4 6 David (d) 35 6 4

Table1: Sele tionofpersonnelproblem.

The HR department expresses its opinion about the andidates. Evidently, the

preferen es a  and b  d are immediately suggested; but the other omparisons

are notsoobviousbe ausethe asso iated prolesinterla e. After the analysis of the

problem,theHRdepartmentexpressesthefollowingpreferen es:

andidatea andidateb and andidated andidate

FortheHRdepartment,thereasonisthatastheagein reases,theyearsof

profes-sionalexperien ebe omemoreimportantthantheyearsofhigheredu ation. However,

when the andidate is younger, it is more important to have a high edu ation than

experien e. Inthis asetheprofessional experien eand thenumberof yearsof higher

edu ation are not preferentially independent of the age of the andidate. Thus, the

rsttwo riteriaarenotpreferentiallyindependentofthethird. Therefore,inthis ase,

thereisnoadditivemeasurethat ouldreprodu etheHRdepartmentranking.

A wellknown exampleof dependen e in the sense of preferen es is thepreferen e

forwhitewineorredwinedependingonwhetheryouareeatingsh ormeat.

Inordertotakeintoa ountthebehaviorof riteria: the orrelationanddependen e

in the sense of preferen es and to have a exible representation of these intera tions

between riteria,it isusefulto introdu e the on eptoffuzzy measuresand integrals.

Thus, in thenextse tions weintrodu ethese main on epts, andweshow howfuzzy

measuresprovideameansofrepresentingtherelationshipbetween riteria.

3 Fuzzy measures

Theadditivitypropertyofthe\measure",oneofthemostimportant on eptsin

math-emati s,isoftenin exibleortoorigidtorepresentthemanyfa etsofhumanreasoning.

Therefore,tobeabletoexpresshumansubje tivity,Sugeno[13℄proposedtorepla ethe

additivitypropertybyaweakerone,themonotoni ity,andhe alledthesenon-additive

monotoni measures\fuzzymeasures".

Beforetheintrodu tionoffuzzymeasuresbySugenoin1974(togeneralizeadditive

measures),this on epthasbeenintrodu ed in1953byChoquet[1℄as\ apa ities".

Thus,thefuzzymeasuresarealso allednon-additivemeasures,or apa ities.

Denition3.1 A dis rete \fuzzy measure" on N is a set fun tion : 2 N

! [0;1℄

i) (;)=0; (N)=1

ii) ST N =) (S)(T)

Afuzzymeasure issaidto be:

i) Additiveif(S[T)=(S)+(T)wheneverS\T =;.

ii) Super-additiveif(S[T)(S)+(T)wheneverS\T =;.

iii) Sub-additiveif(S[T)(S)+(T)wheneverS\T =;.

Tounderstandthe meaningof fuzzymeasures in pra ti alappli ations wegive an

exampleproposedbyMurifushiandSugeno[12℄:

Example2 LetN beasetofworkersinaworkshop,andsupposetheyprodu ethe

sameprodu ts. SupposethatagroupofworkersAN worksinthemosteÆ ientway.

Let (A) bethe number of produ tsmade by A in one hour.  an be onsideredas a

measureofprodu tivity,and anbenormalizeddividingby(N). Then,thenormalized

fun tion of the set fun tion :2 N

!IR

+

is monotoneand vanishes atthe empty set,

andtherefore itisafuzzymeasure.

Observethatin thissituation, maybenon-additive,sin etwogroupsof workers

AandB together anprodu emore(orless)thaniftheyworkseparately. IfAandB

work separately, then (A[B)= (A)+(B). Nevertheless, generally ifthey work

jointly, theyintera tand theequalitymaynotbekept. Thus,anee tive ooperation

ofmembersofA[B givestheinequality(A[B)>(A)+(B). Ontheotherhand,

the in ompatibility between the groups of workersA and B ould give the opposite

inequality(A[B)<(A)+(B).

Inade isionaidframework,thefuzzymeasure(T)representstheweightof

impor-tan eofthesubsetof riteriaT. Therefore,inadditiontotheusualweightson riteria

takenintoa ountseparately,weightsonallthe ombinationsof riteriawillbedened

aswell.

In the asewhere the fuzzy measure is additive, we an onsider that there is no

intera tionbetween riteriaanditsuÆ estodenenweightsf(1);:::;(n)gtodene

themeasure entirely. Whenthefuzzymeasureissuper-additive,thismeansthatthere

isapositiveintera tionbetween riteria(synergy). Otherwise,ifthefuzzymeasuresis

sub-additiveitmeansthatthereisanegativeintera tionbetween riteria(redundan y).

