A critical analysis of the eigenvalue method used to derive priorities in AHP

Acriticalanalysisoftheeigenvaluemethod

usedtoderiveprioritiesinAHP

CarlosA.BanaeCosta

aa,b,*,1,Jean-ClaudeVansnick

cCEG-IST,CentreforManagementStudiesofIST,TechnicalUniversityofLisbon,Lisbon,PortugalbDepartmentofManagement-OperationalResearchGroup,LondonSchoolofEconomics,UK

c´deMons-Hainaut,F.W.S.E.,PlaceduParc,20-7000Mons,BelgiumUniversite

Availableonline2January2007

Abstract

AlotofresearchhasbeendevotedtothecriticalanalysisoftheAnalyticHierarchyProcess(AHP),fromvariousper-spectives.However,asfarasweknow,noonehasaddressedafundamentalproblem,discussedinthispaper,concerning

themeaningofthepriorityvectorderivedfromtheprincipaleigenvaluemethodusedinAHP.TheroleofAHP’sconsis-tencyratioisalsoanalysed.

Ó2006ElsevierB.V.Allrightsreserved.

Keywords:Decisionanalysis;AnalyticHierarchyProcess;Eigenvaluemethod;Conditionoforderpreservation

1.IntroductionandobjectiveoftheanalysisSinceSaaty(1977,1980)introducedtheAnalyticHierarchyProcess(AHP),manyapplicationsinreal-worlddecision-makinghavebeenreported(cf.Zahedi,1986;Goldenetal.,1989;Shim,1989;Var-gas,1990;Saaty,2000;FormanandGass,2001;GoldenandWasil,2003;VaidyaandKumar,2006).Inparallel,AHPhasoftenbeencriticisedintheliterature,fromseveralperspectives(see,forexample,WatsonandFreeling,1982,1983;BeltonandGear,1983,1985;French,1988;Holder,1990;

*Correspondingauthor.Address:CEG-IST,CentreforMan-agementStudiesofIST,TechnicalUniversityofLisbon,Lisbon,Portugal.

E-mailaddresses:c.bana@lse.ac.uk(C.A.BanaeCosta),vansnick@umh.ac.be(J.-C.Vansnick).1ThisauthorwassupportedbyPOCTIandLSE.

Dyer,1990a,b;BarzilaiandGolany,1994;SaloandHa¨ma¨la¨inen,1997).AdebateaboutthemaincriticismsofAHPcanbefoundinBeltonandStew-art(2002)andSmithandvonWinterfeldt(2004).Saatyhasfrequentlycontestedthesecritics(see,forexample,Saatyetal.,1983;SaatyandVargas,1984;Saaty,1990,1997;SaatyandHu,1998)and,inessence,hasnotmodifiedhisoriginalmethod(seeSaaty,2005).Independentlyofouragreementwithsomeofthosecriticisms,theanalysisofwhichisbeyondthescopeofthispaper,webelievethattheelicitationofpairwisecomparisonjudgementsandthepossibilityofexpressingthemverballyarecor-nerstonesofthepopularityofAHP.

Thereis,however,akeyproblemthat,asfarasweknow,hasneverbeforebeenaddressedinthelit-erature.Itconcernsthemeaningofthepriorityvec-torderivedfromtheprincipaleigenvaluemethodusedinAHP.The‘‘AHPusesaprincipaleigenvalue

0377-2217/$-seefrontmatterÓ2006ElsevierB.V.Allrightsreserved.doi:10.1016/j.ejor.2006.09.022

C.A.BanaeCosta,J.-C.Vansnick/EuropeanJournalofOperationalResearch187(2008)1422–14281423

method(EM)toderivepriorityvectors’’(SaatyandHu,1998,p.121).FollowingSaaty,thepriorityvec-torhastwomeanings:‘‘Thefirstisanumericalrankingofthealternativesthatindicatesanorderofpreferenceamongthem.Theotheristhattheorderingshouldalsoreflectintensityorcardinalpreferenceasindicatedbytheratiosofthenumeri-calvalues(...)’’(Saaty,2003,p.86).Thissecondmeaningrequires,inourview,thattheseratiospre-serve,wheneverpossible,theorderoftherespectivepreferenceintensities,whichisnotalwaysthecaseforAHPpriorityvectors.Indeed,theratiosofAHPpriorityvaluescanviolatethisorderalbeittheratiosofalternativepriorityvalues,derivedfromthesamepairwisecomparisons,preserveit.Fromourdecision-aidperspective,thisisabasicdrawbackofAHP.Considerthefollowingcondition:

