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10
MarcinAnholcer
Iftheaboveconditionsaresatisfied,thepairwisecomparisonmatrixAis
calledconsistent.Thecondition(1)israthereasytofulfillinpractice(the
decisionmakermaye.g.fillonlytheelementsofAabovethediagonalandthen
theremainingonesareeasilycalculated).Thecondition(2)ismuchmore
difficulttosatisfyandisthemainsourceoftheinconsistency.
ItiseasytoprovethatthematrixAisconsistentifandonlyifthereexist
positiveweightsw1,w2,…,wn(formingthevectorw)suchthat
a.j.
W.
Wj
,..1,2,…,n,j.1,2,…,n
(3)
Theelementsofwareinterpretedastheexplicitvaluesrepresentingthe
prioritiesofthedecisionvariants.Findingtheirvaluesisthusessential.Note
thatifsomevectorwdefinesthematrixAthenalsothevectorAWforevery
A>0.
1.Problemformulation
Asinreal-lifeproblemsthematrixAisveryoftennotconsistent,itis
impossibletofindthevectorw(infact,itdoesnotexist).Insuchasituationthe
goalistofindthevectorWthatdefinesthematrixBwhichisascloseas
possibletotheoriginalpairwisecomparisonmatrixA.
ThedistancebetweenmatricesAandBmaybecalculatedinvarious
ways.OneofthemethodsistocalculateSaaty3sinconsistencyindexusingthe
eigenvaluesofthe(relative)estimationerrormatrix,whichcanbeapproximated
bytherow-wisegeometricmeans[seee.g.Saaty1980;MogiandShinohara
2009].Estimationerrorsarecalculatedasthequotientsordifferencesofthe
respectiveelementsofAandB.Anotherapproach,basedontheadditivePCM
(aformulationequivalenttotheonediscussedinthispaper),maybefounde.g.
inFedrizzi,Giove[2007].
Anotherapproachistocalculatesomekindofaverageoferrors.The
mostpopularmeasureisthesquaremeancalculatedaccordingtotheformula
1
G2(A,v).(
l
π
∑
π
.Żl
∑
π
jŻl
(a.j−
vj
vi
)
2
)
2
.
(4)
Thismethodoftheinconsistencymeasurement(calledleastsquare
method,LSM)wasintroducedinthiscontextbyChuetal.[1979]andusede.g.
byAnholceretal.[2011],Bozóki[2008],Fülöp,KoczkodajandSzarek[2010],
Fülöp[2008],BozókiandRapcsàk[2008],MogiandShinohara[2009].
