Bayesian best-worst method

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Delft University of Technology

Bayesian best-worst method

A probabilistic group decision making model

Mohammadi, Majid; Rezaei, Jafar

DOI

10.1016/j.omega.2019.06.001

Publication date

2019

Document Version

Final published version

Published in

Omega (United Kingdom)

Citation (APA)

Mohammadi, M., & Rezaei, J. (2019). Bayesian best-worst method: A probabilistic group decision making

model. Omega (United Kingdom), 96, [102075]. https://doi.org/10.1016/j.omega.2019.06.001

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ARTICLE

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JID:OME [m5G;August31,2019;6:6]

Omega xxx (xxxx) xxx

ContentslistsavailableatScienceDirect

Omega

journalhomepage:www.elsevier.com/locate/omega

Bayesian

best-worst

method:

A

probabilistic

group

decision

making

model

R

Majid

Mohammadi

∗

,

Jafar

Rezaei

Faculty of Technology, Policy and Management, Delft University of Technology, The Netherlands

a

r

t

i

c

l

e

i

n

f

o

Article history: Received 10 December 2018 Accepted 3 June 2019 Available online xxx Keywords: Best-worst method Multi-criteria decision-making Bayesian hierarchical model Generalizability

Group decision-making

a

b

s

t

r

a

c

t

The best-worst method (BWM) is a multi-criteria decision-making method which finds the optimal weightsofaset ofcriteriabasedonthepreferencesofonlyone decision-maker(DM)(or evaluator). However,itcannotamalgamatethepreferencesofmultipledecision-makers/evaluatorsintheso-called groupdecision-makingproblem.AtypicalwayofaggregatingthepreferencesofmultipleDMsistouse theaverageoperator,e.g.,arithmeticorgeometricmean.However,averagesaresensitivetooutliersand providerestrictedinformationregardingtheoverallpreferencesofallDMs.Inthispaper,aBayesianBWM isintroducedtofindtheaggregatedfinalweightsofcriteriaforagroupofDMsatonce.Tothisend,the BWMframeworkismeaningfullyviewedfromaprobabilisticangle,andaBayesianhierarchicalmodelis tailoredtocomputetheweightsinthepresenceofagroupofDMs.Wefurtherintroduceanew rank-ingschemefordecisioncriteria,calledcredalranking,where aconfidencelevelisassignedtomeasure theextenttowhichagroupofDMsprefersonecriterionoveroneanother.Aweighteddirectedgraph visualizesthecredalrankingbasedonwhichtheinterrelationofcriteriaandconfidencesaremerely un-derstood. Thenumerical example validatesthe resultsobtained bythe Bayesian BWMwhileit yields muchmoreinformationincomparisontothatoftheoriginalBWM.

1. Introduction

Multi-criteria decision-making (MCDM) is a sub-discipline of Operations Research, which has growingly gained momentum since its genesis.In atypical MCDM problem, anumber of alter-natives are evaluated basedon ahandful numberofcriteria. The evaluationisusuallyperformedbasedontheelicitationof prefer-encesofadecisionmaker(DM)andcommonlyresultsinsorting, ranking, or selecting the alternative(s). Inorder to do the evalu-ation, we need to ﬁnd the performance of the alternatives with respect to the criteria, which is called the performance matrix, and the importance (weight) of the criteria. Finding the perfor-mancematrix usually followsa simpleyet crucialdatacollection approach.Weightdeterminationisusuallydonebasedonthe pref-erences of the actual DM. There exist several preference elicita-tionmethodstoinfertheweightsofthedecisioncriteriabasedon the preferences of the DM,including the analytic hierarchy pro-cess (AHP) [1], the analytic network process (ANP) [2], the sim-plemulti-attributeratingtechnique(SMART)[3,4],Swing[5],FARE

R_{This manuscript was processed by Associate Editor Triantaphyllou.} ∗ _{Corresponding author.}

E-mail addresses: m.mohammadi@tudelft.nl (M. Mohammadi), J.rezaei@tudelft.nl

(J. Rezaei).

[6],CILOSandIDOCRIW[7],tonamejustafew(see [8]formore MCDMmethods). One ofthe mostrecentlydeveloped preference elicitationmethodsisthebest-worstmethod(BWM)developedby Rezaeiin2015[9,10],whichisapairwisecomparison-basedMCDM method.

