195
4, XXII, 2019
In o ma ion Managemen
10.15240/ ul/001/2019-4-013
In oduc ion
P esumably complex sys ems can be be e
unde s ood when hey a e b oken down in o
hei cons i uen elemen s and s uc u ed
hie a chically. Then, judgmen s abou hese
elemen s can be syn hesized on he basis
o hei ela i e impo ance a each le el o
he hie a chy in o a se o o e all p io i ies.
By b eaking down a eali y in o homogenous
clus e s and subdi iding hem in o smalle
ones, i is possible o in eg a e la ge amoun s
o in o ma ion in o he s uc u e o a p oblem
and o m a mo e comp ehensi e pic u e o he
whole sys em.
The e is a decision suppo me hodology
(DSM) which con o ms o he abo e
p esc ip ion. I is called he Analy ic Hie a chy
P ocess (AHP) and was de ised a he Wha on
School o Business by Thomas Saa y (1980).
I s con empo a y applica ions can be ound,
o example in Lidinska and Jablonsky (2018),
Abdelmaguid and El ashidy (2016), K amulo á
and Jablonský (2016), and Ponis e al. (2015).
This DSM is based on he pai wise judgmen s
echnique which comes om an in l uen ial pape
o Ma quis de Condo ce (1785), who used his
echnique in he elec ion p ocess (Young, 1988),
and which was popula ized by Thu s one (1927)
hanks o his i s con empo a y applica ion a
he beginning o he 20 h cen u y.
When g oup decision-making (GDM) is
aken in o conside a ion, he AHP seems
a pa icula ly a ac i e me hodology, and
al hough i has been examined nume ous imes
om he pe spec i e o i s e ec i eness and
applicabili y in GDM p ocesses (see e.g. Scala
e al., 2016; Saa y & Peniwa i, 2008; Saa y &
Va gas, 2012; Agua ón e al., 2014; Hosseinian
e al., 2012; Mo eno-Jiménez e al., 2005,
2008; Al uza a e al., 2010; Sun & G eenbe g,
2006), s ill a esea ch gap could ha e been
iden i i ed. Thus, his pape examines judgmen s
consis ency in l uence on he c edibili y
o p io i y a ios (PRs) wi hin a pa icula
p io i y ec o (PV) de i ed om inconsis en
pai wise judgmen s made by a decision make
(DM). Examina ion esul s gene alize o he
syn hesized pai wise compa ison ma ix ha
is ob ained on he basis o indi idual pai wise
compa ison ma ices o all g oup membe s.
Ha ing in mind ha a consis ency index o he
PCM deno ing g oup p e e ences canno be
g ea e han he consis ency index o he mos
inconsis en indi idual PCM i became possible
o designa e he c edibili y o he p io i y
ec o o he g oup on he basis o he mos
inconsis en indi idual PCM.
The a icle is o ganized a ound h ee
main sec ions: he in oduc o y sec ion
which elabo a es on pai wise judgmen s,
AHP, and GDM wi h applica ion o AHP; he
me hodological sec ion de o ed o he esea ch
me hodology, comp ising an illus a i e
example o he p oblem and selec ed pi alls
du ing p io i y a ios es ima ion p ocess
which builds on p eselec ed measu es o
es ima ion e o s; he in es iga ional sec ion
encompassing he esea ch ou come, i s
con ibu ion o he esea ch i eld and he
examina ion b eak h ough om he iewpoin
o o he esea ch pape s. The i nal pa o he
a icle cons i u es he sec ion ‘Conclusions‘
which summa izes examina ion i ndings.
1. Backg ound
The AHP can be conside ed o be bo h
a desc ip i e and p esc ip i e model o decision
making. I p omo es pai wise judgmen s (i.e.
alua ion on he basis o pai wise compa isons)
o c i e ia and al e na i es wi h espec o
a c i e ion. Genuinely (as p oposed by he
c ea o o AHP), he compa ison p ocess
p oceeds wi h he applica ion o a undamen al
scale o absolu e numbe s ha has been
PAIRWISE JUDGMENTS CONSISTENCY
IMPACT ON QUALITY OF MULTI-CRITERIA
GROUP DECISION-MAKING WITH AHP
Pawel Tadeusz Kazibudzki, Jiří Křupka
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196 2019, XXII, 4
In o ma ion Managemen
p o en in p ac ice i.e. Saa y’s nume ical scale
which comp ises o he in ege s om one
(equi alen o he e bal judgmen - ’equally
p e e ed‘) o nine (equi alen o he e bal
judgmen - ’ex emely p e e ed‘), and hei
ecip ocals. O he nume ical scales ha e been
conside ed also see e.g. Dong e al. (2008).
The me hodology o AHP is based on he well-
de i ned ma hema ical s uc u e o consis en
ma ices and hei associa ed p incipal igh
eigen ec o ’s (REV) abili y o gene a e ue o
app oxima e weigh s, see e.g. Me kin (1979),
Saa y and Va gas (1984).
Gene ally, he p oblem o de i ing PRs om
a pai wise compa ison ma ix (PCM) de no ed
as nxnij
aA ][ wi h elemen s jiij aaa , is o
es ima e w = [w1, w2, w3,…, wn]T on he basis o
ma ix A which comp ises a decision make ’s
pai wise judgmen s (deno ing DM p e e ences)
conce ning he impo ance o a gi en bina y
se o al e na i es. Commonly PRs wi , whe e
I = 1,…, n, a e selec ed o be posi i e and
no malized o uni y 
n
ii
w1, and he elemen s
aij o ma ix A a e hen he DM’s judgmen s
abou he PRs jiij www , whe e i, j = 1,…,n,
and n is he numbe o all al e na i es being
conside ed. In a pe ec judgmen case hen,
he p oblem can be designa ed as:
wwA 

