Some Interval Neutrosophic Dombi Power Bonferroni Mean Operators by Florentin Smarandache

SomeIntervalNeutrosophicDombiPower BonferroniMeanOperatorsandTheirApplication inMulti–AttributeDecision–Making

QaisarKhan 1,PeideLiu 2,* ,TahirMahmood 1 ,FlorentinSmarandache 3 and KifayatUllah 1

1 DepartmentofMathematicsandStatistic,InternationalIslamicUniversity,Islamabad44000,Pakistan; qaisarkhan421@gmail.com(Q.K.);tahirbakhat@iiu.edu.pk(T.M.);kifayat.phdma72@iiu.edu.pk(K.U.)

2 SchoolofManagementScienceandEngineering,ShandongUniversityofFinanceandEconomics, Jinan250014,Shandong,China

3 DepartmentofMathematicsandSciences,UniversityofNewMexico,705GurleyAve,Gallup,NM87301, USA;fsmarandache@gmail.com

* Correspondence:peide.liu@gmail.com;Tel.:+86-531-8222-2188

Received:16August2018;Accepted:28September2018;Published:2October2018

Abstract: ThepowerBonferronimean(PBM)operatorisahybridstructureandcantakethe advantageofapoweraverage(PA)operator,whichcanreducetheimpactofinappropriatedatagiven bytheprejudiceddecisionmakers(DMs)andBonferronimean(BM)operator,whichcantakeinto accountthecorrelationbetweentwoattributes.Inrecentyears,manyresearchershaveextendedthe PBMoperatortohandlefuzzyinformation.TheDombioperationsofT-conorm(TCN)andT-norm (TN),proposedbyDombi,havethesupremacyofoutstandingflexibilitywithgeneralparameters. However,intheexistingliterature,PBMandtheDombioperationshavenotbeencombinedfor theaboveadvantagesforinterval-neutrosophicsets(INSs).Inthisarticle,wefirstdefinesome operationallawsforintervalneutrosophicnumbers(INNs)basedonDombiTNandTCNand discussseveraldesirablepropertiesoftheseoperationalrules.Secondly,weextendthePBMoperator basedonDombioperationstodevelopaninterval-neutrosophicDombiPBM(INDPBM)operator,an interval-neutrosophicweightedDombiPBM(INWDPBM)operator,aninterval-neutrosophicDombi powergeometricBonferronimean(INDPGBM)operatorandaninterval-neutrosophicweighted DombipowergeometricBonferronimean(INWDPGBM)operator,anddiscussseveralpropertiesof theseaggregationoperators.Thenwedevelopamulti-attributedecision-making(MADM)method, basedontheseproposedaggregationoperators,todealwithintervalneutrosophic(IN)information. Lastly,anillustrativeexampleisprovidedtoshowtheusefulnessandrealismoftheproposedMADM method.ThedevelopedaggregationoperatorsareverypracticalforsolvingMADMproblems,asit considerstheinteractionamongtwoinputargumentsandremovestheinfluenceofawkwarddata inthedecision-makingprocessatthesametime.Theotheradvantageoftheproposedaggregation operatorsisthattheyareflexibleduetogeneralparameter.

Keywords: intervalneutrosophicsets;Bonferronimean;poweroperator;multi-attributedecision making(MADM)

1.Introduction

Whiledealingwithanyrealworldproblems,adecisionmaker(DM)oftenfeelsdiscomfortwhen expressinghis\herevaluationinformationbyutilizingasinglerealnumberinmulti-attributedecision making(MADM)ormulti-attributegroupdecisionmaking(MAGDM)problemsduetotheintellectual fuzzinessofDMs.Forthiscause,Zadeh[1]developedfuzzysets(FSs),whichareassignedbya Symmetry 2018, 10,459;doi:10.3390/sym10100459 www.mdpi.com/journal/symmetry

symmetry SS Article

truth-membershipdegree(TMD)in [0,1] andareabettertooltopresentfuzzyinformationforhandling MADMorMAGDMproblems.AftertheintroductionofFSs,differentfuzzymodellingapproaches weredevelopedtodealwithuncertaintyinvariousfields[2–4].However,insomesituations,it isdifficulttoexpresstruth-membershipdegreewithanexactnumber.Inordertoovercomethis defectandtoexpressTMDinamoreappropriateway,Turksen[5]developedintervalvaluedFSs (IVFSs),inwhichTMDisrepresentedbyintervalnumbersinsteadofexactnumbers.SinceonlyTMD wasconsideredinFSsorIVFSsandthefalsity-membershipdegree(FMD)cameautomaticallyby subtractingTMDfromone,itishardtoexplainsomecomplicatedfuzzyinformation,forexample,for theselectionofDeanofafaculty,iftheresultsreceivedfromfiveprofessorsareinfavor,twoareagainst andthreeareneitherinfavornoragainst.Then,thistypeofinformationcannotbeexpressedbyFSs.So, inordertohandlesuchtypesofinformation,Atanassov[6]developedintuitionisticfuzzysets(IFSs), whichwereassignedbyTMDandFMD.Atanassovetal.[7]furtherenlargedIFSsanddeveloped theintervalvaluedIFS(IVIFSs).However,theshortcomingofFSs,IVFSs,IFSsandIVIFSsarethat theycannotdealwithunreliableorindefiniteinformation.Tosolvesuchproblems,Smarandache[8,9] developedneutrosophicsets(NSs).Inneutrosophicset,everymemberofthedomainsethasTMD, anindeterminacy-membershipdegree(IMD)andFMD,whichcapturevaluesin]0 ,1+[.Duetothe containmentofsubsetsof]0 ,1+[inNS,itishardtoutilizeNSinrealworldandengineeringproblems. TomakeNSshelpfulinthesecases,someauthorsdevelopedsubclassesofNSs,suchassinglevalued neutrosophicsets(SVNSs)[10],intervalneutrosophicsets(INSs)[11,12],simplifiedneutrosophicsets (SNSs)[13,14]andsoforth.Inrecentyears,INSshavegainedmuchattentionfromtheresearchers andagreatnumberofachievementhavebeenmade,suchasdistancemeasures[15–17],entropiesof INS[18–20],correlationcoefficient[21–23].ThetheoryofNSshasbeenextensivelyutilizedtohandle MADMandMAGDMproblems.

Forthelastmanyyears,informationaggregationoperators[24–27]havestimulatedmuch awarenessofauthorsandhavebecomeverydominantresearchtopicofMADMandMAGDM problems.Theconventionalaggregationoperators(AGOs)proposedbyXu,XuandYager[28,29]can onlyaggregateagroupofrealnumbersintoasinglerealnumber.NowtheseconventionalAGOswere furtherextendedbymanyauthors,forexample,Sunetal.[30]proposedtheintervalneutrosophic numberChoquetintegraloperatorforMADMandLiuetal.[31]developedprioritizedordered weightedAGOsforINSsandappliedthemtoMADM.Inaddition,somedecision-makingmethods werealsodevelopedforMADMproblems,forexample,Mukhametzyanovetal.[32]developeda statisticallybasedmodelforsensitivityanalysisinMADMproblems.Petrovicetal.[33]developed amodelfortheselectionofaircraftsbasedondecisionmakingtrialandevaluationlaboratoryand analytichierarchyprocess(DEMATEL-AHP).Royetal.[34]proposedaroughrelationalDEMATEL modeltoanalyzethekeysuccessfactorofhospitalquality.Sarkaretal.[35]developedanoptimization techniquefornationalincomedeterminationmodelwithstabilityanalysisofdifferentialequationin discreteandcontinuousprocessunderuncertainenvironment.Thesemethodscanonlygivearanking result,however,AGOscannotonlygivetherankingresult,butalsogivethecomprehensivevalueof eachalternativebyaggregatingitsattributevalues.

Itisobviousthat,differentaggregationoperatorshavedistinctfunctions,afewofthemcanreduce theimpactofsomeawkwarddataproducedbypredisposeDMs,suchaspoweraverage(PA)operator proposedbyYager[36].ThePAoperatorcanaggregatetheinputdatabydesignatingtheweight vectorbasedonthesupportdegreeamongtheinputarguments,andcanattainthisfunction.Nowthe PAoperatorwasfurtherextendedbymanyresearchersintodifferentenvironments.Liuetal.[37] proposedsomegeneralizedPAoperatorforINNs,andappliedthemtoMADM.Consequently,some aggregationoperatorscanincludetheinterrelationshipbetweentheaggregatingparameters,suchas theBonferronimean(BM)operatorsdevelopedbyBonferroni[38],theHeronianmean(HM)operator introducedbySykora[39],MuirheadMean(MM)operator[40],Maclaurinsymmetricmean[41] operators.Inaddition,theseaggregationoperatorshavealsobeenextendedbymanyauthorstodeal withfuzzyinformation[42–46].