WhenwehaveasingleDM,theelicitedpreferencesaredirectly usedinthedecisionanalysiswhileincorporatingtheelicited pref-erencesisnota straightforwardstepwhenthereareseveralDMs. ThelattercaseisusuallycalledgroupMCDM[11–13].Wecan clas-sify group MCDMproblemsinto two categories.Inthe first cate-gory,whichhasanormativeapproach,agroupofDMsseeksa so-lutionwhichsomehowrepresentstheopinionofthewholegroup. Inthesecondcategory,whichisofadescriptiveapproach,wewant tohave a clearunderstanding of thepreferences ofthe DMs. An example of the first category is when a number of DMs from a supply chain management department of a company decides on selectingthe best suppliersforsome materials used in the com-pany[14], whileanexample fromthesecond categoryiswhen a researchertriestounderstandtheimportanceofthecriteriawhich define the quality of passenger transport transit nodes [15]. The mainfocus ofthisstudy ison group MCDM,where wehave the preferencesofagroupofDMs,whetheritisusedforanormative oradescriptiveapproach.

https://doi.org/10.1016/j.omega.2019.06.001

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Fortheweight elicitationmethods thatare basedonthe pair-wise comparison(PC),there are two classesof techniqueswhich can be used to reconcile the discrepancy among DMs [16,17]. Theﬁrstapproach istheaggregation ofindividualjudgment(AIJ)

[18,19],inwhichthePCs ofdifferentDMsarefirstintegratedinto one,andtheresultingaggregatedPCisthentreatedasasingleDM problemandevaluationis performedaccordingly.Theother class istheaggregationofindividualpriorities(AIP)[20–24].IntheAIP, aweightvectorisfirstcalculatedforeachDM,andtheconsequent weightsarecombinedtoresultinasingleweightvector.Themost popular technique to find the optimal weight forthe AIP is the arithmeticmean[25](forothertechniquesofaggregation,see,for instance[26]).BothAIJandAIPapproachesresultinaweight vec-torwhichrepresentsthepreferencesofthewholegroup.Although botharepractically simple,we losemuchinformationduetothe aggregation.Thatistosay,weusethecentralityfeatureandignore thedispersion property. On top ofthat, averages are sensitive to outliers.Therefore,evenifone decision-makerhasdifferent pref-erencesfrom the entiregroup, he/shewill significantly influence theoverallaggregatedpreferencesofallDMs.

In this study,we propose a novel approach forgroup MCDM. TheproposedapproachisparticularlypresentedfortheBWMdue toitsparticularfeatures. Thepairwise comparisonvectors associ-atedwitheachDMintheBWMcontainintegersonly;hence,they canbe modeled usingthemultinomial distribution. Nevertheless, the proposed approach can be extended for other MCDM meth-odswithsome efforts. Morespecifically,theBayesianBWMis in-troducedwhich can solve the group MCDM problem. The inputs totheBayesian BWMareidenticalto thoseoftheoriginal BWM, whichare the pairwise comparisons.The output is, onthe other hand,theoptimalaggregatedfinalweightsreflectingthetotal pref-erencesofallDMsalongwiththeconfidencelevelforrankingthe criteria.

SincetheBayesianBWMisstochastic,theinputsandoutputsof themethodneedtobemodeledusingprobabilitydistributions.In particular,wemodelthepairwisecomparisonsusingthe multino-mialdistribution,andtheﬁnalaggregatedweightsbytheDirichlet distribution.We further demonstratethat such modeling, though different,is identicalto what is expectedin theMCDM, and the BWMinparticular.

Based on the inputs and required outputs, a Bayesian hierar-chicalmodelisdevelopedto ﬁndtheoptimalweights ofallDMs andthe aggregated ﬁnal weight at once. The proposed model is distinctfromthatoftheAIJinwhichvariousPCMsare combined toreacha consensus matrix.Inthe AIJ,one needs toaccept that someDMscompromiseinordertogetaunanimousranking. How-ever,we merelyviewvarious DMsasstatisticalsamplesbasedon whichthecriteriaareprobabilisticallyevaluated.Thecredalranking

isfurtherintroducedinwhicheachpairofcriteriahasarelation, e.g., < or >,withaconfidencelevel.Theconfidencelevel repre-sentstheextenttowhichonecanbecertainaboutthesuperiority ofacriterion overoneanother. Theconfidencelevel iscomputed basedontheBayesiantestthatisespecially-tailoredbasedonthe proposedhierarchicalmodel.Aweighteddirectedgraphvisualizes theoutcomeofthecredalranking.

The main contributionof thisstudyisto propose a novel ap-proachingroupMCDMandtoapplyBayesianstatisticstoMCDM. ThisapproachisusedfortheBWM,whichisasigniﬁcant empow-ermentforthemethodforitsuseinthecontextofgroup decision-making.TheproposedBayesianBWMisparticularlyverypowerful when the goalis to describe the preferences ofa group ofDMs (whocanbetheactualDMs,experts,orusers).

The remainder of this article is structured as follows.

