A-peer-IF-TOPSIS-based-syst-packaging-machine-selection_2014_Expert-Systems-with-Applications

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ApeerIF-TOPSISbaseddecisionsupportsystemforpackagingmachine

selection

DavideAloini?,RiccardoDulmin,ValeriaMininno

UniversityofPisa,http://wendang.chazidian.comzzarino,Pisa,Italy

articleinfoabstract

Selectingtheappropriatemanufacturingmachineisaveryimportantandcomplexproblemfor?rmswhichusuallyhavetodealwithbothqualitativeandquantitativecriteriaandinvolvedifferentdecisionmakerswhoseknowledgeisoftenvagueandimprecise.

Thispaperproposesapeer-basedmodi?cationtointuitionisticfuzzymulti-criteriagroupdecisionmakingwithTOPSISmethod(peerIF-TOPSIS)andappliesittoapackagingmachineselectionproblem.Intuitionisticfuzzyweightedaveraging(IFWA)operatorhasbeenselectedbothtoobtainthegroupopin-ionontherelevanceofthesingledecisionmakersandtoaggregateindividualopinionsofdecisionmak-ersforratingtheimportanceofcriteriaandalternatives.

Acasestudyillustratestheapplicationofthemodi?edIF-TOPSISmethodinordertoselectaVerticalFormFillandSeal(VFFS)forDoubleSquareBottomBag(DSBB)machineinfoodpackaging.

Ó2013ElsevierLtd.Allrightsreserved.

Keywords:

IntuitionisticfuzzysetTOPSISmethod

PeergroupdecisionmakingMachineselection

1.Introduction

Manufacturingcompaniesworldwideareforcedtoundergotransformationprocessesinordertoimprovetheirabilitytosuc-ceedwiththeirproductsonextremelycompetitiveinternationalmarkets.Inthisperspective,anadequateselectionoftheappropri-atemachinetoolsifoftencrucialbutverydif?culttoachieve.Advancedmanufacturingtechnology,infact,requiresahighle-velofinitialinvestmentandusuallydealswithbothqualitativeandquantitativebene?tswhichmakethetraditionalinvestmentmodelbasedonReturnOnInvestment(ROI),CashFlowAnalysis(CF),Pay-Back(PB)andNetPresentValue(NPV)notreallysuitable.Arguablythesemodelsemphasizequantitativeand?nancialanal-ysis,butfailtocapturemanyofthe‘‘intangible’’bene?tssuchasgreatermanufacturing?exibility,improvedproductquality,quickresponsetocustomerdemandandbetteremployeesafetyandmotivation(Abdel-Kader,1997;Chen&Small,1996;Kaplan,1986)whicharetypicallymorechallengingtomeasureandmon-etize,ortheyarenotatall.

Thehighinvestmentriskinherentinadvancedmanufacturingtechnologyoftenleadstotheuseofarbitrarilyhighhurdledis-countsrates(Accola,1994;Kaplan,1986;Kaplan&Atkinsons,1989).Moreover,adjustmentstothediscountrateareaffectedCorrespondingauthor.Address:DepartmentofEnergy,System,LandandConstructionEngineering,UniversityofPisa,http://wendang.chazidian.comzzarino,56122Pisa,Italy.Tel.:+39502217088;fax:+39502217333.

E-mailaddresses:Davide.Aloini@dsea.unipi.it(D.Aloini),Riccardo.Dulmin@d-sea.unipi.it(R.Dulmin),Valeria.Mininno@dsea.unipi.it(V.Mininno).

0957-4174/$-seefrontmatterÓ2013ElsevierLtd.Allrightsreserved.http://wendang.chazidian.com/10.1016/j.eswa.2013.09.014

bydecisionmaker’sattitudetowardthespeci?criskratherthanbyanexplicitrepresentationoftherisksinherentintheinvest-mentalternatives(Accola,1994).Finally,alsowhenriskisexplic-itlyassessedandsystematicallyincludedintotheinvestmentevaluation,asforexamplebyusinginnovativeevaluationpara-digmssuchasRealOptionApproach(ROA),theproblemofesti-matingreturnsfromintangiblesstillremainunsolved.Thisconditionoftenresultsinasevereandsometimeirreparableeval-uationbiasaffectingthe?naldecision.

