AnInnovativeGreyApproachforGroupMulti-Criteria DecisionAnalysisBasedontheMedianofRatingsby UsingPython
DragišaStanujki´c 1 ,DarjanKarabaševi´c 2,* ,GabrijelaPopovi´c 2 ,PredragS.Stanimirovi´c 3 , FlorentinSmarandache 4 ,MuzaferSaraˇcevi´c 5 ,AlptekinUluta¸s 6 andVasiliosN.Katsikis 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 MathematicsandScienceDivision,GallupCampus,UniversityofNewMexico,705GurleyAve., Gallup,NM87301,USA;smarand@unm.edu
5 DepartmentofComputerSciences,UniversityofNoviPazar,DimitrijaTucovi´cabb,36300NoviPazar,Serbia; muzafers@uninp.edu.rs
6 DepartmentofInternationalTradeandLogistics,FacultyofEconomicsandAdministrativeSciences,Sivas CumhuriyetUniversity,Sivas58140,Turkey;aulutas@cumhuriyet.edu.tr
7 DivisionofMathematicsandInformatics,DepartmentofEconomics,NationalandKapodistrianUniversityof Athens,Panepistimiopolis,15784Ilissia,Greece;vaskatsikis@econ.uoa.gr
* Correspondence:darjan.karabasevic@mef.edu.rs
Citation: Stanujki´c,D.;Karabaševi´c, D.;Popovi´c,G.;Stanimirovi´c,P.S.; Smarandache,F.;Saraˇcevi´c,M.; Uluta¸s,A.;Katsikis,V.N.An InnovativeGreyApproachforGroup Multi-CriteriaDecisionAnalysis BasedontheMedianofRatingsby UsingPython. Axioms 2021, 10,124. https://doi.org/10.3390/ axioms10020124
AcademicEditors:Goran Cirovi´cand DraganPamuˇcar
Received:24May2021 Accepted:16June2021 Published:19June2021
Publisher’sNote: MDPIstaysneutral withregardtojurisdictionalclaimsin publishedmapsandinstitutionalaffiliations.
Abstract: Somedecision-makingproblems,i.e.,multi-criteriadecisionanalysis(MCDA)problems, requiretakingintoaccounttheattitudesofalargenumberofdecision-makersand/orrespondents. Therefore,anapproachtothetransformationofcrispratings,collectedfromrespondents,ingrey intervalnumbersformbasedonthemedianofcollectedscores,i.e.,ratings,isconsideredinthis article.Inthisway,thesimplicityofcollectingrespondents’attitudesusingcrispvalues,i.e.,by applyingsomeformofLikertscale,iscombinedwiththeadvantagesthatcanbeachievedbyusing greyintervalnumbers.Inthisway,agreyextensionofMCDAmethodsisobtained.Theapplication oftheproposedapproachwasconsideredintheexampleofevaluatingthewebsitesoftourism organizationsbyusingseveralMCDAmethods.Additionally,ananalysisoftheapplicationofthe proposedapproachinthecaseofalargenumberofrespondents,doneinPython,ispresented.The advantagesoftheproposedmethod,aswellasitspossiblelimitations,aresummarized.
Keywords: MCDA;greyintervalnumbers;groupdecision-making;Python
1.Introduction
Copyright: ©2021bytheauthors. LicenseeMDPI,Basel,Switzerland. Thisarticleisanopenaccessarticle distributedunderthetermsand conditionsoftheCreativeCommons Attribution(CCBY)license(https:// creativecommons.org/licenses/by/ 4.0/).
Multi-criteriadecision-making(MCDM),ormulti-criteriadecisionanalysis(MCDA), hassofarbeenusedforsolvingalargenumberofnumerousdifferentdecision-makingproblems[1 4].Therefore,MCDAisdealingwithsolvingcomplexreal-worldproblemsofthe greatestinteresttotheorganizationthatcannotbesolvedbyconventional methods[5 8] Induecourseoftime,manymulti-criteriaanalysis(MCA)methodswereproposed,primarilyduetothedynamicandrapiddevelopmentofthefieldofoperationalresearch.The followingcanbementionedassomeofthemostcitedarticlesfromthisarea:Hajkowicz andCollins[9],HajkowiczandHiggins[10],Kaklauskasetal.[11],Kostrevaetal.[12],and BeltonandVickers[13].
