Filling in pattern designs for incomplete pairwise comparison matrices: (quasi-)regular graphs with minimal diameter

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Omega

journalhomepage:www.elsevier.com/locate/omega

Filling in pattern designs for incomplete pairwise comparison matrices: (Quasi-)regular graphs with minimal diameter ^R

Zsombor Szádoczki

^a,b,∗

, Sándor Bozóki

^a,b

, Hailemariam Abebe Tekile

^c

aResearch Laboratory on Engineering & Management Intelligence, Institute for Computer Science and Control (SZTAKI), Eötvös Loránd Research Network (ELKH), Budapest, Hungary

bDepartment of Operations Research and Actuarial Sciences, Corvinus University of Budapest, Hungary

cDepartment of Industrial Engineering, University of Trento, Italy

a rt i c l e i nf o

Article history:

Received 20 July 2020 Accepted 2 October 2021 Available online 8 October 2021 Keywords:

pairwise comparison

incomplete pairwise comparison matrix graph

diameter regular graph

a b s t r a c t

Pairwisecomparisonshavebecomepopularinthetheoryandpracticeofpreferencemodellingandquan- tification.Incaseofincompletedata,thearrangementsofknowncomparisonsarecrucialforthequality ofresults.Wefocusondecisionproblemswherethesetofpairwisecomparisonscanbechosenandit isdesignedcompletelybeforethedecisionmakingprocess,withoutanyfurtherpriorinformation.The objectiveofthispaperistoproviderecommendationsforfillingpatterns ofincompletepairwisecom- parisonmatricesbasedontheirgraphrepresentation.Theproposedgraphsareregularandquasi-regular oneswithminimaldiameter(longestshortestpath).Regularitymeansthateachitemiscomparedtooth- ersforthesamenumberoftimes,resultinginakindofsymmetry.Agraphonanoddnumberofvertices iscalledquasi-regular,ifthedegreeofeveryvertexisthesameoddnumber,exceptforonevertexwhose degreeislargerbyone.Wedrawattentiontothediameter,whichismissingfromtherelevantliterature, inordertoremaintheclosesttodirectcomparisons.Ifthediameterofthegraphofcomparisonsisas lowaspossible(amongthegraphsofthesamenumberofedges),wecandecreasethecumulatederrors thatarecausedbytheintermediatecomparisonsofalongpathbetweentwoitems.Contributionsofthis paperincludealistcontaining(quasi-)regulargraphswithdiameter2and3upuntil24vertices.Exten- sivenumericaltestsshowthattherecommendedgraphsindeedleadtobetterweightvectorscompared tovarious othergraphswith thesame numberofedges. Itis alsorevealedby examplesthatneither regularitynorsmall diameterissufficient onitsown, bothpropertiesareneeded. Boththeorists and practitionerscan utilizetheresults,giveninseveralformats intheappendix:plottedgraph,adjacency matrix,listofedges,‘Graph6’code.

© 2021TheAuthor(s).PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBYlicense(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Pairwise comparisons form the basis of preference measure- ment, ranking,psychometrics anddecisionmodelling [1–3].Mul- ticriteriaDecision Making isindeedan importanttool both atan individual andatan organizationallevel.Wecan thinkaboutdif- ferentkindofrankingofalternatives orweightingofcriteria,like tenders,selectionamongschoolsorjoboffers,selectionamongthe implementationofdifferentprojectsinanenterprise,etc.

Oneofthemostcommonlyusedtechniquesinconnectionwith Multi-criteriaDecisionMakingisthemethodofthepairwisecom-

R Area: Decision Analysis and Preference-Driven Analytics. This manuscript was processed by Area Editor Luis Dias.

∗Corresponding author.

E-mail address: szadoczki.zsombor@sztaki.hu (Zs. Szádoczki).

parisonmatrices.Onecanapply thistechniquebothfordetermin- ing theweights of thedifferentcriteria andfortherating ofthe alternatives accordingtoa criterion.Usuallywe denotethenum- berofcriteriaoralternativesbyn,whichmeansthepairwisecom- parisonmatrixisann×nmatrix,oftendenotedbyA.Inthiscase thei j-thelementoftheAmatrix,a_{i j} showshowmanytimesthe i-thitemislarger/betterthanthe j-thelement.

Formally, matrix A is called a pairwise comparison matrix (PCM)ifitispositive(a_{i j}>0for

∀

^{iand j)andreciprocal(1}/a_{i j}= a_jifor

∀

^{iand j)[4]}, which also indicates thata_ii=1 for

∀

^i.

Dealingwithincompletedatagetsmoreandmoreattentionin theliterature.Whensome elementsofaPCMaremissingwecall itanincompletePCM.Therecouldbemanydifferentreasonswhy theseelementsareabsent,some datacould havebeenlostorthe comparisonsaresimplynotpossible(forinstanceinsports[5]).

https://doi.org/10.1016/j.omega.2021.102557

0305-0483/© 2021 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license ( http://creativecommons.org/licenses/by/4.0/ )

The mostinteresting case forusiswhen the decisionmakers do not have time, willingness or the possibility to make all the n(ⁿ−1)/2comparisons.

Inthisarticlewewouldliketostudywhichcomparisonsshould bemade, ormorepreciselywhatpatternsofcomparisonsarerec- ommended in order to get good approximation of the decision makers’preferencescalculatedfromthewholesetofcomparisons.

Thegraphrepresentationofthepairwisecomparisonsisanatural andconvenienttooltoexamineourquestion,thuswewillusethis throughoutthepaper.

Inmanycasesthesetofcomparisonscanbeadaptive,i.e.,the next questionsdepend ontheanswers totheprevious onesasin, e.g., [6–8].However, we assume inthe paper that the wholeset of comparisons is designed completely before the decision mak- ing process, and we do not have any further prior information abouttheitemstobecompared.Thusthe‘confidencelevel’ofev- erysinglecomparisonisthesameinourproblems,theprobability of their ‘errors’is identical.For instancethe (pre-)compilation of questionnaires in connection withdecisionmaking problems can be named asan indeedcommon practical example that satisfies theseconditions.

There are already known special structures proposed for in- completepairwisecomparisonmatricesintheliterature,whichin- clude:

(i) spanning tree, in particular if one row/column is filled in completely(itsassociatedgraphisthestargraph)

(ii) two rows/columns are filled in completely (its associated graphistheunionoftwostargraphs)[9]

(iii)amethodof2-cyclicdesigns,theunionoftwoedge-disjoint n-cycles,hasbeenalsorecommendedtoselect2npairedcompar- isonsfromnnumberofobjects[10]

(iv)more orlessregular graphs,forexample theregularity of thecomparisons’graphappearsinthedesignsof[11]and[12].