ConditionofOrderPreservation(COP):Forallalternativesx1,x2,x3,x4suchthatx1dominates2x2andx3dominatesx4,iftheevaluator’sjudgementsindicatetheextenttowhichx1dominatesx2isgreaterthattheextenttowhichx3dominatesx4,thenthevectorofprioritieswshouldbesuchthat,notonlyw(x1)>w(x2)andw(x3)>w(x4)(preserva-tionoforderofpreference)butalsothatw(x1)/w(x2)>w(x3)/w(x4)(preservationoforderofinten-sityofpreference).

Forinstance,ifx1stronglydominatesx2andx3moderatelydominatesx4,itisfromourviewfunda-mentalthat,wheneverpossible,thevectorofprior-itieswbesuchthatw(x1)/w(x2)>w(x3)/w(x4);indeed,thesejudgementsindicatethattheintensityofpreferenceofx1overx2ishigherthantheinten-sityofpreferenceofx3overx4.

WewillprovewithsimpleexamplesthattheAHPpriorityvectordoesnotnecessarilysatisfytheCOP,eventhoughitispossibletorespectthiscondition.Insuchcases,alternativepriorityvaluesthatsatisfyCOPcaneasilybefoundbyamathe-maticalprogramincludingCOPconstrains.Theparticularprogramthatweusedisnotimportantinthescopeofthispaper,sinceourintentionisnotatalltosuggestanalternativeproceduretoAHP.

NotethatanumericalscalethatsatisfiestheCOPdoesnotalwaysexist.Inourconstructiveperspec-tive,itisessentialtodetectthesesituationsanddis-

2Inthispaper,‘‘dominance’’isusedinthesenseof‘‘strictpreference’’.

cussthemwiththeevaluatorbeforeproposinganypriorityscale.AcomplementaryobjectiveofthispaperistoanalyseiftheconsistencyratiousedinAHPcanrevealsuchsituations.

Therestofthispaperisorganisedinthefollowingmanner:inSection2,wereviewtheprincipaleigen-valuemethodusedinAHPtoderivepriorityvectors;inSections3and4,wepresentsomeexamplesinwhichitwouldbepossibletosatisfytheCOP,how-ever,theAHPpriorityvectorsviolateit;inSection5,weshowthattheAHPconsistencyratioisnotsuit-ablefordetectingtheexistence(orthenon-existence)ofanumericalscalesatisfyingtheCOP;abriefconclusionispresentedinSection6.

2.Overviewoftheprincipaleigenvaluemethod(EM)LetX={x1,x2,...,xn}beasetofelementsand}‘‘apropertyorcriterionthattheyhaveincommon’’(Saaty,1996,p.24)–forexample,Xcouldbeasetofcarsand}theircomfort.Howcanwehelpaper-sonJquantifytherelativepriority(orimportance)thattheelementsofXhaveforher,intermsof}?TheEMusedinAHPtoderiveprioritiesfortheelementsofXrequiresthatanumber–denotedwij–beassignedtoeachpairofelements(xi,xj)repre-senting,intheopinionofJ,theratioofthepriorityofthedominantelement(xi)relativetothepriorityofthedominatedelement(xj)(Saaty,1997).Jisinvitedtocomparetheelementspairwiseandcanexpressherjudgementsintwodifferentways:•eithernumerically,bygivingarealnumberbetween1(inclusive)and10(exclusive)(Saaty,1989)–forexample,ifxiisaChevroletandxjaLadaandifJjudgestheChevrolettobesixtimesmorecomfortablethantheLada,thanwij=6;

•orverbally,bychoosingoneofthefollowingexpressions:equalimportance,moderatedomi-nance,strongdominance,verystrongdomi-nance,extremedominance,oranintermediatejudgementbetweentwoconsecutiveexpressions;eachverbalpairwisecomparisonelicitedisthenautomaticallyconvertedintoanumberwijasexhibitedinTable1–forexample,ifxiisaPeu-geotandxjanOpelandifJjudgesthePeugeottobemoderatelymorecomfortablethantheOpel,thenwij=3.

