The Analytic Hierarchy and Analytic Network Measurement Processes

76 Pages·2010·1.85 MB·English

Chapter 4
The Analytic Hierarchy and Analytic Network
Measurement Processes: The Measurement of
Intangibles
Decision Making under Benefits, Opportunities, Costs
and Risks
ThomasL.SaatyandMariyaSodenkamp
AbstractMulticriteriathinkingdemonstratesthatinordertomakeabestchoicein
a decision,discussionandcause-effectreasoningareinadequateto learnwhatthe
bestoveralloutcomeis.TheAnalyticHierarchyProcess(AHP)anditsgeneraliza-
tiontodependenceandfeedback,theAnalyticNetworkProcess(ANP),providea
comprehensivestructureandmathematicstoincorporatemeasurementsfortangible
criteriaandderiveprioritiesforintangiblecriteriatoenableonetochooseabestal-
ternativeforadecision.Itovercomesso-calledboundedrationalitythatisbasedon
theassumptionoftransitivitybyincludinginitsstructuresandcalculations,thesen-
sitivityanddepthoffeelingsassociatedwithunderstandingandtheimaginationand
awarenessneededtoaddressalltheconcerns.TheAHPcancopewiththeinherent
subjectivityinalldecisionmaking,andmakeitexplicittothestakeholdersthrough
relativequantitativepriorities.Italsoprovidesthemeanstovalidateoutcomeswhen
measurementsareavailabletoshowthatitdoesnotdonumbercrunchingwithout
meaningfuljustification.Itcandealwiththebenefits,opportunities,costsandrisks
separatelyandbringthemtogethertodeterminethebestoveralloutcome.Onecan
alsoperformdynamicsensitivityanalysisofchangesinjudgmentstoensurethatthe
bestoutcomeisstable.
In an award from the Institute for OperationsResearch and the ManagementSci-
ences (INFORMS) given to the author in October 2008 it is written: “The AHP
hasrevolutionizedhowweresolvecomplexdecisionproblems...theAHPhasbeen
applied worldwide to help decision makers in every conceivable decision context
acrossboththepublicandprivatesectors,withliterallythousandsofreportedappli-
cations.”
ThomasL.Saaty
UniversityofPittsburgh,USAe-mail:[email protected](thischapterwhichoriginallyappeared
inInt.J.AppliedDecisionSciences,Vol.1,No.1,2008[7](Interscienceretainsthecopyright),is
basedonmaterialprovidedbythisauthorandonhisresearch)
MariyaSodenkamp
UniversityofPaderborn,Germanye-mail:[email protected](visitingstudentwho
assistedinpreparationofpaper)
C. Zopounidis and P.M. Pardalos (eds.), Handbook of Multicriteria Analysis, Applied 91
Optimization 103, DOI 10.1007/978-3-540-92828-7_4, © Springer-Verlag Berlin Heidelberg 2010
92 T.L.Saaty,M.Sodenkamp
4.1 Introduction
The Analytic Hierarchy Process (AHP) and its generalization to dependence and
feedback,theAnalyticNetworkProcess(ANP)arepsychophysicaltheoriesofmea-
surement.Thismeansthattheymaketheassumptionthatjudgmentsaboutsubjec-
tive feelings and understandingare essentially not verydifferentthan and depend
onjudgmentsaboutthephysicalworldinwhichweacquireourexperienceandun-
derstanding.Inthephysicalworldwerespondtointensitiesofoccurrence,suchas
thevaryingintensitiesofsight, soundandsmell.Theseintensities fall indifferent
threshold intervals of just noticeable differences because we are unable to detect
changeinintensityuntilastimulusisincreasedbyanoticeableamount.Judgments
mustreflectnotonlyknowledgeaboutinfluences,butalsothestrengthswithwhich
these influencesoccur.These strengthsare expressedbyus, andespeciallybyex-
pertswhohaveexperiencedthecomplexitywithwhichweareconcerned,through
judgmentsfromwhichprioritiesarederivedinrelativeformthatreflectnumerical
intensitiesthatcanbevalidatedinthosecaseswherewehavemeasurementinorder
toimproveourconfidenceintheapplicabilityofourquantifiedjudgmentsinthose
cases where measurements are not available. Measurements in science are made
onscaleswitharbitraryunitsandneedinterpretationthroughjudgmenttoindicate
thedegreetowhichtheyserveourvaluesystems.Occasionallythemeasurements
are used directly in normalizedform as priorities that reflect our values if indeed
wethinktheydo.Indecisionmakingwehavetotrade-offdifferentkindsofmea-
surementbyfilteringthemthroughourjudgmentsfromwhichprioritiesarederived
therebyreducingamultidimensionalscalingproblemtoaone-dimensionalscaleof
prioritiesof the importanceof influenceson whichouractions are based.All this
tellsusthatitisnotenoughtoadvocatetheuseofatheorywithnumbersasajusti-
fiablewaytomakedecisionsbecausejudgmentsaresubjectiveanyway.Therehas
to be validationof theprocess througha varietyof examplestomake it a science
basedonreason,quantityandmathematics,notareligionbasedonthestrengthof
authority,beliefandlots of statistics devoidofunderstanding,tradeoffsand inter-
pretation.