Regularity results in a kind of symmetry that is also desir- able in case of sport competitions [13], where the number of matches played equals for every player or team, at least in the firstphase(beforetheknockoutstages).Thisalsoappearsinother sporttournaments,wheretheyusetheso-calledSwisssystem, in which besides a lotof other requirements,every player orteam plays the same number of matches (if possible) [14–16]. Thus the resulting representing graph of the comparisons is regular [17].

Aspecialtypeandextensionofregulargraphsisconsideredby [18].Theyproposed the(quasi-)stronglyregulardesigns basedon (quasi-)stronglyregulargraphsinordertoselectpairstobecom- pared within incomplete information. A graph is called strongly regular with parameters (ⁿ,k,

λ

,

μ

)^{,if each ofthe nvertices has} degreek,and(i)foranypairofadjacentverticesuand

v

,thenum- berofverticesadjacenttobothuand

v

is

λ

^{;(ii)foranypairofnot}

adjacentverticesuand

v

,thenumberofverticesadjacenttoboth uand

v

is

μ

^{.Sincethesepropertiesarerather}restrictive, alinear algebraicgeneralization,thesocalledquasi-stronglyregulargraphs arealsotakenintoconsideration.Bysimulation,they showedthat both designs give better results(based ona logarithmic distance functiondefinedontheweightvectors)thanotherrandomdesigns ofthesamecardinality.

[19] create an incompleteness indexbased on the number of missing pairwise comparisons andtheir arrangements. Using dif- ferent kindofMonteCarlo simulationsthey concludethat incon- sistencyandincompletenessbothhavecrucialeffectonsensitivity, andthe regularityofthePCM alsohasahuge effectbothon the quantitativeandthequalitativeresults.

Notethat thefirst threeexamplesabove lackregularity.Regu- larity means that each item is compared to others for the same number of times (if the cardinality of the items to compare is

odd,one of thedegreescan be smalleror greater– in our anal- ysis, greater – by one), resulting in a kind of symmetry, as we mentioned earlier. Despite the fact that regularity has been rec- ognized as an important property in connection with the rep- resenting graph of the comparisons, the above-mentioned ex- amples do not examine it as generally as we do, their defini- tions onregularityismorerestrictive andtheirinstances areless systematic.

Diameter,theotherkeyconceptofthepaperbesidesregularity, showshowfaritemscanbefromeachotherinthesensethathow manycomparisons are neededinorder to havean indirectcom- parisonbetweenthem.The wellknown telephonegameoreffect [20], also known asThe Whisper Game [21] shows small errors arecumulatedalongasufficientlylongseries.Ifamessagepasses through a lineofpeople, in awhisper, the original andthefinal versionsdiffer a lot, despitethe neighboringversions areusually quitesimilar. Aclassicalexample forthe non-transitivityofindif- ference[22] istheadditionofverysmallportionsofsugartothe samecup ofcoffee.No one can distinguishbetweentwo consec- utivesteps,however,ifthissequenceislongenough,theindiffer- encedisappears[23].

Inthesetofconnectedgraphs,diametercanbeconsideredasa measureofcloseness,orastrongertypeofconnectedness.Itisnot properlystudiedintheliterature,however,forinstancein[24]the estimationofthematrixofcomparisonprobabilitiesisinvestigated forseveralgraphstructures andsome researchquestions,e.g., on apossiblerelationofthegraph’sdiameterandtheworst-caseap- proximationerror,areraised.Oneofournotablefindingsistode- terminethediameteroftherepresentinggraphasacrucialprop- ertyforfillinginpatterndesignsofincompletePCMs.

Notethat regular graphs canhave largediameter, e.g., acycle on n vertices is 2-regular and has diameter d=

ⁿ/2

^{. The star}

graph,mentioned among the examples, hasminimal diameter2, but it is far from being regular. Our aim is to find the graphs, among(quasi-)regularones,withminimaldiameter.Weareespe- ciallyinterestedin thesmallestnontrivialvaluesof thediameter, namelyd=2andd=3.Intuitionsuggests,anditisconfirmedby the graphs found, that fora fixed n, higherregularity, i.e., more edges,makesthediametersmaller.

The rest of the paper is structured as follows. Basic mathe- maticalconceptsareintroduced inSection 2.Lateronwe assume that we know the number nof alternatives or criteria, it is also a key assumptionthrough our paperthat the graphrepresenting theMCDMproblemisk-(quasi-)regularandwealsoknow(orwith thehelpoftheotherinputswecandetermine) thediameterdof the graph. In Section 3 (which is complemented by Appendix A (online)) we provide a systematic collectionof suggestedincom- pletepairwise comparisons’patterns withthehelp ofthe above- mentionedinputsandall/somegraphsfortheexaminedcases.We wouldliketoemphasizethatthislistisamajorcontributionofour paper.Section4presentsamotivationalexampleshowingthatthe diameterofaregulargraphcanbelargeandtheresultcanbevery sensitivetotheerrorsofthematrixelements.Awiderangeofnu- mericalsimulations, usingthedistancesofthe weightscomputed withdifferentfillinginpatternsrespecttotheweights calculated fromthecompletePCMs,isalsoprovidedinordertovalidateour recommendations. Finally, Section 5 concludes and provides fur- ther research questions closely connectedto the discussed topic.

ResultsofSections3and4aregiveninmoredetailsintheappen- dices.B(online)includestherecommendedgraphsthemselves.For practitioners,thislistmightserve asa‘recipe’ indesigningques- tionnairesbasedonpairwisecomparisons.AppendixD(online)in- cludestheresultsofthecomparisonsofweightvectorscalculated fromthedifferentgraphs.AppendicesA,B,CandDcanbe found intheonlinesupplementarymaterial.

Fig. 1. Graph representation example

2. Basicconceptsofthegraphrepresentation

The graph representation of paired comparisons has already been used inthe 1940s [25]. Ofcourse after thewidespread ap- plication ofPCMs andincompletePCMs ithasbecome acommon methodintheliterature,seeforinstance[26],[27]or[28].

