A New Grey Approach for Using SWARA and PIPRECIA Methods in a Group Decision-Making Environment

ANewGreyApproachforUsingSWARAandPIPRECIA MethodsinaGroupDecision-MakingEnvironment

DragišaStanujki´c 1 ,DarjanKarabaševi´c 2,* ,GabrijelaPopovi´c 2 ,PredragS.Stanimirovi´c 3 , MuzaferSaraˇcevi´c 4 ,FlorentinSmarandache 5 ,VasiliosN.Katsikis 6 andAlptekinUluta¸s 7,*

1 TechnicalFacultyinBor,UniversityofBelgrade,VojskeJugoslavije12,19210Bor,Serbia; dstanujkic@tfbor.bg.ac.rs

2 FacultyofAppliedManagement,EconomicsandFinance,UniversityBusinessAcademyinNoviSad, Jevrejska24,11000Belgrade,Serbia;gabrijela.popovic@mef.edu.rs

3 FacultyofSciencesandMathematics,UniversityofNiš,Višegradska33,18000Niš,Serbia;pecko@pmf.ni.ac.rs

4 DepartmentofComputerSciences,UniversityofNoviPazar,DimitrijaTucovi´cabb,36300NoviPazar,Serbia; muzafers@uninp.edu.rs

5 MathematicsandScienceDivision,GallupCampus,UniversityofNewMexico,705GurleyAve., Gallup,NM87301,USA;smarand@unm.edu

6 DepartmentofEconomics,DivisionofMathematicsandInformatics,NationalandKapodistrianUniversityof Athens,Panepistimiopolis,15784Ilissia,Greece;vaskatsikis@econ.uoa.gr

7 DepartmentofInternationalTradeandLogistics,FacultyofEconomicsandAdministrativeSciences, SivasCumhuriyetUniversity,58140Sivas,Turkey

* Correspondence:darjan.karabasevic@mef.edu.rs(D.K.);aulutas@cumhuriyet.edu.tr(A.U.)

Citation: Stanujki´c,D.;Karabaševi´c, D.;Popovi´c,G.;Stanimirovi´c,P.S.; Saraˇcevi´c,M.;Smarandache,F.; Katsikis,V.N.;Uluta¸s,A.ANewGrey ApproachforUsingSWARAand PIPRECIAMethodsinaGroup Decision-MakingEnvironment. Mathematics 2021, 9,1554. https:// doi.org/10.3390/math9131554

AcademicEditors:SvajoneBekešiene andIevaMeidute-Kavaliauskiene

Received:31May2021 Accepted:30June2021 Published:1July2021

Publisher’sNote: MDPIstaysneutral withregardtojurisdictionalclaimsin publishedmapsandinstitutionalaffiliations.

Abstract: Theenvironmentinwhichthedecision-makingprocesstakesplaceisoftencharacterized byuncertaintyandvaguenessand,becauseofthat,sometimesitisveryhardtoexpressthecriteria weightswithcrispnumbers.Therefore,theapplicationoftheGreySystemTheory,i.e.,greynumbers, inthiscase,isveryconvenientwhenitcomestodeterminationofthecriteriaweightswithpartially knowninformation.Besides,thecriteriaweightshaveasignificantroleinthemultiplecriteria decision-makingprocess.Manyordinarymultiplecriteriadecision-makingmethodsareadapted forusinggreynumbers,andthisisthecaseinthisarticleaswell.Anewgreyextensionofthe certainmultiplecriteriadecision-makingmethodsforthedeterminationofthecriteriaweightsis proposed.Therefore,thearticleaimstoproposeanewextensionoftheStep-wiseWeightAssessment RatioAnalysis(SWARA)andPIvotPairwiseRelativeCriteriaImportanceAssessment(PIPRECIA) methodsadaptedforgroupdecision-making.Intheproposedapproach,attitudesofdecision-makers aretransformedintogreygroupattitudes,whichallowstakingadvantageofthebenefitthatgrey numbersprovideovercrispnumbers.Themainadvantageoftheproposedapproachinrelationtothe useofcrispnumbersistheabilitytoconductdifferentanalyses,i.e.,consideringdifferentscenarios, suchaspessimistic,optimistic,andsoon.Byvaryingthevalueofthewhiteningcoefficient,different weightsofthecriteriacanbeobtained,anditshouldbeemphasizedthatthisapproachgivesthesame weightsasinthecaseofcrispnumberswhenthewhiteningcoefficienthasavalueof0.5.Inaddition, inthisapproach,thegreynumberwasformedbasedonthemedianvalueofcollectedresponses becauseitbettermaintainsthedeviationfromthenormaldistributionofthecollectedresponses.The applicationoftheproposedapproachwasconsideredthroughtwonumericalillustrations,basedon whichappropriateconclusionsweredrawn.

Keywords: greynumbers;groupdecision-making;PIPRECIA;SWARA

Copyright: ©2021bytheauthors. LicenseeMDPI,Basel,Switzerland. Thisarticleisanopenaccessarticle distributedunderthetermsand conditionsoftheCreativeCommons Attribution(CCBY)license(https:// creativecommons.org/licenses/by/ 4.0/).