Section2 containstheoriginal best-worstmethod andthe corre-spondingoptimization problem to obtain the optimalweights of thecriteriaforoneDMonly. InSection 3,weprovide the

proba-bilisticinterpretation of theBWMinputs andoutputs andjustify thatsuchaninterpretationwouldpreservetheunderlyingideasin theoriginalBWM.Section4isdedicatedtotheproposedBayesian model,andwepresentthecredalrankinginSection5.The numer-ical exampleregardingtheproposed model isgiveninSection 6, andthearticleisconcludedinSection7.

2. Best-worstmethod

The BWM is a relatively new MCDM method [9,10]. One of the most popular pairwise comparison-based MCDM methods istheAHP[1]whichneedstohavethepairwisecomparisonofall thendecisioncriteriatogether,i.e.,n(n-1)/2pairwisecomparisons. Incontrast,theBWMneeds onlytheso-calledreferencepairwise comparisons,i.e., 2n-3pairwise comparisons.Otherthan this fea-ture of the BWM, which makes it a more data eﬃcient method comparedtoAHP,ithasseveralotherinterestingfeatures.Byﬁrst selecting the best andthe worst criteriaandthen comparing all the other criteria withthese two criteria, it gives a structure to theproblem.SuchstructurehelpstheDMtoprovidemorereliable pairwisecomparisons [9].Furthermore,theparticularstructureof theBWMleadstotwovectorscontainingonlyintegers,which pre-ventsafundamentaldistanceproblemassociated withtheuseof fractions inpairwise comparisons [27].The original BWMis pre-sentedasanon-linearoptimizationproblem[9],whilethereexists alinearapproximation[10],amultiplicativeversion[28],andsome hybridversionssuchasBWM-MULTIMOORA[29]andBWM-VIKOR. The method has also been extensively used in many real-world applications including, butnot limitedto, transport andlogistics

[30–32], supply chain management [33–39], technology manage-ment[40],riskmanagement[41],scienceandresearchassessment

[42,43],andenergy[44,45](see [46] formorerecentadvancesin theBWManditsapplications).

SincethetwovectorsprovidedbyeachDMintheBWMmight representdifferentcomparisons(withdifferentbestsandworsts), the AIJ is not a proper way of aggregating the preferences of a group of DMs forthis method.Almost all applications presented inexistingliteratureusetheAIPfortheaggregation,i.e.,the arith-meticmeanoftheweightsofthecriteriaobtainedfromthe indi-vidual DMs.There exists anumber ofresearcherswho have pro-poseddifferent waysfor thecaseof group decision-makingwith theBWM[47,48].However, noneofthem hasproposeda wayto ﬁnd the overall weights of the group in a probabilistic environ-ment.

ThestepsrequiredfortheoriginalBWMareasfollows[9].

Step1: TheDMneedstoprovideasetofdecisioncriteriaC=

{

c1,c2,...,cn

}

.

Step2: TheDMselectsthebest(cB)andtheworst(cW)criteria

fromC.

Inthisstep,theDMonlyselectsthebestandtheworst fromthecriteriasetCidentiﬁed intheﬁrst step.The DMdoesnotconductanypairwisecomparisoninthis stage.BasedontheDM’spreference,thebestcriterion isthemostimportantorthemostdesirablewhilethe worst criterion istheleast importantortheleast de-sirablecriterionamongothers.

Step3: The DM conducts the pairwise comparison between thebest(cB)andtheothercriteriafromC.

In this step,the DM calibrates his/her preferences of the best criterion to the other criteria by a number betweenone andnine, whereone means equally im-portant and nine means extremely more important. Thepairwisecomparisonleadstothe“Best-to-Others”

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vectorAB as AB=

aB1,aB2,...,aBn

(1)

whereaBjrepresentsthepreferenceofthebest(cB)to

thecriterionc_j_∈C.

Step4: The DM conducts the pairwise comparison between theworst(cW)andtheothercriteriafromC.

Similar to Step 3, the DM needs to calibrate his/her preferences ofthe other criteriaover theworst crite-rionbyanumberbetweenoneandnine.Theresultof thisstepisthe“Others-to-Worst” vectorAWas AW=

a1W,a2W,...,anW

_T

(2)

where ajW represents the preference of the criterion cj∈Covertheworst(cW).