(1)
and w can be compu ed by sol ing he
eigen ec o equa ion (1). In a pe ec case
(ma ix A is consis en ) λ is he only nonze o
eigen alue o A i.e. he nonze o solu ion o he
cha ac e is ic equa ion:

0de  IA

(2)
whe e I deno es he iden i y ma ix o o de n.
In his case, also λ = n. On he o he hand,
when he case is no pe ec (ma ix A is no
consis en ) an es ima e o he ue w is he
no malized p incipal igh eigen ec o (REV)
associa ed wi h he maximal eigen alue. Thus,
in o de o ob ain he es ima e i is needed o
sol e he gene al eigen ec o equa ion:
wwA  max

(3)
whe e λmax deno es he p incipal eigen alue
which is no smalle han n, is simple and i s
exis ence is gua an eed by he Pe on-F obenius
Theo em, see e.g. Saa y and Va gas (1984).
The ma ix o a ios A = (w
i./.w
j) is consis en ,
i and only i n is i s p incipal eigen alue and
A ∙ w = n ∙ w. Fu he , w > 0 is unique o wi hin
a mul iplica i e cons an .
I he elemen s o a ma ix A sa is y he
condi ion wij = 1/wji o all i, j = 1,…, n hen
he ma ix A is said o be ecip ocal. I i s
elemen s sa is y he condi ion wikwkj = wij o all
i, j, k = 1,…, n and he ma ix is ecip ocal, hen
i is called ca dinally ansi i e o consis en .
Ma ix A can also be only ansi i e i he
ollowing condi ions hold: (i) i o any i = 1,…, n,
an elemen wij is no less han an elemen wik
hen wij ≥ wik o i = 1,…, n, and (ii) i o any
i = 1,…, n, an elemen wji is no less han an
elemen wki hen wji ≥ w
ki o i = 1,…, n. In
he case o ecip ocal PCMs – which a e he
only accep ed PCMs o he AHP al hough
coun e a gumen s exis in li e a u e (see e.g.
Lina es e al., 2016) he wo condi ions (i) and
(ii) a e equi alen .
Fundamen ally, all heo ies a e based on
axioms, so is he AHP. I s c ea o Saa y (2006)
de i nes i e condi ions o good app oxima ions:
ecip oci y, homogenei y ( he elemen s being
compa ed mus be o he same o de o
magni ude), independency (judgmen s abou ,
o he p io i ies o , he elemen s in a hie a chy
canno depend on lowe le el elemen s), nea
consis ency and uni o m con inui y (elemen s
wi, I = 1,…, n should be ela i ely insensi i e
o small changes in he elemen s aij, only hen
good app oxima ions o aij emain wi / wj a ios).
The cen al poin o AHP and he key issue
o a heo y o choice ha is based upon AHP
is he me hodology o cap u ing (in)consis ency
o PCMs wi hin he AHP. In o de o de i e
c edible p io i y ec o s (PV) wi hin AHP, i is
necessa y o impose some bounda ies on (in)
consis ency o PCMs in ol ed in he p ocess.
Indeed, signi i can iola ion o he PCM (in)
consis ency may mislead he ue alues o
p io i y a ios wi hin he PV making he en i e
me hodology i sel useless. On he o he hand,
i does no mean ha a high consis ency o PCM
gua an ees c edible alues o PV because e en
pe ec ly consis en PCMs may no be e o ee,
see e.g. G zybowski (2016) and Temesi (2011).
Tha is why es ablishing some ela ions be ween
(in)consis ency o PCM and he c edibili y o
p io i y a ios es ima es seems so impo an .
The AHP genuine measu e o PCM (in)
consis ency belongs o Saa y (1980) and is
s ic ly ela ed o he REV, which makes i
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4, XXII, 2019
In o ma ion Managemen
especially a ac i e. I does no mean ha he
o he PCM inconsis ency measu es (called
consis ency o inconsis ency indices) do no
exis . To he con a y, a numbe o o he indices
can be ound in li e a u e, see e.g. Mizuno
(2019), Peláez, Ma ínez and Va gas (2018),
Dixi (2018), Fed izzi and Fe a i (2017),
p oposed qui e ecen ly.
A de ailed analysis o all consis ency indices
a ailable in li e a u e is beyond he scope o his
esea ch. Howe e , a eade in e es ed in a ious
app oaches o consis ency measu emen du ing
pai wise compa isons may wan o e iew hose
e e ences. I behoo es o men ion ha pai wise
judgmen s consis ency measu emen was also
a opic o mo e c oss-sec ional su eys e.g.
B unelli (2018), Kou e al. (2016). Ha ing he
pe spec i e on a scale o esea ch de o ed
o a ious ways o pai wise compa isons
consis ency iden i i ca ion, i emains o men ion
ha Saa y’s concep o pai wise compa isons
consis ency measu emen p oposed o AHP
is cu en ly sys ema ically c i icized, see e.g.
Xu e al. (2008), Koczkodaj and Szwa c (2014),
Koczkodaj and U ban (2018).
Howe e , aking in o accoun ha he
AHP c ea o ’s concep is s ill applied in he
way i was p oposed a ew decades ago,
in e es ed eade s in a de ailed pe spec i e o
Saa y’s concep , as well a mo e undamen al
analysis o he whole AHP app oach, may
wan o s udy a mo e de ailed examina ion o
his me hodology o ins ance in Wu and Kou
(2016), Kou e al. (2016), and Saa y (2008b).
Fu he discussion wi hin his a ea, o easons
o b e i y, is delibe a ely omi ed. Ins ead, some
key issues o g oup decision making wi h he
applica ion o AHP will be b ie l y depic ed.
I is a ac ha mos o eal-li e decisions
a e no made by indi iduals, bu by g oups o
indi iduals e.g. commi ees, councils, e c. F om
ha pe spec i e he ela ion be ween he quali y
o pai wise judgmen s made by indi iduals and
he quali y o he ep esen a i e judgmen o
a g oup o indi iduals is o g ea impo ance.
Thus, he p esc ip ion exis s o an indi idual
judgmen s agg ega ion in a way which enables
ob aining a ep esen a i e g oup judgmen .
The ecip ocal p ope y o he AHP plays an
impo an ole om ha pe spec i e. Gene ally,
judgmen s ha e o be combined in such a way
ha ecip ocals o he syn hesized judgmen s
a e equal o he syn heses o hese judgmen
ecip ocals.
I has been deduced ha he only unique way o
do ha is o apply he geome ic mean p ocedu e.
I can be done in wo ways (Saa y, 2008b):
 i expe s a e appoin ed as decision make s,
hen a he han combining hei indi idual
judgmen s, hei i nal ou come om
a hie a chy is syn hesized wi h applica ion
o a geome ic mean;
 on he o he hand, i he indi iduals
hemsel es ha e di e en deg ees o
impo ance i.e. o ing powe s, hei
indi idual judgmen s a e aised o hei
o ing powe and he g oup ou come is
es ablished on he basis o hei indi idual
judgmen s i.e. he weigh ed geome ic
mean is o med (Fo mulae 7 and 8).