Tomakecomplexriskydecisionsweneednotonlyjudgmentsbutalsostructures
thatrepresentourbestunderstandingoftheflowofinfluences.Thebasicstructurein
doingthisisahierarchyfortheAHPandaninfluencenetworkofclustersandnodes
containedwithintheclustersfortheANP.PrioritiesareestablishedintheAHPand
ANP using pairwise comparisons and judgment. Many decision problems cannot
bestructuredhierarchicallybecausetheyinvolvetheinteractionanddependenceof
higher-levelelements such as objectivesandcriteria in a hierarchyon lower-level
elements.Notonlydoestheimportanceofthecriteriadeterminetheimportanceof
thealternativesas inahierarchy,butalsotheimportanceofthealternativesthem-
selvesdeterminestheimportanceofthecriteriaasinanetworkTwobridges,both
strong,butthestrongerisalsouglier,wouldleadonetochoosethestrongbutugly
oneunlessthecriteriathemselvesareevaluatedintermsofthebridges,andstrength
receives a smaller value and appearance a larger value because both bridges are
4 TheAnalyticHierarchyandAnalyticNetworkMeasurementProcesses 93
strong.Feedbackenablesustofactorthefutureintothepresenttodeterminewhat
wehavetodotoattainadesiredfuture.
Thefeedbackstructuredoesnothavethetop-to-bottomformofahierarchybut
looks more like a network, with cycles connecting its components of elements,
which we can no longer call levels, and with loops that connect a component to
itself.Italsohassourcesandsinks.Asourcenodeisanoriginofpathsofinfluence
(importance)andneveradestinationofsuchpaths.Asinknodeisadestinationof
paths of influence and never an origin of such paths. A full network can include
sourcenodes;intermediatenodesthatfallonpathsfromsourcenodes,lieoncycles,
or fall onpaths to sink nodes;andfinally sinknodes.Some networkscancontain
onlysourceandsinknodes.Stillotherscanincludeonlysourceandcyclenodesor
cycleandsinknodesoronlycyclenodes.A decisionprobleminvolvingfeedback
arisesfrequentlyinpractice.Itcantakeontheformofanyofthenetworksjustde-
scribed.Thechallengeistodeterminetheprioritiesoftheelementsinthenetwork
andinparticularthealternativesofthedecisionandevenmoretojustifythevalidity
oftheoutcome.Becausefeedbackinvolvescycles,andcyclingisaninfiniteprocess,
theoperationsneededtoderivetheprioritiesbecomemoredemandingthaniswith
hierarchies.
4.2 Paired Comparisons, The Fundamental Scale, Eigenvectors,
Consistency, Homogeneity
HowtomeasureintangiblesisthemainconcernofthemathematicsoftheAHP.In
theendwemustfit ourentireworldexperienceintooursystemofprioritiesifwe
needtounderstanditinbothitsdetailsanditsgeneralworkings.Aswesaidabove,
theAHPreducesamultidimensionalproblemintoaonedimensionalone.Decisions
aredeterminedbyasinglenumberforthebestoutcomeorbyavectorofpriorities
thatgivesanorderingofthedifferentpossibleoutcomes.Wecanalsocombineour
judgmentsorourfinalchoicesobtainedfromagroupwhenwewishtocooperateto
agreeonasingleoutcome.