There are severalrepresentations ofafuzzy measure. A ording to Grabis h[4℄ a

representationofafuzzymeasure ouldbedenedby\anysetfun tionfromwhi h it

ispossibletore overwithoutlossof information". Themostutilizedrepresentations

arethefollowing:

 Theusual(measure)representation;thisissimply()itself.

 TheMobiusrepresentationorpolynomialrepresentation(w).

4 Fuzzy integral

\Fuzzyintegral"is ageneri termfor integralswith respe t tofuzzy measures. There

aremany kindsof fuzzy integrals: TheintegralsofChoquet, 

Sipos,Sugeno,t- onorm

integral,et . Inthispaperwewillmainly on entrateontheChoquetintegral.

Murofushi and Sugenoproposed the so- alled Choquetfuzzy integral, referringto

afun tion dened by Choquetinadierent ontext. TheChoquetintegralisafuzzy

integralbasedonafuzzymeasure that providesalternative omputationals hemefor

aggregatinginformation.

Infa t, thisaggregation operator,in ade isionaid ontext, takesinto a ount, in

theevaluationpro ess,notonlytheimportan eofea h riteria,butalsotheimportan e

ofallsubsets. Theweightsarerepresentedbythe oeÆ ientsofthefuzzymeasure.

Denition4.1 Let,  be a fuzzy measure on N. The dis rete Choquet integral of a

fun tion g:N !IR withrespe ttoisdenedby:

C  (g (1) ;:::;g (n) )= n X i=1 (g (i) g (i 1) )(A (i) ) (3) where (:)

indi ates that the indi es have been permutated, su h that f0  g

(1)  g (2) :::g (n) g; A (i) =f(i);:::;(n)gandg (0) =0.

Figure1: TheChoquetintegral.

Forinstan e, if g 3  g 1 g 2 , then g (1) = g 3 , g (2) = g 1 and g (3) = g 2 , the fuzzy measures(A (1) )=(312);(A (2) )=(12)and(A (3)

)=(2) andthefuzzyintegral

willbe: C  (g 1 ;g 2 ;g 3 )=(g 3 g (0) )(3;1;2)+(g 1 g 3 )(1;2)+(g 2 g 3 )(2)

5 Properties of the Choquet integral

Inordertoshowthesuitabilityoffuzzyintegralsinthemulti riteriade isionaid

frame-work,wepresentin thisse tionthemainpropertiesforaggregationofthisoperator.

TheChoquetintegralisamonotoni in reasingfun tion C



dened from [0;1℄ n

in

[0;1℄andhasthefollowingproperties:

i) Boundary onditions: Thispropertyexpressesthebehavioroftheaggregation

operatorin theworstandin thebest ase.

C



(0;:::;0)=0 C



(1;:::;1)=1

ii) Idempoten e: Forallg

j

identi al(=a),thisaggregationoperatorrestitutesthe

ommonvalue(a).

C



(a;:::;a)=a

iii) Continuity : Thispropertymeansthat theChoquetintegralis ontinuouswith

respe ttoea hofitsvariables. Itmeansthat,thisoperatordoesnotpresentany

haoti rea tiontoasmall hangeinthearguments.

iv) Monotoni ity : This operatorisnon-de reasingwithrespe t toea hvariable.

g i g j =)C  (g 1 ;:::;g i ;:::g n )C  (g 1 ;:::;g j ;:::g n )

v) De omposability : Thispropertymeansthatanysubsetofelementsg2R n

an

berepla edbytheirpartialaggregationwithout hangingtheglobalaggregation.

C (n)  (g 1 ;:::;g k ;g k +1 :::;g n )=C (n)  (g;:::;g;g k +1 ;:::;g n ) whereg=C (k )  (g i ;:::;g k )

vi) Stability under the samepositive linear transformations: This property

translatesastabilityoftheoperatorfora hangeofmeasurements ale.

C  (rg 1 +t;:::;rg n +t)=rC  (g 1 ;:::;g n )+t 8r>0;8t2IR :

Fromthis property, we an say that hangingthe s aledoesnot hange the

re-sult. This is an extremely important property in utility theory, given that the

evaluationsofana tiona ordingto ea h riterionaredened withrespe tto a

positivelineartransformation. Theglobalutilityofthea tionhavethustokeep

thisproperty.