InordertosolvethisgapandaccordingtoLe?ey(1996),wethinkamoresophisticated(andnot?nancial)approachisneededtotheappraisalofamachineselection,whichcouldtakeintoac-countthestrategicnatureandthefullbene?tsfromsuchinvest-ments.ThemostadoptedproceduresinliteratureareMultiCriteriaDecisionAnalysis(MCDA)orMultiCriteriaDecisionMak-ing(MDCM)methods,oftencombinedwithfuzzylogicorsubse-quentevolutionoffuzzysettheoryinordertodealwiththevaguenessandimprecisioninherentwithadvancedmanufacturingtechnologyselectionproblem.

Thispaperproposesamodi?edversionofBoran,Genç,Kurt,andAkay(2009)fuzzymulti-criteriadecisionmakingwithTOPSISmethodwhichisinspiredbyapeer-basedviewofjudgments.Dif-ferentlyfromthepreviousversionofthealgorithmweadvanceapeerprocedurefordeterminingtheweightsofDecisionMakers’opinions.Thus,IntuitionisticFuzzyWeightedAveraging(IFWA)operatorisusedtoobtainthegroupopinionontherelevanceofthesingledecisionmakers.Inahighuncertainenvironment,infact,asinglesupervisorcanbesubjectedtoasigni?cantbiaswhenassessingweightstothesubjectsinvolvedinthedecisionprocess,

2158D.Aloinietal./ExpertSystemswithApplications41(2014)2157–2165

asinBoranetal.(2009).ApeervotingprocedureamongDMssup-portedbyIFWAoperatorcouldbeveryimportanttosupporttheaggregationoftheexpertopinionsand?nallyachieveconsensusingroupdecisionmakingproblems.Themethodisthendemon-stratedthroughanapplicationtothecaseofselectionofaninno-vativemachineforfoodpackaging.

Thefollowingsectionsaregoingtopresent:afocusedtheoret-icalbackgroundontheevolutionofTOPSISmethodsandFuzzysophistications(Section2),thedecisioncontextforthemachineselection(Section3),theoryandapplicationtothecasestudyofthemodi?edIF-TOPSISprocedure(Section4),and?nallyconclu-sion(Section5).

2.Theoreticalbackground:afocusedreviewonTOPSISAmongthenumerousMCDA/MCDMmethods,TechniqueforOrderPreferencebySimilaritytoIdealSolution(TOPSIS)continuestoworksatisfactorilyindifferentapplicationareas.ThestandardTOPSISmethodaimstochoosealternativesthatsimultaneouslyhavetheshortestdistancefromthepositiveidealsolutionandthefarthestdistancefromthenegativeidealsolutionaccordinglytoprede?nedevaluationcriteriawhichareusuallydividedintocostandbene?tcriteria(theyarerespectivelyminimized/maxi-mizedormaximized/minimizedinthetwocases).SeeBehzadian,Otaghsara,Yazdani,andIgnatius(2012)foranin-depthreviewoftheliteratureonTOPSISmethodsandrelatedapplications.Themainreasonofasowideacceptanceisbecauseitsconceptisrea-sonable,easytounderstandandcomparedwithotherMCDMmethods,likeAHPandELECTREI,itrequireslesscomputationalef-forts,andthereforecanbeappliedeasily(Kim,Park,&Yoon,1997;Shih,Shyur,&Lee,2007).

Then,FuzzySetTheorystartedtobeadoptedconjointlywithTOPSISinordertodealwithuncertaintyandrelatedconceptssuchasriskandambiguitywhichareprominentintheliteratureondecision-makingandlinguisticexpressionsregardedasthenaturalrepresentationofthejudgment.EvidenceofanextensiveuseofFuzzy-TOPSISisavailableagaininBehzadianetal.(2012).