Besidesthisresearch,therearemanystudiesinthisarea,suchas:researchand developmentprojectportfolioselection[14](MavrotasandMakryvelios,2021),assessing nationalenergysustainability[15],energyconsumptionanalysisofhigh-speedtrains[16],
evaluationoftransportemissionsreductionpolicies[17],planningrenewableenergyuse andcarbonsaving[18],andsoforth.
MCDAhasalsobeenusedsuccessfullyforsolvingdecision-makingproblemsthat arerelatedtouncertaintiesorrequireagroupdecision-makingapproachforsolving them[19 25].Assomeexamplesofsuchapproaches,thefollowingcanbementioned: agreyabsolutedecisionanalysis[26],amultiplecriteriadecisionanalysisframework forthedispersedgroup[27],afuzzymulti-criteriaanalysis[28 30],andcollaborative decision-makinginthemulti-actormulti-criteriaanalysis[31].
Fromtheaforestated,itisclearthatsomedecision-makingproblemscanbemore adequatelysolvedifalargernumberofrespondentstakepartinsolvingthem.Insuch cases,thequestionthatarisesishowtotransformtheattitudescollectedfromrespondents intogroupattitudes.
Theapproachbasedontheuseofafive-pointLiker’sscale,orsimilar,canbementionedasoneoftheprobablysimplestapproachesforcollectingtherespondents’attitudes. Sofar,innumerousarticlespublishedinscientificjournals,numerousapproacheshave beenproposedforthetransformationofindividualattitudesacquiredinthiswayinto groupattitudes.Theresultsobtained,theadvantages,aswellastheweaknessesofthese approaches,arealsopresentedinthesejournals.
Inthisarticle,anapproachtothetransformationofcrispratings,collectedfrom respondents,asgreyintervalnumbersformbasedonthemedianofcollectedscores,i.e., ratings,isconsidered.Therefore,thearticleproposesthetransformationofindividual ratingscollectedfromrespondentsintogreyintervalswiththeaimofperformingMCDA withminimallossofinformationinrelationtocaseswhencrispratingsaretransformed intocrispgroupratings.Theapplicationoftheproposedapproachwasconsideredon theexampleofevaluatingthewebsitesoftourismorganizationsbyusingseveralMCDA methods,andalsoananalysisoftheapplicationoftheproposedapproachinthecaseofa largenumberofrespondentswasdoneinPythonanddescribed.Additionally,themain ideaofthearticlewastoproposeasimpleprocedureforgatheringrespondent’sattitudes insteadofacomplexprocedurethatissometimesdifficulttounderstandbyordinary respondents/decision-makerswhoarenotfamiliarwithMCDMandfuzzylogic.
Therefore,therestofthisarticleisorganizedasfollows:InSection 2,somebasic definitionsaboutgreynumbersaregiven,whileanewapproachisproposedinSection 3.In Section 4,anumericalillustrationispresentedinordertohighlightthebasiccharacteristics oftheproposedapproach,whileinSection 5 ananalysisoftheobtainedresultsisperformed. Finally,conclusionsaregivenattheendofthearticle.
2.Preliminaries
Definition1. Greynumber[32].Agreynumber ⊗x issuchanumberwhoseexactvalueis unknown,buttherangeinwhichvaluecanlieisknown.
Definition2. Intervalgreynumber[32].Anintervalgreynumberisagreynumberwithaknown lowerboundx andupperbound x,butwiththeunknownvalueofx,anditisshownasfollows:
Definition3. Thewhiteningfunction[33 35].Thewhiteningfunctiontransformsaninterval greynumberintoacrispnumberwhosepossiblevaluesliebetweentheboundsoftheintervalgrey number.Forthegivenintervalgreynumber,thewhitenedvaluex(λ) ofintervalgreynumber ⊗x is definedas x(λ) = (1 λ)x + λx,(2) where ∈ [0,1] denotesthewhiteningcoefficient.