1.Introduction

Multiple-CriteriaDecision-Making(MCDM)aimstoprovidethedecision-makerwith achoiceofthebestoptionsfromthefinalsetofalternativesbyanalyzingthemfromseveral angles,i.e.,throughseveralcriteria/attributes.Criteriaweightscanhaveasignificant impactontherankingofalternativesortheselectionofthemostacceptablealternative

inMultiple-CriteriaDecision-Making(MCDM).Therefore,manymethodshavebeenproposedsofarfordeterminingcriteriaweights—Entropymethod[1],AnalyticHierarchy Process(AHP)introducedbySaaty[2],CRiteriaImportanceThroughIntercriteriaCorrelation(CRITIC)proposedbyDiakoulakietal.[3],AnalyticNetworkProcess(ANP) proposedbySaatyandVargas[4],Step-wiseWeightAssessmentRatioAnalysis(SWARA) proposedbyKersulieneetal.[5],Best-WorstMethod(BWM)proposedbyRezaei[6],FUll COnsistencyMethod(FUCOM)proposedbyPamucaretal.[7],etc.

Besidesthesemethods,thereisalsoanewgenerationofprominentmethods,such astheCharacteristicObjectsMethod(COMET)anditsextensions[8 12],thepreference rankingorganizationmethodforenrichmentevaluation(PROMETHEE)—partialranking andPROMETHEEII—completeranking[13,14].Thesuccessofthesemethodsisattributed totheircorrectmathematicalsettingsandeaseofapplicationinpractice.

Besidesdeterminingcriteriaweights,theSWARAmethodwasusedtosolvemany decision-makingproblems,aswellastodeterminetheweightsofcriteriaintheseproblems. AlthoughitcanbeobservedasarelativelynewMCDMmethod,theSWARAmethod hassofarbeenusedfor—prioritizingsustainabilityassessmentindicators[15],product designfrominternationalandlocalperspectives[16],evaluationofexteriorwallinsulation[17],evaluationofpackagingdesign[18],technologyforesightaboutR&Dproject evaluation[19],regionallandslidehazardassessment[20],personnelselection[21],copper prospectingmapping[22],riskassessment[23],engineoperatingparameteroptimization[24],evaluationofsmartcardsystemsinpublictransportation[25],marineenergy plantlocationselectionproblem[26],evaluationofsustainabilityindicatorsforrenewable energy[27],evaluationofahouseplanshape[28],selectionoflandfillsite[29],supplier selection[30,31],andsoforth.

Also,manyextensionshavebeenproposedfortheSWARAmethodthatextendsits applicationtousewithgrey[32,33],fuzzy[34],androughsets[35].

BeforeapplyingtheSWARAmethod,itisnecessarytorankthecriteriaaccordingto theirexpectedsignificance,whichinsomecasesmaybealimitationforitsuse.Therefore,Stanujkicetal.[36]extendedtheSWARAmethodintendingtoenabledetermining criteriaweightsevenincaseswhenthecriteriaarenotrankedaccordingtotheexpected significance,andtheyproposedthenamePIvotPairwiseRelativeCriteriaImportance Assessment(PIPRECIA)methodforthementionedextension.

UnliketheSWARAmethod,thePIPRECIAmethodhasbeenusedforsolvinga significantlyfewernumberofdecision-makingproblems.Thefollowingarticlescanbe mentionedassomeexamplesoftheuseofthismethod—determiningthesignificanceof criteriafortheimplementationofhigh-performancecomputinginDanuberegioncountries[37],greensupplierselection[38],assessingthequalityofe-learningmaterials[39,40], safetyevaluationofrailwaytraffic[41],assessmentofroadtransportationrisks[42],stackersselection[43],evaluationoftheICTadoption[44],andsoforth.Besides,thePIPRECIA methodhasalsobeenextendedforusinggrey[45]andfuzzy[46]numbers.

ManyextensionsofMCDMmethodsthatenableusingofgrey,fuzzyandneutrosophic setshavebeendonetoenablesolvingcomplexdecision-makingproblemsoftenassociated withtheinaccuracyandunreliabilityoftheinformationusedtomakedecisions.However, manydecision-makingproblemsoftenrequiretheparticipation,orexaminationofthe attitudes,ofmorethanonedecision-makerorrespondent.Therefore,inthisarticle,anew greyextensionofSWARAandPIPRECIAmethodsadaptedforgroupdecision-makingis proposed.Inthisextension,theattitudesoftherespondentscollectedusingcrispvalues arefurthertransformedintogreyintervalnumbers,thusretainingthepossibilitiesof differentanalyses.

Themaincontributionisthattheproposedapproachallowstheattitudesofalarger numberofrespondents,expressedincrispnumbers,tobetransformedintograyintervals basedonthemedianvalueofthecollectedresponses.Theproposedapproachcanbeused asasubstituteforapproachesinwhichgroupdecisionmatricesareformedbasedonthe meanvalueoftheanswersreceivedfromallrespondents.Therefore,themainadvantage

research methodology is presented with some basic elements calculation procedures of the SWARA and PIPRECIA methods 3, a new approach for transforming respondents’ ratings numbers is presented, while Section 4 presents two numerical proposed approach is applied. Finally, conclusions are given.

oftheproposedapproachinrelationtotheuseofcrispnumbersistheabilitytoconduct differentanalyses,i.e.,consideringdifferentscenarios,suchaspessimistic,optimistic,and soon.

2.

Research Methodology

2.1. Grey Numbers

Therefore,theremainingpartofthisarticleisorganizedasfollows—inSection 2, researchmethodologyispresentedwithsomebasicelementsofgreysettheoryandthe calculationproceduresoftheSWARAandPIPRECIAmethodsarepresented.In Section 3, anewapproachfortransformingrespondents’ratingsintoappropriategreyinterval numbersispresented,whileSection 4 presentstwonumericalillustrationsinwhichthe proposedapproachisapplied.Finally,conclusionsaregiven.