Duringtheelicitationprocess,apositiverecipro-calmatrix,inwhicheachelementx1,x2,...,xnofX

1424C.A.BanaeCosta,J.-C.Vansnick/EuropeanJournalofOperationalResearch187(2008)1422–1428

Table1

Converting‘‘verbaljudgements’’into‘‘numbers’’VerbalexpressionsaCorrespondingnumbersEqual

1Equaltomoderate2Moderate

3Moderatetostrong4Strong

5Strongtoverystrong6Verystrong

7Verystrongtoextreme8Extreme

9

aInSaaty(1996,2005)theverbalexpressions‘‘equaltomod-erate’’,‘‘moderatetostrong’’,‘‘strongtoverystrong’’and‘‘verystrongtoextreme’’arereplacedby‘‘weak’’,‘‘moderateplus’’,‘‘strongplus’’and‘‘very,verystrong’’,respectively.

isassignedonelineandonecolumn,canbefilledbyplacingthecorrespondingnumberattheintersec-tionofthelineofxiwiththecolumnofxj8>>>wijifxidominatesxj;<1=wijifxjdominatesxi

;

>>>:

1ifxidoesnotdominatexjandxjdoesnotdominatexi:Forexample,assumingthatforalli,j2{1,2,...,n}xidominatesxjifandonlyifi0

B

1w12ÁÁÁw1n

1W¼BB

1=w12

1ÁÁÁw2nCBCBB

ÁÁÁÁÁÁÁÁÁÁÁÁCC@ÁÁÁ

ÁÁÁÁÁÁÁÁÁCC:

1=wA1n1=w2nÁÁÁ1Inordertoassigna‘‘priority’’(ora‘‘weight’’)toeachelementxi–anumericalvaluethatwewilldenotew(xi)–theprincipaleigenvaluekmaxofmatrixWanditsnormalisedeigenvectorarecalcu-lated:thecomponentsofthisvectorarethew(xi).Thisprocedurehasaveryinterestingproperty:ifthejudgementsofJaresuchthatwijÆwjk=wikforalliHowever,cardinalconsistencyisseldomobservedinpractice.Therefore,AHPmakesuseofa‘‘consistencytest’’thatpreventsprioritiesfrombeingacceptediftheinconsistencylevelishigh.InordertomeasurethedeviationofmatrixWfrom‘‘consistency’’,aconsistencyindexCIisdefinedas

kmaxÀn/(nÀ1)andarandomindexRI(ofordern)iscalculatedastheaverageoftheCIofmanythousandsreciprocalmatrices(ofordern)randomlygeneratedfromthescale1to9,withreciprocalsforced.ThevaluesofRIformatricesofsize1,2,...,10canbefoundinSaaty(2005,p.374).TheratioofCItoRIforthesameordermatrixiscalledtheconsistencyratioCR.AccordingtoSaaty(1980,p.21),‘‘aconsistencyratioof0.10orlessisconsideredacceptable’’.Thatis,aninconsistencyisstatedtobeamatterofconcernifCRexceeds0.1,inwhichcasethepairwisecomparisonsshouldbere-examined.

Iftheelementsaretobecomparedaccordingtoseveral},theAHPproposesthatahierarchybebuiltwiththegeneralgoalontop,theelementsatthebottomandthe}atintermediatelevels.Theproceduredescribedaboveisthenrepeatedlyappliedbottom-up:tocalculateavectorofprioritiesfortheelementswithrespecttoeach}situatedatthebottomintermediatelevel;tocalculateavectorofweightsforeachclusterof}atthedifferentlev-els.Allthisjudgementalinformationisthensyn-thesisedfrombottomtotopbysuccessiveadditiveaggregations,inordertoderiveavectorofoverallprioritiesfortheelements.

3.ExamplesinwhichtheCOPisviolatedbythepriorityvectorderivedfromtheEM

WepresentinthissectiontwoexamplesprovingthattheCOPmaybeviolatedbythepriorityvectorgivenbytheEMforeachoneofthem,althoughscalesexistthatdorespectit.Example1involvesverbaljudgementsandExample2involvesnumeri-caljudgements.