Intheparticularcase λ =0.5,thewhitenedvaluebecomesthemeanoftheinterval greynumber,asfollows:
x(0.5) = 0.5(x + x .(3)
3.TheNewlyProposedApproach
Supposethatthedecisionmatrixispresentedintheform
D =[xk ij ],(4)
where: xk ij denotestheevaluationofalternative i tocriterion j statedbythedecision-maker k; i =1, ,m,and m denotesthenumberofalternatives; j =1, ,n,and n denotesthe numberofcriteria; k =1... K,and K denotesthenumberofdecision-makers.
Suchathree-dimensionalmatrixcanbetransformedintoagrouptwo-dimensional matrixasfollows:
D =[x ij ],(5) with x ij = K ∑ k=1 xk ij /K.(6)
Essentially, x ij denotesratingofalternative i tocriterion j.Suchdefined x ij isactually themeanvalueofallassessmentsofthealternative i inrelationtothecriterion j. However,thematrixshownusingEquation(4)canbealsotransformedintoagrey groupdecisionmatrix,asfollows:
D =([xij, xij ]),(7) with xij = ∑ k∈k xk ij /n ;(8) xij = ∑ k∈k+ xk ij /n+.(9)
In(8), k denotesthesetofelementswhosevaluesarelessthanorequaltothemedian valueof xk ij,and n denotesthenumberofelementsinthisset.Similarly, k+ in(9)denotes asetofelementswhosevaluesaregreaterthanorequaltothemedianvalueof xk ij and n+ denotesthenumberofelementsinthisset.
Example
Let S beasequenceof10integersfrominterval[1,5]and S =(1,2,3,1,5,3,3,1,4,5). Then,themeanandmedianof S areasfollows: mean =2.80and median =3.00.The meanvalueofanumberwhichislessthanorequaltothemedian(1,2,3,1,3,3,1)is xl = 2.00andthemeanvalueofanumbergreaterorequaltothemedian(3,5,3,3,4,5)is xu =3.83.
Themeanvalueofsuchinterval[2.00,3.83],determinedusingEquation(3),is2.915, andthedistancebetweenitandthemeanis2.915 2.80=0.115,thatisinpercentages4.11%.
Theresultsobtainedbasedonseveralsequencesofrandomlygeneratednumbersfrom interval[1,10]areshowninTable 1.ThecalculationwasdoneinPythonusingtheseed(1).
Table1. Differencebetweenthemeanvalueofthesequenceofnumbersandthevalueobtainedby theproposedapproach.
Sample MeanMedianxl xu xm =(xu xl)/2 d =abs(Mean xm) d (%) 55.605.004.007.005.500.101.79 107.107.505.408.807.100.000.00 155.136.002.887.505.190.051.06 206.507.005.088.006.540.040.64 254.524.002.436.504.460.061.23 505.205.003.077.145.110.091.81 1005.015.002.937.095.010.000.02 1505.165.003.056.914.980.183.45
FromTable 1,itcanbeseenthatthedifferencebetweenthemeanvalueofthesequence ofnumbersandthevalueobtainedbytheproposedapproachisnotlarge.
4.ANumericalIllustration
Inthissection,theuseoftheproposedapproachispresentedinthecaseofevaluating websitesoftouristorganizationsfromEasternSerbia.Theevaluationwasperformedon thewebsitesof5touristorganizationsfromtheTimokfrontier,ormorepreciselytourist organizationsoftheMunicipalitiesofBoljevac,Bor,Majdanpek,NegotinandKladovo (Itisimportanttostatethattheorderofmunicipalitiesdoesnotcorrespondtotheorderof alternatives,becausetheaimofthisarticleisnottofavoranyoftheabove-mentionedtourist organizations.).Theevaluationisperformedbasedonthefollowingcriteria:Visualdesign— C1,Structureandnavigability—C2,Content—C3,Innovation—C4,Personalization—C5
TheevaluationwasperformedusingARAS[36],WASPAS[37],CoCoSo[38]and WISP[39]methods.Inthefirstcase,theevaluationwasperformedusingordinaryMCDA methodsandthemeanvalueofthecollectedratings,whileinthefirstcase,theevaluation wasperformedusingtheproposedapproach.