4.2.1 PairedComparisonsand the Fundamental Scale
To maketradeoffsamongthe manyobjectivesandcriteria,the judgmentsthatare
usuallymadeinqualitativetermsareexpressednumerically.Todothis,ratherthan
simplyassigningaseeminglyarbitraryscoreoutofaperson’smemorythatappears
reasonable,onemustmakereciprocalpairwisecomparisonsinacarefullydesigned
scientific way. In paired comparisons the smaller or lesser element is used as the
unit, and the largeror greaterelement is estimated as a multiple of that unit with
respect to the commonpropertyor criterionforwhichthe comparisonsare made.
Inthissensemeasurementwithjudgmentsis mademorescientificallythanbyas-
94 T.L.Saaty,M.Sodenkamp
signingnumbersmoreorlessarbitrarily.Becausehumanbeingsarelimitedinsize
andthefiringsoftheirneuronsarelimitedinintensity,itisclearthatthereisalimit
ontheirabilitytocomparetheverysmallwiththeverylarge.Itispreciselyforthis
reasonthatpairwisecomparisonsaremadeonelementsoralternativesthatareclose
orhomogeneousandthemoreseparatedtheyare,themoreneedthereistoputthem
indifferentgroupsandlinkthesegroupswitha commonelementfromonegroup
toanadjacentgroupofslightlygreaterorslightlysmallerelements.Inthiswayone
cangraduallycomparegrainsofsandofvaryingsizes increasingtosmallpebbles
andlargerstones.Whendoneproperly,thelargestelementinonegroupisusedas
thesmallestoneinthenextgroup,andintheendeachgroupiscomparedseparately
andthemeasurementcombined.
Fromallthepairedcomparisons,onederivesascaleofrelativevaluesforthepri-
orities.Asweshallseebelow,duetoinevitableinconsistencyamongthejudgments
andmoreimportantlybecauseoftheneedfortheinvarianceofpriorities,itismath-
ematicallynecessarytoderivetheprioritiesintheformoftheprincipaleigenvector
ofthematrixofpairedcomparisons.
WelearnfrommakingpairedcomparisonsintheAHPthatifAis5timeslarger
than B and B is 3 times larger than C, then A is 15 times larger than C and A
dominatesC 15times.That is differentfromA having5 dollarsmorethanB and
Bhaving3dollarsmorethanCimpliesthatAhas8dollarsmorethanC.Defining
intensityalongthearcsofagraphandraisingthematrixtopowersmeasuresthefirst
kindof dominanceprecisely andneverthe second.It has definite meaningand as
weshallseebelow,inthelimititismeasureduniquelybytheprincipaleigenvector.
Thereisausefulconnectionbetweenwhatwedowithdominanceprioritiesinthe
AHPandwhatisdonewithtransitionprobabilitiesbothofwhichusematrixalgebra
tofindtheiranswers.Probabilitiesoftransitionsbetweenstatesaremultipliedand
added.To compose the priorities for the alternatives of a decision with respect to
differentcriteria,itisalsonecessarythattheprioritiesofthealternativeswithrespect
toeachcriterionbemultipliedbythepriorityofthatcriterionandthenaddedover
allthecriteria.
TheFundamentalScaleusedforthejudgmentsappliedtocomparehomogeneous
(close)elementsisgiveninTable4.1.Judgmentsarefirstgivenverballyasindicated
inthescaleandthenacorrespondingnumberisassociatedwiththatjudgment.
Judgments that represent dominance belong to an absolute scale of numbers
which unlike interval and ratio scales that can be transformedto other interval or
ratioscales respectivelyandyielddifferentnumbersthat meanthesame thing,an
absolutescaleisinvariantundertheidentitytransformationthatisitsnumberscan-
not be changed to other numbers and mean the same thing. From such numbers
prioritiescanbederivedwhichalsobelongtoanabsolutescaleofrelativenumbers
whosetotalsumisequaltoone.
Table4.2exhibitsanexampleinwhichthescaleisusedtocomparetherelative
consumptionofdrinksintheUnitedStates(donebyanaudiencemanyyearsago).