Example1(Caseofverbaljudgements).LetX={x1,x2,x3,x4,x5}beasetofalternativesbetweenwhichthefollowingpairwisecomparisonswereformulatedbyapersonJ:

{x1,x2}:x1dominatesx2,equaltomoderatedominance.

{x1,x3}:x1dominatesx3,moderatedominance.{x1,x4}:x1dominatesx4,strongdominance.{x1,x5}:x1dominatesx5,extremedominance.{x2,x3}:x2dominatesx3,equaltomoderatedominance.

{x2,x4}:x2dominatesx4,moderatetostrongdominance.

{x2,x5}:x2dominatesx5,extremedominance.

C.A.BanaeCosta,J.-C.Vansnick/EuropeanJournalofOperationalResearch187(2008)1422–14281425

{x3,x4}:x3dominatesx4,equaltomoderatedominance.

{x3,x5}:x3dominatesx5,verystrongtoextremedominance.

{x4,x5}:x4dominatesx5,verystrongdominance.FromTable1,thecorrespondingpositivereci-procalmatrixis0

123B

12B1=2

B

B1=31=21BB

@1=51=41=2

54211=7

19C9CC8CCC7A1

wÃðx1Þ¼0:385;wÃðx2Þ¼0:275;wÃðx3Þ¼0:195;wÃðx4Þ¼0:125;wÃðx5Þ¼0:020;

respectstheCOP,asshowninTable2.LetusalsopointoutthatthevalueoftheconsistencyratioforthejudgementsinExample1is0.05,signifi-cantlysmallerthanthe0.10threshold;therefore,inAHP’sperspectivethejudgementsneednotberevised.

Example2(Caseofnumericaljudgements).LetX={x1,x2,x3,x4}beasetofalternativesbetweenwhichthefollowingpairwisecomparisonswerefor-mulatedbyapersonJ:{x1,x2}:{x1,x3}:{x1,x4}:{x2,x3}:{x2,x4}:{x3,x4}:

x1x1x1x2x2x3dominatesdominatesdominatesdominatesdominatesdominates

x2x3x4x3x4x42.5times.4times.9.5times.3times.6.5times.5times.

1=91=91=8

forwhichthenormalisedeigenvectorcorrespondingtoitsprincipaleigenvalueis

10

0:426

CB

B0:281C

CB

B0:165C:

CB

CB

@0:101A0:027Consequently,giventhejudgementsofJ,thepri-oritiesobtainedthroughtheEMarewðx1Þ¼0:426;wðx2Þ¼0:281;wðx3Þ¼0:165;wðx4Þ¼0:101;wðx5Þ¼0:027:

Then,inparticular,w(x1)/w(x4)%4.218andw(x4)/w(x5)%3.741,thatis,w(x1)/w(x4)>w(x4)/w(x5).GiventhatJjudgedthatx4verystronglydominatesx5andx1stronglydominatesx4,thepri-orityvectorgivenbytheEMviolatestheCOP.Yet,forexample,thescalew*Table2

Example1–valuesoftheratiosw*(xi)/w*(xj)PossibleverbaljudgementsEqualtomoderateModerate

ModeratetostrongStrong

StrongtoverystrongVerystrong

VerystrongtoextremeExtreme

Thecorrespondingpositivereciprocalmatrixis01B1=2:5BB

@1=41=9:5

2:511=31=6:5

4311=5

19:56:5CCC5A1

forwhichthenormalisedeigenvectorcorrespondingtoitsmaximaleigenvalueis010:533B0:287CBCBC:@0:139A0:041

(xi,xj)pair(s)andrespectivew*(xi)/w*(xj)ratios(x1,x2):(x1,x3):(x2,x4):(x1,x4):;

(x4,x5):(x3,x5):(x2,x5):

1.40(x2,x3):1.41(x3,x4):1.561.972.203.08

6.259.75

13.75(x1,x5):19.25

1426C.A.BanaeCosta,J.-C.Vansnick/EuropeanJournalofOperationalResearch187(2008)1422–1428

Table3

Example2–valuesofwijandw(xi)/w(xj)

wijw(xi)/w(xj){x1,x4}9.513{x2,x4}6.57{x3,x4}53.39{x1,x3}43.83{x2,x3}32.06{x1,x2}

2.5

1.86

Consequently,giventhejudgementsofJ,thepri-oritiesobtainedthroughtheEMarewðx1Þ¼0:533;

wðx2Þ¼0:287;wðx3Þ¼0:139;wðx4Þ¼0:041:

Foralli,j2{1,2,3,4}suchthatiw13butw(x3)/w(x4)respectstheCOP.Indeed,

wÃ

ðx1ÞÃ

Ã

Ã

wÃðx¼12>wðx2ÞÞ¼8>wðx3Þwðx1Þ

wÃðx¼4>4ÞwÃðx44ÞwÃðx3Þ

¼3>wÃðx2ÞwÃ

ðx¼2>wÃðx1Þ

ÃÞ

¼1:5:3Þwðx2Moreover,thevalueoftheconsistencyratiofor

thejudgementsinExample2is0.05,significantlysmallerthanthe0.10threshold;thereforeinAHP’sperspectivethejudgementsneednotberevised.4.AnalysisofoneofSaaty’sexamples

Example3.Inthissection,weanalysetheviolationoftheCOPinoneoftheexamplespresentedin

Saaty(1977,pp.254–256)andSaaty(1980,pp.40–41)toempiricallyvalidatetheEM.WerefertotheexampleofpairwisecomparisonsoftheGNPofseveralcountries,inwhich,foragivenmatrixofverbaljudgements,theprioritiesgivenbytheAHPareremarkablyclosetothenormalisedGNPvalues.Thecountriesare(Saaty’snotation)‘‘US,USSR,China,France,UK,JapanandW.Germany’’andthematrixofjudgementspresentedis

0

USUSSRChinaFranceUKJapanW:Germany1

USBUSSRBB1496655CBB1=4175534CCBCChinaFranceBB1=91=711=51=51=71=5CCBUKB1=61=55111=31=3CCBCJapanBB1=61=55111=31=3CC@1=51=373312CCAW:Germany

1=5

1=4

5

3

3

1=2

1

ThecorrespondingprioritiesarewðUSÞ¼0:427;wðUSSRÞ¼0:230;wðChinaÞ¼0:021;wðFranceÞ¼0:052;wðUKÞ¼0:052;wðJapanÞ¼0:123;wðW:GermanyÞ¼0:094:

ThesearetheprioritiesappearinginSaaty(1980),whicharealittledifferentfromthoseinSaaty(1977):0.429,0.231,0.021,0.053,0.053,0.119,and0.095,respectively.Nevertheless,inbothofthesepriorityvectorsthesamefiveviolationsoftheCOPcanbeobserved.Wewillanalysetwoofthesehereafter.

(1)Accordingtothematrixofjudgements,US

dominatesUSSR(4times)morethanJapandominatesFrance(3times).But,w(US)/w(USSR)%1.857andw(Japan)/w(France)%2.365,thatis,w(US)/w(USSR)(2)Accordingtothematrixofjudgements,Japan

dominatesChina(7times)morethanUSdominatesUK(6times).But,w(Japan)/w(China)%5.857andw(US)/w(UK)%8.212,thatis,w(Japan)/w(China)C.A.BanaeCosta,J.-C.Vansnick/EuropeanJournalofOperationalResearch187(2008)1422–1428

1427

Table4

VerificationoftheCOPPossibleverbal(xi,xj)pair(s)andrespectivew*(xi)/judgementsw*(xj)ratios

Equaltomoderate(Japan,W.Germany):1.23Moderate

(W.Germany,France):1.38(W.Germany,UK):1.38(Japan,France):1.70(Japan,UK):1.70(USSR,Japan):1.85Moderatetostrong(US,USSR):1.91

(USSR,W.Germany):2.28Strong

(USSR,France):3.14(USSR,UK):3.14(US,Japan):3.54(UK,China):3.63(France,China):3.63(US,W.Germany):4.36Strongtoverystrong(US,France):6.00(US,UK):6.00Verystrong(Japan,China):6.16(USSR,China):11.42VerystrongtoB

extremeExtreme

(US,China):21.79

wÃðUSÞ¼0:414;wÃðUSSRÞ¼0:217;wÃðChinaÞ¼0:019;wÃðFranceÞ¼0:069;wÃðUKÞ¼0:069;wÃðJapanÞ¼0:117;wÃðW:GermanyÞ¼0:095:

Letusalsopointoutthatthevalueoftheconsis-tencyratioforthejudgementsofthisexampleis0.08.