Step5: Obtainingtheoptimalweightsw∗₌

(

w∗₁_,w∗₂_,_._._._,w∗_n

)

. Given AB and AW, a weight vector w∗ must be

com-puted. The weight vector must be in the neighbor-hood ofthe equationswB/wj=aB j and wj/wW=ajW

for j=1,2,.,n.Thus,onecanminimizethemaximum absolute differences

|

wB

wj − aB j

|

and

|

w_j

wW − ajW

|

for all

j₌1_,2_,_._._._,n.Besides,thenon-negativityand sum-to-onepropertyoftheweightvectormustbefulﬁlled.As a result, the following optimization problemcan ﬁnd theoptimalweightvectorw∗[9]

min w maxj

wB wj − aB j

,

wj wW − ajW

s.t. n j=1 wj=1, wj≥ 0

∀

j=1,2,...,n. (3)

Similarly, the weightvector can alsobe calculated by thefollowingproblem[10]

min ξ,w

ξ

s.t.

wB wj − aB j

≤

ξ ∀

j=1,2,...,n

wj wW − ajW

≤

ξ ∀

j=1,2,...,n n j=1 wj=1, wj≥ 0

∀

j=1,2,...,n. (4)

To check the reliability of the optimal weights, the veracitybetween theinput pairwise comparisonsand their associated weight ratios are checked using the followingconsistencyratio(CR):

CR=

ξ

∗

CI (5)

where

ξ

∗ isthe optimalobjectivevalue ofmodel (4), and CI (consistency index) is a ﬁxed value per aBW,

whichcanbereadfromTable1.

CRisanumberbetween0and1,where0meansfull consistencyandbyincreasingthevalueofCRthe con-sistencyofthepairwisecomparisonsystemis decreas-ing.

3. ProbabilisticinterpretationofBWM

Inthissection,weprovide aprobabilistic interpretationofthe BWMinputsandoutputs,andthenreviewtwoschoolsofthoughts intheprobability estimation,e.g.,frequentist andBayesian,inthe contextoftheBWM.

3.1. Modelinginputsandoutputs:multinomialandDirichlet distributions

Asstatedbefore,thetypicaloutcomeoftheMCDMmethodsis theweightvectorw₌[w1,...,wn]suchthat wj≥ 0,nj=1wj=1.

Themagnitude ofeach wj indicates the importanceof the

corre-spondingcriteriacj.

From a probabilistic perspective, the criteria are seen as the random events,andtheir weights are thus their occurrence like-lihoods.Mathematicallyspeaking,suchaninterpretationisinline with the MCDM since w_j_{≥ 0} and n_j₌₁w_j₌1 according to the probabilitytheoryaswell.Itisfurtheroftheessencetoillustrate that probabilistic modeling makes sense froma decision-making pointofview.

Fortheprobabilisticreasoning,one needs tomodelall the in-putsandtheoutputsastheprobabilitydistributions.First,consider

ABandAWwhicharetheinputstotheBWM.Fromamathematical

pointofview,themultinomial distributioncanmodelthevectors sincealloftheirelementsareintegers.Theprobabilitymass func-tion(PMF)ofthemultinomialdistributionforagivenAw is[49] P

(

AW

|

w

)

=

n j=1ajW

! n j=1ajW! n j=1 wajW j (6)

wherewistheprobabilitydistribution.

Inthemultinomialdistribution,theweight vectoristhe prob-abilitydistribution andAW containsthenumber ofoccurrenceof

eachevent.Apparently,itiscompletelydifferentfromwhatis ex-pected for the BWM represented in Section 2. Interestingly, we showthatmodelingwithmultinomialwouldfulﬁlltheunderlying ideaoftheBWM.

Based on the multinomial distribution, the probability of the eventj isproportionate tothenumberofoccurrenceoftheevent tothetotalnumberoftrials,i.e.,

wj∝ ajW n

i=1aiW

,

∀

j=1,...,n. (7)

Similarly,onecanwritethesameequationfortheworst crite-rionas wW∝ aWW n i=1aiW =n1 i=1aiW (8)

UsingEqs.(7)and(8),oneobtains

wj wW ∝

ajW,

∀

j=1,...,n, (9)

whichisprecisely therelationwe seekintheoriginal BWM pre-sentedinStep5ofSection2.

Similarly, AB can be modeled using the multinomial

distribu-tion.However,AB isdifferentfromAWsincetheformerrepresents

the preferencesof the best over theother criteriawhile the lat-terdenotesthepreferencesoftheothersovertheworst.Thus, AB

Table 1

Consistency Index (CI) Table [9] .

aBW 1 2 3 4 5 6 7 8 9

Consistency Index (CI) 0.00 0.44 1.00 1.63 2.30 3.00 3.73 4.47 5.23

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yieldstheinverseoftheweight,i.e.,

AB∼ multinomial

(

1/w

)

(10)

where w is the probability distribution, and / represents the element-wise division operator. Identical to the worst criterion, onecanwrite

1 wj ∝ aBj n i=1aBi , _w1 B ∝ aBB n i=1aBi =n1 i=1aBi ⇒wB wj ∝ aBj,

∀

j=1,...,n, (11)

whichisagaintheexactrelationweseekintheBWM.