One compares a drink indicated on the left with another indicated at the top and
answers the question: How many times more, or how strongly more is that drink
consumed in the US than the one at the top? More simply which drink of a pair
4 TheAnalyticHierarchyandAnalyticNetworkMeasurementProcesses 95
Table4.1 Thefundamentalscaleofabsolutenumbers
Intensityof
Importance Definition Explanation
1 Equalimportance Two activities contribute equally to
theobjective
2 Weakorslight
3 Moderateimportance Experienceandjudgmentslightlyfa-
voroneactivityoveranother
4 Moderateplus
5 Strongimportance Experienceandjudgmentstronglyfa-
voroneactivityoveranother
6 Strongplus
7 Verystrongordemonstratedimportance An activity is favored very strongly
over another; its dominance demon-
stratedinpractice
8 Very,verystrong
9 Extremeimportance The evidence favoring one activity
overanotherisofthehighestpossible
orderofaffirmation
1.1–1.9 When activitiesare very close adecimal Perhaps a better way than assigning
isadded to1toshow theirdifference as thesmalldecimalsistocomparetwo
appropriate closeactivitieswithotherwidelycon-
trastingones,favoringthelargeronea
littleoverthesmalleronewhenusing
the1–9values.
Reciprocals Ifactivityihasoneoftheabovenonzero Alogicalassumption
ofabove numbers assigned to it when compared
withactivity j,then j has the reciprocal
valuewhencomparedwithi
dominatestheotherandhowstrongly?Ingeneral,oneusestheverbalexplanation
to developa judgmentandthenenters its numericalvalue:forexampleenter9 in
the(coffee,wine)positionmeaningthatcoffeeconsumptionisextremelymorethan
wine consumption.It is automaticthat 1/9is whatone needsto use inthe (wine,
coffee)position.Notethatwaterisconsumedalittle morethancoffee,sooneen-
ters 2 in the (water, coffee) position, and 1/2 in the (coffee,water) position. One
alwaysentersthewholenumberinitsappropriatepositionandautomaticallyenters
itsreciprocalinthetransposeposition.
Thepriorities,(obtainedinexactformbyraisingthematrixtolargepowersand
summingeachrowanddividingeachbythetotalsumofalltherows,orapproxi-
matelybyaddingeachrowofthematrixanddividingbytheirtotalandtakingthe
averageoftheresultingcolumns)areshownat thebottomofthe tablealongwith
thetruevaluesexpressedinrelativeformbydividingtheconsumptionofeachdrink
(volume)bythesumoftheconsumptionofalldrinks.Theinformationaboutactual
consumptionwasobtainedfromtheUSStatisticalAbstracts.Weseetheanswersare
verycloseandpair-wisecomparisonjudgmentsofsomeonewhoknowscanleadto
96 T.L.Saaty,M.Sodenkamp
accurateresultsofdrinkconsumption.Therearenumerousexamplesofthiskindof
validation.
Table4.2 WhichdrinkisconsumedmoreintheU.S.?
Anexampleofestimationusingjudgments:
DrinkconsumptionintheU.S.
Coffee Wine Tea Beer Sodas Milk Water
Coffee 1 9 5 2 1 1 1/2
Wine 1/9 1 1/3 1/9 1/9 1/9 1/9
Tea 1/5 2 1 1/3 1/4 1/3 1/9
Beer 1/2 9 3 1 1/2 1 1/3
Sodas 1 9 4 2 1 2 1/2
Milk 1 9 3 1 1/2 1 1/3
Water 2 9 9 3 2 3 1
Very early in the history of the subject, T.L. Saaty and M. Khouja did the fol-
lowingexerciseonanairplanein1973.Theysimplyusedtheircommonknowledge
abouttherelativeinfluenceandstandingofthesecountriesintheworldandwithout
referringtoanyspecificeconomicdatarelatedtoGNPvalues.Thetworesultsare
closeanddemonstratethatthegeneralunderstandinganinterestedpersonhasabout
a problemcan beused to advantageto makefairlygoodestimates throughpaired
comparisons.
Table4.3givesthejudgmentsusingtheAHP 1–9scaleandTable4.4provides
thederivedpriorities,theactualandrelativeGNPvalues.