5.Discussionabouttheconsistencyratio(CR)Example4.Inthissection,wepresentanexampleinwhichitisimpossibletofindanumericalscalesatisfyingtheCOPandanalysethevalueoftheCR.LetX={x1,x2,x3,x4,x5}beasetofalternativesbetweenwhichthefollowingpairwisecomparisonjudgementswereformulatedbyapersonJ:{x1,x2}:x1dominatesx2,equaltomoderatedominance.

{x1,x3}:x1dominatesx3,strongdominance.{x1,x4}:x1dominatesx4,verystrongdominance.

{x1,x5}:x1dominatesx5,extremedominance.{x2,x3}:x2dominatesx3,equaltomoderatedominance.

{x2,x4}:x2dominatesx4,moderatedominance.{x2,x5}:x2dominatesx5,verystrongdominance.{x3,x4}:x3dominatesx4,moderatedominance.{x3,x5}:x3dominatesx5,strongdominance.{x4,x5}:x4dominatesx5,equaltomoderatedominance.

Forthissetofjudgements,itisimpossibletosatisfytheCOP.Indeed,oneshouldsimultaneouslyhave:

(1)w(x1)/w(x3)>w(x2)/w(x4),because,according

toJ’sjudgements,x1dominatesx3(strongdominance)morethanx2dominatesx4(mod-eratedominance),and

(2)w(x3)/w(x4)>w(x1)/w(x2),because,according

toJ’sjudgements,x3dominatesx4(moderatedominance)morethanx1dominatesx2(equaltomoderatedominance).Thisisimpossiblebecausetheproduct,membertomember,ofthesetwoinequalitiesgivesw(x1)/w(x4)>w(x1)/w(x4).

Inourview,thisshowsthatweareinfaceofarealcaseofjudgementalinconsistencybecause,contrarytoExamples1–3,thesetofjudgementsinthepresentexampleisincompatiblewithanumer-icalrepresentationthatguaranteesorderpreserva-tion.Andyet,thevalueoftheCRcorrespondingtothesejudgementsisverysmall(0.03),whichmeans,intheAHP’sperspective,thatthesejudgementswouldnotnecessitatetoberevised.Moreover,0.03issmallerthanthevaluesoftheconsistencyratiosforExamples1–3(0.05,0.05and0.08)inwhich,asshowninSections3and4,scalesexistthatsatisfytheCOP,unliketothepresentexampleinwhichaninconsistencyproblemundoubtedlyexists.ThisshowsthattheCRusedinAHPisnotsuitablefordetectingtheexistence(orthenon-existence)ofanumericalscalesatisfyingtheCOP.6.Conclusion

Inthisarticle,wehaveaddressedthefoundationsofAHP,byanalysingtheeigenvaluemethod(EM)usedtoderiveapriorityvector.Ourmainconclu-sionisthat,althoughtheEMisveryelegantfromamathematicalviewpoint,thepriorityvector

1428C.A.BanaeCosta,J.-C.Vansnick/EuropeanJournalofOperationalResearch187(2008)1422–1428

derivedfromitcanviolateaconditionoforderpres-ervationthat,inouropinion,isfundamentalindecisionaiding–anactivityinwhichitisessentialtorespectvaluesandjudgements.Inlightofthat,andindependentlyofallothercriticismspresentedintheliterature,weconsiderthattheEMhasaseri-ousfundamentalweaknessthatmakestheuseofAHPasadecisionsupporttoolveryproblematic.AsSaaty(2005,p.346)pointsout,‘‘thepurposeofdecision-makingistohelppeoplemakedecisionsaccordingtotheirownunderstanding’’,and‘‘...methodsofferedtohelpmakebetterdecisionsshouldbeclosertobeingdescriptiveandconsider-ablytransparent’’.

Finally,itisworthwhiletonotethatthecriticismoftheEM,presentedinthispaper,isalsovalidforanyothermethodthathasbeen(ormaybe)conceivedtoderiveavectorofprioritiesfromapairwisecomparisonmatrixonthebasisofamath-ematicaltechniquethatdoesnotintegratewhatwecalltheCOP,ordoesnotautomaticallyguaranteeitssatisfaction.References

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