Usuallyinthesearticlestheauthorsusedirectedgraphsforthe representation, because they distinguish the preferreditem from thelesspreferredoneinevery pair.Inourapproachtheonlyim- portant thing is whetherthere exists a comparison betweenthe two elements. This means that we useundirected graphs, where theverticesdenotethecriteriaorthealternatives.Thereisanedge betweentwoverticesifandonlyifthedecisionmakersmadetheir comparisonforthetworespectiveitems(theappropriateelement ofthePCMisknown).Inordertounderstandtheconceptssofar, thereisasmallexamplebelow:

Example1. Letusassumethatthereare4criteria(n=4)andour decision maker already answered some questions, denoted their locationsinthematrixby•andtheirreciprocalvaluesby◦,which leadtothefollowingincompletePCM:

A=

⎡

⎢ ⎣

1 • •

◦ 1 •

◦ 1 •

◦ ◦ 1

⎤

⎥ ⎦

ThisincompletePCMisrepresentedbythegraphinFigure1. Aswecanseethereisnoedgebetweenthefirstandthefourth vertices,where thePCMhasmissingvalues andthere isnoedge betweenthesecond andthird vertices,wherethesituationisthe same.There isan edgebetweeneveryother pair,wherewe have nomissingvaluesinthePCM.

Itisimportanttoemphasizethatastheknownelementsofthe PCMdeterminetherepresentinggraph,itisalsotrueintheother wayaround.Thus,thegraphinFigure1showswhichcomparisons areknowninthePCM.Thisisthekeypropertythatweuseinthis paper, as we presentthe representing graphs that show the fill- inginpatterns,thecomparisonsthatshouldbemade.Weassume that the representinggraphs are connectedand k-(quasi-)regular through ourpaper,thus weneed some definitionstomake these conceptsclear.

Definition 1(Connectedgraph). Inanundirectedgraph,two ver- ticesuand

v

arecalledconnectedifthegraphcontainsapathfrom uto

v

.Agraphissaidtobeconnectedifevery pairofverticesin thegraphisconnected.

Definition 2. (k-regular graph)A graphiscalledk-regular ifev- eryvertexhaskneighbours,whichmeansthatthedegreeofevery vertexisk.

Definition 3. (k-quasi-regular graph) A graph is called k-quasi- regular if exactly one vertex has degree k+1, and all the other verticeshavedegreek.

Thek-regularitybasicallymeansthattheverticesarenotdistin- guished,there isnoparticular vertexas, forexample,inthecase ofthestargraph,thuswewouldliketoavoidthecaseswhenthe eliminationofrelativelyfewverticeswouldleadtothedisintegra- tionofthe wholecomparisonsystem[29].Besidesregularity, the connectednessoftherepresenting graphisindeedimportant,be- cause to approximate the decision makers’ preferences well, we need tohave atleastindirect comparisonsbetweenthe different criteria, otherwisewe cannot sayanythingabouttherelation be- tweencertainelements[30].

However,itisalsonotablethatwewouldliketoavoidthecases whentwoitemsarecomparedonlyindirectlythroughaverylong path, because this could aggregate the small, tolerable errors of the differentcomparisons and we could end up with an intoler- ably large error in the relation between the two elements. Such an example was found in [29], wherethe graphgenerated from thetabletennisplayersmatchesincludedalongshortestpathbe- tweentwovertices(players),andthecalculatedresultappearedto be misleading. The diameter of the representing graphis a very suitablemathematicaltooltomeasurethisproblem:

Definition4(Thediameterofagraph). Thediameter(denotedby d)ofagraphGisthelengthofthelongestshortestpathbetween anytwovertices:

d= max

u,v∈V(G)

(

u,

v ₎

,

whereV(^G)^{denotesthesetofverticesofGand}(.,.)^isthegraph distancebetweentwo vertices,namelythe length oftheshortest pathbetweenthem.

Wealsodefine herethetwisted product,agraph construction methodthatisusedbyusextensivelytofindtheproposedgraphs:

Definition5(Twistedproductoftwographs). ([31])

Let G=(^V,E) ^{and G}=(^V,E) ^{be two undirected graphs,} whereV andV are the vertexsets,while E andE arethe edge setsoftherespectivegraphs.Let−→

E denotethesetofarcsinanar- bitraryorientation ofG.Foreacharc(ⁱ,j)∈−→

E,let

π

(ⁱ,j)be aone to one mapping fromV to itself. The twisted product of graphs GandG,denotedby G∗G isdefinedasfollows:itsvertexsetis the Cartesian product V×V, andthere is an edge betweenver- tices (ⁱ,i)^and(^j,j)^ifeither[i= j and(ⁱ,j)∈E]or[(ⁱ,j)∈−→ E and j=

π

(i,j)(ⁱ)^].

Notethat the twisted product with

π

=identity resultsin the Cartesianproduct.

Briefly from now on we will examine graphs representing MCDMproblemsdefinedbythefollowinginputs:(ⁿ,k,d)^,wheren isthenumberofvertices(criteria),kshowsthelevelofregularity ofthegraphanddisthediameterofthegraph.

3. (Quasi-)regulargraphswithminimaldiameter

Inthissection,wepresentoneofthemostimportantfindings ofthepaper,theexamined(quasi-)regulargraphsthemselves.First of all, it is a key step to determine which cases are interesting for us considering our inputs. It is important to emphasize that wedealwithunlabelledgraphs,becausewearetryingtofindout whatkindofpatternsareneededinthecomparisonsfordifferent

instances.Thus ifwe exchangethe‘names’oftwocriteria(forin- stance‘1’and‘2’inExample1)thepatternwouldbethesame.

Thenwecanconsidertheregularityparameterk.Thek=1case ispossibleonlywhenniseven,buttheyarenotconnectedexcept forn=2,sothisisnotinterestingforus.Whenk=2thereisonly one connectedgraphforeveryn,namelythecycle,forwhichd=

ⁿ/2

^{asalreadymentionedinthe}introduction.

The larger regularity parameters could be interesting, but of courseweneedareasonableupperboundforthenumberofcrite- ria, n,whichisalsoanindirectupperboundfork.Inourresearch we examined then=5,6,...,24cases,becauseon theone hand forlargernparameters,somecomputationsbecomeverydifficult, andon theother handthe largest5-regular graphwithdiameter 2contains24vertices,sothisisanicetheoreticalbound,aswell.

It isalsotruethatinthemajorityofthefieldsofapplicationitis sufficienttoexaminethenumberofalternatives(vertices)upuntil 24.