2.ResearchMethodology

2.1.GreyNumbers

TheGreySystemsTheoryisaneffectivemethodologythatcanbeusedforsolving uncertainproblemswithpartiallyknowninformation[47].Thebasicconceptofthistheory isthatallinformationisclassifiedintothreecategories,dependingontheiruncertainty, andtheappropriatecolorsareassignedtothemasfollows—knowninformationislabeled aswhite,unknowninformationislabeledasblack,andtheuncertaintyinformationis labeledgrey.Thewhite,blackandgreynumbersareshownonFigure 1.

The Grey Systems Theory is an effective methodology uncertain problems with partially known information theory is that all information is classified into three categories, uncertainty, and the appropriate colors are assigned information is labeled as white, unknown information uncertainty information is labeled grey. The white, black and Figure 1.

white number black number

grey number

Figure1. White,blackandintervalgreynumbers.

Figure 1. White, black and interval grey numbers.

Definition1. Greynumber[47].Agreynumber ⊗x issuchanumberwhoseexactvalueis unknown,buttherangeinwhichvaluecanliesisknown.

Definition 1. Grey number [47]. A grey number ⨂�� is such unknown, but the range in which value can lies is known.

Definition2. Intervalgreynumber[47].Intervalgreynumberisagreynumberwithknown lowerboundx andupperbound x,butwithunknownvalueofx,anditisshownasfollows:

Definition 2. Interval grey number [47]. Interval grey number lower bound �� and upper bound �� , but with unknown value of

Definition3. Basicoperationsofgreynumbers[34].Let ⊗x1 =[x1, x1] and ⊗x2 =[x2, x2] be twogreynumbers.Thebasicoperationsofgreynumbers ⊗x1 and ⊗x2 aredefinedasfollows:

Definition 3. Basic operations of grey numbers [34]. Let 1x ⊗ two grey numbers. The basic operations of grey numbers 1x ⊗

Definition4. Thewhiteningfunction[48].Thewhiteningfunctiontransformsanintervalgrey numberintoacrispnumberwhosepossiblevaluesliebetweentheboundsoftheintervalgrey number.Forthegivenintervalgreynumber,thewhitenedvaluex(λ) ofintervalgreynumber ⊗x is definedas

x(λ) = (1 λ)x + λx,(6) where λ denotesthewhiteningcoefficient,and λ ∈ [0,1]

Intheparticularcase,when λ =0.5,thewhitenedvaluebecomesthemeanofthe intervalgreynumber,asfollows:

x(0.5) = 0.5(x + x) (7)

2.2.TheSWARAandthePIPRECIAMethods

ThebasicstepsofthecomputationalproceduresoftheSWARAandPIPRECIAmethodsfordeterminingcriteriaweightsbasedontheattitudesofarespondent,i.e.,decisionmakerorexpert,arepresentedinthissection.

2.2.1.TheSWARAMethod

Theprocedurefordeterminingweightsof n criteriabyapplyingtheSWARAmethod canbesummarizedasfollows[5,18]:

Step1.Identificationandselectionofcriteria; Step2.Sortingcriteriaaccordingtotheirexpectedsignificance,indescending order;and Step3.Determiningtherelativesignificanceofcriteria.Inthisstep,therelative significanceofthefirstcriterionissetto1,whiletherelativesignificanceoftheothers sj is determinedaccordingtothepreviousone.

IntheSWARAmethod, sj 1 hasavalue0ifcriterion j 1 hasthesamesignificance asthecriterion j, sj ∈ [0,1],andthelowervalueof sj 1 indicatesahighersignificanceof criterion j inrelationtocriterion j 1.

Step4.Calculatingthevalueofthecoefficient kj as: kj = 1 ifj = 1 sj + 1 ifj > 1 (8)

Step5.Calculatingthevalueoftherecalculatedsignificance qj asfollows: qj = 1 ifj = 1 qj 1 kj ifj > 1 (9)

Step6.Calculatingthecriteriaweights wj asfollows: wj = qj ∑n j=1 qk (10)

ThecalculationprocedureoftheSWARAmethod,fordeterminingthecriteriaweights, issummarizedinFigure 2.

Criteria sorting

Criteria identification j = 1; kj = 1; qj = 1; j < n �� ≺��

Criteria comparation

False j += 1 sj = 0 �� ; �� ∈ 0,1 �� =�� +1 �� = �� −1 ��

True

�� = �� ∑ ��

Figure 2. Flowchart of the SWARA method.

Figure2. FlowchartoftheSWARAmethod.

2.2.2.ThePIPRECIAMethod

2.2.2. The PIPRECIA Method

The procedure for determining criteria weights by applying the PIPRECIA method can be summarized as follows [36]:

TheprocedurefordeterminingcriteriaweightsbyapplyingthePIPRECIAmethod canbesummarizedasfollows[36]:

Step1.Identificationandselectionofcriteria;

Step 1. Identification and selection of criteria; Step 2. Sorting criteria according to their expected significance, in descending order. This step is not strictly required in the PIPRECIA method. Unlike the SWARA method, the PIPRECIA method can be used even when it is not possible to assess the significance of the criteria in advance; and

Step2.Sortingcriteriaaccordingtotheirexpectedsignificance,indescendingorder. ThisstepisnotstrictlyrequiredinthePIPRECIAmethod.UnliketheSWARAmethod, thePIPRECIAmethodcanbeusedevenwhenitisnotpossibletoassessthesignificanceof thecriteriainadvance;and

Step3.Determiningtherelativesignificanceofcriteria.Inthisstep,therelative significanceofthefirstcriterionissetto1,whiletherelativesignificanceoftheothers sj is determinedconcerningthepreviousone.