Inthisperspective,however,itisalsovaluabletonoticethatthehybridizationoftraditionalMCDMapproacheswiththefuzzylogicsuffersfromtheepistemological?awconsistinginconsideringafuzzyintervalasasimplesubstitutetoaprecisenumber,wherebyadirectextensionofthestandardmethodconsistjustinreplacingnumbersbyfuzzyintervalsandrunningasimilarcomputationasintheprecisecase(Dubois,2011).TosolvethisproblemtheFuzzySetconceptshavebeenextendedintroducingtheIntuitionisticFuzzySet(IFS)theory(Atanassov,1986)consideredasuitablewaytodealwithvaguenessandsolvemanydecision-makingprob-lemsunderuncertainenvironment.Accordinglytothe‘‘intuitionis-ticapproach/philosophy’’,vaguenessreferstolackofde?niteorsharpdistinction,whereasambiguityisduetouncleardistinctionofvariousalternatives,whichisfurtherdividedintodiscord(con-?ict)andnon-speci?city(Klir&Yuan,1995).IFSgeneralizestheFuzzySetTheoryintroducinganotherdegreeoffreedom(non-memberships)intothesetdescriptiontogetherwithathirdparameterpA(x)whichisknownastheIntuitionisticIndexofHes-itation(Tamalika&Raya,2008).Inthiswayitisexpectedcopingbetterwiththepresenceofvaguenessandhesitancyoriginatingfromimpreciseknowledgeorinformation.

Followingthismajortrendinresearch,IFStheoryisconsideredhavingenormouschancesofsuccessformulti-criteriadecisionmakingproblemsduetothegreatsuperiorityondealingwithvagueness,sothatithasbeenappliedinmanyareassuchasdeci-sion-makingproblems(Atanassov,Pasi,&Yager,2005;Chen&Tan,1994;Hong&Choi,2000;Liu&Wang,2007;Szmidt&Kacprzyk,2002,2003;Wang,Cheng,&Huang,2009;Xu,2007a,2007b,2007c;Xu&Yager,2006,2008).

Onlymorerecently,somestudieshavecombinedIFSwithTOP-SISinordertobetterfacewithsubjectivity,imprecision,andvaguenessingroupdecision-makingproblemundermultiplecrite-ria.Boranetal.(2009),as?rst,developedIF-TOPSISbasedonXu’s(2007d)IntuitionisticFuzzyWeightedAveraging(IFWA)aggregat-ingoperatorandadaptedtosupplierselectionproblem.Then,theyalsoappliedIF-TOPSIStoapersonnelselectionproblemconcerningwithidentifyinganindividualfromapoolofcandidatessuitableforavacantposition(Boran,Genç,Kurt,&Akay,2011)andtoeval-uationofrenewableenergytechnologies(Boran,Boran,&Menlik,2012).

DrawingonBoranetal.(2009)thispaperextendstheapplica-tionofIF-TOPSIStoanotherchallengingdecisionproblemusuallysubjectedtouncertaintyandevaluationfrommultipleexperts:selectingtheadequatemanufacturingmachine.Italsoproposesaninnovativepeer-basedprocedureforvotingtherelevanceofthesingledecisionmakersintothegroupdecisionprocess.ThisisinordertoskipacentralizedassignmentofDMsweights.3.Problemde?nition

Selectingtheadequatemanufacturingmachineisacomplexdecision.Multipledecisionmakers,withdifferentperspectiveandexpertise,areusuallyinvolvedintheprocessandhavetodealwithuncertaintyandambiguity.Theseuncertaintiesentailincom-pleteinformation,inadequateunderstanding,andundifferentiatedalternatives(Lipshitz&Strauss,1997;Ahn,Park,Han,&Kim,2000).Ata?nanciallevel,thishappensbecauseofthedif?cultyinestimatingtheimpactofunexpectedchangesoncash?ows(Franz,Duke,&Omer,1995;Sutardi&Goulter,1995).Moreover,itisoftendif?culttomeasurethepositiveimpactoncash?owsbroughtaboutbytheincreaseinqualityand?exibilitywhichwouldallowquickerreactionstochangesinthemarket(Franzetal.,1995;Kaplan,1986).