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ForaggregatingINNs,someAGOsaredevelopedbyutilizingdifferentT-norms(TNs)and T-conorms(TCNs),suchasalgebraic,EinsteinandHamacher.Usually,theArchimedeanTNandTCN arethegeneralizationsofvariousTNsandTCNssuchasalgebraic,Einstein,Hamacher,Frank,and Dombi[47]TNsandTCNs.DombiTNandTCNhavethecharacteristicsofgeneralTNandTCNbya generalparameter,andthiscanmaketheaggregationprocessmoreflexible.Recently,severalauthors definedsomeoperationallawsforIFSs[48],SVNSs[49],hesitantfuzzysets(HFSs)[50,51]basedon DombiTNandTCN.Inpracticaldecisionmaking,wegenerallyneedtoconsiderinterrelationship amongattributesandeliminatedtheinfluenceofawkwarddata.Forthispurpose,someresearchers combinedBMandPAoperatorstoproposesomePBMoperatorsandextendedthemtovarious fields[52–55].ThePBMoperatorshavetwocharacteristics.Firstly,itcanconsidertheinteraction amongtwoinputargumentsbyBMoperator,andsecondly,itcanremovetheeffectofawkwarddata byPAoperator.TheDombiTNandTCNhaveageneralparameter,whichmakesthedecision-making processmoreflexible.Fromtheexistingliteratures,weknowthatPBMoperatorsarecombinedwith algebraicoperationstoaggregateIFNs,orIVIFNs,andthereisnoresearchoncombiningPBMoperator withDombioperationstoaggregateINNs.

Inaword,byconsideringthefollowingadvantages.(1)SinceINSsarethemoreprécisedclassby whichonecanhandlethevagueinformationinamoreaccuratewaywhencomparedwithFSsandall otherextensionslikeIVFSs,IFSs,IVIFSsandsoforth,theyaremoresuitabletodescribetheattributes ofMADMproblems,sointhisstudy,wewillselecttheINSsasinformationexpression;(2)DombiTN andTCNaremoreflexibleinthedecisionmakingprocessduetogeneralparameterwhichisregarded asdecisionmakers’riskattitude;(3)ThePBMoperatorshavethepropertiesofconsideringinteraction betweentwoinputargumentsandvanishestheeffectofawkwarddataatthesametime.Hence,the purposeandmotivationarethatwetrytocombinethesethreeconceptstotaketheabovedefined advantagesandproposedsomenewpowerfultoolstoaggregateINNs.(1)wedefinesomeDombi operationallawsforINNs;(2)weproposesomenewPBMaggregationoperatorsbasedonthesenew operationallaws;(3)wedevelopanovelMADMbasedonthesedevelopedaggregationoperators.

Thefollowingsectionsofthisarticleareshownasfollows.InSection 2,wereviewsomebasic conceptsofINSs,PAoperators,BMoperators,andGBMoperators.InSection 3,wereviewbasic conceptofDombiTNandTCN.Afterthat,weproposesomeDombioperationsforINNs,anddiscuss someproperties.InSection 4,wedeﬁneINDPBMoperator,INWDPBMoperator,INDPGBMoperator andINWDPGBMoperatoranddiscusstheirproperties.InSection 5,weproposeaMADMmethod basedontheproposedaggregationoperatorswithINNs.InSection 6,weuseanillustrativeexample toshowtheeffectivenessoftheproposedMADMmethod.TheconclusionisdiscussedinSection 7

2.Preliminaries

Inthispart,somebasicdeﬁnitions,propertiesaboutINSs,BMoperatorsandPAoperators arediscussed.

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expressedby

NS = v, t NS v , i NS v , f NS v v ∈ Ω

where, t NS v , i NS v and f NS v

v ∈

NS.

v ∈

t NS v , i NS v f NS v ∈ ]0

0 ≤ t NS

2.1.TheINSsandTheirOperationalLaws Deﬁnition1. Let Ω bethedomainset[8,9],withanon-speciﬁcmemberin Ω

v ANS NS in Ω is expressedby

,(1)

respectivelyexpresstheTMD,IMDandFMDoftheelement

U to theset

Foreachpoint

U,wehave,

,1+ [ and

v + i

v + f NS v ≤ 3+ TheNSwaspredominantlydevelopedfromphilosophicalperspective,anditishardtobeapplied toengineeringproblemsduetothecontainmentofsubsetsof ]0 ,1+ [.So,inordertouseitmoreeasily

inreallifeorengineeringproblem,Wangetal.[8]presentedasubclassofNSbychanging ]0 ,1+ [ to [0,1] andwasnamedSVNS,andisdeﬁnedasfollow:

Deﬁnition2. Let Ω bethedomainset[10],withanon-speciﬁcmemberin Ω expressedby v ASVNS SV in Ω isexpressedby

SV = u, t SV v , i SV v , fSV v v ∈ Ω ,(2) where t SV v , i SV v and fSV v expresstheTMD,IMDandFMDoftheelement v ∈ Ω totheset SV respectively.Foreachpoint v ∈ Ω,wehave, t SV v , i SV v , fSV v , ∈ [0,1] and 0 ≤ t SV v + i SV v + fSV v ≤ 3.

Inordertodefinemorecomplexinformation,Wangetal.[9]furtherdevelopedINSwhichis defineasfollows: Definition3. Let Ω bethedomainsetand v ∈ Ω [11].ThenanINS INin Ω isexpressedby

IN = v, TRIN v , IDIN v , FLIN v v ∈ Ω ,(3) where, TRIN v , IDIN v and FLIN v respectively,expresstheTMD,IMDandFMDoftheelement v ∈ Ω totheset IN Foreachpoint v ∈ U,wehave, TRIN v , IDIN v , FLIN v ⊆ [0,1] and 0 ≤ maxIDIN v + maxIDIN v + maxFLIN v ≤ 3.

Forcomputationalsimplicity,wecanuse in = TR L , TR U , ID L , ID U , FL L , FL U to expressanelement in inanINS,andtheelement in iscalledanintervalneutrosophicnumber(INN). Where TR L , TR U ⊆ [0,1], ID L , ID U ⊆ [0,1], FL L , FL U ⊆ [0,1] and 0 ≤ TR U + ID U + FL U ≤ 3.

Deﬁnition4. Let in1 = TR L 1 , TR U 1 , ID L 1 , ID U 1 , FL L 1 , FL U 1 and in2 =

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TR L 2 , TR U 2 , ID L 2 , ID U 2 , FL L 2 , FL U 2 beanytwoINNs[12],and ζ > 0.Thentheoperational lawsofINNscanbedefinedasfollows: (1) in1 ⊕ in2 = TR L 1 + TR L 2 TR L 1 TR L 2 , TR U 1 + TR U 2 TR U 1 TR U 2 , ID L 1 ID L 2 , ID U 1 ID U 2 , FL L 1 FL L 2 , FL U 1 FL U 2 ; (4) (2) in1 ⊗ in2 = TR L 1 TR L 2 , TR U 1 TR U 2 , ID L 1 + ID L 2 ID L 1 ID L 2 , ID U 1 + ID U 2 ID U 1 ID U 2 , FL L 1 + FL L 2 FL L 1 FL L 2 , FL U 1 + FL U 2 FL U 1 FL U 2 ; (5) (3) in ζ 1 = TR L 1 ζ , TR U 1 ζ , 1 1 ID L 1 ζ ,1 1 ID U 1 ζ , 1 1 FL L 1 ζ ,1 1 FL U 1 ζ ; (6) (4) ζin1 = 1 1 TR L 1 ζ ,1 1 TR U 1 ζ , ID L 1 ζ , ID U 1 ζ , FL L 1 ζ , FL U 1 ζ (7) Definition5. Let in = TR L , TR U , ID L , ID U , FL L , FL U [42],beanINN.Thenthescorefunction S in andaccuracyfunctionA in canbedefinedasfollows: (i) S in = TR L + TR U 2 + 1 ID L + ID U 2 + 1 FL L + FL U 2 ;(8)

(ii) A in = TR L + TR U 2 + 1 ID L + ID U 2 + FL L + FL U 2 .(9)

InordertocomparetwoINNs,thecomparisonrulesweredeﬁnedbyLiuetal.[36],whichcanbe statedasfollows.