1
1
1













n
kk
k
w
n
k
w
ijkij aa
(7)











n
k
k
n
k
w
ijkij waa k
1
1
lnexp
(8)
whe e wk deno es a p io i y o impo ance o
he indi idual.
In he la e case, he i nal ou come o
a hie a chy is compu ed wi h he applica ion
o he s anda d AHP agg ega ion i.e. wi h
applica ion o he weigh ed a i hme ic mean
( he p io i y o he pa icula al e na i e unde
i s c i e ion is weigh ed by he p io i y o i s
c i e ion(s), hen he o al p io i y o he gi en
al e na i e is de e mined by he sum o hei
weigh ed p io i ies).
I has been p o en ha applica ion o
he geome ic mean p ocedu e o indi idual
p e e ences agg ega ion is he only one which
sa is i es a numbe o impo an p ope ies
(Aczel & Saa y, 1983). I has been also p o en,
(see e.g. Liu, Zhang, & Wang, 2012; G ošelj
& S i n, 2012; Escoba , Agua ón, & Mo eno-
Jimenez, 2004; Xu, 2000), ha he CI(A) o
he g oup p e e ences canno be g ea e han
he CI(A*) o he mos inconsis en indi idual
PCM = A*, i.e.:
Cl(A) ≤ max{Cl(A1), Cl(A2),...CI(An)}.
Howe e , despi e he ele ance o
he i ndings s a ed abo e, i needs o be
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198 2019, XXII, 4
In o ma ion Managemen
s essed ha ela i ely consis en indi idual
pai wise judgmen s do no gua an ee 100%
p e e ences c edibili y de i ed he eo , see
e.g. Temesi (2011). Thus, a he hen ocusing
on inconsis ency o he g oup p e e ences,
p ima ily he consis ency o indi idual pai wise
judgmen s mus be me iculously con olled om
he pe spec i e o i s ela ionship wi h p io i y
a ios es ima ion e o s which can dis o
c edibili y o a pa icula p io i y ec o . The
necessi y o esea ch in his a ea seems o be
pa amoun .
2. Resea ch Me hodology
I is emphasized, ha ew esea ch pape s
ha e deal in dep h wi h he abo e p esen ed
p oblem i.e. he ela ion be ween a le el o
he pai wise judgmen s (in)consis ency and
he ange o possible es ima ion e o s o
es ablished p io i y a ios, see e.g. G zybowski
(2016) and Kazibudzki (2019a).
Howe e , despi e o he ele ance o i ndings
published he ein, hose esea ch s udies
concen a e on a e age es ima ion e o s wi hin
pa icula p io i y ec o s. The consequence o
such a pe spec i e is a aci assump ion ha
es ima ion e o s o pa icula p io i y a ios a e
mo e o less he same as he mean e o o
he pa icula p io i y ec o . I u ns ou , ha i
is no necessa ily ue, especially when ela i e
measu es o es ima ion e o s a e conside ed.
The examina ion o hese issues is in o de and
will be made b ie l y in he below subsec ion
en i led ‘P oblem illus a ion’, and hen
ho oughly in he pape ’s subsec ion en i led
‘Examina ion b eak h ough’. The p oblem
exempli i ca ion is i s o be conside ed.
2.1 P oblem Illus a ion
The ollowing hypo he ical no malized
p io i y ec o ( he ec o o p io i y a ios) is
conside ed: PV(w) = [0.0625, 0.1042, 0.1458,
0.1875, 0.2292, 0.2708]. I is assumed ha
he ec o e l ec s ‘ ue’ (no es ima ed) he
DM‘s ela i e p e e ence owa d six objec s
whose ela i e cha ac e is ics a e known e.g.
he s eng h o p e e ences owa d he objec s
is associa ed wi h he size o hese objec s
( hei mass, olume, ci cum e ence e c.), so
hei ela i e impo ance can be calcula ed by
di iding he pa icula objec ’s size by he o al
size o all objec s. On he basis o his ‘ ue’
PV(w), he PCM(w) is o med deno ed as A(w)
wi h elemen s wij = wi / wj:





