Alsowhennot?nancialapproachesareadoptedandintangiblecriteriaareincludedintothedecisionprocess,uncertaintyisstillhigh.Decisionmaking,infact,involvesintangiblecriteriausedtorankthealternativeswhicharehardtoquantifyorhavenomea-surementstoserveasaguide.Asaconsequence,creatingprioritiesforthecriteriathemselvesinordertoweightheprioritiesofthealternativesandaddoverallthecriteriatoobtain,forexample,thedesiredoverallranksofthealternatives,isachallengingtask.Moreover,aspreviouslystated,thisisoftenagroup-decisionwhichneedstoachieveconsensusamongdifferentplayers,sothatassessingtherelativeimportanceofsingleDMs’opinionisanotherdif?cultdimensionoftheproblem.

Inthiscontext,ourstudyisspeci?callyaimedtotheselectionoftherightDBSS-VFFSpackagingmachineamonganumberofiden-ti?edalternatives.

VerticalForm,Fill,andSeal(VFFS)machinesareusedintheconsumerproductsindustryforawidevarietyofpackagingappli-cations.Variousproductslikesalt,tea,sugar,spices,snackfoods,wafers,detergentandcandiesareplacedintoformedpouchesandthensealed.Thepouchmaterialis?exibleandtypicallyheat-sealableplastic.Paperisalsousedandsealedbyglue.Accord-ingtoaprocessperspective,theVFFSmachinecanbedividedintofourfunctionalareas:(1)mixing,weighing,dosing;(2)forming,(3)feeding,aligning,registration;and(4)closing,sealing,cutting.Fig.1showsthefunctionalschemeofthemachine.

DBSS-VFFSmachineextendtheVFFSprocessbyaddingare?ne-mentoperationtotheclosingsealingandcuttingphaseinordertorealizeaDoubleSquareBottomBagsothatthepackagecanstanduponthesupermarketshelf.

Thechoiceconsistsofamulti-facedproblemwhichissubjectedtonumerousdimensionsofanalysis,andparticularlysafety,ef?-ciency,?exibilityandinnovationoftheprocess.

Packaging,asanoperationactivity,haveclearlytorespondtoproductionrequirementsintermshighproductionoutput,consis-tentreliabilityandproductqualitywithlowmanpowerrequire-mentsandlowmaintenancecosts.Themachines,however,alsoneedtobe?exibleenoughtoadapttovariationsinbagdimensionsandsophisticateddesigns.Packaging,infact,alsoplaysanimpor-tantroleasamediuminthemarketingmix,inpromotioncam-paigns,asapricingcriterion,inde?ningandcommunicatingthecharacterofnewproducts,creatingbrandidentity.Insuchacom-petitiveenvironment,customersreceivestimulationforthebuyingdecisionwhenstandinginfrontofthesupermarketshelf,oftenevenpreferringthistootherformsofcommunication.Theyhavetobeinformedandinspiredsothatpackaginghastoabsolvealsopersuasivefunctions,transferemotionsandsensualimpressions.Asaconsequence?rms’attentionduringthedecisionprocessisfocusednotonlyonsafety,reliabilityandproductivityofproduc-tion,butalsotoanumberofcriteriawhicharerelatedtothewholeprocess?exibilitythemachinewouldallow:numberoftypesandformats,shortset-up,guaranteeandassistance,upgradingoptionsandspaceoptimization.