Table4.3 Pairedcomparisonsoftherelativedominanceinwealthofsevennations
U.S U.S.S.R China France U.K Japan W.Germany
U.S 1 4 9 6 6 5 5
U.S.S.R 1/4 1 7 5 5 3 4
China 1/9 1/7 1 1/5 1/5 1/7 1/5
France 1/6 1/5 5 1 1 1/3 1/3
U.K 1/6 1/5 5 1 1 1/3 1/3
Japan 1/5 1/3 7 3 3 1 2
W.Germany 1/5 1/4 5 3 3 1/2 1
The reader may now want to know how the foregoing integer-valued scale of
responseusedinmakingpairedcomparisonjudgmentscanbederivedmathemati-
callyfromthewell-knownpsychophysicallogarithmicresponsefunctionofWeber-
Fechner.Foragivenvalueofthestimulus,themagnitudeofresponseremainsthe
sameuntilthevalueofthestimulusis increasedsufficientlylargeinproportionto
thevalueofthestimulus,thuspreservingtheproportionalityofrelativeincreasein
stimulus for it to be detectable for a new response. This suggests the idea of just
noticeabledifferences(jnd),wellknowninpsychology.
4 TheAnalyticHierarchyandAnalyticNetworkMeasurementProcesses 97
Table4.4 Outcomeofestimatedrelativewealthandtheactualandrelativevalues
Normalized ActualGNP Normalized
eigenvector (1972) GNPvalues
U.S 0.427 1,167 0.413
U.S.S.R 0.230 1,635 0.225
China 0.021 1,120 0.043
France 0.052 1,196 0.069
U.K 0.052 1,154 0.055
Japan 0.123 1,294 0.104
W.Germany 0.094 1,257 0.091
Toderivethevaluesinthescalestartingwithastimuluss successivemagnitudes
0
ofthenewstimulitaketheform:
Δs
s =s +Δs =s + 0s =s (1+r)≡s α
1 0 0 0 0 0 0
s
0
s =s +Δs =s (1+r)=s (1+r)2=s α2
2 1 1 1 0 0
.
.
.
sn=sn−1α=s0αn (n=0,1,2,...)
Weconsidertheresponsestothesestimulitobemeasuredonaratioscale(b=0).
AtypicalresponsehastheformM =alogαi,i=1,...,n,oroneafteranotherthey
i
havetheform:
M =alogα, M =2alogα,..., M =nalogα
1 2 n
We take the ratios M/M i=1,...,n of these responses in which the first is
i 1
thesmallestandservesastheunitofcomparison,thusobtainingtheintegervalues
1,2,...,nofthefundamentalscaleoftheAHP.Itappearsthatnumbersareintrinsic
toourabilitytomakecomparisons,andthattheywerenotaninventionbyourprim-
itiveancestors.Wemustbegratefultothemforthediscoveryofthesymbolism.In
alessmathematicalvein,wenotethatweareabletodistinguishordinallybetween
high,mediumandlowatonelevelandforeachoftheminasecondlevelbelowthat
alsodistinguishbetweenhigh,mediumandlowgivingusninedifferentcategories.
We assign the value one to (low,low) which is the smallest andthe valuenine to
(high,high)whichisthehighest,thuscoveringthespectrumofpossibilitiesbetween
twolevels,andgivingthevaluenineforthetopofthepairedcomparisonsscaleas
comparedwiththelowestvalueonthescale.Becauseofincreaseininconsistency
whenwecomparemorethanabout7elements,wedon’tneedtokeepinmindmore
than7±2elements.ThiswasfirstconjecturedbythepsychologistGeorgeMillerin
the1950’s.Finally,wenotethatthescalejustderivedisattachedtotheimportance
weassigntojudgments.Ifwehaveanexactmeasurementsuchas2.375andwant
98 T.L.Saaty,M.Sodenkamp
touseitasitisforourjudgmentwithoutattachingsignificancetoit,wecanuseits
entirevaluewithoutapproximation.
InthejudgmentmatrixA,insteadofassigningtwonumbersw andw andform-
i j
ingtheratiow/w weassignasinglenumberdrawnfromtheFundamentalScale
i j
ofabsolutenumberstorepresenttheratio(w /w )/1.Itisanearestintegerapprox-
i j
imationtotheratiow/w .Thederivedscalewillrevealwhatw andw are.Thisis
i j i j
acentralfactabouttherelativemeasurementapproach.Itneedsafundamentalscale
toexpressnumericallytherelativedominancerelationship.Thegeneraleigenvalue
formulationisobtainedbyperturbationofthefollowingconsistentformulation:
⎡ ⎤⎡ ⎤ ⎡ ⎤
w1 ··· w1 w w
⎢w.1 w.n ⎥⎢ .1⎥ ⎢ .1⎥
Aw=⎣ .. .. ⎦⎣ .. ⎦=n⎣ .. ⎦=nw
wn ··· wn w w
w1 wn n n
whereAhasbeenmultipliedontherightbythetransposeofthevectorofweights
w=(w ,...,w ).Theresultofthismultiplicationisnw.Thus,torecoverthescale
1 n
fromthematrixofratios,onemustsolvetheproblemAw=nwor(A−nI)w=0.