Step 3. Determining the relative significance of criteria. In this step, the relative significance of the first criterion is set to 1, while the relative significance of the others �� is determined concerning the previous one.

Unlike the SWARA method, in the PIPRECIA method, �� =1 if criterion �� −1 has the same significance as the criterion �� , �� ∈(1,2 if criterion �� −1 has higher significance than criterion �� , and �� ∈(1,0 if criterion �� −1 has a lower significance than criterion �� .

UnliketheSWARAmethod,inthePIPRECIAmethod, sj 1 = 1 ifcriterion j 1 has thesamesignificanceasthecriterion j, sj ∈ (1,2] ifcriterion j 1 hashighersignificance thancriterion j,and sj ∈ (1,0] ifcriterion j 1hasalowersignificancethancriterion j

Step4.Calculatingthevalueofthecoefficientasfollows: kj = 1 ifj = 1 2 sj ifj > 1 (11)

Step 4. Calculating the value of the coefficient as follows: �� = 1���� �� =1 2−�� ���� �� >1 (11)

Step 5. Calculating the value of the recalculated significance as: �� = 1���� �� =1 ���� �� >1 (12)

Step5.Calculatingthevalueoftherecalculatedsignificanceas: qj = 1 ifj = 1 qj 1 kj ifj > 1 (12)

Step 6. Calculating the criteria weights as: �� = ∑ (13)

Step6.Calculatingthecriteriaweightsas: wj = qj ∑n j=1 qk (13)

The calculation procedure of the PIPRECIA method is shown in Figure 3.

ThecalculationprocedureofthePIPRECIAmethodisshowninFigure 3.

Criteria

Criteria sorting

Criteria comparation

Figure 3. Flowchart of the PIPRECIA method.

Figure3. FlowchartofthePIPRECIAmethod.

From the previously described procedures, it can be concluded that the basic difference between SWARA and PIPRECIA methods is in the way of assigning the relative importance of the criteria, which is manifested in Equation (11) of the calculation procedure of the PIPRECIA method.

Fromthepreviouslydescribedprocedures,itcanbeconcludedthatthebasicdifference betweenSWARAandPIPRECIAmethodsisinthewayofassigningtherelativeimportance ofthecriteria,whichismanifestedinEquation(11)ofthecalculationprocedureofthe PIPRECIAmethod.

InthecaseofapplyingtheSWARAmethod, sj showshowmuchlessimportanta criterionisthanthepreviousone,whichiswhyitisnecessarytopre-rankthecriteria accordingtotheirexpectedsignificance.Incontrast,thePIPRECIAmethodallowsthe relativesignificanceofthecriteriatobelessthan,equalto,orgreaterthanthesignificanceof thepreviouscriterion,whichiswhyinthecaseofapplyingthismethoditisnotnecessary torankthecriteriaaccordingtotheirexpectedsignificance.

In the case of applying the SWARA method, sj shows how much less important a criterion is than the previous one, which is why it is necessary to pre-rank the criteria according to their expected significance. In contrast, the PIPRECIA method allows the relative significance of the criteria to be less than, equal to, or greater than the significance of the previous criterion, which is why in the case of applying this method it is not necessary to rank the criteria according to their expected significance.

The use of the SWARA method has already been proven in numerous published papers. In contrast, the PIPRECIA method is proposed as an extension of the SWARA method that can be used when it is not possible to determine in advance the expected significance of the criteria, as in the case of surveying the attitudes of a large number of respondents using electronic questionnaires.

TheuseoftheSWARAmethodhasalreadybeenproveninnumerouspublished papers.Incontrast,thePIPRECIAmethodisproposedasanextensionoftheSWARA methodthatcanbeusedwhenitisnotpossibletodetermineinadvancetheexpected significanceofthecriteria,asinthecaseofsurveyingtheattitudesofalargenumberof respondentsusingelectronicquestionnaires.

Thereareseveralwaystoapplythepreviouslydescribedsingle-userprocedures SWARAandPIPRECIAmethodsinthecaseofsolvinggroupdecision-makingproblems. Nevertheless,anewapproachfortransformindividualcrispattitudesintoagreygroup attitudeisproposedanddiscussedbelow.

There are several ways to apply the previously described single-user procedures SWARA and PIPRECIA methods in the case of solving group decision-making problems. Nevertheless, a new approach for transform individual crisp attitudes into a grey group attitude is proposed and discussed below.

3.ANewGreyApproachforUsingSWARAandPIPRECIAMethodsinaGroup Decision-MakingEnvironment

Asmentionedabove,decision-makersorexpertsinvolvedingroupdecision-making assessthevaluesoftherelativesignificanceofcriteriaandexpressthesevaluesusingcrisp numbers.Asaresultofgroupevaluation, K listsofrelativesignificancesareformed,where K denotesthenumberofdecision-makersorexperts.

Insuchcases,inordertodeterminethecriteriaweights,itispossibleto:

• Calculatethegroupsignificanceasfollows: sj = 1 k ∑K k=1 sk j ,(14)

andthenapplytheSWARAorPIPRECIAmethod.