Inordertorespondtotheneedforacompleteandeffectiveevaluationprocess,ina?rststep,thedecisiongrouphasconsid-eredtheVFFSmachine’stechnical-economicalspeci?cationstoidentifythecriteriafortheranking.Theoutputoftheprocesshasbeenthelistofvariablesasfollows:A.Bene?tcriteria:

1.Speed:itaccountsthenumberofboxesproducedbytheVFFSmachineovertimeintotheselectedDSBBformat;2.Mix?exibility:itisacombinedmeasureofmachine?exibil-itywhichreferstothenumberofdifferenttypesofpackagesproducedbythemachine,thetimeneedingforchangingtypesandformats(set-up),thenumberofworkingprograms;

3.Safety:itisaboutsecuritysystemsforpeopleworkingwith/aroundthemachineandproductwhichiscontainedbytheboxes;

4.Technologicalparameters:theymainlyrefertothetypeofsealingprocess,theprocessdata(IPC)collectiononthetouch-screenmonitor,andthecontrolofthemachineasforexampleconnectiontocompanyEthernet,remotecon-trolandalerts.

5.Easeofuse:itmostlyconcernsthesimplicityinprogram-mingpackagingtypeandsizes(automatedornot;fastbutnoteasy;withorwithoutsupportingtools);

6.Accessories:theyconsideradditionalfunctionalitieswhichareprovidedbythemachineasforexamplegas?ushing,evacuationandaromaprotection,vacuumetc.

7.Upgrading(expansion?exibility):itincludesthepossibilityofextendingmachineresourcesinashorttime,andcontainingcosts,inordertosatisfytheproductdemand’sgrowth.

8.Guarantee,assistanceandafter-saleservices(G.A.As.):theyrefertothelevelofguaranteeandassistanceincludedintotheagreementintermsoflength(years),availabilityoftheservice(hoursforday),supportlanguage,remoteassistancefordiagnosis,ordinaryandextraordinarymaintenanceetc.B.Costcriteria:9.Price

10.Wear,maintenance,replacementandrepair(W.M.R.R.):they

strictlydependonthesealingtechnologyandconsideralsotheaccessibilitytotheworkstationsandtotheelectricpanel.

11.Electricconsumption:itisreferredtotheglobalexpenditure

ofelectricenergyforafulldayVFFSproductioncycleintheDSBBformat.Thisishighlydependentonthepresenceofenergyrecoverysystems.

12.Dimension:itdealswiththeoverallmachinedimensionsand

totheadaptabilitytoaspeci?cavailablespace(forexampleleveragingonmachinemodularity).Bothquantitativeandqualitativevariableshavebeensubjectedtoapersonaljudgmentbytheexpertsandcodi?edinanIFSscale.Moreover,sincethechosencriteriarequiredifferentexpertisefortheirassessmentandthustheinvolvementofagroupofdeci-sionmakerswhichwererepresentativeofthediversefunctionsandhavingspeci?cknow-how,thedecisionmakingprocesshasbeenmanagedbyagroupof?vedecisionmakers.

Also,weassumedthateachDMcansigni?cantlyassesshisevaluationonalltheselectedcriteria,buttherelevanceofeachDMshouldbeassessedbyapeervotingprocedureamongtheDMs.Thus,eachdecisionmakervotedautonomouslywithoutthein?uenceoftheothersboththeweightsoftheotherDMs,theweightsofthecriteriaand?nallythelevelofperformanceachievedbyeachevaluatedmachine.

Thisinordertoavoidevaluationbiasandalsotoparallelizethevotingprocessesreducingthetimeconsumingeffects.

4.Themethod

TheadvancedmethodmostlydrawsontheBoranetal.(2009)procedureimplementingIF-TOPSISinagroupdecisionprocesswherethelinguisticvariablesmappedbyintuitionisticfuzzynumbersareusedtoassesstheindividualweightsofallcriteriaandtheratingsofeachalternativewithrespecttoeachcriterion.Themethodconsideredthatthegroupinvolvesmultiple

actors

2160D.Aloinietal./ExpertSystemswithApplications41(2014)2157–2165

(DecisionMakers),eachwithdifferentskills,experienceandknowledgerelatingtodifferentaspects(criteria)oftheproblemsothattheauthorsusedintuitionisticfuzzyweightedaveraging(IFWA)operatortoaggregateindividualopinionsofdecisionmak-ersforratingtheimportanceofcriteriaandalternatives.