The smaller the d parameter is, the more stableor trustwor- thy oursystemof comparisonsis.This means thatin an optimal case we would like to minimize this parameter, while the num- ber of the criteria (n) is always a fixed exogenous parameter in ourMCDMproblems.Aswementionedabove,kiscrucialtoavoid thecaseswhensomecriteria(vertices)wouldbetooimportantin thesystem,howeveritalsoshowsushowmanycomparisonshave tobe made,becauseeveryvertexhasadegreeofk,whichmeans the numberof edges is nk/2. Thus ifour decisionmakers would like tospendthe shortesttimewiththecreation ofthePCM,we shouldchooseasmallkparameter.But,ofcourse,asusuallyhap- pensinthesesituations,thereisatradeoff betweentheparame- ters,because formanycriteria(largen) thesmallerregularity(k) willcauseabiggerdiameter(d), namely,a morefragilesystemof comparisons.

In thispaperwe would liketo providea listof graphs which shows the patterns of thecomparisons that have to be made in case of different parameters. We used computational and con- structingmethodstodeterminethegraph(s)withthesmallestdi- ameter(dparameter)foragiven(ⁿ,k)^{pair.Withthehelpofthese} results it waseasy to determine which k is the smallest that is neededtoreach agivendforagivenn.Wefoundthat, withthe chosen upperboundofn(24)theinterestingvaluesfortheregu- larity arek=3,4,5,whilethe interestingvaluesforthediameter ofthegraphared=2,3.Ofcoursed=1wouldmeanacomplete graphthatisnotreachableformany(ⁿ,k)^{pairs,anditrepresents} a complete PCM, thus it is not interesting for us. For a general MCDM problem probablyinstead of k, it would give more infor- mation ifwe considered an indicator that showshow farwe are fromthe ‘extreme’case, whenthedecisionmakers havetomake all thecomparisons.Thiswouldmeann(ⁿ−1)/2comparisonsin- steadofournk/2incaseofregulargraphsor(^nk+1)/2incaseof quasi-regular graphs, thereforethe completion ratiois definedas follows:

c=

_nk/2

n(ⁿ−1)/2 ifnor kis even (^nk+1)/2

n(n−1)/2 ifnandkareodd thatwewillcalculateforeveryinstance.

Herewewillpresentthegraphswiththesmallestdiameterfor a given(ⁿ,k) ^{pair,it isimportantto emphasizethat it isrecom-} mended toreadthissectiontogether withAppendixA,asalarge part of ourlist (Tables 2, A1a,A1b, A2,A3, A4a,and A4b) takes placethere,becauseofthelengthofthetables.Thefindingforthe different graphs inour list consistedof severalmethods, sources andlayers:

1. As a startingreference point, we checked the builtin graphs in Wolfram Mathematica[32], which areeven complete cata-

Table 1

The summary of our list of graphs: the different sets of graphs based on the regularity level k and the number of vertices n can be found in the indicated tables, from which Table 2 can be found in the main text, while the other ta- bles take place in Appendix A in the supplementary mate- rial. Lightgray denotes d = 2 and gray denotes d = 3 .

k

n 3 4 5

n = 5 , . . . , 10 Table 2

n = 11 , . . . , 15 Table A1a Table A2

n = 16 , . . . , 20 Table A1b Table A4a n = 21 , . . . , 24 Table A3 Table A4b

loguesin case of small number of vertices, thus we selected theoneswithminimaldiameteramongthem.

2. Forsmallerandmiddle-sizedgraphs,whenMathematica’sbuilt inexamples coveronly a sampleofthe cases,we usednauty andTraces[33]andIGraph/M[34] togenerateallthepossible (quasi-)regulargraphsandselecttheneededones.

3. Ourresultscontain manywell knowngraphs aswell,like the Petersengraph[35],thatwecollectedfromdifferentkindofar- ticlesindicatedintherespectivetablesas‘Source’.Wealsocol- lectedfurtherinformation,likeuniqueness,aboutthosegraphs thatwegotwiththehelpofMathematicaandarewellknown cases.Wecitetheseinformationas‘Seealso’inourtables.

4. Forlarger graphs we were not able to generateall the possi- bleregularcases,thusweusedseveralconstructiontechniques such as the twisted product, integer linear programming or mergingandextendingmethodswiththehelpofsomealready knowngraphs.Manyofthesecaseswerechallengingandtime- consumingtofind,thesameideararelyworkedtwice.

5. It is also importantthat ask-quasi-regularitywas defined by us,all ofthequasi-regulargraphs areourfindings (oratleast wearethefirsttousetheminthiskindofcontext),butwedo notdenotethisseparatelyinthetables.

Table1presentsatableoftablesthatprovidesan overviewof ourlistofgraphs.

Table2showsthecaseswhen k=3andd=2isthe minimal value ofthe parameter. Itis importantto note that k=3 isonly possiblewhenniseven,butwhenitisodd,weexamine3-quasi- regulargraphs,whereallverticeshavedegree3exceptonewhere ithas4,becausethesearetheclosestto3-regularity.

Wecanseethatwithk=3theminimaldiametercanbe2un- tilwe have10vertices.Ofcourseforn≤3the3-regularityisnot possible,and forn=4 the diameteris 1, becausethis isa com- pletegraph,thatiswhywe skipthosein thetable.It isalsono- tablethatthecompletionratio(c)evenreaches1/3whenwehave 10vertices(itisobviouslydecreasinginn).Weshouldemphasize thefactthatthereareonlyafewgraphsforevery (ⁿ,k)^pairwith the minimal diameter.Some ofthem are bipartite graphs, which havespecialspectralproperties[27,Lemma4,Theorem2,Propo- sition3], andthey alsoindicatethat there aretwo groupswhich arealwayscomparedthroughtheotherones.

Ifwegoontolargergraphs(n>10),thenwewillfindthatthe smallest reachable diameterchanges to d=3, butit is also true thatatfirstwe havesomanygraphsthatsatisfytheseproperties.

However, aswe examine the n=18 orthe n=20cases, we can seethatthereisonlyone graphthatfulfilsourassumptions [36]. Theresultsincaseoflargergraphs,with3-regularityand3asthe minimaldiametercanbefoundinTablesA1aandA1binAppendix A.

Aswecanseethecompletion ratioisstill decreasinginnand onlargergraphsitcanbetakenbelow0.2.Itisalsotruethat we

Table 2

k = 3 -(quasi-)regular graphs on n vertices with minimal diameter d = 2 .

k = 3 Graph Further information

n = 5 •c = 8 / 10 = 0 . 8

•Unique graph

n = 6

3-prism graph ( C 3×K 2)

•c = 9 / 15 = 0 . 6

•2 graphs

•Source: [36]

•The other solution is the bipartite graph K 3,3

n = 7 •c = 11 / 21 ≈0 . 524

•4 graphs

n = 8

Wagner graph

•c = 12 / 28 ≈0 . 429

•2 graphs

•See also: [37]

•The other solution is the X 8graph [31]

n = 9 •c = 14 / 36 ≈0 . 389

•2 graphs

n = 10

Petersen graph

•c = 15 / 45 ≈0 . 333

•Unique graph

•See also:[38]

do notneedto answerformorethan 30questionsforanMCDM problemevenwith20criteria,whichcanbeindeeduseful.