Deﬁnition6. Let in1 = TR L 1 , TR U 1 , ID L 1 , ID U 1 , FL L 1 , FL U 1 and in2 = TR L 2 , TR U 2 , ID L 2 , ID U 2 , FL L 2 , FL U 2 beanytwoINNs[42].Thenwehave:

(1) IfS in1 > S in2 , then in1 isbetterthan in2,anddenotedby in1 > in2; (2) IfS in1 = S in2 , andA in1 > A in2 , then in1 isbetterthan in2,anddenotedby in1 > in2; (3) IfS in1 = S in2 , andA in1 = A in2 , then in1 isequalto in2,anddenotedby in1 = in2

Deﬁnition7. Let in1 = TR L 1 , TR U 1 , ID L 1 , ID U 1 , FL L 1 , FL U 1 and in2 = TR L 2 , TR U 2 , ID L 2 , ID U 2 , FL L 2 , FL U 2 beanytwoINNs[15].ThenthenormalizedHamming distancebetweenn1 andn2 isdescribedasfollows. D in1, in2 = 1 6 TR L 1 TR L 2 + TR U 1 TR U 2 + ID L 1 ID L 2 + ID U 1 ID U 2 + FL L 1 FL L 2 + FL U 1 FL U 2 (10)

2.2.ThePAOperator

ThePAoperatorwasﬁrstpresentedbyYager[36]anditisdescribedasfollows. Deﬁnition8. Forpositiverealnumbers ℘h (h = 1,2,..., l) [36],thePAoperatorisdescribedas PA(℘1, ℘2,..., ℘l ) =

l ∑ h=1 (1 + T(℘h ))℘h l ∑ h=1 (1 + T(℘h )) ,(11) where, T(℘h ) = l ∑ y=1,h=y sup ℘h, ℘y , and sup ℘h, ℘y isthedegreetowhich ℘h supports ℘y Thesupport degree(SPD)satisﬁesthefollowingproperties. (1) sup ℘h, ℘y = sup ℘y, ℘h ; (2) sup ℘h, ℘y ∈ [0,1]; (3) sup ℘h, ℘y ≥ sup(℘c, ℘d ), if ℘h ℘y ≤ |℘c ℘d |

2.3.TheBMOperator

TheBMoperatorwasinitiallypresentedbyBonferroni[38],anditwasexplainedasfollows: Deﬁnition9. Fornon-negativerealnumbers ℘h (h = 1,2, , l),and x, y ≥ 0 [38],theBMoperatoris describedas BMx,y (℘1, ℘2,..., ℘l ) = 1 l2 l l ∑ h=1

l ∑ s=1,h=s ℘x h ℘y s

1 x+y .(12)

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TheBMoperatorignorestheimportancedegreeofeachinputargument,whichcanbegivenby decisionmakersaccordingtotheirinterest.ToovercomethisshortcomingofBMoperator,Heetal.[52] deﬁnedtheweightedBonferronimean(WBM)operatorswhichcanbeexplainedasfollows:

Deﬁnition10. Forpositiverealnumbers ℘h (h = 1,2, , l) and x, y ≥ 0 [52],thentheweightedBMoperator (WBM)isdescribedas

WBMx,y (℘1, ℘2,..., ℘l ) = 1 l2 l l ∑ h=1

TheWBMoperatorhasthefollowingcharacteristics:

l ∑ s=1,h=s

κhκs 1 κh ℘x h ℘y s

1 x+y ,(13) where κ = (κ1, κ2,..., κl )T istheimportancedegreeofevery ℘h (h = 1,2,..., l)

Theorem1. (Reducibility)Iftheweightvectoris κ = 1 l , 1 l ,..., 1 l T ,then WBMx,y (℘1, ℘2,..., ℘l ) = 1 l2 l l ∑ h=1

l ∑ s=1,z=s ℘x h ℘y s

1 x+y = BMx,y (℘1, ℘2,..., ℘m ) (14)

Theorem2. (Idempotency)Let ℘h = ℘, (h = 1,2,..., l).ThenBMx,y (℘1, ℘2,..., ℘l ) = ℘

Theorem3. (Permutation)Let (℘1, ℘2,..., ℘l ) beanypermutationof Z1 , Z2 ,..., Zl .Then WMBx,y Z1 , Z2 ,..., Zl = WBM(℘1, ℘2,..., ℘l ).(15)

Theorem4. (Monotonicity)Let ℘h ≥ Kh (h = 1,2,..., l).Then WBMx,y (℘1, ℘2,..., ℘l ) ≥ WBMx,y K1 , K2 ,..., Kl .(16)

Theorem5. (Boundedness)TheWBMx,y liesintheminandmaxoperators,thatis, min(℘1, ℘2,..., ℘l ) ≤ WBMx,y (℘1, ℘2,..., ℘l ) ≤ max(℘1, ℘2,..., ℘l ).(17)

SimilartoBMoperator,thegeometricBMoperatoralsoconsidersthecorrelationamongtheinput arguments.Itcanbeexplainedasfollows:

Deﬁnition11. Forpositiverealnumbers ℘h (h = 1,2, , l) and x, y ≥ 0 [53],thegeometricBMoperator (GBM)isdescribedas GBMx,y (℘1, ℘2,..., ℘l ) = 1 x + y

l ∏ h=1

l ∏ s=1,h=s (x℘h + y℘s ) 1 l2 l .(18)

TheGBMoperatorignorestheimportancedegreeofeachinputargument,whichcanbegiven bydecisionmakersaccordingtotheirinterest.InasimilarwaytoWBM,theweightedgeometric

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The GBM operator ignores the importance degree of each input argument, which can be given by decision makers according to their interest. In a similar way to WBM, the weighted geometric BM (WGBM) operator was also presented. The extension process is same as that of WBM, so it is omitted here. The definition of power Bonferroni mean (PBM) and power geometric Bonferroni mean (PGBM) operators are given in Appendix A.

1 ,,....,. ll xy ll lhs hshs GBMxy xy    

(1,2,...,) h hl ,0xy      2 1 , 12 11,

BM(WGBM)operatorwasalsopresented.TheextensionprocessissameasthatofWBM,soitis omittedhere.

3. Some Operations of INSs Based on Dombi TN and TCN

Dombi

TN and TCN

ThedeﬁnitionofpowerBonferronimean(PBM)andpowergeometricBonferronimean(PGBM) operatorsaregiveninAppendix A.

3.SomeOperationsofINSsBasedonDombiTNandTCN

Dombi operations consist of the Dombi sum and Dombi product.

DombiTNandTCN

are explained as follows:

DombioperationsconsistoftheDombisumandDombiproduct.

be any two real numbers [47] Then the Dombi TN and TCN among 

(19)

and

Deﬁnition12. Let and ℵ beanytworealnumbers[47].ThentheDombiTNandTCNamong and ℵ are explainedasfollows: TD ( , ℵ)= 1 1 + 1− l + 1−ℵ ℵ l 1 l ;(19) TD ∗( , ℵ)= 1 1 1 + 1− l + ℵ 1−ℵ l 1 l ,(20) where,l ≥ 1, and ( , ℵ) ∈ [0,1] × [0,1] AccordingtotheDombiTNandTCN,wedevelopafewoperationalrulesforINNs. Deﬁnition13. Let in = TR L , TR U , ID L , ID U , FL L , FL U , in1 = TR L 1 , TR U 1 , ID L 1 , ID U 1 , FL L 1 , FL U 1 and in2 = TR L 2 , TR U 2 , ID L 2 , ID U 2 , FL L 2 , FL U 2 beanythreeINNsand Φ > 0.Then,basedonDombiTNandTCN,thefollowingoperationallawsaredeveloped forINNs.

Symmetry 2018, 10,459 7of32



1 1 (,); 11 1 D

          

* 1 1 (,)1, 1 11 DT           

1, 

      ,0,10,1.