So far, we showed that the multinomial distribution could meaningfullymodeltheinputstotheBWM.Theproblemof ﬁnd-ingthe weights inthe MCDM problemisthus transferred to the estimationofaprobabilitydistribution.Therefore,onecanusethe statisticalinferencetechniquestoﬁndwinthemultinomial distri-bution.

Aweightvector fortheMCDMmustsatisfythenon-negativity andsum-to-oneproperties.Therefore,an appropriate distribution tomodelthe weightsis tousethe Dirichletdistribution.Givena parameter

α

∈Rn_,_the_Dirichlet_distribution_of_the_weights_w_is

de-ﬁnedas[49] Dir

(

w

|

α

)

= 1 B

(

α

)

n j=1 wαj−1 j . (12)

The distribution has only a vector parameter

α

,and wmeets theconstraintsofanoptimalweightvectorofMCDMsinceitisa probabilitydistribution.

3.2.Estimationoftheweightvector:statisticalinference

Foramoment,assumethatthereisonlyAWintheBWM,then

weconsidertwowidely-acceptedinferencetechniques:frequentist andBayesian. The underlying idea of thefrequentist approach is thatthereisapreciseyetunknownoptimalpoint,andtheeffortis toestimate itbasedontheobservations.Asaresult,theoutcome ofthe frequentistinferenceisa preciseweightvector fora setof criteria.Themaximumlikelihoodestimation(MLE)isarguablythe mostpopularinference techniquewhichﬁndsthe optimalweight vectorusingthefollowingoptimization

w=arg max

w,n j=1wj=1

P

(

AW

|

w

)

. (13)

Theoptimumof(13)yieldsat

w∗_j=najW i=1aiW

,

∀

j=1,...,n, (14)

whichisindeedthenormalizedAW.Thesamesolutionwillbe

ob-tainedbytheBWMifthe preferencesoftheDMare fully consis-tent.Thus, (14)showsthatthe MLEbearsthesame resultasthe BWMunderspeciﬁccircumstances.

The second approach isthe Bayesianestimation, inwhichthe parameters are approximated by using a distribution ratherthan aprecise pointasis intheMLE.Thus, we ﬁrstneed tospecifya priordistributionfortheweightvector.IntheBayesianinference, theDirichletdistribution isusedasthe priorto themultinomial. TheDirichletdistribution can perfectlyrepresentthe weight vec-torsince itsatisﬁesbothitsnon-negativityandsum-to-one prop-erties.Using Dirichletasthe prior andmultinomial as the likeli-hood,the posteriordistribution would alsobe Dirichletwiththe posteriorparameter

α

post=

α

+AW.Sincetheprior shouldbe

un-informativesothatitsimpactontheposteriorisminimal,we set thepriorparameter

α

=1.

AsaresultoftheBayesianestimation,thevaluesofwisshown bya Dirichletdistribution.Themode oftheposteriordistribution

μ

∈Rn _with_the_parameter

_α

post

μ

j=

α

postj− 1 n i=1

α

posti− n = 1+ajW− 1 n i=1

(

aiW+1

)

− n (15) = ajW n i=1aiW ,

∀

j=1,...,n.

Thus,themodeoftheposteriordistributionwouldprovidethe exact MLE.As a result, theBayesian paradigm wouldyield more informationregardingtheeventsunderstudysinceitsoutcomeis a distribution, not apoint. The standard deviation ofsuch a dis-tribution,forinstance,isanindicatorofuncertaintyregardingthe inferenceproblem,whichcanhavedistinctinterpretationswith re-specttotheproblemunderstudy.

So far, we merely considered AW for estimating the weights;

however,itiscriticaltousebothABandAWaccordingtotheBWM.

TheMLEinferencecontainingbothABandAWdoesnotbearan

an-alyticalsolutionduetothecomplexityofthecorresponding opti-mization problem. Further, the simpleDirichlet-multinomial con-jugate cannot encompass the AB and AW together. The problem

compounds when itcomes to havingthe preferences ofmultiple DMs. Considering these issues, a Bayesian hierarchical model is presented in the next section to estimate the optimal weight of thecriteriaconsideringbothAB andAWofmultipleDMs.

4. Bayesianbest-worstmethod

ThissectionpresentsaBayesianhierarchicalmodeltoﬁndthe optimal weights of a set of criteria based on the preferences of multipleDMsusingthebest-worstframework.

4.1. Groupdecision-making:ajointprobabilitydistribution

Assume that the kth _DM, _k₌₁_,_._._._,_K_, _evaluates _the _criteria

c1,...,cn by providing the vectors Ak_B and Ak_W. We show the set

ofall vectorsofK DMsbyA1:K

B andA1:WK.Fromnowon,the

super-script 1:K _would _indicate _the _total _of _all _vectors _in_the _base. _We

alsorepresenttheoveralloptimalweightbywagg_.

