Pairwise judgments consistency impact on quality of multi-criteria group decision-making with AHP[S1]

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 EM_4_2019.indd 195EM_4_2019.indd 195 13.12.2019 12:44:4813.12.2019 12:44:48 196 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 EM_4_2019.indd 196EM_4_2019.indd 196 13.12.2019 12:44:4913.12.2019 12:44:49 197 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 EM_4_2019.indd 197EM_4_2019.indd 197 13.12.2019 12:44:4913.12.2019 12:44:49 198 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 EM_4_2019.indd 198EM_4_2019.indd 198 13.12.2019 12:44:4913.12.2019 12:44:49 199 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 EM_4_2019.indd 200EM_4_2019.indd 200 13.12.2019 12:44:5013.12.2019 12:44:50 201 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. 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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] EM_4_2019.indd 211EM_4_2019.indd 211 13.12.2019 12:44:5313.12.2019 12:44:53 212 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. EM_4_2019.indd 212EM_4_2019.indd 212 13.12.2019 12:44:5313.12.2019 12:44:53