Fromboundary onditionsi),idempoten yii)andmonotoni ityiv)properties,the

Lemma 1 The Choquet integrals are omprised between the minimun and the maximunvalue. minfg 1 ;:::;g n gC  (g)maxfg 1 ;:::;g n g

Proof: From boundary onditionsi),idempoten yii)and monotoni ityiv)

prop-ertieswehave: Forg=fg 1 ;:::;g n g Let g f

=minfggbethe minimumvalueof g, and g

k

=maxfggbethemaximum

valueofg g f g i 8i2f1;:::;ng g k g i 8i2f1;:::;ng fg f ;:::;g f gfg 1 ;:::;g n gfg k ;:::;g k g

Fromthemonotoni itypropertywehave:

C  (g f ;:::;g f )C  (g 1 ;:::;g n )C  (g k ;:::;g k )

Fromtheidempoten yproperty

g f C  (g)g k asg f =min(g)andg k =max(g)then min(g)C  (g)max(g) 

Fromthislemma,we ansaythatthisaggregationoperatorisa\ ompromise

oper-ator".

Inthisse tionwementionedthepropertiesofthefuzzyintegralsrelatedtothe

rep-resentationofintera tionsbetween riteria. Now,we anillustrate,whatisunderstood

byintera tionsandhowthey anbemodelledbyfuzzyintegral,withasimpleexample.

Example2 Considerthesameexampleofthesele tionofpersonnelproblemshown

in se tion 2.2, with other andidates, A, B and C. The HR department has alist of

three riteria: age of the andidate (g

1

), number of years of professional experien ein

arelevantrelatedjob(g

2

)andnumberofyearsofhigheredu ation(g

3

). Theevaluation

involvingthese andidates isdes ribed inthe Table 2.

The HRdepartmentwantswellequilibrated andidateswithoutweakevaluations.

Thismeans,thattheideal andidateshouldbeyoung,withvariousyearsofprofessional

experien eandwithahigh levelofedu ation.

The normalized matrix 1

and the weight sum results are shown in Table 3. The

rankingofthistableisnotfullysatisfa toryfortheHRdepartment,sin ethe andidate

1

The riteriong1 shouldbe minimized,and g2;g3 maximized. We hoosehere to maximizethe

inversevalues 1

g1

andforthenormalization,wedivideg

i

bythetotalsum,i.eg 0 i = g i P n i g i

Candidate g 1 g 2 g 3 A 28 7 3 B 36 2 7 C 29 5 5

Table 2: Sele tionofpersonnelproblem.

Candidate g 1 g 2 g 3 Globalevaluation (weighted sum) A 0,36 0,50 0,20 0,36 B 0,28 0,14 0,47 0,29 C 0,35 0,36 0,33 0,35 Weight 0,35 0,35 0,3

Table3: Normalizedmatrixandweightedsumresults.

Ahasaweaknessinthenumberofyearsofhigheredu ation(seeTable2),buthasbeen

onsidered better than andidate C, whohas no weak point. The reason is that too

mu h importan eisgivento ageandthenumberofyearsofprofessionalexperien eof

the andidates,whi hareredundant. Usually, theageandtheprofessionalexperien e

haveastronglink. SolvingthisproblemwiththeChoquetintegralandanappropriate

fuzzymeasurewehave:

i) FortheHRdepartment,theageandtheprofessionalexperien earemore

impor-tantthanthenumberofyearsofhigheredu ation.

(fg 1 g)=(fg 2 g)=0:35 (fg 3 g)=0:3

ii) Be ause the riteria g

1 and g

2

are redundant, the weight attributed to the set

fg

1 ;g

2

gshould belessthanthesumoftheweightsofthepairof riteria.

(fg

1 ;g

2

g)=0:45<0:35+0:35

iii) Sin e, the riteria g

1 ;g 3 and g 2 ;g 3

have a positive intera tion, the weight

at-tributed to theset fg

1 ;g 2 gandfg 2 ;g 3

gshould begreaterthanthe sumof

indi-vidualweights. (fg 1 ;g 3 g)=0:8>0:35+0:3 (fg 2 ;g 3 g)=0:8>0:35+0:3 And(fg 1 ;g 2 g 3 g)=1.

Table4showstheresultsapplyingthisfuzzymeasureto the andidates.

TheHR ansee thattheChoquetintegralgivestheexpe tedresults.