Definition 13. Let ,,,,,, LULULU inTRTRIDIDFLFL     111111 1 ,,,,, LULULU inTRTRIDIDFLFL    

222222 2 ,,,,, LULULU inTRTRIDIDFLFL     be

0  . Then,

Definition 12. Let Dombi TN

 and the following operational laws are developed for INNs. (1) 111 121212 121212 111 1,1,, 11 111 11111 LLUULL LLUULL TRTRTRTRIDID TRTRTRTRIDID                 111 121212 121212 111 ,, 111111 111 v UULLUU UULLUU IDIDFLFLFLFL IDIDFLFLFLFL                                   ;         (21)(21) (2) in1 ⊗ in2 =      1 1+ 1 TR L 1 TR L 1 γ + 1 TR L 2 TR L 2 γ 1 γ , 1 1+ 1 TR U 1 TR U 1 γ + 1 TR U 2 TR U 2 γ 1 γ      ,      1 1 1+ ID L 1 1 ID L 1 γ + ID L 2 1 ID L 2 γ 1 γ , 1 1 1+ ID U 1 1 ID U 1 γ + ID U 2 1 ID U 2 γ 1 γ      ,      1 1 1+ FL L 1 1 FL L 1 γ + FL L 2 1 FL L 2 γ 1 γ , 1 1+ FL U 1 1 FL U 1 γ + FL U 2 1 FL U 2 γ 1 γ      ; (22)

and 

(20) where,

According to the Dombi TN and TCN, we develop a few operational rules for INNs.

and

any three INNs and

based on

and TCN,

(24) Now,basedonthesenewoperationallawsforINNs,wedevelopsomeaggregationoperatorsto aggregateINinformationintheprecedingsections. 4.TheINPBMOperatorBasedonDombiTNandDombiTCN

Inthispart,basedontheDombioperationallawsforINNs,wecombinePAoperatorandBM tointroduceintervalneutrosophicDombipowerBonferronimean(INDPBM),intervalneutrosophic weightedDombipowerBonferronimean,intervalneutrosophicDombipowergeometricBonferroni mean(INDPGBM)andintervalneutrosophicweightedDombipowerBonferronimean(INWDPGBM) operatorsanddiscusssomerelatedproperties.

4.1.TheINDPBMOperatorandINWDPBMOperator

Symmetry 2018, 10,459 8of32 (3) Φin =     1 1 1+ Φ TR L 1 TR L γ 1 γ ,1 1 1+ Φ TR U 1 TR U γ 1 γ     ,     1 1+ Φ 1 ID L ID L γ 1 γ , 1 1+ Φ 1 ID U ID U γ 1 γ         1 1+ Φ 1 FL L FL L γ 1 γ , 1 1+ Φ 1 FL U FL U γ 1 γ     ;

(4) in Φ =     1 1+ Φ 1 TR L TR L γ 1 γ , 1 1+ Φ 1 TR U TR U γ 1 γ     ,     1 1 1+ Φ ID L 1 ID L γ 1 γ ,1 1 1+ Φ ID U 1 ID U γ 1 γ         1 1 1+ Φ FL L 1 FL L γ 1 γ ,1 1 1+ Φ FL U 1 FL U γ 1 γ    

(23)

Deﬁnition14. Let ini = TR L i , TR U i , ID L i , ID U i , FL L i , FL L i , (i = 1,2, , l),beagroupofINNs, andx, y ≥ 0.If INDPBMx,y in1, in2,..., inl =          1 l2 l          l ⊕ i, j = 1 i = j         l 1 + T(ini ) l ⊕ u 1 1 + T(inu ) ini          x ⊗D     l 1 + T(inj ) l ⊕ u 1 1 + T(inu ) inj     y             1 x+y

then INDPBMx,y issaidtobeINDombipowerBonferronimean(INDPBM)operator,where T inz = l ⊕ s=1,s=z Sup inz, ins . Sup inz, ins isthesupportdegreefor inz from ins, whichsatisfiesthefollowing axioms:(1) Sup inz, ins ∈ [0,1]; (2) Sup inz, ins = Sup ins, inz ; (3) Sup inz, ins ≥ Sup ina, inb , if D inz, ins < D ina, inb ,inwhich D ina, inb isthedistancemeasurebetweenINNs ina and inb defined inDefinition7. InordertosimplifyEquation(25),wecangive Λz = 1 + T inz l ⊕ z=1 1 + T inz

(25)

(26)

(27)

a group of INNs. Then the value obtained



1 ,1  

 l U ij j ij U j

 

                                                                                 

(28) Proof. ProofofTheorem6isgiveninAppendix B

(28)

FL FL

Proof. Proof of Theorem 6 is given in Appendix B. □

In order to determine the PWV 

, we firstly need to determine the support degree among INNs. In general, the similarity measure among INNs can replace the support degree among INNs. That is,

InordertodeterminethePWV Λ,weﬁrstlyneedtodeterminethesupportdegreeamongINNs. Ingeneral,thesimilaritymeasureamongINNscanreplacethesupportdegreeamongINNs.Thatis, Sup ini, inm = 1 D ini, inm (i, m = 1,2,..., l).(29) Example1. Let in1 = [0.3,0.7], [0.2,0.4], [0.3,0.5] , in2 = [0.4,0.6], [0.1,0.3], [0.2,0.4] and in3 = [0.1,0.3], [0.4,0.6], [0.2,0.4] beanythreeINNs, x = 1, y = 1, γ = 3.ThenbyTheorem6 inEquation(28),wecanaggregatethesethreeINNsandgeneratethecomprehensivevalue in = TR L , TR U , ID L , ID U , FL L , FL U whichiscalculatedasfollows:

9of32

Λl

INDPBM

=        1 l2 l l ⊕ i

i =

lΛ

       1

L i

, xy INDPBM     1, ,. l zzs ssz TinSupinin     , zs

   

zsSupinin     

zssz

     ,, zsab SupininSupinin      ,, zsab DininDinin    , abDinin ain bin In order to simplify Equation (25), we can give         1 1 1 z z l z z Tin Tin     (26) and call   12,,..., T l is the power weight vector

such that 1 0,1. l zz z 

Equation (25) into the following form       1 , 12 2 ,1 1 ,,....,. xy xy l xy lij iDj ij ij INDPBMinininlinlin ll           

Theorem 6. Let ,,,,,,(1,2,...,) LULULL iiiiii i inTRTRIDIDFLFLil     be

by utilizing Equation

is expressed as   , 12,,...., xy e

1 22 1111,1111 1111                           LLUU ijij ijij LLU iji llxyllxy xyxyTRTRTRTR llll TRTRTR   1 ,1 , 11                                               

    

 



                                       l ij ij U j LL ij ijLL ij TR llxy IDID ll IDID    11 2 ,1,1 ,11111 11                                               ll UU ijij ij ijij ijUU ij llxy xy IDID ll IDID   ,                                                 



LLU iji

Symmetry 2018, 10,459

andcall Λ = (

2,...,

)T isthepowerweightvector(PWV),suchthat Λz ≥ 0, l ⊕ z=1 Λz = 1.It turnsEquation(25)intothefollowingform

x,y in1, in2,..., inl

, j

iini x ⊗D lΛjinj y

x+y .(27) Theorem6. Let ini = TR

U i , ID

i , ID U i , FL L i , FL L i , (i = 1,2, ... , l) beagroupofINNs. ThenthevalueobtainedbyutilizingEquation(25)isexpressedas

Supinin

in , sin

,0,1;

,,;

SupininSupinin

(PWV),

It turns

(25)

INDPBMininin





 



1 22 11111,11111 1111

                         LLU iji ijii

llxyllxy xyxyFLFLFL llll FLFLFL



and

be any three INNs, 1,1,3xy  

           

which is calculated as follows: Step 1. Determine the supports

. Then by Theorem (6) in Equation (28), we can aggregate these three INNs and generate the comprehensive value ,,,,, LULULU inTRTRIDIDFLFL 



0.1,0.3,0.4,0.6,0.2,0.4 in

be any three INNs, 1,1,3xy 

Step1. Determinethesupports Sup ini, inj , i, j = 1,2,3 byusingEquation(29),andthen weget Sup in1, in2 = Sup in2, in1 = 0.9, Sup in1, in3 = Sup in3, in1 = 0.933, Sup in2, in3 = Sup in3, in2 = 1. Step2. DeterminethePWVBecauseT inz = 3 ∑ s=1,s=z Sup inz, ins , and

. Then by Theorem (6) in Equation (28), we can aggregate these three INNs and generate the comprehensive value ,,,,,

        

 

by using Equation (29), and then we get

 