If we go on to larger graphs, the minimal diameter would change to d=4, however, in this paper we only consider the graphswithd≤3,sowediscussedtheinterestingcasesfork=3. The former results mean that, if we would like to examine the graphswherek=4,itisobviousthattheminimaldiameterwould be2untiln=10,butitisnotsoimportanttomakesomanycom- parisonsbecausethispropertycanbereachedwithk=3,aswell.

Thusfork=4theinterestingcasesstartabove10vertices,andthe questionisifwecanreachasmallerdiameter(amorestablesys- temofcomparisons)withtheriseoftheansweredquestions.We foundthatwithk=4wecanget2astheminimaldiameteruntil n=15,butforlargervaluesofn,itwillbe3again,whichcanbe alsoreachedbyk=3,thuswewouldnotrecommend thesecom- binations of parameters. The results for (¹¹≤n≤15,k=4) ^are showninTableA2.Itisalsoimportanttonotethatk=4ispossi- bleincaseofbothoddandevenvaluesofn,thusnowwedonot havetopayspecialattentiontothis.

As we can see,the completion ratio isincreasing ink,so we cannot getsosmallc valuesasinTableA1a,howeverthesystem ofcomparisonswillbemorestableevenonmanyvertices,because thesmallestdiameteris2here.Itisalsointerestingthat,forlarger graphs andregularitylevels, thenumber ofconnectedgraphs in- creasesveryrapidly.Forinstance,whenwehave15vertices,there are805491connected4-regulargraphs(thatmeans805491pos- siblefillingpatternsofthePCM), andonlyonehas2asits diam- eter. Ourresults andmethodologyhas astrong relationship with the so-calleddegree/diameter problemthat iswell known inthe literature of mathematics ([39], [40], [41]), but they are looking for the largest possible nfor a given diameterand a givenlevel of maximum degree. Several construction techniques have been proposedforgraphsinconnectionwiththedegree/diameterprob- lem [31,42,43], and one can also find extended tables with the knownresults[44].Foranindeedextensivesummaryoftheprob-

lem,see[45].The scientificresults inthisfield supportourfind- ings, too,because for(^k=3,d=2) ^{the largestn is10, while for} (^k=3,d=3)^{itis20.Inthecaseof}(^k=4,d=2)^{thelargestnis} 15,butfor(^k=4,d=3)^{itisproventhatthelargestgraphismuch} aboveourbound,whiletheoptimalnumberoftheverticesinthis caseisstillanopenquestion.

Aswementionedearlier,thereisnopointinfinding4-regular graphs when 16≤n≤20, thus Table A3 contains the 4-regular graphsfor21≤n≤24forwhichthediameteris3.Whentheta- blescontain‘≥...graphs’,thatmeanswehavenotcheckedallthe possiblecaseswithminimaldiameter,butinconnectionwithdeci- sionmakingproblems,itisenoughtoseethatthereisonepattern thatsatisfiestheneededproperties.

Finally, we can increase the regularity level to 5 in order to findoutifweareabletoget2asthesmallestdiameterforlarger graphs.The answeris yes,actually itis alsoproven that d=2is reachablefor5-regular graphsuntil24vertices,butofcourse we areinterestedinthespecificgraphsthat couldhelpusdetermine theadequatecomparisonpatterns.OurresultscanbefoundinTa- blesA4aandA4b.Thek=5parameterisonlypossiblewhennis evenagain, so whenitis odd,we letone vertextohave 6asits degree.

The5-quasi-regulargraphon21verticeshasbeenfoundbyus asa twisted product K₃∗X₇, where X₇ is a graph withdiameter 2 on 7vertices, in which all vertices havedegree 3,except one, whereithas2.The5-regulargraphon22verticeshasbeenfound by[46]withthehelpofthefollowingintegerlinearprogramming problem:

LetN=

{

¹,...,22

}

^{bethenodes,andletP}=

{

ⁱ∈N,j∈N:i<

j

}

^{be thesetofnode pairs.For}(ⁱ,j)∈P,letbinarydecision variable X_i,j indicate whether (ⁱ,j) ^{is an edge. For} (ⁱ,j)∈P and k∈N

\ {

ⁱ,j

}

^{, let binary decision variable Y}i,j,k indicate whetherkisacommonneighborofiand j.For(ⁱ,j)∈Plet

binarydecisionvariableSLACK_i,jbeaslackvariable.

min (ⁱ,j)∈P

SLACKi,j (1)

(i,j)∈P:k∈{i,j}

X_i,j=5 fork∈N (2)

Xi,j+

k∈N\{ⁱ,j}

Y_i,j,k+SLACKi,j≥1 for

(

i,j

)

∈P (3)

Yi,j,k≤[i<k]Xi,k+[k<i]Xk,i for

(

^i,j

)

^∈Pandk∈N

\ {

^i,j

}

(4)

Y_i,j,k≤[j<k]X_j,k+[k<j]X_k,j for

(

ⁱ,j

)

∈Pandk∈N

\ {

^i,j

}

(5) Constraint (2) enforces 5-regularity. Constraint (3) enforces diameter2.Constraints(4)and(5)enforcethatY_i,j,k=1im- plieskisaneighbor ofi and j,respectively.Adesiredgraph existsifandonlyiftheintegerlinearprogramhasasolution withSLACK_i,j=0for

∀

(ⁱ,j)∈P.

Theauthorsofthispaperarestilllookingfora5-quasi-regular graphon23verticeswithdiameter2,butmanagedtofindagraph, which has23vertices,andits diameteris2,butithasone more edge than itshould, namelythree verticeshave degree6 andall theothershave5.

As we can see in Tables A4a and A4b there are higher com- pletion ratios again, andfor instance when we have 24 vertices, thedecisionmakersshouldmake60comparisons,whichincertain situations can be too many.One can also note that in thistable wereport thattherearesome graphswiththeneededproperties, butneverindicatethenumberofthem.Thereasonbehindthisis simple:theveryhighnumberofthepotentialconnected5-regular graphs(forinstanceinthecaseofn=24thereareroughly2·10²² possibilities).