DifferentlyfromthetraditionalIF-TOPSISmethodwhichas-sumesthatthereisaspecialactor(super-decisionmaker)withauthorityfordeterminingvotingpowerstothegroupmembersonthedifferentcriteria,inourmodeldecisionmakersconsistofagroupofpeerexpertswhichisintendedtoretain‘‘suf?cient’’knowledge,experiencesandresourcestoevaluatethealternativesincompleteautonomy.Accordingly,ourpurposeistogiveeachdecisionmakertheopportunitytoratetheimportanceofeachotherincompleteautonomywithoutincreasingthecomplexityofthemodelandcoherentlywiththelinguisticprocedurewhichchar-acterizestheIF-TOSPISapproach.Thus,thestep1ofthefollowingproceduremodi?estheBoran’sapproachintroducingademocraticpeer-basedvotingsystembetweenthedecisionmakers.

Thereby,letA={A1,A2,...,Am}beasetofalternativesandX={X1,X2,...,Xn}beasetofcriteria,theprocedurethatweproposeisasfollows:

Step1:ConstructtheaggregatedImportanceIFdecisionmatrix.EachdecisionmakervotetheimportanceofeachoftheotherdecisionmakersaccordingtoanIFscale,letiKbetheIFNassociatedwiththisimportance:

The‘‘Opinion’’ofeachdecisionmaker(listhenumberofdeci-sionmakersinvolvedintothedecisionprocess)hasthesamevalueanditisrepresentedbythecoef?cient/(wecallit‘‘ValueoftheOpinion’’)inEq.(1):

/1

1¼/2¼ÁÁÁ¼/l¼l

:

ð1Þ

Ingroupdecisionmakingprocess,alltheindividualdecisionopin-ionsneedtobefusedintoagroupopinion.InordertodothatwedecidetouseIFWAoperator(Xu,2007d)inEq.(2):

Dk¼/ð1Þð2ÞðlÞ

"1i1È/2i2ÈÁÁÁÈ/lil

¼1ÀYlð1ÀlYlðmYl/kÞ/;ð1ÀlY

l#

kÞ/;kÞÀðm/kÞ;

ð2Þ

k¼1

k¼1

k¼1

k¼1

whereDk=[lk,mk,pk]istheintuitionisticfuzzynumberwhichrep-resentstheaggregateimportanceofthedecisionmakerkth.

LinguistictermsusedfortheratingsoftheDMs’importance(andalsoforcriteria)aregiveninTable1.

EachDMvotedtheimportanceoftheotherDMs.Table2showsDMs’opinionusingIFnumbers.

OpinionswerefusedintoagroupopinionusingIFWAoperator(Table3)proposedbyXu(2007d)where:/1¼/2¼ÁÁÁ¼/l¼1l¼1

5

Step2:Determinetheweightofthedecisionmakers.

Theweightkkofthek-thdecisionmakercanbeobtainedastheclassicmethod:

??

??

l

klkkk

k¼l

kþpkÁ

lð3Þ

k¼1

lkþpkÁ

kkk

Table1

LinguistictermsforratingtheimportanceofcriteriaandtheimportanceoftheDMs.Linguisticterms

Intuitionisticfuzzynumbers

l

m

p

Veryimportant(VI)

0.90.050.05Important(I)0.650.250.1Medium(M)0.50.40.1Unimportant(U)

0.350.550.1Veryunimportant(VU)

0.15

0.8

0.05

Table2

TheratingoftheDMs.

DM1

DM2DM3DM4DM5DM1IVIIIVIDM2IMMMIDM3MMMUMDM4MUVIMVIDM

5

U

M

M

M

I

Table3

TheaggregateimportanceofthedecisionmakerDk.

Dk

l

m

p

DM10.5430.3530.104DM20.6180.2810.101DM30.6630.2400.097DM40.5090.3880.103DM

5

0.772

0.144

0.083

and

X

lkk¼1:ð4Þ

k¼1

Table4showstheweightsofthedecisionmakers.