11.18181.44441.85712.64.3333
0.846211.22221.57142.23.6667
0.69230.818211.28571.83
0.53850.63640.777811.42.3333
0.38460.45450.55560.714311.6667
0.23080.27270.33330.42860.61
)(wA
Then, A(w) is pe u bed by pe u ba ion
ac o e which single alue in his example
is gi en e = 0.5. This echnique allows o
emula e inconsis ency du ing DMs judgmen s
conce ning objec s and is widely accep ed
o his pu pose since i s i s applica ion i.e.
Zahedi (1986). In his way he pe u bed ma ix
A(x) is ob ained, whe e xij = wije o i ≠ j and
i, j  N = {1,…,6}.





















10.59090.72220.92861.32.1667
0.423110.61110.78571.11.8333
0.34620.409110.64290.91.5
0.26920.31820.388910.71.1667
0.19230.22730.27780.357110.8333
0.11540.13640.16670.21430.31
)(xA
Fu he , he uppe iangle elemen s o A(x)
a e ounded o he closes alue o Saa y’s
scale and ecip oci y is imposed (only he
uppe iangle elemen s i.e. elemen s abo e
A(x) diagonal a e conside ed o ounding while
he lowe iangle elemen s a e compu ed as
ecip ocals o he uppe iangle elemen s).
In his way he scaled and ecip ocal A( )
is ob ained which e l ec s DMs judgmen s
conce ning objec s exp essed wi h applica ion
o he pa icula p e e ence scale, in his
example Saa y’s scale. O he scales can be
applied also, o e e ences see e.g. Dong e al.
(2008).





















123459
0.513347
0.33330.33331346
0.250.33330.3333135
0.20.250.250.333313
0.11110.14290.16670.20.33331
)( A
I should be emphasized ha A( ), on
he basis o Saa y’s consis ency philosophy,
should be conside ed as accep ably consis en
because i s CI(A( )) = 0.0670, RI(6) = 1.24
hus CR(A( )) = 0.0541 which in o ms o
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199
4, XXII, 2019
In o ma ion Managemen
an accep able le el o consis ency (Saa y
sugges ed ha CR < 0.1).
On he basis o A( ), PV( ) is calcula ed,
in his example, wi h he applica ion o he
REV me hod as he genuine AHP p io i iza ion
echnique (PT). I behoo es o men ion ha
o he PTs, which we e sugges ed in li e a u e o
his pu pose, can be also applied. Fo b e i y,
hey will no be discussed in his esea ch.
Howe e , he in e es ed eade may wan o i nd
e e ences whe e hese PTs a e sc u inized.
The mos ecen a e O bán-Mihálykó e al.
(2017), Kazibudzki (2016b), Kułakowski (2015).
Con inuing he main s eam o he
esea ch, on he basis o A( ) he ollowing
PV( ) is ob ained wi h applica ion o he REV
me hod: PV( ) = [0.0277, 0.0566, 0.1027,
0.1714, 0.2679, 0.3736], which is di e en
han PV(w) = [0.0625, 0.1042, 0.1458, 0.1875,
0.2292, 0.2708]. Ha ing hose wo ec o s
o p io i y a ios, i is possible o compu e
de ia ions among hei elemen s i.e. maximal
absolu e de ia ion (MaxAD), mean absolu e
de ia ion (MAD), minimal absolu e de ia ion
(MinAD), maximal ela i e de ia ion (MaxRD),
mean ela i e de ia ion (MRD), and minimal
ela i e de ia ion (MinRD), see o mulae in
Tab. 1.
F om he pe spec i e o his esea ch,
de o ed o designa ing he p io i y a ios
es ima e c edibili y, ou de ia ions p esen ed
in Tab. 1 i.e. MaxAD, MaxRD, MAD and MRD,
become especially signi i can because hey
enable designa ion o con i dence in e als o
‘ ue‘ p io i y a ios (as in he classic s a is ical
es ima ion heo y).
In his exclusi ely illus a i e example, he
con i dence o hose in e als is pu ely hypo he ic
because i canno be designa ed only on he basis
o one case. Howe e , i e a ions o simila cases
a e possible using Mon e Ca lo simula ions which
esul s can p o ide meaning ul da a in his ma e .
This issue is sc u inized u he in he a icle’s
subsec ion ‘Examina ion me hodology’.
Re u ning o he p oblem’s illus a ion
issue, he alues o ou de ia ions especially
signi i can o he conside ed p oblem s udy
a e p esen ed in Tab. 2. On he basis o hese
alues (Tab. 2), he hypo he ic con i dence
in e als o p io i y a ios es ima es (PRE)
can be es ablished and illus a ed (Fig. 1–2).
As can been no iced (Fig. 1–2), he hypo he ic
con i dence in e als o p io i y a ios ha e
di e en ea u es i.e. ange and symme y in
ela ion o he es ima ed p io i y a ios alues.
Those ea u es depend upon applied de ia ion.