This meansthat we haveexaminedall the casesthat wepre- viously called interesting. According toour results,if we usethe (ⁿ,k,d)parameters,thenforsmallerMCDMproblemsthek=3is enough toget2asthediameteroftherepresentinggraph,which leadstoasmallcompletion ratioandastablesystemofthecom- parisons. In larger problems,when we have morealternatives or criteria,wecanchooseifweusek=3,whenthecompletionratio issmaller,butourapproximationcanbeunstable,orchoosehigher level of regularity (and completion ratio) with more reliable re- sults.Wealsoshowedexamplesandgraphswiththeneededprop- erties forthe differentcases,whichcan help anyonein aMCDM problemto decidewhich comparisonshaveto be made. Onecan find the summary of our results in Table 3, which shows how manygraphsweknowforgiven(ⁿ,k,d)parameters.Itisalsotrue that ifthereisagraphfor (ⁿ,k,d)^{inthetable, then,ontheone} hand,nographexistswiththeparameters(ⁿ,k,d−1)^,and,onthe otherhand,graphsfor(ⁿ,k,D)^,whereD>d,arenotcounted,and thecorrespondingcellsareleftempty.Weomittedthecaseswhen k=4andn≤10,becausetheminimaldiameteristhesameasit wasinthecaseofk=3.Thereisthe samereasoningbehindthe emptiness of the table when k=5 and n≤15. We havenot in- cluded the cases whenk=4and16≤n≤20,because d=3can beachievedby3-regulargraphs,butford=2atleast5-regularity is needed. We also not included the k=3 andn≥20 cases,be- causewewereexamininggraphswithd=2and3only.

Table 3

The summary of the results: the number of k -(quasi-)regular graphs on n nodes with diameter d. Lightgray denotes d = 2 and gray denotes d = 3 , ‘ ≥’ means that there are at least as many graphs as indicated, but we could not check all the pos- sible cases.

k

n 3 4 5

5 1

6 2

7 4

8 2

9 2

10 1

11 134 37

12 34 26

13 353 10

14 34 1

15 290 1

16 14 ≥3

17 51 ≥1

18 1 ≥1

19 4 ≥1

20 1 ≥1

21 ≥3 ≥1

22 ≥1 ≥1

23 ≥1 ?

24 ≥1 ≥1

All the graphs in Tables 2, A1a, A1b, A2, A3, A4a, and A4b aregiveninseveralformsinAppendixB:graph,adjacencymatrix (that directly shows which comparisons should be made, which PCMelementsarerequired),listofedgesand‘Graph6’format.The listofedgesalsopresenttheneededcomparisons,forinstancethe graph on 5 vertices in Figure B1 in Appendix B (see it also in Table2) showsthat the decisionmaker shouldfill inthe follow- ing elementsofthe PCM: a₁₂, a₁₃,a₁₄,a₁₅, a₂₃, a₂₄, a₃₅ anda₄₅. Upon request the other graphs of each family are available from theauthorsintheseandotherforms,aswell.

4. Numericalexampleandsimulations

Theregularityofthe representinggraphshasbeenextensively studiedin connectionwithincompletepairwise comparisons’de- signs, while the diameterhas only been investigated partially in theliterature,asitwasmentionedintheintroduction.Wewould liketo presentwhat kindofproblemscanoccur evenwithregu- largraphs,ifwe donottake intoaccountthediameter,througha motivationalexample.

Awiderangeofsimulationshasalsobeenperformedinorder to validate our recommendations, the applied methodology and thegainedresultsarediscussedinmanydetailsbelow.Wewould liketoemphasizethat,inthissectionwerelyontheframeworkof the pairwise comparison matrices, though, our recommendations canbeadoptedinmanyotherfields,aswell.

4.1. Simulationmethodology

It is important to see if the filling in pattern designs recom- mendedby usaretrulyuseful,thus weappliedextensivesimula- tionstohaveabetterunderstandingoftheproblem.Asforthecal- culationtechniquesoftheweightsderivedfromthePCMs,weused thewell-knownLogarithmicLeastSquaresMethod(LLSM)andthe Eigenvector Methodbased on theCR-minimal completion (CREV) [30]. We applied two metrics to determine the differences from the weights calculated from the complete PCMs, that is the Eu- clideandistance(deuc) andthemaximumabsolutedistance(dmax, alsoknownasChebyshevdistance),givenbythefollowingformu-

Fig. 2. The scaling on different ranges

las:

deuc

(

u,

v )

=

n i=1

(

^ui−

v

i

)

²

dmax

(

^u,

v ₎

= max

i∈1,...,n

|

^ui−

v

i

|

,

whereudenotestheweightvectorcalculatedfromacertainfilling in design,while

v

isthe weight vector calculatedfromthe com- pletePCM.uand

v

arenormalizedbyn

i=1u_i=1andn i=1

v

i=1, respectively, while

v

_iandu_idenotetheithelement ofthe appro- priatevectors.

The process ofthesimulation fora given(ⁿ,k)^{pair consisted} ofthefollowingsteps:

1. We generated random n×n complete and consistent pair- wise comparison matrices. The elements of these matrices weregivenasa_{i j}=w_i/w_j,wherew_i∈[1,9]isauniformlydis- tributedrandomrealnumberfor

∀

^i.

2. Then we perturbed the elements of our consistent matrices threedifferentways,togetinconsistentPCMswiththreedistin- guishableinconsistencylevels.Wecalltheselevelsweak,mod- estandstronggivenwiththefollowingformulas:

b_ij=max

₁

2, a_ij+

∈[−1,1] (weak)

b_ij=max

₁

2, a_ij+

∈[−2,2] (modest)

b_ij=max

₁

3, a_ij+

∈[−3,3] (strong)

Whereb_{i j} istheelementoftheperturbedmatrix,a_{i j} istheel- ementoftheconsistentmatrix,a_{i j}≥1,and^{isuniformlydis-} tributedinthegivenranges.Themotivationbehindthisstruc- ture is the following, we can get perturbed data even from an ordinal point of view, when b_{i j}<1. However, in order to getmeaningfulresults,we shoulduseadifferentscale forthe rangeof(0,1)comparedtotherangeof(1,9]inconnectionwith PCMs,asFigure2suggests.Thatiswhythemaximumfunction andthelowerbounds(1/2,1/2and1/3,respectively)appearin thedefinition.Theseelement-wiseperturbationmethodscorre- latewiththewellknownConsistencyRatio(CR),asitisshown inFigure3.Wetestedseveralcombinationsofparameters,and foundthatthese,moreorlessbalancedperturbationsaround1, resultinthemostrelevantlevelsofinconsistency.

Fig. 3. The connections between CR and our element-wise perturbations. Each point shows the average CR of 10 0 0 randomly generated perturbed pairwise com- parison matrices.