Thisillustrationdoesnotshowallthepossibilitiesthattheproposedapproachprovidesintermsofanalysis.Themaingoalwastocomparetheresultsobtainedbyapplying themeanvalueofallassessmentsandtheproposedapproach,wherethetransformationof greynumberswasperformedusingEquation(3)and λ =0.5.
Theratingobtainedfrom10respondentsisshowninTables 2 6
Table2.
Table4. Ratingsofalternative A3 inrelationtotheevaluationcriteriaobtainedfrom10respondents.
A3 IIIIIIIVVVIVIIVIIIIXX C1 1222222321 C2 3554223444 C3 1442224322 C4 1133114211 C5 3553443444
Table5. Ratingsofalternative A4 inrelationtotheevaluationcriteriaobtainedfrom10respondents.
A4 IIIIIIIVVVIVIIVIIIIXX C1 4445444454 C2 5545334553 C3 5544334555 C4 4455535543 C5 4453443544
Table6. Ratingsofalternative A5 inrelationtotheevaluationcriteriaobtainedfrom10respondents.
A4 IIIIIIIVVVIVIIVIIIIXX C1 4355535343 C2 4445435444 C3 5444334435 C4 4453243533 C5 3444434544
Thegroupdecisionmatrix,formedonthebasisoftheresponsesofallrespondents, isshowninTable 7.Theelementsofthismatrixrepresentthemeanvalueoftheratings obtainedfromtherespondents.
Table7. Groupdecision-makingmatrix.
CriteriaAlternatives C1 C2 C3 C4 C5
A1 2.903.603.802.702.20 A2 3.604.403.503.703.60 A3 1.903.602.601.803.90 A4 4.204.204.304.304.00 A5 4.004.103.903.603.90
AsimilardecisionmatrixisshowninTable 8,wheretheelementsofthatmatrix representthemedianofratingsobtainedfromtherespondents.
Table8. Themedianofratingsobtainedfromtherespondents.
CriteriaAlternatives C1 C2 C3 C4 C5
A1 2.963.813.922.702.11 A2 3.794.403.503.823.81 A3 1.953.792.311.403.96 A4 4.104.204.304.304.00 A5 4.004.053.963.603.95
TheresultsoftheevaluationperformedusingordinaryARAS,WASPAS,CoCoSoand WISPmethods,weightingvector wi =(0.25,0.24,0.22,0.20,0.10),andthedatafromTable 7, areshowninTable 9
Table9. RankingofalternativesusingordinaryARAS,WASPAS,CoCoSoandWISPmethods.
ARASWASPASCoCoSoWISP
Alternatives Si Rank Si Rank Si Rank Si Rank
A1 0.7340.7341.8040.874
A2 0.8830.8832.1730.953 A3 0.6150.6051.4950.815 A4 0.9910.9912.4311.001 A5 0.9220.9222.2520.962
Inthesecondcase,basedondatafromTable 8 aswellasratingsfromTables 2 6,a greydecisionmatrixwasformedasshowninTable 10
Table10. Greygroupdecision-makingmatrix.
CriteriaAlternatives C1 C2 C3 C4 C5
A1 [2.50,3.43][3.44,4.17][3.50,4.33][1.60,3.80][1.71,2.50]
A2 [3.25,4.33][3.80,5.00][2.80,4.20][3.14,4.50][3.44,4.17]
A3 [1.78,2.13][3.25,4.33][1.83,2.78][1.00,1.80][3.62,4.29]
A4 [4.00,4.20][3.40,5.00][3.60,5.00][3.60,5.00][3.75,4.25] A5 [3.33,4.67][3.88,4.22][3.63,4.29][2.80,4.40][3.78,4.13]
TheevaluationresultsgeneratedusingthegreyARAS,WASPAS,CoCoSoandWISP methods,andthedatafromTable 10,areshowninTable 11.Itshouldbenotedagainthat thegreynumbersfromTable 10 weretransformedintocrispvalues,usingEquation(3)and λ =0.5,beforetheevaluation.