Theestimationofwagg _entails_using_several_auxiliary_variables.

Inparticular,wagg _is_computed_based_on_the_optimal_weights_of_K

DMsshownbywk_,_k₌₁_,_._._._,_K._Thus,_the_proposed_Bayesian_model

wouldsimultaneouslycompute wagg _and_w1:K_._Prior_to_conducting

anystatisticalinference,itisrequiredtowritethejointprobability distributionofallrandomvariablesgiventheavailabledata.Inthe group decision-makingwithin the BWM, A1:K

B andA1:WK aregiven,

andw1:K _and_wagg _must _be _estimated _accordingly. _Thus, _the

fol-lowingjointprobabilitydistributionissought

P

wagg_,_w1:K

_A1:K B ,A1:WK

. (16)

If the probability in (16) is computed, then the probability of each individual variable can be computedusing the following probabilityrule

P

(

x

)

= y

P

(

x,y

)

(17)

wherexandyaretwoarbitraryrandomvariables.

4.2. Bayesianhierarchicalmodel

Todevelop aBayesianmodel,we ﬁrstneed toidentifythe in-dependenceandconditionalindependenceofvariables.Fig.1plots

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k

Fig. 1. The probabilistic graphical model of the Bayesian BWM.

the graphical model corresponding to the proposed method. The nodesinthegrapharethevariables.Asaconvention, the rectan-glesaretheobservedvariables,whicharetheinputstothe origi-nalBWM. Thecircularnodesarethevariableswhichmustbe es-timated.Also, arrowsdenotethatthenode inoriginisdependent onthenodeattheotherend.Thatistosay,thevalueofwk is

de-pendentonAk

W andAkB,andthevalueofwaggisalsodependenton wk.

The plate,whichcoversa setofvariables,meansthat the cor-responding variablesare iteratedforeach DM,andwagg _is _not_in

theplatesincethereisonlyonewagg _for_all_DMs.

The conditional independence between various variables is clearbasedonFig.1.Forinstance,Ak

W isindependentofwagggiven w_k,i.e., P

Ak W

wagg,wk

=P

Ak W

wk

(18)

Consideringallindependenceamongdifferentvariables, apply-ingtheBayesruletothejointprobability(16)follows

P

wagg_,_w1:K

_A1:K B ,AW1:K

∝P

A1:K B ,AW1:K

wagg,w1:K

P

wagg_,_w1:K

=P

(

wagg

)

K k=1 P

AWk

wk

P

AkB

wk

P

wk

wagg

. (19)

where the last equality is obtained using the probability chain rule andthe conditional independenceofdifferent variables, and thefactthateach DMprovideshis/herpreferencesindependently. SincetheestimationoftheparametersinEq.(19)isreliantonthe estimationofothervariables,thereisachainbetweendifferent pa-rameters.The existenceofthechainisthereasonthatthemodel iscalledhierarchical.

Wenowneedtospecifythedistributionsofeachandevery el-ementinEq.(19).WehavealreadyshownthatAB andAWcanbe

perfectlymodeledusingthemultinomialdistributioninthesense thatitpreservestheunderlyingideaoftheBWM.Thereisonlyone differencebetweenAB andAWsince theformershowsthe

prefer-ence of all the criteriaover the worst, while the latter contains thepreferenceofthebestoveralltheothercriteria.Thus,onecan modelthemas

AkB

|

wk∼ multinomial

1/wk

_,

_∀

_k₌₁_,_._._._,_K, Ak

W

|

wk∼ multinomial

wk

_,

∀

_k₌₁_,_._._._,_K. ₍₂₀₎

Given wagg_, _one _can _expect _that _each _and _every _wk _be _in _its

proximity. To this end, we reparameterize the Dirichlet distribu-tionwithrespectto itsmeanandaconcentration parameter.The

modelsofwk_given_wagg_are

wk

_|

_wagg_{∼ Dir}

₍

_γ

_{× w}agg

₎

_,

_∀

_k₌₁_,_._._._,_K, ₍₂₁₎

wherewagg_is_the_mean_of_the_distribution_and

_γ

_is_the

concentra-tionparameter.

Theequationin(21)saysthattheweightvector wk _associated

with each DM must be in the proximity of wagg _since _it _is _the

meanof the distribution, andtheir closeness isgoverned by the non-negativeparameter

γ

. Such a technique is used in different Bayesian models as well [50]. The concentration parameter also needstobe modeledusinga distribution.Areliableoptionisthe gammadistributionwhichsatisﬁesthenon-negativityconstraints, i.e.,

γ

∼ gamma

(

a,b

)

, (22)

wherea andb are the shapeparameters of thegamma distribu-tion.