ni
ii w wMaxAD
...,1
max,












ni
i
ii
w
w
wMaxRD
...,1
max,
















n
i
ii w
n
wMAD
1
1
,





n
ii
ii
w
w
n
wMRD
1
1
,

ni
ii w wMinAD
...,1
min,












ni
i
ii
w
w
wMinRD
...,1
min,












Sou ce: own
Tab. 1: Fo mulae o de ia ions among ‘ ue’ and es ima ed ec o s o p io i y a ios
MAD MaxAD MRD MaxRD
0.0472 0.1028 0.3239 0.5568
Sou ce: own
Tab. 2: De ia ions among ‘ ue’ and es ima ed ec o s o p io i y a ios
EM_4_2019.indd 199EM_4_2019.indd 199 13.12.2019 12:44:4913.12.2019 12:44:49

200 2019, XXII, 4
In o ma ion Managemen
No iceably, a highe sp ead o hypo he ic
con i dence in e als and hei highe
asymme y is obse ed when ela i e and/o
maximum de ia ions a e applied. Fo ela i e
de ia ions, he sp ead o hypo he ic con i dence
in e als also depends on he alue o he
pa icula p io i y a io i.e. highe alues o
p io i y a ios en ail a la ge sp ead o hei
hypo he ic con i dence in e als: he p oblem
is clea ly isible o MaxRD (Fig. 2). I is e y
impo an o no ice ha al hough he sp ead
and asymme y o he hypo he ic con i dence
in e als o ela i e de ia ions a e signi i can ,
he hypo he ic con i dence in e als o he i s
and second p io i y a io i.e.
]041.0,0209.0[
1 , ]0837.0,0427.0[
2 ,
es ablished wi h he applica ion o MRD do no
encompass he ‘ ue’ alues o he i s and
second p io i y a io which equal espec i ely
x1 = 0.0625, x2 = 0.1042.
I bea s men ioning ha he MRD has also
ano he e y una ac i e ea u e i.e. i can
mask signi i can dispe sion among pa icula
p io i y a ios de ia ion.
Fo example, le wo no malized
hypo he ic ec o s o p io i y a ios be gi en as
PV(z) = [0.2262, 0.2729, 0.1390, 0.3619], and
i s es ima e PV(s) = [0.2143, 0.3571, 0.1429,
0.2857]. In his case PV(z) is he ‘ ue‘ p io i y
ec o in ela ion o which a mean ela i e
Fig. 1: Hypo he ic con i dence in e als o PRE se wi h applica ion
o MAD and MaxAD
Sou ce: own
Fig. 2: Hypo he ic con i dence in e als o PRE se wi h applica ion
o MRD and MaxRD
Sou ce: own
0.421
0.315
0.219
0.150
0.104
0.075
0.326
0.221
0.124
0.056
0.009
0.000
0.028
0.057
0.103
0.171
0.268
0.374
0.476
0.371
0.274
0.206
0.159
0.130
0.271
0.165
0.069
0.000
0.000
0.000
0.374
0.268
0.171
0.103
0.057
0.028
0.553
0.396
0.254
0.152
0.084
0.041
0.282
0.202
0.129
0.078
0.043
0.021
0.374
0.268
0.171
0.103
0.057
0.028
0.843
0.605
0.387
0.232
0.128
0.063
0.240
0.172
0.110
0.066
0.036
0.018
0.374
0.268
0.171
0.103
0.057
0.028
MAD
MRD
MaxAD
MaxRD
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201
4, XXII, 2019
In o ma ion Managemen
de ia ion o p io i y a ios is calcula ed. In
such a si ua ion, one ecei es MRD = 0.15
which seems a a he small ela i e de ia ion.
Howe e , as can be no iced, singula ela i e
de ia ions (SRD) among p io i y a ios
wi hin PV(z) and PV(s) equal, espec i ely
SRD = [0.0527, 0.3087, 0.0280, 0.2106],
and a e highly di e gen . In consequence,
a e e sal o p io i y a ios anking is no ed i.e.
PV(z) = {3, 2, 4, 1} and PV(s) = {3, 1, 4, 2}.
This is he exempla y si ua ion which needs
p e en ion, hus i is a gued o wi hd aw ela i e
de ia ions om u he applica ion o simila
p oblems.
Ne e heless, because he abo e
p oposi ion is based only on a hypo he ic
illus a i e example, i is ho oughly examined
u he in his pape in he subsec ion en i led
‘Examina ion b eak h ough’ o mo e c edible
conclusions.
2.2 Examina ion Me hodology
Con inuing he main s eam o he esea ch,
in eal AHP applica ions, he ‘ ue’ ec o
o p io i y a ios (in he illus a i e example
deno ed as PV(w)) is unknown. The en i e
AHP concep assumes i can be es ima ed wi h
he applica ion o he selec ed p io i iza ion
echnique (PT), which in classic AHP is he REV
me hod desc ibed ea lie in his pape .
As can be no iced om he ea lie p o ided
example, an es ima e o he unknown PV can be
mo e o less c edible. In gene al, his c edibili y
gene ally depends on he applied p e e ence
scale, PT, and he consis ency o PCM on he
basis o which he es ima e o unknown PV is
de i ed. An examina ion conce ning di e ences
be ween a ious p e e ence scales and PTs
in ela ion o c edibili y o PVs ob ained wi h
hei applica ion is beyond he scope o his
esea ch. Howe e , he ela ion be ween he
consis ency o PCM and he PV es ima e
c edibili y seems pa icula ly a ac i e. The
examina ion p oceeds wi h he applica ion o
Mon e Ca lo simula ions coded and pe o med
in Wol am Ma hema ica So wa e.
Taking in o accoun he ac ha Saa y’s
concep o PCM consis ency measu emen was
se iously ques ioned (see e.g. Xu e al., 2008;
G zybowski, 2012 Koczkodaj & Szwa c, 2014;
G zybowski, 2016; Koczkodaj & U ban, 2018),
and he c edibili y o REV as he PT is sligh ly
unde mined (see e.g. Kazibudzki, 2019b; Bana
e Cos a & Vansnick, 2008; Schone & Wedley,
1989; Budescu e al., 1986; Bel on & Gea ,
1983; Johnson e al., 1979), o he Mon e Ca lo
simula ions in his esea ch, he Loga i hmic
Leas Squa es Me hod (LLSM) de eloped by
C aw o d and Williams (1980, 1985) is applied,
as he oldes al e na i e o he REV (Fo mulae
9 and 10), as well LLSM based consis ency
index CI(LLSM) p oposed by he same au ho s
(Fo mula 11), and examined by Agua ón and
Mo eno-Jimenez (2003).