Table11. RankingofalternativesusinggreyWS,WP,WASPASandCoCoSomethods.
ARASWASPASCoCoSoWISP
Alternatives Si Rank Si Rank Si Rank Si Rank
A1 0.7640.4641.9040.744 A2 0.9130.5532.2930.893 A3 0.5950.3651.4750.625 A4 0.9910.6012.4810.991 A5 0.9220.5622.3220.922
FromTables 9 and 11,itcanbeseenthatdifferencesinrankingordersofalternatives achievedonthebasisofthemeanvalueofallassessmentsandtheproposedapproach werenotobserved.Ofcourse,itshouldbereiteratedherethattheproposedapproach providessignificantlygreateropportunitiesintermsofanalyzingvariousscenarios,such aspessimisticoroptimistic.
5.AnalysisandDiscussion
Inordertoverifytheproposedapproach,thissectionpresentstheresultsofthe evaluationbasedontheassessmentsofanumberofvirtualrespondents.Foreasier evaluation,thescoresweregeneratedasrandomnumbersfromtheinterval[1,10],using aprogramwritteninthePythonprogramminglanguage,inwhichallcalculationswere alsoperformed.Inthisanalysis,randomnumbersaregeneratedwiththeseed(1).The resultsobtainedonthebasisofseriesof10,50,100and150virtualrespondentsareshown inTables 12 20.Theweightingvector wi =(0.2,0.2,0.2,0.2,0.2)isusedinthisevaluation. Thecalculationdetailsobtainedonthebasisof10virtualrespondentsareshownin Tables 12 and 13.AscanbeseenfromTables 12 and 13,inthiscase,thesameranking ordersareobtainedbyapplyingallmethodsandapproaches.
Alternatives
Table12. Rankingofalternativesonthebasisof10virtualrespondentsandcrispapproach.
WSWPWASPASCoCoSoWISP
Si Rank Pi Rank Qi Rank Ki Rank Si Rank
A1 0.8350.7550.7551.7750.595
A2 1.0010.9110.9112.1611.001
A3 0.9820.8720.8822.0920.882
A4 0.8440.7540.7641.7940.614
A5 0.8930.8030.8131.9130.713
Table13. Rankingofalternativesonthebasisof10virtualrespondentsandtheproposedgreyapproach.
WSWPWASPASCoCoSoWISP
Alternatives
Si Rank Pi Rank Qi Rank Ki Rank Si Rank
A1 0.8250.7450.7551.7850.585
A2 1.0010.9210.9212.1811.001
A3 0.9520.8420.8622.0420.802
A4 0.8340.7540.7541.7940.594
A5 0.8830.8030.8131.9130.693
Table14. Rankingofalternativesonthebasisof50virtualrespondentsandcrispapproach.
WSWPWASPASCoCoSoWISP
Alternatives Si Rank Pi Rank Si Si Ki Rank Si Rank
A1 0.9730.9430.9431.9030.913
A2 0.9240.8940.8941.8040.784 A3 1.0010.9710.9711.9711.001 A4 0.9150.8850.8951.7950.775 A5 0.9720.9420.9521.9120.912
Table15. Rankingofalternativesonthebasisof50virtualrespondentsandtheproposedgreyapproach.
WSWPWASPASCoCoSoWISP
Alternatives Si Rank Pi Rank Qi Rank Ki Rank Si Rank
A1 0.9820.9420.9421.9020.942 A2 0.9250.8950.8951.7950.805 A3 1.0010.9610.9611.9411.001 A4 0.9340.8940.8941.8140.824 A5 0.9730.9330.9331.8730.903
Table16. Rankingordersofalternativesobtainedonthebasisof50virtualrespondents.