Weﬁnallysupplythepriordistributionoverwagg_using_an

un-informativeDirichletdistributionwiththeparameter

α

=1as

wagg_{∼ Dir}

₍

_α

₎

_. ₍₂₃₎

The speciﬁed modeldoes not bear a closed-form solution.As aresult,Markov-chain MonteCarlo(MCMC)techniques[51]must beusedtocomputetheposteriordistribution.FortheMCMC sam-pling,weusethe“justanotherGibbssampler” (JAGS)[52],which isoneofthebestavailableprobabilisticlanguagestodate,to sam-pleandcomputetheposteriordeterminedin(19).Theuseful out-comeofthemodelistheposteriordistributionofweights for ev-erysingleDMandtheaggregatedwagg_.

Theproposed Bayesian modelwillreplaceStep 5ofthe origi-nalBWMexplainedinSection2.Infact,theoptimizationproblem issubstitutedwithaprobabilistic modelwhile theinputstoboth methods are identical.However, theproposed modelwould pro-videmoreinformationregardingtheconﬁdenceoftherelation be-tweeneach pairofcriteria. Theexcessive informationisobtained bydevisinganewBayesiantestbasedontheapproximated distri-butionfromthemodel,whichisexplainedinthenextsection.

5. Credalranking

The modus operandi in the MCDM is to say one criterion is more important than one another merely if its weight, or the weightaverageforthegroupdecision-making,ishigherthanone another. Assume that there are three criteriac1, c2,and c3 with

the weightvector w=[0.49,0.50,0.01]. According tothe MCDM,

c2 is superior to both c1 andc3.However, theconﬁdence of the

superioritycannotbedeterminedbysolelycomparingtwoﬁgures. Thisisevenmuchmoreimportantwhentheweightvector repre-sentsthepreferencesofagroupofDMs.Todate,therearevarious rankingschemessuchasinterval-basedranking[53],fuzzyranking

[54,55],andrankingbasedongreyrelationalanalysis(GRA)[56]. Thenotionofcredalrankingisnowintroduced,whichcan cali-bratethedegreetowhichonecriterionissuperiortooneanother. Havingtheposteriordistributionofweightswouldhelpgaugethe conﬁdenceoftherelationsbetweenvariouscriteria.Thedifference betweenthe credal ranking andother ranking schemes is that a conﬁdenceiscomputedinthecredalrankingbasedonone distri-bution,i.e., theDirichletdistribution ofwagg_,_while _other _ranking

methods usually take two numbers/intervals and try to ﬁnd the extenttowhichoneissuperior.

Weﬁrstdeﬁnethecredalordering,whichisthebuilding-block ofcredalranking.

Deﬁnition5.1(CredalOrdering). Forapairofcriteriaciandcj,the

credalorderingOisdeﬁnedas

O=

(

ci,cj,R,d

)

(24)

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Table 2

Comparison of the original BWM and the Bayesian BWM on the mobile phone selection criteria based on the preferences of 50 students.

rankingofcriteriaisobtainable byapplying athresholdof0.5to thecredalranking.

6. Numericalexamples

In this section, a real-world example is analyzed using the BayesianBWM,andthecorrespondingcredalrankingiscomputed andvisualized by usinga weighteddirected graph. The MATLAB implementation of the proposed model can be found at http:// bestworstmethod.com/software/.

Thereal-worldapplicationistheselectionofthemobilephone, towhichtheBWMhasbeenalreadyapplied[9].Theproblem im-plicates the selection of one from a set of mobile phones based ondifferent criteria. Six differentcriteria that studied andfound intheliteratureisusedtoevaluatethemobilephonealternatives. Thecriteriaarebasicrequirement,physicalcharacteristics, techni-calfeatures,functionality,brandchoice,andcustomerexcitement.

The datacollectedin[9]wasfrom50universitystudentswho completelygot familiar withdifferent selected criteriathrough a provideddocument.Variouscharacteristics(e.g.,price,dimension, weight, display) of four particular mobile phones were given to theparticipants.Throughaquestionnaire,studentsﬁlledinaform

togetthe informationrequiredforthe originalBWM, i.e.,AB and AW.

Thefirstapproach,whichwasemployedin[9],wastofindthe optimal weight vector separately foreach student, and then ag-gregatethemusingthearithmetic meantocompute thefinal ag-gregatedweightvector.ThefirstrowofTable2tabulatesthefinal aggregatedweightsobtainedbytheBWM.Weparticularlyconsider thearithmeticmeantovalidatetheproposedBayesianmodelsince the average of50 participants isa reliable measureaccording to

centrallimittheorem.