 








n
i
n
ji
j
ijLLSM w
w
aw
11
2
lnmin
(9)

n
n
i
n
j
ij
n
n
j
ijLLSMi aaw
/1
11
/1
1  
















(10)

 











ji i
jij
LLSM w
wa
nn
2
log
21
2
Cl
(11)
To p ope ly examine he p oblem om he
gi en pe spec i e, he ollowing simula ion
scena io is conside ed. I behoo es o men ion
ha i s assump ions come om G zybowski
(2016) who i s de ised i s amewo k o
a simila analysis. Thus, he ollowing s eps in
he scena io a e conside ed:
S ep 1: Fo he assumed n, andomly
gene a e a ‘ ue’ [n × 1] p io i y ec o
w = [w1,…, wn]T and he co esponding ‘genuine’
PCM(w) = G(w).
S ep 2: Randomly selec an elemen wxy o
x < y o G(w), and eplace i wi h wxyeB, whe e eB
is a ela i ely signi i can e o , andomly d awn
(wi h applica ion o uni o m dis ibu ion) om he
in e al eB[2;4]. E o s o ha magni ude a e
basically conside ed as ela i ely “signi i can ”,
see e.g. Dijks a (2013), G zybowski (2016).
S ep 3: Fo e e y elemen wij, i < j ≤ n,
o he han wxy, andomly selec a alue eij o
he ela i ely small e o in acco dance wi h he
gi en p obabili y dis ibu ion  (applied in equal
p opo ions as gamma, log-no mal, unca ed
no mal, and uni o m dis ibu ion) and eplace
he elemen wij wi h he elemen wijeij whe e eij
is andomly d awn om he in e al eij[0,5;1,5]
wi h applica ion o uni o m dis ibu ion.
S ep 4: Fo all i, j such ha i < j, ound all
alues o wijeij o G(w) o he closes alue om
he selec ed scale.
EM_4_2019.indd 201EM_4_2019.indd 201 13.12.2019 12:44:5013.12.2019 12:44:50
202 2019, XXII, 4
In o ma ion Managemen
S ep 5: Replace all elemen s wij o i > j o
G(w) wi h 1/wij. The pe u bed PCM(w) in S eps
2–5 deno e as P( ).
S ep 6: On he basis o P( ) compu e he
alue o he examined consis ency index CI as
well as he es ima e o he ec o w deno ed as
de i ed om P( ) wi h applica ion o assigned
p io i iza ion echnique. Then calcula e MaxAD1
and MaxAD2 i.e. he maximum and he second
maximum absolu e de ia ion be ween p io i y
a ios o w and in acco dance wi h o mula
p esen ed in Tab. 1. Sa e he alues compu ed
in his s ep as one eco d.
S ep 7: Repea S eps 2–6 NP imes.
S ep 8: Repea he scena io NT imes.
S ep 9: Sa e all he eco ds as one da abase
i le.
The abo e p esen ed simula ion amewo k
enables examina ion o ela ions be ween
pe o mance o a gi en consis ency index
and g ea es de ia ions be ween a ‘ ue’ and
es ima ed ec o o p io i y a ios. This way
i is possible o associa e a ious alues o
a gi en consis ency index wi h highes po en ial
es ima ion e o s o ob ained p io i y a ios.
Fo o mali y, he simula ion amewo k
p esen ed abo e exac ly emula es s eps
sc u inized in he example p o ided ea lie in
his pape in subsec ion ‘P oblem illus a ion’.
All pa ame e s o he applied p obabili y
dis ibu ions in he simula ion amewo k
i.e. gamma, log-no mal, unca ed no mal,
and uni o m, a e se in such a way ha he
expec ed alue EV(eij) = 1. In his way he
simula ion examina ion and i s esul s e l ec
he easonable assump ion conce ning human
na u e i.e. decision make s judgmen s a e
mo e o less de ia ed om op imal ou come bu
‘close’ o i .
3. Resul s and Discussion
Fo b e i y i was decided o sc u inize he
examina ion esul s o n = 4. I behoo es
o men ion ha o n = 3, di ec in e ela ion
be ween consis ency indices is obse ed, see
e.g. Bozóki and Rapcsák (2008) and/o Dijks a
(2013).
3.1 Resea ch Ou come
The simula ion esul s a e p esen ed in Tab.
3 and 4. They a e based on NP = 100, and
NT = 1000.
3.2 Examina ion Con ibu ion
Ha ing he empi ical dis ibu ion o maximal
absolu e de ia ions be ween ‘ ue‘ and
es ima ed p io i y ec o s, he empi ical
con i dence in e als o pa icula p io i y a ios
can be es ablished e.g. wi h he applica ion
o ‘a e age maximum absolu e de ia ion’
es ablished du ing simula ions. On he basis o
selec ed s a is ics, he c edibili y o he p io i y