Thisisasystemofhomogeneouslinearequations.Ithasanontrivialsolutionifand
only if the determinant of A−nI vanishes, that is, n is an eigenvalue of A. Now
Ahasunitranksinceeveryrowisa constantmultipleofthefirst row.Thusallits
eigenvaluesexceptonearezero.Thesumoftheeigenvaluesofamatrixisequalto
itstrace,thatis,thesumofitsdiagonalelements.InthiscasethetraceofAisequal
ton.Thusnis aneigenvalueofA,andonehasa nontrivialsolution.Thesolution
consistsofpositiveentriesandisuniquetowithinamultiplicativeconstant.
Theforegoingmatrixofratiosofmeasurementsisconsistent.Itsentriessatisfy
the relationshipa a =a forall i,j,k. Note that theratio oftwo readingsfrom
ij jk ik
a ratio scale is an absolute (dimensionless) number. If we were to use judgment
we would estimate this absolute number by using the Fundamental Scale of Ta-
ble 4.1.When we use judgmentwe no longercan ensure consistency.It becomes
importantforustoknowhowinconsistentweareandwhicharethemostinconsis-
tent judgmentsand howthey can be changedto improvethe consistency.But our
knowledgemaynotbeadequatetocorrectourinconsistencyasneeded.Iftheincon-
sistency remainsveryhighdespite the changeswe make that are compatiblewith
our understanding,we cannot make a decision. The priorityweights are obtained
directly by adding and normalizing to one the sum of the rows of the matrix, or
anyofitscolumns.Theintransitivityofinfluences(howmuchAdominatesBand
howmuchBdominatesCandthenhowmuchCdominatesA)cannotoccurwhen
thejudgmentsareconsistent.However,whenthejudgmentsareinconsistent,such
dominancemayhappenalongwiththefactthata a =a foralli,j,k nolonger
ij jk ik
holds.Itisknownthatthedifferentordertransitivityofinfluencescanbemeasured
byraisingthematrixtodifferentpowers.Eachpowerofthematrixyieldsaset of
prioritiesobtainedasthenormalizedsumofitsrows.Itisnotdifficulttoshowthat
theaveragepriorityoftheallthesepriorityvectorsistheirCesarosumthatleadsto
takingthelimitingpowerofthematrix.Perron’stheoryaboutpositivematricestells
usthatthislimitistheprincipaleigenvectorofthematrixthusrequiringustosolve
4 TheAnalyticHierarchyandAnalyticNetworkMeasurementProcesses 99
theprincipaleigenvalueproblemforourpositivematrix.Thisshowsthattheprin-
cipal eigenvectoris a necessaryconditionforderivingprioritiesfrominconsistent
judgments.
Associatedwiththeweightsisaninconsistencyindex.Theconsistencyindexof
a matrix is given by C.I.= λmax−n ≡ μ. The consistency ratio (C.R.) is obtained
n−1
byformingtheratioofC.I.andtheappropriateoneofthefollowingsetofnumbers
showninTable4.5,eachofwhichisanaveragerandomconsistencyindexcomputed
forn≤10forverylargesamples.Theycreaterandomlygeneratedreciprocalmatri-
cesusingthescale1/9,1/8,...,1/2,1,2,...,8,9andcalculatetheaverageoftheir
eigenvalues.ThisaverageisusedtoformtheRandomConsistencyIndexR.I.Table
4.5showsthevaluesobtainedfromonesetofsuchsimulationsandalsotheirfirst
orderdifferences,formatricesofsize1,2,...,15.Ofcoursewedonotrecommend
comparingmorethan7itemsinanysinglematrix.