The obtained inputs from 50 participants in this experiment werealsogiventotheBayesianBWM,andtheoutcomeofthetest isobtained. Sincetheoutputoftheaggregatedweightisa distri-butionintheBayesianBWM,itisnotpossibletocomparethetwo methodsdirectlyandverifyiftheoutputoftheBayesianBWMis valid.Tohaveameaningfuldiscussionandvalidation,however,we usetheaverage oftheDirichletdistributionofwagg _to_be _able_to

comparethe twomethods. Thesecond rowofTable 2showsthe averageofthe ﬁnal aggregatedweight computedbythe Bayesian BWM.

Table 2 indicates that the estimation based on the proposed Bayesian model yields a meaningful result since the average of the estimateddistribution iscentered around the overall average ofeachindividualpreferences.Theexampleshowsthattheoutput of the Bayesian BWM is valid and makes perfect sense. Keep in mindthattheBWMobtainstheweightofeachindividualﬁrstand then aggregate themby the arithmetic mean,whilethe Bayesian BWMcomputestheaggregateddistributionandalltheindividual preferencesatonceusingprobabilisticmodeling.Theessential dif-ference betweentheBWMandtheBayesianBWMisthatwe can comparethecriteriacolorfully.Thecurrentpracticeistosaya cri-terion is more important than one another ifits average weight hasahighervalue;therefore,itisa blackandwhite(orzero-one) decision.

We compare all pairs of criteria with each other using the credalranking and visualizeits outcome using a weight directed graph.Fig.2displaysthecredalrankingofcriteriaforselectingthe cellphones.Thenodesinthisgrapharethecriteriaandeachedge

A_→d BtellsthatcriterionAismoreimportantthanBwiththe con-ﬁdenced.

Based on Fig. 2, functionality is the most important criterion based on the opinions of all participants. At the other extreme, customerexcitement andbrandchoicearetheleastdesirable fea-tures forthe participants in thisexperiment. Further, the degree of certainty about the relationof criteria is alsoevident. For in-stance,technicalfeaturesiscertainlymoreimportantthancustomer excitement,butitismoredesirablethanbasicrequirementwiththe conﬁdenceof0.71.

Asmentionedbefore,thecredalrankingvisualizedinFig.2can be changedintotheconventional rankingmerelyby applyingthe threshold of 0.5 to the obtained conﬁdences. The threshold can vary from one problem to one another, and it is entirely to the DMs’volitiontoopt fora particularthresholdvalue. Forinstance, theconﬁdence0.71betweentechnicalfeatureandbasicrequirement

might be strongenough for themobile selection problem, butit couldnotberegardedasstrongifthestudywasonanother prob-lem.Inother words,credalranking could beshaped torepresent

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ARTICLE

IN

PRESS

Fig. 2. The visualization of the credal ranking for the example of mobile phone selection criteria.

therankingofcriteriaindifferentproblemsbasedontheDMs’ de-siredconﬁdence.

7. Conclusion

This paperpresents a Bayesian modelfor thegroup decision-makingwithintheBWM.Theproposedmethodmodelstheinputs of theBWMusing themultinomial distribution anditis demon-stratedthatsuchadistributionwouldpreservetheunderlyingidea oftheoriginal BWM.Further, theweightvector ismodeledusing theDirichletdistribution.Theproposed Bayesianmodelisableto compute the weight distribution of each individual in the group decision-making,andanaggregatedfinaldistributionrepresenting theoverallpreferencesofallDMs.Thecredalrankingofcriteriais developedbasedonwhicheach pairofcriteriaareassigneda re-lationandaconfidence.Theconfidencewhichiscomputedbased on aproposed Bayesian modelshowsthe extentto whichone is certain abouttherelationofthecorresponding pairofcriteria.In addition,thecredalrankingisvisualizedusingaweighteddirected graphwhichshowstheinterrelationofcriteriaclearly.

TheproposedBayesianBWMisapromisingmethodinthe con-text ofgroupdecisionmakingwhereone isinterestedinthe col-lectiveopinion ofa group,butatthesametime,one couldcheck therankingoftheweightsinaprobabilisticsense.Thegroupwill bemorecertainabouttherelationoftwocriteriaifitisassociated with a highconﬁdence level while the relations with low conﬁ-dencelevelshouldbeinterpretedmorecarefully.

There are several avenuesto extend thecurrent research. We aimtoapply such modelingsinother importantMCDMmethods. It isalsointerestingto investigatetheroleofoutliers indifferent groupBWM.Finally,itwouldbeinterestingtoworkonsomeother featuresoftheBayesianBWMsuchasconsistencymeasure.

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Bayesian best-worst method

Delft University of Technology