ec o can also be designa ed wi h he selec ed
ank o a quan ile. Thus, one can expec
a con i dence in e al wi h an a e age le el o
ce ain y o maximal absolu e de ia ion when
he a e age maximum absolu e de ia ion is
applied. In addi ion, one can expec a con i dence
in e al no ed by he ank o he quan ile, when
quan iles o maximum absolu e de ia ions
a e applied. No iceably, con i dence in e als
es ablished on he basis o quan iles will be
sligh ly exagge a ed because he maximum
absolu e de ia ion among gi en p io i y a ios
wi hin wo p io i y ec o s canno epea i sel
( emaining de ia ions mus be smalle ). Tha is
why ano he app oach is p oposed.
No iceable, in mul ic i e ia decision making
p ocesses, a decision make (DM) is usually
in e es ed in he mos a ac i e al e na i e. So,
he p obabili y o he highly anked al e na i e
e e sal is o g ea impo ance. Thus, i is
p oposed o apply a maximum absolu e
de ia ion and he second maximum absolu e
de ia ion o examina ion, i he isk o ank
e e sal o he i s wo al e na i es exis s, and
o examina ion pu poses, how high he isk is.
To exempli y, he ollowing hypo he ic
no malized ec o o p io i y a ios is conside ed:
PV( ) = [0.62, 0.24, 0.1, 0.04]. The PV( )
designa es he ollowing anks o e alua ed
op ions: A1 ≺ A2 ≺ A3 ≺ A4. I is assumed ha
he PV( ) was de i ed om he PCM o which
CI(LLSM) = 0.109736. In his case, a decision
make may wonde abou he p obabili y o he
highly anked op ion e e sal. In ligh o he
esea ch ou come, he answe o his inqui y
depends on he le el o ce ain y assumed by
a decision make .
When his le el equals 95%, hen he
0.95-quan ile o he maximum absolu e de ia ion
dis ibu ion o CI(LLSM) = 0.109736 equals
0.208885 (Tab. 3), and 0.95-quan ile o he
second maximal absolu e de ia ion dis ibu ion
equals 0.150579. Thus, i a di e ence be ween
he i s wo p io i y a ios o he hypo he ic PV( ) is
highe han 0.150579 + 0.208885 i.e. 0.359464
EM_4_2019.indd 202EM_4_2019.indd 202 13.12.2019 12:44:5013.12.2019 12:44:50
203
4, XXII, 2019
In o ma ion Managemen
ii h in e al
o CILLSM
A e age
CILLSM
wi hin i h
in e al
p–quan iles o MaxAD1 be ween w and
o i h in e al o CILLSM
A e age
MaxAD1
be ween
w and
p = 0.8 p = 0.9 p = 0.95 p = 0.98 p = 0.99
1 [0.0, 0.0202) 0.009789 0.055462 0.081919 0.123431 0.194786 0.241852 0.042064
2 [0.0202, 0.081) 0.050333 0.083198 0.127897 0.173985 0.230384 0.266555 0.058667
3 [0.081, 0.141) 0.109736 0.128978 0.167430 0.208885 0.257923 0.292974 0.086182
4 [0.141, 0.201) 0.170316 0.135142 0.175823 0.215673 0.260070 0.291170 0.097059
5 [0.201, 0.261) 0.230957 0.139692 0.181055 0.217441 0.260860 0.294106 0.100639
6 [0.261, 0.322) 0.290961 0.148930 0.185320 0.216985 0.258141 0.293103 0.104673
7 [0.322, 0.382) 0.351898 0.152956 0.183949 0.214520 0.257819 0.291483 0.108191
8 [0.382, 0.442) 0.411428 0.154353 0.184168 0.216551 0.261614 0.295555 0.111088
9 [0.442, 0.503) 0.472071 0.151550 0.181402 0.215974 0.259817 0.291782 0.111199
10 [0.503, 0.563) 0.532404 0.150774 0.184129 0.220709 0.265578 0.302564 0.111745
11 [0.563, 0.623) 0.591957 0.151667 0.187019 0.224346 0.271002 0.304387 0.112368
12 [0.623, 0.684) 0.652322 0.152885 0.189522 0.229630 0.275033 0.309352 0.112625
13 [0.684, 0.744) 0.713164 0.156414 0.196321 0.236259 0.286920 0.318705 0.114256
14 [0.744, 0.804) 0.773112 0.159058 0.201239 0.240192 0.290268 0.326163 0.115300
15 [0.804, 0.865) 0.833323 0.161285 0.202429 0.240903 0.290718 0.326226 0.116094
16 [0.865, 0.925) 0.893977 0.160628 0.203376 0.242114 0.294927 0.334319 0.116101
17 [0.925, 0.985) 0.954481 0.167689 0.210992 0.248777 0.297410 0.334572 0.119510
18 [0.985, 1.046) 1.014420 0.171211 0.215302 0.257097 0.307838 0.342969 0.120865
19 [1.046, 1.106) 1.075330 0.175970 0.219679 0.259016 0.312604 0.352029 0.122409
20 [1.106, 1.166) 1.135290 0.177132 0.222577 0.262351 0.312678 0.345093 0.123456