3. We deleted the respective elements of the matrices in order toget the filling inpatternthat we were examining,and ap- plied the LLSM and the CREV techniques to get the weights.

We always computed the certain designs’ distances from the weightsthatwecalculatedfromthecompleteinconsistentma- trices.Weused1000PCMsforeverylevelofinconsistencyand appliedthefollowingfillinginpatternstocomparethemwith eachother:

(i) Ourrecommendations:k-(quasi-)regulargraphs ofminimal diameter,detailedinSection3andAppendixA,

(ii)Randomconnectedgraphswiththesamenumberofedges as our recommendation (1000 graphs per inconsistency level persimulation),

(iii)Connectedk-(quasi-)regulargraphs,butnotofminimaldi- ameter(1000graphspersimulation),

(iv)Randomlygenerated,connected, ofminimal diameter,but not regular graphs with the same number of edges (1000 graphspersimulation),

(v)Minimal diameter, modified/extendedstargraphs withthe samenumberofedges(1000graphsperinconsistencylevelper simulation).

4. Finally,wesaved themeanandstandard deviationofthedis- tancesforthedifferentweight calculationmethods andfilling indesigns.

Fig. 4. The graph representation of two 3-regular designs

We restricted the connected k-(quasi-)regular graphs to the Hamiltonian ones during the generation. With this we excluded the k-(quasi-)regular graphs with the largest diameters as well.

This wasalso interesting,because all ofour recommendations in Section 3 and Appendix A are Hamiltonian except the Petersen graphandtheTietzegraph,butthesetwoare well-knownexcep- tions[47,48].

Incaseof(iv),webasicallygeneratedrandomconnectedgraphs andselectedtheoneswithminimal diameter(the samediameter asourrecommendation),until wehad1000such graphs,atleast inthecaseswherewehavefoundsomanyinstancesinareason- ablylongtime.

As for (v), when thediameter ofour recommendationwas 2, thenwegeneratedarandomstargraph,andcomplementeditwith theneedednumberofrandomedges.Whileincaseofdiameter3, we did the same,butat theend, we deletedone edge from the starandreplaceditwithanotherone,sothat thediameterofthe graphbecame3.

Itisimportanttonote thatweconsidered onlythegraphpre- sented inSection 3andAppendix Afora given(ⁿ,k) ^pairin(i), and not all the k-(quasi-)regular graphs with minimal diameter.

Thisisduetothefactthatinmanycaseswewereabletofindone graph withtheneededproperties,butcould notfindall ofthem orevencouldnotdeterminetheexactnumberofsuchgraphs.

Before the results ofthe simulations, we show a motivational example,inwhichwecomparetwodifferentfillinginpatternde- signs similarly as in the case of the simulations. This numerical instance showsthat it isalso importantto take intoaccount the minimaldiameterproperty,andnotjustregularity.

4.2. Motivationalexample

Letusdemonstrate thesimulation process,aswell asthe im- portance ofthe diameter, whenwe have 10 alternatives, andwe examineonlytwodifferentfillinginstructures.

We generate 1000 n×n consistent PCMs with elementsa_{i j}= w_i/w_j,wherew_i,w_j∈[1,9]areuniformlydistributedrandomreal numbers.ThenweperturballoftheelementsofthesePCMsthree differentwaysasdescribedinEquationsweak,modestandstrong. We would like to compare the differences of the calculated weightsfromtheonesthatwegetfromthesecompleteperturbed PCMs,whenweconsiderthetwofillinginpatternsrepresentedby thegraphsinFigure4.Thefillingstructuresrelatedtothesegraphs can beseeninTable4,whichmeans thatwe deleteall theother elements, when we compute the weights according to the given pattern.

Asforthetworepresentinggraphs,thePetersengraphhasmin- imal diameteramong 3-regular graphs on10 vertices, its diame- ter is2, whiletheAlternative 3-regulargraph’s diameteris5. As onecanseetherearecommonelementsofthetwo fillinginpat- terns, as for instance the bridge-edge between vertices 1 and 6 (a₁₆,bridge set[49]), whichconnectsthe twosymmetric compo- nentsoftheAlternativegraph.Itisalsoworthtomentionthatthe specialstructure ofthisgraph(also highlightedby thetwo sepa- ratepartsoftherelatedPCMinTable4)ensures thattheweights of1and6arealwaysdeterminedexactlybyb₁₆.

Table5summarizesthemean(denotedbyM)andthestandard deviation (

σ

^{) of distances (deuc and dmax) of the weights calcu-}

latedfromthetwofillingpatternsrespecttothecompletecasefor the three inconsistency (perturbation) levels (Weak, Modest and Strong).

One can see that there are significant contrasts between the outcomesofthe examined fillinginpatterns.In caseofboth the Euclidean and maximum absolute (Chebyshev) metrics, the dis- tancesoftheweightscomputedfromtheAlternativegraphrespect totheoneswegotfromthecompletePCMare approximately1.5 timeslarger,than thesameforthePetersengraph,howeverboth therelativeandabsolutedifferencesaresmallerincaseoftheab- solutemaximumdistance.Thesameresultsaretruewhenwecon- siderthestandarddeviationofthedistances.Thismeans thatthe Petersengraphtendstoprovidesmallerrorsandaconsistentper- formance (small standard deviation) depending on the perturba- tions,compared tothefilling patternrepresented bythe Alterna- tivegraph.

We think that this example can give a deeper understanding ofthesimulation method.Besidesthat,the mainmessageofthis sub-sectionisthat,oneshouldconsiderthediameterofthegraph asanimportantparameterinthesedesigns, becauseevenamong regulargraphs,therecanbelargedifferences.

4.3. Simulationresults

The resultsof the simulations seemto mainly depend on the valueofk,andbarelyonn,aswellasthepatternsoftheoutcomes seemtobethesameforeverycase.

The tables for all parameters (ⁿ,k,d) ^{calculated are available} in Appendix D, while we have chosen to visualize only the fol- lowingrepresentativeexamples:(ⁿ=16,k=3,d=3),(ⁿ=11,k= 4,d=2) ^and(ⁿ=24,k=5,d=2)^{.Thefirst one isthelargest 3-} regularcase,wherewecouldapply(iv),anditistheonlyonethat can be found in the main text due to the length of the figures.

The second one isthe smallest4-regular, andthe last oneis the largest5-regularcasethat weexamined.The resultsofthesimu-

Table 4

The known elements of the given PCM in case of the two different filling in patterns represented by the graphs in Figure 4 . The design related to the Alternative graph can be seen to the left, while the filling structure of the Petersen graph is shown in the PCM to the right.