0.2,0.4

TinSupinin

    3

123 1 0.3269, 111 Tin TinTinTin

0.3346, 111 Tin TinTinTin

then Λ1 = T in1 + 1 T in1 + 1 + T in2 + 1 + T in3 + 1 = 0.3269, Λ2 = T in2 + 1 T in1 + 1 + T in2 + 1 + T in3 + 1 = 0.3346, Λ3 = T in3 + 1 T in1 + 1 + T in2 + 1 + T in3 + 1 = 0.3385. Step3. Determinethecomprehensivevalue in = TR L , TR U , ID L , ID U , FL L , FL U byusing Equation(28),wehave

Similarly, we can get

Symmetry 2018, 10,459 10of32

Symmetry 2018, 10, x 10 of 32

(29)

 

  

 



           

PWV

    3 1, ,,zzs ssz TinSupinin    and                 112132212333132



        1 1

                2 2



             3 3

 

     

,,,,, LULULU inTRTRIDIDFLFL     by using Equation (28), we have 3 333333 ,1 1213 1213 1213 1212 111 333333 111111 LLLLLL ij ij ij ijLLLLLL ij xy TRTRTRTRTRTR TRTRTRTRTRTR                 3333 2331 2331 2331 1 33 1212 11 3333 1111 LLLL LLLL TRTRTRTR TRTRTRTR                     33 32 32 32 12 10.1281 33 11 LL LL TRTR TRTR               1 3 2 3 ,1 3311 11110.2590 11 33 11 LL ij ij ij ijLL ij TRTR TRTR                                                                           

     



(29)

    ,1,(,1,2,....,). imim SupininDininiml 

Example 1. Let             120.3,0.7,0.2,0.4,0.3,0.5,0.4,0.6,0.1,0.3,

inin  and  

  3





LULULU inTRTRIDIDFLFL 

which is calculated as follows: Step 1. Determine the supports

,,,1,2,3ij Supininij

by using Equation (29), and then we get

122113312332 ,,0.9,,,0.933,,,1. SupininSupininSupininSupininSupininSupin in  Step 2. Determine the

Because

,,1.833,,,1.9,,,1.933,TinSupininSupininTinSupininSupininTinSup ininSupinin

then

123 1 0.3269, 111 Tin TinTinTin 

123 1 0.3346, 111 Tin TinTinTin 

 

123 1 0.3385. 111 Tin TinTinTin



Step 3. Determine the comprehensive value

0.2590,0.5525,0.2221,0.4373,0.2334,0.4365 n

    ,1,(,1,2,....,). imim SupininDininiml 

Example 1. Let

120.3,0.7,0.2,0.4,0.3,0.5,0.4,0.6,0.1,0.3, 0.2,0.4 inin 

3 0.1,0.3,0.4,0.6,0.2,0.4 in 

 

,,,1,2,3ij Supininij 

       122113312332

,,0.9,,,0.933,,,1. SupininSupininSupininSupininSupininSupin in 

ssz

 

                 

        1 1

                 2 2 123 1

                 3 3







,,,,,

   by

3 333333 ,1

ij ij

                33 21 21 21 3333 2331 2331 2331 12 1 33 11 1212 11 3333 1111 LL LL LLLL LLLL TRTR TRTR TRTRTRTR TRTRTRTR                         33 32 32 32 12 10.1281 33 11 LL LL TRTR TRTR               1 3 2 3 ,1 3311 11110.2590 11 33 11 LL ij ij ij ijLL ij TRTR TRTR                                                                            Similarly, we can get      0.2590,0.5525,0.2221,0.4373,0.2334,0.4365 n  . 1/          1 +          32 3 1 + 1 × 1/          3 ∑ i, j = 1 i = j 1/       1 3Λi TR L i 1 TR L i γ + 1 3Λj TR L j 1 TR L j γ                       1 3          = 0.2590

= [

]

] .

= TR L i , TR U i , ID L i , ID U i FL L i , FL L i

Step 2. Determine the PWV Because ID L

1, ,,zzs

 and

112132212333132 ,,1.833,,,1.9,,,1.933,TinSupininSupininTinSupininSupininTinSup ininSupinin 

then

123 1 0.3385. 111 Tin TinTinTin 

 



  Step 3. Determine the comprehensive value

LULULU inTRTRIDIDFLFL 

using Equation (28), we have

1213 1213 1213 1212 111 333333 111111 LLLLLL

ij ijLLLLLL ij

TRTRTRTRTRTR TRTRTRTRTRTR

Similarly,wecanget n

0.2590,0.5525

, [0.2221,0.4373], [0.2334,0.4365

Theorem7. (Idempotency)Let ini

, (i = 1,2, , l),beagroupofINNs,ifall ini (i = 1,2,..., l) areequal,thatis ini = in = TR L , TR U ,

, ID U , FL L , FL U , (i = 1,2,..., l),.Then INDPBMx,y in1, in2,..., inl = in.(30)

Proof. ProofofTheorem7isgiveninAppendix C.

  , be a group of INNs, if all   1,2,..., i inil  are equal, that is

  , 12,,...,. xy l INDPBMinininin  (30) Proof. Proof of Theorem 7 is given in Appendix C □ Theorem 8. (Commutativity) Assume that u in is any permutation of   1,2,...,, u inul  then

1 x+y , and INDPBMx,y in1, in2,..., inl =

 ,, 1212 ,,...,,,...,. xyxy ll INDPBMinininINDPBMininin   (31) Proof. From Definition (14), we have       1 12 2 ,1 1 ,,...,, xy l xy xy eij iDj ij ij INDPBMinininlinlin ll          

     

and

lΛiini x ⊗D lΛjinj y 

     

1 x+y Because, l ∑ i, j = 1 i = j

      1 , 12 2 ,1 1 ,,...,. xy l xy xy lij iDj ij ij

lΛ iin i x ⊗D lΛ jin j y = l ∑ i, j = 1 i = j

,, 1212 ,,...,,,..., xyxy ll INDPBMinininINDPBMininin

         Because,         ,1,1 llxyxy ijij iDjiDj ijij ijij linlinlinlin    , hence,

Now,weshallstudyafewspecialcasesofthe INDPBMx,y withrespectto x and y. (1)When y → 0, γ > 0,thenwecanget Symmetry 2018, 10, x 11 of 32 Theorem 7. (Idempotency) Let ,,,,,(1,2,...,) LULULL iiiiii i inTRTRIDIDFLFLil 

            . Then   12,,...,. l inINDPBMinininin   (32) Proof. Proof of Theorem 9 is given in Appendix D □ Now, we shall study a few special cases of the , xy INDPBM with respect to x and y (1) When 0,0, y   then we can get   ,0 12,,...., x lINDPBMininin

be a group of INNs, and

Symmetry 2018, 10,459 11of32



Theorem8. (Commutativity)Assumethat in u isanypermutationof inu (u = 1,2,..., l), then INDPBMx,y in 1, in 2,..., in l = INDPBMx,y in1, in2,..., inl .(31) Proof. FromDeﬁnition14,wehave INDPBMx,y in 1, in 2,..., in e =

   

     

   □

  ,,,,,,(1,2,...,) LULULL

  1 l2 l l ∑ i, j = 1 i = j    

lΛ iin i x ⊗D lΛ jin j y  111 111



 1 l2 l l ∑ i, j = 1 i = j



lΛiini x ⊗D lΛjinj y , Hence, INDPBMx,y in 1, in 2,..., in l = INDPBMx,y in1, in2,..., inl Theorem9. (Boundedness)Let ini = TR L i , TR U i , ID L i , ID U i , FL L i , FL L i , (i = 1,2, , l) beagroupofINNs,and in + = l max i=1 TR L i , TR U i , l min i=1 ID L i , ID U i , l min i=1 FL L i , FL U i , in = l min i=1 TR L i , TR U i , l max i=1 ID L i , ID U i , l max i=1 FL L i , FL U i ,.Then in ≤ INDPBM in1, in2,..., inl ≤ in + .(32) Proof. ProofofTheorem9isgiveninAppendix D

  ,,,,,,1,2,...,, LULULU i ininTRTRIDIDFLFLil     . Then

INDPBMinininlinlin ll

Theorem 9. (Boundedness) Let

iiiiii i inTRTRIDIDFLFLil

max,,min,,min,,min,,max,,max,, llllll LULULULULULU iiiiiiiiiiii iii iii inTRTRIDIDFLFLinTRTRIDIDFLFL

(37)

Proof. Proof of Theorem 10 is similar to Theorem 6. □

Similar to the INDPBM operator, the INWDPBM operator has the properties of boundedness, idempotency and commutativity.