Table17. Rankingordersofalternativesobtainedonthebasisof100virtualrespondents.
A1
A2
A3
A4
A5
Table18. Rankingofalternativesonthebasisof150virtualrespondentsandcrispmethods.
WSWPWASPASCoCoSoWISP
Alternatives Si Rank Pi Rank Si Si Ki Rank Si Rank
A1 0.99320.96020.96121.84020.9772
A2 0.97530.94430.94431.80830.9273
A3 0.97150.93850.93951.79950.9145
A4 1.00010.96810.96811.85411.0001 A5 0.97340.94240.94241.80540.9234
Table19. Rankingofalternativesonthebasisof150virtualrespondentsandgreymethods.
WSWPWASPASCoCoSoWISP
Alternatives Si Rank Pi Rank Qi Rank Ki Rank Si Rank
A1 0.98820.95420.95521.80620.9652
A2 0.98830.95330.95431.80530.9633
A3 0.98740.95240.95341.80440.9604
A4 1.00010.96610.96611.82811.0001 A5 0.98550.95250.95251.80150.9585
Table20. Rankingordersofalternativesobtainedonthebasisof150virtualrespondents.
CrispGreyApproach
AlternativesWSWP WASPAS CoCoSoWISPWSWP WASPAS CoCoSoWISP
A1 2222222222 A2 3333333333 A3 5555544444 A4 1111111111 A5 4444455555
Thecalculationdetailsobtainedonthebasisof50virtualrespondentsareshownin Tables 14 16.Inthiscase,thereweresomediscrepanciesintheorderofthesecondand third-placedalternatives,whichcanbeclearlyseeninFigure 1
Figure1. Rankingordersofalternativesobtainedonthebasisof50virtualrespondents.
Figure 1. Ranking orders of alternatives obtained on the basis of 50 virtual respondents.
It can be seen from Table 14 and Table 15 that differences between second-placed and third-placed alternatives are not high, which is why it can be expected that the same ranking order of alternatives could be obtained by using another weight vector.
Ranking orders of alternatives, obtained on the basis of 100 virtual respondents, are shown in Tabe 17, and presented in Figure 2. This case is similar to the previous one.
Table 17. Ranking orders of alternatives obtained on the basis of 100 virtual respondents.
Ranking orders of alternatives, obtained on the basis of 100 virtual respondents, are shown in Tabe 17, and presented in Figure 2. This case is similar to the previous one.
Table 17. Ranking orders of alternatives obtained on the basis of 100 virtual respondents.
Crisp
Grey Approach
Alternatives WS WP WASPAS CoCoSo WISP WS WP WASPAS CoCoSo WISP
ItcanbeseenfromTables 14 and 15 thatdifferencesbetweensecond-placedandthirdplacedalternativesarenothigh,whichiswhyitcanbeexpectedthatthesameranking orderofalternativescouldbeobtainedbyusinganotherweightvector.
Rankingordersofalternatives,obtainedonthebasisof100virtualrespondents,are showninTabe17,andpresentedinFigure 2.Thiscaseissimilartothepreviousone.
A1 4 4 4 4 4 4 4 4 4 4 A2 3 3 3 3 3 2 2 2 2 2 A3 2 2 2 2 2 3 3 3 3 3 A4 1 1 1 1 1 1 1 1 1 1 A5 5 5 5 5 5 5 5 5 5 5 WSWPWASPASCoCoSoWISP WSGrey WPGrey WASPAS -Grey CoCoSoGrey WISPGrey A14444444444 A23333322222 A32222233333 A41111111111 A55555555555
0 1 2 3 4 5 6 A1A2A3A4A5
Figure2. Rankingordersofalternativesobtainedonthebasisof100virtualrespondents.
FromTable 17 andFigure 2 itcanbeobservedthatinthiscase,thedifferencesoccur onlyinthecaseofthesecondandthird-placedalternatives.