• Calculatethecriteriaweightsforeachrespondent wk j ,andthencalculatethegroup criteriaweightasfollows: wj = 1 k ∑K k=1 wk j (15)

However,inordertotakeadvantageofthebenefitsthatgreynumbersprovide,i.e., thepossibilityofanalysis,atransformationofrespondents’attitudesintogreyrelative groupsignificanceisproposed,usingthefollowingsteps:

Step1.Determinethemedianvalue mdnj foreachcriterion,asfollows: mdnj = median(sk j ) (16)

Step2.Determinethelower sj andupper sj boundsofgreyrelativegroupsignificance foreachcriterion,asfollows:

sj = mean(sk j |sk j ≤ mdnj ) (17)

sj = mean(sk j |sk j ≥ mdnj ) (18)

Step3.UsingEquation(6)anddifferentvaluesforthecoefficient λ,therelativegroup significancecanbedeterminedafterwhichthecriteriaweightscanbedeterminedusing theSWARAorPIPRECIAmethods.

Usingtheproposedapproach,theattitudesofalargernumberofrespondents,i.e.,a largernumberofcrispratings,canbetransformedintoagreyinterval,i.e.,onlytwocrisp numbers,withoutsignificantlossofinformation.

Thismeansthattheproposedapproachcanprovidesimilarresultstoapproachesin whichcrispnumbersareused,butitalsoprovidesanadditionalpossibilitytoconduct analyses.Thewordsimilarisusedherebecauseinthisapproachtheedgesoftheinterval areformedbasedonthemedianvaluesofthescorescollectedfromalltherespondents. Identicalweightsofthecriteria,whenthewhitenedvalueis0.5,couldbeachievedifthe limitsofthegreynumberwereformedonthebasisofthemeanvalueinsteadofthemedian valueofthecollectedrelativesignificances.However,inthisapproach,formingofthe greynumberbasedonthemedianvalueofcollectedrelativesignificanceswasselected becauseitbettermaintainsthedeviationfromthenormaldistributioninthecollected relativesignificances.

4.ANumericalIllustrations

Topresenttheproposedapproachindetail,aswellasitsverification,thissection presentstheresultsoftwostudiesconductedbyagroupof10masteranddoctoralstudentsfromtheFacultyofAppliedManagement,EconomicsandFinance(MEF),Belgrade, Serbia.Beforebeginningresearchaboutevaluationcriteriaforassessingthequalityof websites,studentsinvolvedintheresearcharethoroughlyintroducedusingtheSWARA andPIPRECIAmethods.

4.1.TheFirstNumericalIllustration

Toverifyandexplaininmoredetailtheproposedapproach,astudywithdoctoral andmasterstudents,familiarwiththeuseofSWARAandPIPRECIAmethods,was conducted.Onthatoccasion,thesignificanceoffivecriteriaforevaluatingtraveland tourismwebsitedesignideas,suchasOntheGrid(https://onthegrid.city/ (accessed on1March2021)[49]),VisitAsia(https://www.visitasiatravel.com/ (accessedon1 March2021)[50]),VisitAustralia(https://www.australia.com/en (accessedon1March 2021)[51]),LiveAfrica(http://www.liveafrica.com/ (accessedon1March2021)[52]), BoraBora(https://www.borabora.com/ (accessedon1March2021)[53]),VisitSerbia (https://www.serbia.travel/en (accessedon1March2021)[54])andNationalParkDjerdap(https://npdjerdap.rs/en/ (accessedon1March2021)[55]),wasdetermined.

Basedon7criteriaproposedbythe2020WebAwardCompetition(http://www. webaward.org/ (accessedon1March2021)[56]),thatare,Design,Innovation,Content, Technology,Interactivity,Copywriting,andEaseofuse,thefollowingfivecriteriaare definedforevaluatingtravelandtourismwebsitedesignideas:

• Cr1,Availabilityofinformation;

• Cr2,Structureandnavigability;

• Cr3,Design;

• Cr4,Personalization;and

• Cr5,Innovations.

Onthatoccasion,ininteractionswiththerespondents,theirmeaningwasprecisely determined.Duringthisevaluation,wegavethefollowingmeaningtothecriteria— Criterion Cr1 referstotheavailabilityofusefulinformationaboutthetouristdestination onthewebsite,whilecriterion Cr2,Structureandnavigability,referstotheorganization ofinformationinsuchawaythatvisitorstothewebsitecaneasilyfindit.Thecriterion Cr3,Design,referstothevisualappearanceofthewebsiteanditssuccesstoconveythe beautyoftouristdestinations.The Cr4 criterionindicatestheabilitytotailorthewebsiteto theneedsofvisitors.Thiscriterionincludestheabilitytochoosethelanguageinwhich theinformationisdisplayed,theuseofsearchbars,aswellasresponsivewebdesign. Finally,thecriterion Cr5,Innovations,referstotheapplicationofcurrenttechnologiesand innovationsinwebdesign.

Afterchoosingthecriteria,andconsideringtheirsignificancesandimpact,duringa touroftheabove-mentionedwebsites,respondentsgavetheirattitudes,whichareshown inTable 1.Itshouldbenotedthattheabovecriteriaarelistedinorderoftheirexpected significance,whichisnecessaryfortheapplicationoftheSWARAmethod.

Table1. AttitudesofrespondentsasresultsoftheSWARAmethod.

E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 Mean

Cr1 1.001.001.001.001.001.001.001.001.001.001.00

Cr2 0.000.050.060.200.150.010.030.180.150.200.10

Cr3 0.150.100.100.200.200.050.100.200.150.200.15

Cr4 0.250.150.100.250.300.100.150.230.170.250.20

Cr5 0.250.200.200.300.400.150.150.250.200.300.24

Table 1 alsoshowsthemeanvalueoftheattitudesofallrespondentsforeachcriterion. Thecalculationdetailsandcriteriaweightsobtainedbasedonthemeanvalue,calculated byapplyingtheSWARAmethod,areencounteredinTable 2

Table2. ComputationaldetailsandcriteriaweightsgeneratedbytheSWARAmethodbasedon meanvalues.