Table4.5 Randomindex
Order 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
R.I. 0 0 0.52 0.89 1.11 1.25 1.35 1.40 1.45 1.49 1.52 1.54 1.56 1.58 1.59
Firstorder 0 0.52 0.37 0.22 0.14 0.10 0.05 0.05 0.04 0.03 0.02 0.02 0.02 0.01
differences
Figure4.1belowisaplotofthefirsttworowsofTable4.5.Itshowstheasymp-
toticnatureofrandominconsistency.
1.8
1.6
y
c 1.4
n
e
t 1.2
s
si
n 1.0
o
c
n 0.8
i
m
o 0.6
d
n 0.4
a
R
0.2
0.0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
Number of elements compared
Fig.4.1 Plotofrandominconsistency
100 T.L.Saaty,M.Sodenkamp
Since it would be pointless to try to discern any priority rankingfrom a set of
random comparison judgments, we should probably be uncomfortableabout pro-
ceedingunlesstheconsistencyindexofapairwisecomparisonmatrixisverymuch
smaller than the correspondingrandomindexvalue in Table 4.5. The consistency
ratio(C.R.)ofapairwisecomparisonmatrixistheratioofitsconsistencyindex∼
tothecorrespondingrandomindexvalueinTable4.5.Thenotionoforderofmag-
nitude is essential in any mathematical considerationof changes in measurement.
Whenonehasanumericalvaluesaybetween1and10forsomemeasurementand
onewishestodeterminewhetherchangeinthisvalueissignificantornot,onerea-
sonsasfollows:Achangeofawholeintegervalueiscriticalbecauseitchangesthe
magnitudeandidentityoftheoriginalnumbersignificantly.Ifthechangeorpertur-
bationinvalueisoftheorderofapercentorless,itwouldbesosmall(bytwoorders
ofmagnitude)andwouldbeconsiderednegligible.Howeverifthisperturbationis
adecimal(oneorderofmagnitudesmaller)wearelikelytopayattentiontomodify
theoriginalvaluebythisdecimalwithoutlosingthesignificanceandidentityofthe
originalnumberaswefirstunderstoodittobe.Thusinsynthesizingnearconsistent
judgmentvalues,changesthataretoolargecancausedramaticchangeinourunder-
standing,and valuesthat are toosmall cause nochangein ourunderstanding.We
areleftwithonlyvaluesofoneorderofmagnitudesmallerthatwecandealwithin-
crementallytochangeourunderstanding.Itfollowsthatourallowableconsistency
ratioshouldbenotmorethanabout0.10foramatrixlargerthan5by5,8fora4by
4matrixand5fora3by3matrix.Thisrequirementcannotbemadesmallersuch
as1or0.1withouttrivializingtheimpactofinconsistency.Butinconsistencyitself
isimportantbecausewithoutit,newknowledgethatchangespreferencecannotbe
admitted.Assumingthatallknowledgeshouldbeconsistentcontradictsexperience
thatrequirescontinuedrevisionofunderstanding.
If the C.R. is larger than desired, we do three things: 1) Find the most incon-
sistentjudgmentinthematrix(forexample,thatjudgmentforwhichε = aijwj is
ij wi
largest),2)Determinetherangeofvaluesto whichthat judgmentcanbechanged
correspondingtowhichtheinconsistencywouldbeimproved,3)Askthejudgeto
consider,ifhecan,changehisjudgmenttoaplausiblevalueinthatrange.Ifheis
unwilling,wetrywiththesecondmostinconsistentjudgmentandsoon.Ifnojudg-
mentischangedthedecisionispostponeduntilbetterunderstandingofthestimuli
is obtained. Judges who understand the theory are always willing to revise their
judgmentsoftennotthefullvaluebutpartiallyandthenexaminethesecondmost
inconsistentjudgmentandsoon.Itcanhappenthatajudge’sknowledgedoesnot
permittheimprovementofconsistencyandmoreinformationisrequiredtoimprove
thatconsistency.
Beforeproceedingfurther,thefollowingobservationsmaybeusefulforabetter
understandingoftheimportanceoftheconceptofalimitonourabilitytoprocessin-
formationandalsochangeininformation.Thequalityofresponsetostimuliisdeter-
minedbythreefactors.Accuracyorvalidity,consistency,andefficiencyoramount
ofinformationgenerated.Ourjudgmentis muchmoresensitiveandresponsiveto
large perturbations. When we speak of perturbation, we have in mind numerical
changefromconsistentratiosobtainedfrompriorities.Thelargertheinconsistency