Step3:ConstructtheaggregatedIFdecisionmatrixbasedontheopinionsofDMs.

LetRðkÞ¼??rðkÞ

ijmxn

beanIntuitionisticfuzzydecisionmatrixof

eachdecisionmaker.

k={k1,k2,...,kl}istheweightofeachdecisionmaker.

AgainweaggregatedalltheindividualdecisionopinionsintoagroupopinionbyIFWAoperator(Xu,2007d).

R¼ðrijÞmxn;

ð5Þ

where

r??ij¼IFWAkrð1Þrð2ÞðlÞ ð1Þð2ÞðlÞ

ij;ij;...;rij¼k1rijÈk2rijÈ...Èklr"ij

l¼1ÀY??l1ÀlðkÞ kkY??lðkÞ kkY??lðkÞ kkY??#

ðkÞ

kkij;mij

;1ÀlijÀmijk¼1

k¼1

k¼1

k¼1

ð6Þ

and

rij¼ðlAiðxjÞ;mAiðxjÞ;pAiðxjÞÞði¼1;2;...;m;j¼1;2;...;nÞ:

ð7Þ

TheaggregatedIFdecisionmatrixcanbede?nedasfollows:

26lAx2ÞÞÁÁÁlA31ðx1Þ;mA1ðx1Þ;pA1ðx1ÞÞlA1ðx2Þ;mA1ðx2Þ;pA1ð1ðxnÞ;mA1ðxnÞ;pA1ðxnÞÞAR¼66

6lA2ðx1Þ;mA2ðx1Þ;pA2ðx1ÞÞl2ðx2Þ;mA2ðx2Þ;pA2ðx2ÞÞÁÁÁlA2

ðxnÞ;mA2ðxnÞ;pA2ðxnÞÞ7.76.74

.

...

...

...

77;l5

Amðx1Þ;mAmðx1Þ;pAmðx1ÞÞlAmðx2Þ;mAmðx2Þ;pAmðx2ÞÞÁÁÁlAmðxnÞ;mAmðxnÞ;pAmðxnÞÞ

ð8Þ

2

r11r12

ÁÁÁr1m3

rR¼666r2122ÁÁÁr2m764.

..

....7

...77:ð9Þ

..5rn1

rn2

ÁÁÁrnm

Table4

Decisionmakers’weight.

DM1

DM2DM3DM4DM5Weightkk

0.176

0.199

0.214

0.165

0.245

D.Aloinietal./ExpertSystemswithApplications41(2014)2157–2165

Table5

Linguistictermsforratingalternatives.Linguisticterms

Intuitionisticfuzzynumbers

2161

?? ð1Þð2ÞðlÞ

wj¼IFWAkwj;wj;...;wj

¼k1wjÈk2wjÈÁÁÁÈklwj

"#

l??l??l??l?? kkY kkY kkY kkYðkÞkÞðkÞ

¼1À1Àlj;mð;1ÀljÀmjðkÞ;j

k¼1

k¼1

k¼1

k¼1

ð1Þ

ð2Þ

ðlÞ

l

Veryverygood(VVG)/veryveryhigh(VVH)

Verygood(VG)/veryhigh(VH)Good(G)/high(H)

Mediumgood(MG)/mediumhigh(MH)Fair(F)/medium(M)

Mediumbad(MB)/mediumlow(ML)Bad(B)/low(L)

Verybad(VB)/verylow(VL)

Veryverybad(VVB)/veryverylow(VVL)

0.90.80.70.60.450.40.30.20.05

m

0.050.10.20.30.40.50.60.70.9

p

0.050.10.10.10.150.10.10.10.05

ð10Þ

W¼½w1;w2;...;wj??:

Here

ð11Þ

wj¼ðlj;mj;pjÞðj¼1;2;...;nÞ:

ð12Þ

ThelinguistictermsforratingthealternativesareshowninTable5.TheevaluationsgivenbytheDMstothe?vealternativesaccordingtotheselectedcriteriaareavailableinAppendixA.