Mean kj qj wj ’ Rank

Cr1 1.00110.2571

Cr2 0.101.100.910.2332

Cr3 0.151.150.790.2033

Cr4 0.201.200.660.1704 Cr5 0.241.240.530.1375

Theindividualcriteriaweightscalculatedontheattitudesofeachrespondentare showninTable 3.Table 3 alsoshowsgroupcriteriaweightscalculatedastheaverageof individualcriteriaweights.

Table3. IndividualandgroupcriteriaweightscalculatedbytheSWARAmethod. we1 we2 we3 we4 we5 we6 we7 we8 we9 we10 wj ”

Cr1 0.240.240.240.280.280.220.230.280.260.280.256

Cr2 0.240.230.220.240.240.220.230.240.230.240.232 Cr3 0.210.210.200.200.200.210.210.200.200.200.203

Cr4 0.170.180.180.160.160.190.180.160.170.160.170 Cr5 0.140.150.150.120.110.160.160.130.140.120.138

FromTables 2 and 3,itcanbeseenthatbothapproachesgivethesamecriteria weights,aswellasthesamerankingorderofcriteriabasedontheirsignificance,thatis Cr1 Cr2 Cr3 Cr4 Cr5, whichconfirmsthattheexpectedsignificanceofthecriteria wascorrectlyassessed,aswellasthatallrespondentsfilledinthequestionnairescorrectly. Itisimportanttonoteherethattheevaluationwasperformedbyrespondentsfamiliar withthemethodsusedandthattheyenteredtheirviewsinaquestionnairemadeinthe spreadsheetprogramwhichimmediatelycalculatedcriteriaweightsanddisplayedvalues numericallyandgraphicallyafterenteringtherelativesignificanceofeachcriterion.Inthis way,therespondentswereabletocorrecttheirgradesifnecessary.Respondentswerealso suggestedthattheycouldmodifytheirresponsesuntiltheygainedcriteriaweightsthat reflectedtheirrealopinion.

DeterminingtheCriteriaWeightsBasedontheProposedGreyApproach

Determiningcriteriaweightsusingtheproposedapproachbeganbydetermining themedianvaluesforeachcriterion,afterwhichthelowerandupperboundsofthegrey numbersweredetermined.Themedian,lowerandupperboundsofgreynumbersare arrangedinTable 4.Table 4 alsoshowsthevaluesofrelativegroupsignificanceobtained byapplyingEquation(6)and λ =0.5,aswellasitsdeviationsfromthemean.

Table4. Median,left,andrightboundsofgreyratings.

mdnj s j s j s0.5 j ∆∆%

Cr1 0.240.240.240.280.280.22 Cr2 0.240.230.220.240.240.22 Cr3 0.210.210.200.200.200.21 Cr4 0.170.180.180.160.160.19 Cr5 0.140.150.150.120.110.16

Duetothecomplexityoftheprocedurefordeterminingtheleftandrightboundsof thegreyinterval,thecalculationwasperformedusingaprogramwritteninthePython programminglanguage.

Thecriteriaweightsdeterminedbasedontheleftandrightboundariesofthegrey intervalareshowninTable 5.ThecriteriaweightscalculatedusingEquation(6)and λ =0.5 arealsoshowninTable 5.

Table5. Thecriteriaweightobtainedbasedongreyratings.

w j wj wj ”’

Cr1 0.2790.2340.257 Cr2 0.2370.2270.233 Cr3 0.2010.2050.203 Cr4 0.1600.1810.170 Cr5 0.1230.1530.137

Table 6 showsthecriteriaweightsobtainedbasedonthemeanvalueofrelative significance(wj ’),themeanvaluesoftheweightsobtainedbasedontherespondents’ attitudes(wj ”),andtheweightsobtainedbyapplyingtheproposedapproach(wj ”’).

Table6. Thecriteriaweightobtainedusingthreeapproaches.

wj’wj”wj ”’

Cr1 0.2570.2560.257 Cr2 0.2330.2320.233 Cr3 0.2030.2030.203 Cr4 0.1700.1700.170 Cr5 0.1370.1380.137

ItcanbenoticedfromTable 6 thatallthreeconsideredapproachesgivealmostthe samecriteriaweights.

Toverifytheapplicabilityoftheproposedapproachinthecaseofapplicationofthe PIPRECIAmethod,thevaluesinTable 1 wererecalculatedasfollows:

sj = 1 ifj = 1 1 sj ifj > 1

Thisrecalculationwasnecessaryduetodifferencesinthecalculationofthecoefficient kj,i.e.,inEquations(8)and(11),intheSWARAandPIPRECIAmethods.Thiswayofvalue transformationwaschosenbecausetheaimofthisresearchwasnottocomparetheresults obtainedusingtheSWARAandPIPRECIAmethodsbuttochecktheapplicabilityofthe proposedprocedure.TheresultsachievedusingSWARAandPIPRECIAcoulddifferto someextentbecausetherelativelyimportantfeatureistheassessmentoftherespondent. Eveninthecaseofre-evaluationusingsomeoftheabovemethods,certaindeviations mayoccur.

TherecalculatedvaluesfromTable 1 areshowninTable 7

Table7. AttitudesofrespondentsadaptedforapplyingthePIPRECIAmethod.