21 [1.166, 1.226) 1.194820 0.183751 0.229224 0.270164 0.324909 0.374050 0.128275
22 [1.226, 1.287) 1.256030 0.179100 0.228531 0.269188 0.317348 0.356733 0.125110
23 [1.287, 1.347) 1.316700 0.184083 0.229390 0.271408 0.324651 0.369775 0.128810
24 [1.347, 1.407) 1.376290 0.192343 0.237683 0.275558 0.337192 0.369280 0.131952
25 [1.407, 1.468) 1.437290 0.193035 0.244625 0.285832 0.336465 0.377465 0.133709
26 [1.468, 1.528) 1.497320 0.199676 0.241012 0.285629 0.345064 0.388863 0.136678
27 [1.528, 1.588) 1.557600 0.197001 0.245495 0.291895 0.348678 0.381236 0.134634
28 [1.588, 1.649) 1.618450 0.202120 0.252202 0.299193 0.359296 0.400384 0.138595
29 [1.649, 1.709) 1.678350 0.198996 0.242366 0.282109 0.342629 0.387303 0.140573
30 [1.709, oo) 2.494600 0.241975 0.296090 0.340766 0.392593 0.431660 0.164883
Sou ce: own
No e: The esul s we e gene a ed o n = 4 on he basis o he p esen ed simula ion amewo k. The ou come is based on
100,000 pe u bed ecip ocal PCMs. The simula ion scena io assumed LLSM as he PT and Saa y’s p e e ence scale.
Tab. 3: Dis ibu ion o MaxAD1 i.e. he maximal absolu e de ia ions o es ima ed
p io i y a ios in ela ion o pe o mance o he consis ency index CILLSM
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210 2019, XXII, 4
In o ma ion Managemen
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J. (2019). De ia ions dis ibu ions. RepOD.
h p://dx.doi.o g/10.18150/ epod.2906388
Ing. Pawel Tadeusz Kazibudzki, PhD.
Opole Uni e si y o Technology
Facul y o Economics and Managemen
Depa men o En e p ise O ganiza ion
and Managemen
Poland
[email p o ec ed]
doc. Ing. Jiří Křupka, PhD.
Uni e si y o Pa dubice
Facul y o Economics and Adminis a ion
Ins i u e o Enginee ing Sys ems
and In o ma ics
Czech Republic
[email p o ec ed]
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212 2019, XXII, 4
In o ma ion Managemen
Abs ac
PAIRWISE JUDGMENTS CONSISTENCY IMPACT ON QUALITY
OF MULTI-CRITERIA GROUP DECISION-MAKING WITH AHP
Pawel Tadeusz Kazibudzki, Jiří Křupka
The scope o his esea ch encompasses issues associa ed wi h g oup decision making (GDM) as
he mos challenging p ocess which en ails a ious iewpoin s and p e e ences o indi iduals ha
mus be aken in o conside a ion and somehow combined in o one meaning ul ou come. When
GDM is aken in o conside a ion, he AHP seems o be a pa icula ly a ac i e me hodology. F om
he pe spec i e o i s applica ions, an exis ing esea ch gap has been iden i i ed and examined in
his esea ch pape . Thus, he inconsis ency o judgmen s impac on p io i y ec o quali y has
been examined om he pe spec i e o g oup decision making. Examina ion esul s gene alize
o he syn hesized pai wise compa ison ma ix ha is ob ained on he basis o indi idual pai wise
compa ison ma ices o all g oup membe s. The examina ion p ocess has p oceeded wi h he
applica ion o Mon e Ca lo simula ions coded and execu ed in Wol am Ma hema ica So wa e.
Ha ing in mind ha a consis ency index o he PCM deno ing g oup p e e ences canno be g ea e
han he consis ency index o he mos inconsis en indi idual PCM i became possible o designa e
he c edibili y o he p io i y ec o o he g oup on he basis o he mos inconsis en indi idual
PCM. I is emphasized ha hus a only a ew pape s ha e deal wi h he p oblem conce ning
he ela ion be ween a le el o he pai wise judgmen s inconsis ency and he deg ee o possible
es ima ion e o s o es ablished ec o o p io i y a ios. This esea ch pape o e comes limi a ions
o o he examina ions which dis inguishes i om o he pape s and emphasizes i s no el y.
Keywo ds: G oup decision making, AHP, p io i iza ion quali y, pai wise judgmen s consis ency.
JEL Classi i ca ion: D70, C02, C15, C44, C63.
DOI: 10.15240/ ul/001/2019-4-013.
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