TheaggregatedIFdecisionmatrixRbasedonaggregationofDMsopinionhasbeenconstructed.Tables6a,6band6cshowresults.

Step4.Determinetheweightsofcriteria.

Since,allcriteriamaynotbeassumedtobeequalimportance,letsWrepresentsasetofgradesofimportance.InordertoobtainW,theindividualdecisionmakeropinionsrelatedtotheimpor-tanceofeachcriterianeedtobefusedintoawholejudgmentasfollows.hi

ðkÞðkÞðkÞðkÞ

Letwj¼lj;mj;pjbeanIFNassignedtocriterionXjbythekthdecisionmaker.

ThentheweightsofthecriteriaarecalculatedbyusingIFWAoperator:

Referringtothecasestudy,theimportanceofthecriteriarepre-sentedaslinguistictermsinTable7areaggregatedinW(Table8)todeterminetheweightofeachcriterion.

ThelinguistictermsforratingthealternativeswerereportedinTable1.

Step5.ConstructtheaggregatedweightedIFdecisionmatrix.TheaggregatedweightedIFdecisionmatrixisconstructedaccordingtothefollowingde?nition(Atanassov,1986):

R??W¼fx;lAiðxÞÁlWðxÞ;mAiðxÞþmWðxÞÀmAiðxÞþmWðxÞjx

2Xg

and

ð13Þ

pAiÁWðxÞ¼1ÀmAiðxÞÀmWðxÞÀlAiðxÞÁlWðxÞþmAiðxÞ

þmWðxÞ:

ð14Þ

Then,theaggregatedweightedIFdecisionmatrixcanbede?nedasfollows:

Table6a

TheaggregatedintuitionisticfuzzydecisionmatrixR.

Speed

Mix?exibility

Safety

Technologicalparameters

l

AlternativeAlternativeAlternativeAlternativeAlternative

ABCDE

0.8850.7460.8500.6620.742

m

0.0570.1500.0750.2360.154

p

0.0570.1040.0750.1010.103

l

0.2640.7000.7410.7790.700

m

0.6420.2000.1550.1190.200

p

0.0950.1000.1030.1020.100

l

0.7000.7000.7250.7000.850

m

0.2000.2000.1720.2000.075

p

0.1000.1000.1030.1000.075

l

0.5680.4500.5070.4380.800

m

0.3220.4000.3630.4220.100

p

0.1110.1500.1310.1390.100

Table6b

TheaggregatedintuitionisticfuzzydecisionmatrixR.

Easeofuse

Accessories

Upgrading

G.A.As

l

AlternativeAlternativeAlternativeAlternativeAlternative

ABCDE

0.4400.4220.4220.4090.800

m

0.4180.4550.4550.4810.100

p

0.1410.1230.1230.1100.100

l

0.3490.4000.4000.2770.243

m

0.5550.5000.5000.6230.657

p

0.0960.1000.1000.1000.100

l

0.4400.4500.4500.8000.800

m

0.4180.4000.4000.1000.100

p

0.1410.1500.1500.1000.100

l

0.6750.5170.5170.7640.517

m

0.2070.3550.3550.1330.355

p

0.1180.1270.1270.1030.127

Table6c

TheaggregatedintuitionisticfuzzydecisionmatrixR.

Price

W.M.R.R.

Electricconsumption

Dimension

l

AlternativeAlternativeAlternativeAlternativeAlternative

ABCDE

0.7590.4420.7460.5170.600

m

0.1350.4150.1500.3550.300

p

0.1050.1430.1040.1270.100

l

0.7500.6840.6660.3660.366

m

0.1440.2150.2330.5080.508

p

0.1060.1010.1010.1260.126

l

0.6150.3770.6000.5170.600

m

0.2660.5230.3000.3550.300

p

0.1190.1000.1000.1270.100

l

0.5620.3610.4500.4210.544

m

0.3080.5390.4000.4560.338

p

0.1300.1000.1500.1230.118