Cr1 1.001.001.001.001.001.001.001.001.001.001.00 Cr2 1.000.950.940.800.850.990.970.820.850.800.90 Cr3 0.850.900.900.800.800.950.900.800.850.800.86

Cr4 0.750.850.900.750.700.900.850.770.830.750.81 Cr5 0.750.800.800.700.600.850.850.750.800.700.76

Thevaluesobtainedusingthethreeapproaches,andthePIPRECIAmethod,aregiven inTables 8 10,whilethecomparisonofweightsobtainedusingthethreeapproachesis summarizedinTable 11

Table8. ComputationaldetailsandthecriteriaweightsobtainedbythePIPRECIAmethodbasedon meanvalues.

Mean kj qj wj ’

Cr1 1.00110.257

Cr2 0.901.100.910.233 Cr3 0.861.150.790.203 Cr4 0.811.200.660.170 Cr5 0.761.240.530.137

Table9. IndividualandgroupcriteriaweightscalculatedusingthePIPRECIAmethod.

e1 we2 we3 we4 we5 we6 we7 we8 we9 we10 wj ”

Cr1 0.240.240.240.280.280.220.230.280.260.280.256 Cr2 0.240.230.220.240.240.220.230.240.230.240.232

Cr3 0.210.210.200.200.200.210.210.200.200.200.203

Cr4 0.170.180.180.160.160.190.180.160.170.160.170 Cr5 0.140.150.150.120.110.160.160.130.140.120.138

Table10. Thecriteriaweightobtainedbasedongreyratings.

mdnj s j s j s0.5 j wj ”’

Cr1 1.0001.0001.0001.0000.257 Cr2 0.8950.8240.9700.8970.233 Cr3 0.8500.8170.8920.8540.203 Cr4 0.8000.7440.8660.8050.170 Cr5 0.7750.7000.8200.7600.137

Table11. Thecriteriaweightobtainedusingthreeapproaches. wj’wj”wj ”’

Cr1 0.2570.2560.257 Cr2 0.2330.2320.233 Cr3 0.2030.2030.203 Cr4 0.1700.1700.170 Cr5 0.1370.1380.137

AsinthecaseofusingtheSWARAmethod,itcanbeseenfromTable 11 thatallthree approachesgivealmostthesamecriteriaweights.

Finally,Table 12 showsacomparisonofcriteriaweightsobtainedusingtheSWARA andPIPRECIAmethods,theproposedgreyapproach,and λ =0.5.

Table12. ThecomparisonofcriteriaweightobtainedusingtheSWARAandPIPRECIAmethods.

SWARAPIPRECIA wj”’wj ”’

Cr1 0.2570.257 Cr2 0.2330.233 Cr3 0.2030.203 Cr4 0.1700.170 Cr5 0.1370.137

AscanbeseenfromTable 12,bothapproachesgivethesamecriteriaweights.

4.2.TheSecondNumericalIllustration

Inthesecondpartoftheresearch,theaforementionedgroupofstudentsdetermined thesignificanceofthecriteriaforevaluatinge-commercewebsites,fromtheviewpointof first-timevisitors,usingthePIPRECIAmethod.Onthisoccasion,thesignificanceofthe followingcriteria,adoptedfromStanujkicandMagdalinovic[57],wasdetermined:

• Cr1,Designofthewebsite;

• Cr2,Easytouse;

• Cr3,Qualityandavailabilityofinformation;

• Cr4,Orderingprocess;and

• Cr5,Securityassurance.

Inthisevaluation,thefollowingmeaningswereassignedtotheabove-mentioned criteria.Criterion Cr1 referstothewebsite’sdesign,whilecriterion Cr2,Easytouse,refers tothepossibilityofeasilyfindingthedesiredproductandthepossibilityofsortingand selectingproducts.Criterion Cr3 referstotheinformationaboutproductsandorganization ofinformationtoenableeasierdecision-makingaboutorderingproducts.Criterion Cr4 referstothecomplexity,intuitiveness,andflexibilityoftheorderingprocess,whilecriterion Cr5,Securityassurance,referstotheassessmentoforderingsecurityfromthewebsitein thesensethattheywillactuallyreceiveproductsofappropriatequality,andthepossibility ofcomplaintsandthesecurityoftheirpersonalinformationandpaymentcardnumbers.

Afterdiscussingthemeaningoftheabovecriteria,alistthatcontainedfivee-commerce websites,thatnoneoftherespondentshadvisitedbefore,wasformed.Thislistofwebsiteswascompiledbythefollowingwebsites—www.eponuda.com (accessedon21March 2021)[58], www.gamecentar.rs (accessedon21March2021)[59], www.mi-srbija.rs (accessedon21March2021)[60], www.3gstore.rs (accessedon21March2021)[61],and www.lalafo.rs (accessedon21March2021)[62].

Theattitudesoftherespondentscollectedduringtheevaluationareshownin Table 13, aswellastheirmean.

Table13. Respondents’attitudesobtainedbasedontheuseofthePIPRECIAmethod.

E1 E2 E3 E4 E5 E6 E7 E8 E9 E10 Mean

Cr1 1.001.001.001.001.001.001.001.001.001.001.00 Cr2 1.201.101.001.001.050.800.901.201.101.151.05 Cr3 1.201.001.001.201.101.201.201.201.201.301.16 Cr4 1.001.100.901.001.000.901.101.101.001.001.01 Cr5 1.101.101.101.001.001.050.901.001.001.001.03

Thecalculationdetailsandthecriteriaweightsobtainedbasedonthemeanvalueare encounteredinTable 14,whilethegroupcriteriaweights,calculatedbasedontheaverage ofindividualcriteriaweights,areshowninTable 15

Table14. ThecomputationaldetailsandthecriteriaweightsobtainedusingthePIPRECIAmethod basedonmeanvalues.