4.2 The INDPGBM Operator and INWDPGBM Operator

In this subpart, we develop INDPGBM and INWDPGBM operators. Definition 16. Let ,,,,,,(1,2,...,) LULULL iiiiii i inTRTRIDIDFLFLil 

bea group of INNs. Then theINDPGBM operator is defined as

l(T(ini )+1) l ∑ z=1 (T(inz )+1) i + yin

l(T(ini )+1) l ∑ z=1 (T(inz )+1) j



 (38) Then, , xy INDPGBM is said to be an interval neutrosophic Dombi power geometric Bonferroni mean (INDPGBM) operator. Where     1, ,, l zzs ssz TinSupinin 

            



  

InordertosimplifyEquation(38),wecandescribe Λz = 1 + T inz l ∑ z=1 1 + T inz

    , zsSupinin isthesupport degreefor z n from , s n which satisfies the following axioms: (1)    ,0,1zsSupinin  ; (2)     ,,; zssz SupininSupinin  (3)     ,, zsab SupininSupinin  , if     ,, zsab DininDinin  , in which   , abDnn is the distance measure between INNs ain and bin defined in Definition (7). In order to simplify Equation (38), we can describe         1

z z l z z

1 , 1

,(39) andcall Λ = (Λ1, Λ2,..., Λl )T isthepowerweightvector(PWV),suchthat Λz ≥ 0, l ∑ z=1 Λz = 1. ThenEquation(38)canbewrittenasfollows:

    (39) and call   12,,..., T l is the power weight vector (PWV), such that 1 0,1. l zz z

Tin Tin

  Then Equation (38) can be written as follows:

        1 2 ,1 11 1111 ()1()1

LL ij

y llx xy

                                                                2 ,1 11 ,1111 ()1()1 (1)(1)11 UU ij ijii ij ij lUlU zzzzij zz llx xy lwTinlwTinTRTR wTinwTinTRTR                                                            2 11 11111 ()1()111 (1)(1) LL ijiiij lLlL ij y llx xy lwTinlwTinIDID wTinwTinIDID                                                                              1 2 ,1 11 ,1111 ()1()1 1 (1)(1) U ij ijii ij lUl llx xy lwTinlwTin ID ID wTinwTin                                                                                                      ,1 2 11111 ()1 ( ij ij llx xy lwTin wTin                                                                                 1 2 ,1 11 ,11 ()1 11 1)(1) LL ij ij ij lLlL ij y ll x lwTin FLFL FLFL wTin                                                                                                       ,1 11 11 ()1()111 (1)(1) UU ij ijii ij ij lUlU ij x y lwTinlwTinFLFL wTinwTinFLFL                                                                              

Symmetry 2018, 10,459 14of32 Symmetry 2018, 10, x 14 of 32

(1)(1)11

ijii

ij lLlL zzzzij zz

lwTinlwTinTRTR wTinwTinTRTR

(37)

 

          2 11

zz zz

  





1 ()1()1 ()1()1

,1 1 ,,...,. ii ll

ll lTinlTin l TinTin xy l ij ij ij INDPGBMinininxinyin xy    

  

        



       l ∏ i

    

Proof. ProofofTheorem10issimilartoTheorem6. SimilartotheINDPBMoperator,theINWDPBMoperatorhasthepropertiesofboundedness, idempotencyandcommutativity. 4.2.TheINDPGBMOperatorandINWDPGBMOperator Inthissubpart,wedevelopINDPGBMandINWDPGBMoperators. Deﬁnition16. Let ini = TR L i , TR U i , ID L i , ID U i , FL L i , FL L i , (i = 1,2, , l) beagroupofINNs. ThentheINDPGBMoperatorisdeﬁnedas INDPGBMx,y in1, in2,..., inl = 1 x + y

, j = 1 i = j

xin

1 l2 l .(38)

Then, INDPGBMx,y issaidtobeanintervalneutrosophicDombipowergeometricBonferronimean (INDPGBM)operator.Where T inz = l ∑ s=1,s=z Sup inz, ins , Sup inz, ins isthesupportdegreefor nz from ns, whichsatisfiesthefollowingaxioms:(1) Sup inz, ins ∈ [0,1];(2) Sup inz, ins = Sup ins, inz ; (3) Sup inz, ins ≥ Sup ina, inb ,if D inz, ins < D ina, inb ,inwhich D(na, nb ) isthedistance measurebetweenINNs ina and inb definedinDefinition7.

(41)









 1 2

l U ij ij j j U j

y TR l TR

, 1

                                                                                       

then

Symmetry 2018, 10,459 15of32 INDPGBMx,y in1, in2,..., inl = 1 x + y         l ∏ i, j = 1 i = j xin lΛi i + yin lΛj j         1 l2 l

Theorem11. Let ini = TR L i , TR U i , ID L i , ID U i , FL L i , FL L i , (i = 1,2, , l) beagroupofINNs. ThentheresultobtainedfromEquation(38)isexpressedas Symmetry 2018, 10, x 15 of 32   2 1 , 12 ,1 1 ,,...,. ij ll l ll xy lij ij ij INDPGBMinininxinyin xy                  

Theorem 11. Let ,,,,,,(1,2,...,) LULULL iiiiii i inTRTRIDIDFLFLil     be

obtained from Equation (38) is expressed as   12

1 22 ,1 11111,1111 111 LLU ij ij iji iji LLU iji llxyllx xyxyTRTRTR lll TRTRTR                             

1111 11

LL ij ijLL ij

llxy xy IDID ll IDID





   

,1,1 ,1111 11 ll UU ijij ijij ij ijUU ij llxy xy IDID ll IDID                                                 

2 ,1

1111 11 LL ij ij ij ijLL ij llxy xy FLFL ll FLFL

  



                            1 2 ,1 ,1111 11 UU ij ij ij iiUU ij llxy xy FLFL ll FLFL                                                                                       

Theorem 12. (Idempotency) Let ,,,,,,(1,2,...,) LULULL iiiiii i inTRTRIDIDFLFLil     be a group of INNs, if all   1,2,..., i inil  are

  

.(40)  

(40) 13. (Commutativity) Assume that u in is any permutation of   1,2,...,, u inul  then     ,, 1212 ,,...,,,..., xyxy ll INDPGBMinininINDPGBMininin  

a group of INNs. Then the result Theorem 14. (Boundedness) Let ,,,,,,(1,2,...,) LULULL iiiiii i inTRTRIDIDFLFLil     be a group of INNs, and 111 111 max,,min,,min,,min,,max,,max,, llllll LULULULULULU iiiiiiiiiiii iii iii inTRTRIDIDFLFLinTRTRIDIDFLFL     

,,....,   12,,...,. l inINDPGBMinininin   (44) Definition 17. Let ,,,,,,(1,2,...,) LULULL iiiiii i inTRTRIDIDFLFLil    

lINDPGBMininin

                                                   



1 ,1

  

                                   1 1

 

                                                              

 

equal, that is   ,,,,,,1,2,...,, LULULU i ininTRTRIDIDFLFLil 

, 12,,..., xy l INDPGBMinininin  . (42) Theorem

(43)

then

INWDPGBM operator is defined as

be a group of INNs, then the

Theorem12. (Idempotency)Let ini = TR L i , TR U i , ID L i , ID U i , FL L i , FL L i , (i = 1,2, , l) beagroupofINNs,ifall ini (i = 1,2,..., l) areequal,thatis ini = in = TR L , TR U , ID L , ID U , FL L , FL U , (i = 1,2,..., l), then

INDPGBMx,y in1, in2,..., inl = in.(42)

Theorem13. (Commutativity)Assumethat in u isanypermutationof inu (u = 1,2,..., l), then INDPGBMx,y in 1, in 2,..., in l = INDPGBMx,y in1, in2,..., inl .(43)

Symmetry 2018, 10,459

Theorem14. (Boundedness)Let ini = TR L i , TR U i , ID L i , ID U i , FL L i , FL L i , (i = 1,2, , l) beagroupofINNs,and in + = l max i=1 TR L i , TR U i , l min i=1 ID L i , ID U i , l min i=1 FL L i , FL U i , in = l min i=1 TR L i , TR U i , l max i=1 ID L i , ID U i , l max i=1 FL L i , FL U i , then in ≤ INDPGBM in1, in2,..., inl ≤ in + .(44) Deﬁnition17. Let ini = TR L i , TR U i , ID L i , ID U i , FL L i , FL L i , (i = 1,2, , l) beagroupofINNs, thentheINWDPGBMoperatorisdeﬁnedas INWDPGBM