Rankingordersofalternativesthatarisefrom150virtualrespondentsarearrangedin Tables 18 20 andpresentedinFigure 3
WSWPWASPASCoCoSoWISP
Figure3. Rankingordersofalternativesobtainedonthebasisof150virtualrespondents.
Figure 3. Ranking orders of alternatives obtained on the basis of 150 virtual respondents.
Table 20 andFigure 3 clearlyshowthatthealternative A4 isbestrankedaccordingto allmethods,withallcrispmethodsgavethesameorderofranking A4, A1, A2, A5, A3,while theproposedgreyapproachgavethefollowingrankingsorder A4, A1, A2, A3, A5.However, fromTable 20 itisobservablethatthereareverysmalldifferencesinoverallperformance
Table 20 and Figure 3 clearly show that the alternative A4 is best ranked according to all methods, with all crisp methods gave the same order of ranking A4, A1, A2, A5, A3, while the proposed grey approach gave the following rankings order A4, A1, A2, A3, A5. However, from Table 20 it is observable that there are very small differences in overall performance between the second-placed, third-placed and fourth-placed alternatives, which is why it can be expected that different ranking orders of alternatives could be obtained by using another weighting vector.
betweenthesecond-placed,third-placedandfourth-placedalternatives,whichiswhyit canbeexpectedthatdifferentrankingordersofalternativescouldbeobtainedbyusing anotherweightingvector.
6.Conclusions
Theadvantagesofusinggreyinsteadofcrispnumbersinmulti-criteriadecisionanalysishavebeenconsideredandproveninanumberofpreviouslypublishedstudies.One oftheadvantageswhichshouldbeemphasizedusinggreynumbersisthepossibilityof consideringvariousscenarios,suchas:pessimistic,realistic,andoptimistic.Theproposed approachallowsthetransformationofcrispratings,collectedbyemployingsurveysbased ontheuseoftheLikertscale,intogreynumbersandthusconsideringdifferentscenarios. Theproposedapproachmaybesuitablewhenitisnecessarytocollectandanalyzethe realisticattitudesofalargernumberofrespondents.Moreover,theproposedtransformationenablesgreaterrobustnessandfurtherpossibilityofanalysisandconsiderationof differentscenarios.
Theresultsofthewebsiteevaluationbasedonthemeanvalueoftheratingsobtained fromallrespondentsandtheproposedapproachdidnotindicateadifferenceinthe rankingordersofalternatives.However,theresultsoftheconductedanalysisindicate thatdifferencesmayarisebetweenthetwoapproaches,especiallyinthecaseofthelowerrankedalternatives.
Somedifferencesintheresultsareexpectedbecausetheproposedapproachisnota substituteforapplyingthemeanvalueofthescoresobtainedfromallrespondents,butan approachthatfurtherallowsthepossibilityofanalysis.Certaindifferencesintheranking resultsusingthenewlyproposedapproachandapplyingthemeanofthescoresobtained inallrespondentscanbecitedasaweaknessofthisapproach.
Finally,considerationofthetransformationofalargernumberofcrispratingsinto correspondingtriangularfuzzynumbersorinterval-valuedtriangularfuzzynumbers canbementionedasoneofthepossibledirectionsforthefurtherdevelopmentofthe proposedapproach.
AuthorContributions: Conceptualization,P.S.S.,D.K.,V.N.K.andG.P.;methodology,D.K.,D.S.and F.S.;validation,A.U.;investigation,A.U.;datacuration,G.P.;writing—originaldraftpreparation, D.S.,V.N.K.andM.S.;writing—reviewandediting,P.S.S.andM.S.;supervision,D.K.;funding acquisition,F.S.Allauthorshavereadandagreedtothepublishedversionofthemanuscript.
Funding: TheresearchpresentedinthisarticlewasdonewiththefinancialsupportoftheMinistry ofEducation,ScienceandTechnologicalDevelopmentoftheRepublicofSerbia,withinthefunding ofthescientificresearchworkattheUniversityofBelgrade,TechnicalFacultyinBor,accordingto thecontractwithregistrationnumber451-03-9/2021-14/200131.
ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.
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