1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10

1 1 b 12 b 14 b 16 1 1 b 13 b 14 b 16

2 _b¹₁₂ 1 b 23 b 25 2 1 b 24 b 25 b 27

3 _b¹₂₃ 1 b 34 b 35 3 _b¹₁₃ 1 b 35 b 38

4 _b¹_{14 b}¹₃₄ 1 b 45 4 _b¹_{14 b}¹₂₄ 1 b 49

5 _b¹_{25 b}¹_{35 b}¹₄₅ 1 5 _b¹_{25 b}¹₃₅ 1 b 510

6 _b¹₁₆ 1 b 67 b 69 6 _b¹₁₆ 1 b 67 b 610

7 _b¹₆₇ 1 b 78 b 710 7 _b¹_{27 b}¹₆₇ 1 b 78

8 _b¹₇₈ 1 b 89 b 810 8 _b¹_{38 b}¹₇₈ 1 b 89

9 _b¹_{69 b}¹₈₉ 1 b 910 9 _b¹_{49 b}¹₈₉ 1 b 910

10 _b¹_{710 b}¹_{810 b910}¹ 1 10 _b¹_{510 b}¹_{610 b}¹₉₁₀ 1

Table 5

The average distances and their standard deviation for the different designs. The following notations are used: M-mean, σ-standard deviation, ’Weak’,

’Modest’ and ’Strong’ refer to the level of perturbation.

Weak LLSM d eucM CREV d eucM LLSM d maxM CREV d maxM LLSM d eucσ ^{CREV d} eucσ ^{LLSM d} maxσ ^{CREV d} maxσ

Petersen 0.0424 0.0422 0.0275 0.0274 0.0285 0.0283 0.0193 0.0191

Alternative 0.0605 0.0604 0.0370 0.0369 0.0468 0.0467 0.0286 0.0285

Modest

Petersen 0.0673 0.0669 0.0450 0.0445 0.0378 0.0376 0.0278 0.0274

Alternative 0.0956 0.0956 0.0604 0.0602 0.0610 0.0611 0.0400 0.0399

Strong

Petersen 0.0967 0.0952 0.0665 0.0652 0.0527 0.0519 0.0402 0.0390

Alternative 0.1318 0.1314 0.0881 0.0877 0.0825 0.0826 0.0590 0.0592

lations forthem are showninFigures 5,C1 andC2,respectively, anditisalsorecommendedtoreadthissectiontogetherwithAp- pendix C,asthelattertwo casesarepresentedthere.The figures show themeanofthedifferentmetrics(M)andthestandardde- viation (

σ

^{) aswell. Wereferto thedifferentlevelsofthepertur-}

bationas‘Weak’,‘Modest’and‘Strong’,asbefore.

It is clear from the outcomes of the simulations that the stronger perturbationcauseslarger distances,andthehigherreg- ularity level leads to smaller differences. As one can see, our recommendations have the smallest means and standard devia- tions among the different designs in case of both metrics and both weightcalculation methodsforevery (ⁿ,k)^{pair,which sug-} geststhat theresults arenot solelydependent onthe usedtech- niques and parameters. The smallest mean shows that the k- (quasi-)regular graphs withminimal diameterprovidethe closest weights to the complete PCM on an average level. On the other hand,thesmalleststandarddeviationalsoimpliesthatourrecom- mendations arecommonlynotconnectedtohugeerrors,andthat thesefilling inpatterndesigns performat avery consistentlevel regardingthedeviationsfromtheresultsofthecompletePCMs.It isalsotruethattherandomlygeneratedminimaldiametergraphs (denoted by (iv)) tend to have smaller means and standard de- viations comparedto the simplerandom graphs.Again, thissug- geststhat,thediameteroftherepresentinggraphisrelevant.The k-(quasi-)regulargraphs(denotedby (iii))alwayshavethesecond smallestmeansandstandarddeviationsintheirdistances,thusthe already known fact, that regularity is a key property, confirmed hereaswell.Itisalsoimportanttonotethatwehaveexcludedthe k-(quasi-)regularcases withthelargestdiameters, becauseofthe Hamiltonianconstructionaswementionedearlier,thusweexpect random (quasi-)regular graphs to have a bit even ‘worse’ results compared toourrecommendations.Thecaseofthemodifiedstar graphs(denotedby(v))isinteresting.Incaseofk=3,theyalways havesmallermeansandstandarddeviationscomparedtothesim- ple random graphs,butfork=4 they always havelarger means,

andin some caseseven their standard deviations are higher.For k=5themodifiedstargraphstendtohavethelargestmeansand standard deviations among the examined designs. This also sug- geststhatconsidering onlythediameterisnot sufficientinthese problems.Finally,we wouldliketoemphasizethatthesepatterns and findings, are the very same for all studied (ⁿ,k) ^{pairs, es-} pecially regarding thedominance of the k-(quasi-)regular graphs, thus our recommendations seem to perform indeed well in the frameworkofpairwisecomparisonmatrices.

5. Conclusionsandfurtherresearch 5.1. Summary

The main contribution of the paperis a systematic collection of recommended filling patterns ofincomplete pairwise compar- isons’ usingthe graph representationof thePCMs. The proposed (quasi)-regulargraphs withminimaldiameterhavenot onlypure graphtheoretical relevance,buttheir importanceinmulti-criteria decisionmakingisalsodemonstratedviathecomparisonstoother incompletefillinginpatternsofthesamecardinality.

Graphs are included in several formats in Appendix B, which canshow practitioners thecomparisons thatshould be made,i.e.

thePCM elementsto be filledin.We presentedourresultsusing thenumbernofcriteriaoralternatives,regularitylevelk anddi- ameter dof the representing graphasparameters. We identified thediameter,thatwasmissingfromtherelevantliterature ofde- cisiontheoryandpreferencemodelling,asanimportantparameter intheseproblems.Ithasbeenshownthatrelativelysmalldiame- tersd=2,3can beachievedwithrelativelysmallcompletion ra- tios, andexamples hasbeen provided forevery caseup until 24 vertices.

We alsovalidated ourrecommendations withthe help ofnu- mericalsimulations.1000perturbedPCMs wereusedincaseof3 differentinconsistency(perturbation)leveltocompareseveralfill-

Fig. 5. The results of the simulation for (n = 16, k = 3, d = 3). The following notations are used: M-mean, σ-standard deviation, d euc-Euclidean distance, d max-maximum absolute distance, Weak, Modest and Strong refer to the level of perturbation. See Table D13 in Appendix D for numerical details.