Theorem15. Let ini = TR L i , TR U i ,

(45)

5. MADM Approach Based on the Developed Aggregation Operator

In this section, based upon the developed INWDPBM and INWDPGBM operators, we will propose a novel MADM method, which is defined as follows. Assume that in a MADM problem, we need to evaluate

alternatives

with respect to

attributes

and the importance degree of the attributes is represented by

16of32

l = 1 x + y         l ∏ i, j =

i = j       xin lwi (T(ini )+1) l ∑ z 1 wz (

) i

lwj (T(ini )+1) l ∑ z 1 wz (T(inz )+1) j               1 l2 l

,y in1, in2,..., in

T(inz )+1

+ yin

L i

ID U i , FL L i

FL L i

Symmetry 2018, 10, x 16 of 32           2 11 1 ()1()1 ()1()1 , 12 ,1 1 ,,...,. ijii ll zzzz zz ll lwTinlwTin l wTinwTin xy l ij ij ij INWDPGBMinininxinyin xy                            (45) Theorem 15. Let ,,,,,,(1,2,...,) LULULL iiiiii i inTRTRIDIDFLFLil     be a group of INNs. Then the aggregated result

  12,,...., xy lINWDPGBMininin 1                     1                               (46)

, (i = 1,2, , l) beagroupofINNs. ThentheaggregatedresultfromEquation(45)isexpressedas

from Equation (45) is expressed as

  12,,..., u



  12



  12,,..., T v   ,

  1 0,1,1. v hh h  

gh mn Dd    

,,,, LULULU ghghghghghgh gh dTRTRIDIDFLFL     is an INN

the

g M

h C   1,2,...,;1,2,...,. guhv 

5.MADMApproachBasedontheDevelopedAggregationOperator Inthissection,baseduponthedevelopedINWDPBMandINWDPGBMoperators,wewill

AssumethatinaMADMproblem,weneedtoevaluate u alternatives M = M1, M2,..., Mu

attributes C = C1, C2,..., Cv

MMMM

,,...,

CCCC

satisfying the condition

The decision matrix for this decision problem is denoted by

, where

for

alternative

with respect to the attribute

(46)

proposeanovelMADMmethod,whichisdeﬁnedasfollows.

withrespectto v

,andtheimportancedegreeoftheattributes

isrepresentedby = ( 1, 2,..., v )T ,satisfyingthecondition h ∈ [0,1], v ∑ h=1 h = 1

Thedecisionmatrixforthisdecisionproblemisdenotedby D = dgh m×n ,where dgh = TR L gh, TR U gh , ID L gh, ID U gh , FL L gh FL U gh isanINNforthealternative Mg withrespecttothe

attribute Ch, (g = 1,2,..., u; h = 1,2,..., v) Thenthemainpurposeistorankthealternativeand selectthebestalternative.

Inthefollowing,wewillusetheproposedINWDPBMandINWDPGBMoperatorstosolvethis MADMproblem,andthedetaileddecisionstepsareshownasfollows:

Step1. Standardizetheattributevalues.Normally,inrealproblems,theattributesareoftwotypes, (1)costtype,(2)beneﬁttype.Togetrightresult,itisnecessarytochangecosttypeofattribute valuestobeneﬁttypeusingthefollowingformula:

dgh = FL L gh, FL U gh , 1 ID U gh,1 ID L gh , TR L gh, TR U gh .(47)

Step2. Calculatethesupports

Supp dgh, dgl = 1 D dgh, dgl , (g = 1,2,..., u; h, l = 1,2,..., v),(48) where, D dgh, dgl isthedistancemeasuredeﬁnedinEquation(10).

Step3. Calculate T dgh

T dgh = u ∑ l = 1 l = h

Supp dgh, dgl , (g = 1,2,..., u; h, l = 1,2,..., v).(49)

Step4. Aggregatealltheattributevalues dgh (h = 1,2, , v) tothecomprehensivevalue Rg by usingINWDPBMorINWDPGBMoperatorsshownasfollows.

Rg = INWDPBM dg1, dg2,..., dgv ;(50) or Rg = INWDPGBM dg1, dg2,..., dgv .(51)

Step5. Determinethescorevalues,accuracyvaluesof Rg (g = 1,2,..., u),usingDeﬁnition5.

Step6. Rankallthealternativesaccordingtotheirscoreandaccuracyvalues,andselectthebest alternativeusingDeﬁnition6.

Step7. End.

ThisdecisionstepsarealsodescribedinFigure 1

Symmetry 2018, 10,459 17of32

Step 5. Determine the score values, accuracy values of using Definition (5).

Step 6. Rank all the alternatives according to their score and accuracy values, and select the best alternative using Definition (6). Step 7. End. This decision steps are also described in Figure 1 Figure 1. Flow chart for developed approach. Figure1. Flowchartfordevelopedapproach.

6.IllustrativeExample

Inthispart,anexampleadaptedfrom[42]isusedtoillustratetheapplicationandeffectivenessof thedevelopedmethodinMADMproblem.

Aninvestmentcompanywantstoinvestasumofmoneyinthebestoption.Thecompany mustinvestasumofmoneyinthefollowingfourpossiblecompanies(alternatives):(1)carcompany M1;(2)foodcompany M2;(3)Computercompany M3;(4)Anarmcompany M4,andtheattributes underconsiderationare(1)riskanalysis C1;(2)growthanalysis C2;(3)environmentalimpactanalysis C3.Theimportancedegreeoftheattributesis = (0.35,0.4,0.25)T .Thefourpossiblealternatives Mg (g = 1,2,3,4) areevaluatedwithrespecttotheaboveattributes Ch (h = 1,2,3) bytheformofINN, andtheINdecisionmatrix D islistedinTable 1.Thepurposeofthisdecision-makingproblemisto rankthealternatives.

Table1. TheINdecisionmatrix D.

Alternatives/Attributes C1 C2 C3

M1 [0.4,0.5], [0.2,0.3], [0.3,0.4] [0.4,0.6], [0.1,0.3], [0.2,0.4] [0.7,0.9], [0.7,0.8], [0.4,0.5] M2 [0.6,0.8], [0.1,0.2], [0.1,0.2] [0.6,0.7], [0.15,0.25], [0.2,0.3] [0.3,0.6], [0.2,0.3], [0.8,0.9] M3 [0.3,0.6], [0.2,0.3], [0.3,0.4] [0.5,0.6], [0.2,0.3], [0.3,0.4] [0.4,0.5], [0.2,0.4], [0.7,0.9] M4 [0.7,0.8], [0.01,0.1], [0.2,0.3] [0.6,0.7], [0.1,0.2], [0.3,0.4] [0.4,0.6], [0.5,0.6], [0.8,0.9]

6.1.TheDecision-MakingSteps

Step1. Since C1, C2 areofbeneﬁttype,and C3 isofcosttype.So, C3 willbechangedintobeneﬁt typeusingEquation(47).So,thenormalizedecisionmatrix D isgiveninTable 2

   

  

 

     

l lh

  

Symmetry 2018, 10,459 18of32

,1,,(1,2,...,;,1,2,...,), ghglghgl SuppddDddguhlv

, ghglDdd

gh Td

1 ,,1,2,...,;,1,2,...,.

ghghgl

TdSuppddguhlv

g R

6.1 The Decision-Making

Steps

Alternatives/Attributes

Step 1. Since 1C

Table2. TheNormalizeINdecisionmatrix D.

is given in Table 2.