Mean kj qj wj ’

Cr1 1.00110.170 Cr2 1.050.951.050.179 Cr3 1.160.841.250.213 Cr4 1.010.991.270.216 Cr5 1.030.981.300.221

Table15. IndividualandgroupcriteriaweightscalculatedbyapplyingthePIPRECIAmethod.

we1 we2 we3 we4 we5 we6 we7 we8 we9 we10 wj ”

Cr1 0.140.170.200.170.180.210.180.140.160.140.170

Cr2 0.180.190.200.170.190.170.170.170.180.160.179

Cr3 0.220.190.200.220.210.220.210.210.220.230.213

Cr4 0.220.210.180.220.210.200.230.240.220.230.216

Cr5 0.240.240.210.220.210.210.210.240.220.230.221

ThecriteriaweightsobtainedusingtheproposedapproachareshowninTable 16. Finally,thecomparisonsofweightsobtainedusingthreeapproachesarepresentedin Table 17.

Table16. Thecriteriaweightobtainedbasedongreyratings.

mdnj s j s j s0.5 j wj ”’

Cr1 1.0001.0001.0001.0000.169 Cr2 0.8950.8240.9700.8970.178 Cr3 0.8500.8170.8920.8540.216

Cr4 0.8000.7440.8660.8050.217 Cr5 0.7750.7000.8200.7600.220

Table17. Thecriteriaweightobtainedusingthreeapproaches.

wj’wj”wj ”’

Cr1 0.1700.1700.169 Cr2 0.1790.1790.178 Cr3 0.2130.2130.216 Cr4 0.2160.2160.217 Cr5 0.2210.2210.220

AscanbeobservedfromTable 17,allconsideredapproachesgavealmostthesame criteriaweights.

5.Conclusions

Aninnovativeapproachfordeterminingcriteriaweightsincaseofgroupdecisionmaking,usingSWARAorPIPRECIAmethods,isproposedinthisarticle.

Inthisapproach,itisproposedtotransformtherespondents’attitudesintoappropriategreyintervals,insteadoftransformingthemintocrispvalues.Usinggrey,instead ofcrisp,numbersprovidesmuchgreaterpossibilities,primarilyforperformingcertain analyses.Therefore,inthisarticle,twonumericalexampleswereconductedandthe justificationoftheproposedapproachisconfirmed.Also,byvaryingthevalueofthe whiteningcoefficient,differentweightsofthecriteriacanbeobtained,anditshouldbe emphasizedthatthisapproachgivesthesameweightsasinthecaseofcrispnumbers whenthewhiteningcoefficienthasavalueof0.5.

Thestudywasconductedwithagroupof10selectedrespondentsfromagroupof 25respondentswhowereinstructedtoapplytheSWARAandPIPRECIAmethods.

Theresultsobtainedbyapplyingtheproposedapproach,intheconsideredcases,are identicaltotheresultsobtainedbytheusedproceduresbasedontheapplicationofcrisp numbers.Besides,theresultsobtainedusingtheSWARAandPIPRECIAmethodsare thesameinallcasesconsidered,whichconfirmsthatthisapproachcanbeusedwiththe SWARAandPIPRECIAmethods.

Theadvantageofthisapproachisreflectedinthepossibilityofcombiningcrispand greynumbers,wherecrispnumbersareusedforcollectingtherespondents’attitudes, whilegreynumbersareusedforexpressingattitudesofallrespondentsthataregroup

attitudes.Inthisway,thepossibilityofasimplecollectionofrespondents’attitudesand theadvantagesofgreynumbersiscombined,primarilyintermsofconductinganalyses.

Inaddition,inthisapproach,thegreynumberwasformedbasedonthemedian valueofcollectedresponsesbecauseitbettermaintainsthedeviationfromthenormal distributionofthecollectedresponses.

Theproposedapproachcanbeusedfordeterminingthecriteriaweightsaswellas forcompletelysolvingtheMCDMproblem.Finally,asadirectionforfutureresearch, theproposedapproachcanbeeasilyadaptedforapplyingtriangularfuzzynumbers,or eveninterval-valuedtriangularfuzzynumbers,i.e.,newextensionsforsolvingawider spectrumofcomplexreal-worldproblemscouldbeproposed.

AuthorContributions: Conceptualization,D.S.;methodology,D.S.andD.K.;validation,D.K.,G.P., P.S.S.andM.S.;formalanalysis,G.P.andP.S.S.;datacuration,M.S.,F.S.andV.N.K.;writing— originaldraftpreparation,F.S.,V.N.K.andA.U.;writing—reviewandediting,F.S.,V.N.K.andA.U.; supervision,D.S.;projectadministration,D.S.andD.K.Allauthorshavereadandagreedtothe publishedversionofthemanuscript.

Funding: TheresearchpresentedinthisarticlewasdonewiththefinancialsupportoftheMinistry ofEducation,ScienceandTechnologicalDevelopmentoftheRepublicofSerbia,withinthefunding ofthescientificresearchworkattheUniversityofBelgrade,TechnicalFacultyinBor,accordingto thecontractwithregistrationnumber451-03-9/2021-14/200131.

InstitutionalReviewBoardStatement: Notapplicable.

InformedConsentStatement: Notapplicable.

DataAvailabilityStatement: Thedatausedinthestudyweretakenfromrelatedstudiesin theliterature.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.Thefundershadnoroleinthedesign ofthestudy;inthecollection,analyses,orinterpretationofdata;inthewritingofthemanuscript,or inthedecisiontopublishtheresults.

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