This example exposes how easy it is to translatethe requirementsof the de ision

on-Candidate g 1 g 2 g 3 Globalevaluation (Choquet integral) A 0,36 0,50 0,20 0,32 B 0,28 0,14 0,47 0,31 C 0,35 0,36 0,33 0,34

Table 4: Choquetintegralresults.

sidersmore than 3 riteria, the evaluation of the fuzzy measure be omes diÆ ult. Is

forthisreasonthatthenextse tionintrodu ethek-orderadditivefuzzymeasureand

integral.

6 k-order additive fuzzy measure and integral

Inade isionaidproblem involvingn riteria, ifwetakeintoa ounttheintera tions

between riteria, we need to dene the 2 n

oeÆ ients of the fuzzy measure 

orre-sponding tothe2 n

subsets ofN. Thede isionmakerisnotableto givesu h amount

ofinformation. Inmost ases,hewill beabletoguesstheimportan eofea h riterion

andofea hpairof riterion.

Toover omethisproblem,Grabis h[5℄proposedtousethe on eptofk-orderfuzzy

measure.

Denition6.1 (Grabis h(97) [5 ℄) A fuzzy measure  dened on N is said to be

k-orderadditive if its orresponding pseudo-Boolean fun tion isamultilinearpolynomial

ofdegreek,i.e. itsMobiustransformw(T)=0forall(T)su hthat(T)>k,andthere

existatleastone subsetT of N ofexa tly k elementssu hthat w(T)6=0.

Thus, for the2-additivefuzzy measures, oeÆ ients of the Mobius representation

aregivenby

(i) = w(i); i2N; (4)

(ij) = w(i)+w(j)+w(ij); fi;jg2N (5)

The monotoni ity onstraints and the normalisation of the oeÆ ients of the

2-additivefuzzymeasureareformulatedasfollows:

8 > > > > > > < > > > > > > : (;)=0; X fi;jg2N (ij)+(n 2) X i2N (i)=1; (i)0; 8i;2N X j2S (ij) X j2S (j) (n 2)(i)0; 8i2N;8SNni: (6)

Intermsof theMobiustransformthese onstraintsare: 8 > > > > > > < > > > > > > : w(;)=0; X i2N w(i)+ X fi;jg2N w(ij)=1; w(i)0; 8i;2N w(i)+ X j2S w(ij)0; 8i2N;8SNni: (7)

Ifwe onsiderthe2-ordermodel,theChoquetintegralbe omes:

C  (g)= X i2N w(i)g i + X fi;jgN w(ij)(g i ^g j ) 8g2IR n (8)

The2-order aseoftheChoquetintegral,seemstobeinterestinginpra ti al

appli- ations. It permits tomodel intera tionbetween riteriawhileremainingverysimple.

Indeed,onlyn+( n

2

)=(n(n+1))=2 oeÆ ientsarerequiredtodenethefuzzymeasure.

Therearemanyappli ationsbasedonthe2-ordermodel[1, 2,3,6,7,8,9,10℄.

Areviewoftheliterature[3,6,7,11℄,showsthattherearemainlythreeapproa hes

tondthefuzzymeasures.

 Identi ation basedonsemanti s

 Identi ationbasedonlearningdata

 Identi ationbasedonthe ombinationofthesemanti sandlearningdata.

Therstapproa hisbasedonthede isionmaker'sinterpretationofthefuzzy

mea-sure. The se ond one is established on the assignment by the de ision makerof the

numeri als oreforea ha tionandea h riterion,andalsoanumeri alglobals orefor

ea h a tion,from this informationit ispossibletobuild alearningsystem tondthe

fuzzymeasure. Finallythethirdapproa hisbasedonthe ombinationofthesemanti s

andlearningdata,indeedthispro edure ombinessemanti al onsiderationsand

learn-ingdatawhi hintrodu esveryusefulinformation,redu esthe omplexityandprovides

moreeÆ ientalgorithmsto ndthefuzzymeasure.

7 MCDA methods and Fuzzy measures and integrals

Aswehaveshowninthispaper,inmanysituations,the riteria onsideredinanMCDA

evaluation pro essarenotindependent,theyintera t. Inthesesituationsthefuzzy

in-tegralbe omestheappropriateaggregationoperatortodealwiththiskindofproblems.

Forthisreasonisveryinterestingtheintrodu tionofthispowerfulaggregationoperator,

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