2 M 3 M 4 M 1C 2C 3C  0.35,0.4,0.25 T   (1,2,3,4) g Mg   1,2,3 h Ch 

Alternatives/Attributes C1 C2 C3

M1 [0.4,0.5], [0.2,0.3], [0.3,0.4] [0.4,0.6], [0.1,0.3], [0.2,0.4] [0.4,0.5], [0.2,0.3], [0.7,0.9] M2 [0.6,0.8], [0.1,0.2], [0.1,0.2] [0.6,0.7], [0.15,0.25], [0.2,0.3] [0.8,0.9], [0.6,0.7], [0.3,0.6] M3 [0.3,0.6], [0.2,0.3], [0.3,0.4] [0.5,0.6], [0.2,0.3], [0.3,0.4] [0.7,0.9], [0.6,0.8], [0.4,0.5] M4 [0.7,0.8], [0.01,0.1], [0.2,0.3] [0.6,0.7], [0.1,0.2], [0.3,0.4] [0.8,0.9], [0.4,0.5], [0.4,0.6]

D D

Step2. Determinethesupports Supp dgh, dgl , (g = 1,2,3,4; h, l = 1,2,3) byEquation(48)(for simplicitywedenote Supp dgh, dgl with Sg gh,gl ),wehave

, 2C

are of benefit type, and 3C

C1 C2 C3

1M      0.4,0.5,0.2,0.3,0.3,0.4      0.4,0.6,0.1,0.3,0.2,0.4      0.7,0.9,0.7,0.8,0.4,0.5

2 M      0.6,0.8,0.1,0.2,0.1,0.2      0.6,0.7,0.15,0.25,0.2,0.3      0.3,0.6,0.2,0.3,0.8,0.9

is of cost type. So, 3C

will be changed into benefit type using Equation (47). So, the normalize decision matrix D

3 M      0.3,0.6,0.2,0.3,0.3,0.4      0.5,0.6,0.2,0.3,0.3,0.4      0.4,0.5,0.2,0.4,0.7,0.9

4 M      0.7,0.8,0.01,0.1,0.2,0.3      0.6,0.7,0.1,0.2,0.3,0.4      0.4,0.6,0.5,0.6,0.8,0.9

Step3. Determine T dgh ; (g = 1,2,3,4; h = 1,2,3) byEquation(49),andweget

Table 2. The Normalize IN decision matrix D

Step 4. (a) Determine the comprehensive value of every alternative using the INWDPBM operator, that is, Equation (50)

R1 = [0.3974,0.5195], [0.1823,0.3023], [0.3353,0.4796] ; R2 = [0.6457,0.7954], [0.1700,0.2885], [0.2044,0.3265] ;

C1 C2 C3

1M      0.4,0.5,0.2,0.3,0.3,0.4      0.4,0.6,0.1,0.3,0.2,0.4      0.4,0.5,0.2,0.3,0.7,0.9

R3 = [0.4846,0.6503], [0.2556,0.3711], [0.3376,0.4394] ;

by Equation (48) (for simplicity we denote   , ghglSuppdd

with , g ghglS

2 M      0.6,0.8,0.1,0.2,0.1,0.2      0.6,0.7,0.15,0.25,0.2,0.3      0.8,0.9,0.6,0.7,0.3,0.6 3 M      0.3,0.6,0.2,0.3,0.3,0.4      0.5,0.6,0.2,0.3,0.3,0.4      0.7,0.9,0.6,0.8,0.4,0.5 4 M      0.7,0.8,0.01,0.1,0.2,0.3      0.6,0.7,0.1,0.2,0.3,0.4      0.8,0.9,0.4,0.5,0.4,0.6

), we have 111111222222 11,1212,1112,1313,1211,1313,1111,1212,11 333333 11,1212,1112,1313,1211,1313,1111 



0.950;0.800;0.85;0.933;0.717;0.683; 0.967;0.733;0.700; SSSSSSSSSSSS SSSSSSS

R4 = [0.6938,0.7953], [0.1062,0.2154], [0.3069,0.4278]

(b) DeterminethecomprehensivevalueofeveryalternativeusingtheINWDPGBM operator,thatisEquation(51), (Assumethatx = y = 1; γ = 3),wehave

by Equation (49), and we get 111222 111213111213 333444 111213111213



R1 = [0.4026,0.5381], [0.1570,0.2977], [0.2998,0.4520] ;

R2 = [0.6654,0.8193], [0.1558,0.2686], [0.1836,0.3035] ;

R3 = [0.5159,0.6732], [0.2366,0.3473], [0.3265,0.4279] ; R4 = [0.5159,0.8193], [0.0938,0.1952], [0.2862,0.4037]

Step5. (a) Determinethescorevaluesof Rg (g = 1,2,3,4) byDeﬁnition5,wehave

1;3 Assumethatxy 

S(R1)= 1.8087, S(R2)= 2.2259, S(R3)= 1.8656, S(R4)= 2.2164;

(b) Determinethescorevaluesof Rg (g = 1,2,3,4) byDeﬁnition5,wehave

1 0.3974,0.5195,0.1823,0.3023,0.3353,0.4796; R

S(R1)= 1.8671, S(R2)= 2.2866, S(R3)= 1.9254, S(R4)= 2.1781;

Step6. (a) Accordingtotheirscoreandaccuracyvalues,byusingDeﬁnition6,therankingorder is M2 > M4 > M3 > M1. Sothebestalternativeis M2, whiletheworstalternative is M1.

Symmetry 2018, 10,459 19of32

Step 2. Determine the supports   ,,(1,2,3,4;,1,2,3)ghgl Suppddghl

444444 ,1212,1112,1313,1211,1313,110.902;0.783;0.752;SSSSS 

 



Step 3. Determine      

        

;(1,2,3,4;1,2,3) gh Tdgh

1.800,1.750,1.650,1.617,1.650,1.400, 1.667,1.700,1.433,1.653,1.685,1.535. TTTTTT TTTTTT

, we have

2 0.6457,0.7954,0.1700,0.2885,0.2044,0.3265; R 

T1 11 = 1.800, T1 12 = 1.750, T1 13 = 1.650, T2 11 = 1.617, T2 12 = 1.650, T2 13 = 1.400, T3 11 = 1.667, T3 12 = 1.700, T3 13 = 1.433, T4 11 = 1.653, T4 12 = 1.685, T4 13 = 1.535.

Step4. (a) DeterminethecomprehensivevalueofeveryalternativeusingtheINWDPBM operator,thatis,Equation(50) (Assumethatx = y = 1; γ = 3),wehave

(b) Accordingtotheirscoreandaccuracyvalues,byusingDeﬁnition6,therankingorder is M2 > M4 > M3 > M1. Sothebestalternativeis M2, whiletheworstalternative is M1.

So,byusingINWDPBMorINWDPGBMoperators,thebestalternativeis M2, whiletheworst alternativeis M1.

6.2.EffectofParameters γ,xandyonRankingResultofthisExample

Inordertoshowtheeffectoftheparameters x and y ontherankingresultofthisexample,weset differentparametervaluesfor x and y,and γ = 3 isﬁxed,toshowtherankingresultsofthisexample. TherankingresultsaregiveninTable 3.

AsweknowfromTables 3 and 4,thescorevaluesandrankingorderaredifferentfordifferent valuesoftheparameters x and y,whenweuseINWDPBMoperatorandINWDPGBMoperator. WecanseefromTables 3 and 4,whentheparametervalues x =1or0and y =0or1,thebestchoiceis M4 andtheworstoneis M1. Insimplewords,whentheinterrelationshipamongattributesarenot considered,thebestchoiceis M4 andtheworstoneis M1. Ontheotherhand,whendifferentvalues fortheparameters x and y areutilized,forINWPBMandINWDPGBMoperators,therankingresultis changed.Thatis,fromTable 4,wecanseethatwhentheparametervalues x = 1, y = 1,theranking resultsarechangedastheoneobtainedfor x =1or0and y =0or1.Inthiscasethebestalternativeis M2 whiletheworstalternativeremainsthesame.

Table3. Rankingordersofdecisionresultusingdifferentvaluesfor x and y forINWDPBM.

ParameterValuesINWDPBMOperatorRankingOrders

x = 1, y = 0, γ = 3

x = 1, y = 5, γ = 3

S(R1)= 1.9319, S(R2)= 2.4172, S(R3)= 2.0936, S(R4)= 2.4222; M4 > M2 > M3 > M1

S(R1)= 1.8338, S(R2)= 2.2684, S(R3)= 1.9049, S(R4)= 2.2666; M2 > M4 > M3 > M1

x = 3, y = 7, γ = 3 S(R1)= 1.8169, S(R2)= 2.2398, S(R3)= 1.8777, S(R4)= 2.2327; M2 > M4 > M3 > M1

x = 5, y = 10, γ = 3 S(R1)= 1.8143, S(R2)= 2.2354, S(R3)= 1.8738, S(R4)= 2.2275; M4 > M2 > M3 > M1 x = 1, y = 10, γ = 3 S(R1)= 1.8501, S(R2)= 2.2966, S(R3)= 1.9355, S(R4)= 2.3012; M4 > M2 > M3 > M1

x = 10, y = 4, γ = 3 S(R1)= 1.8182, S(R2)= 2.2419, S(R3)= 1.8796, S(R4)= 2.2352; M4 > M2 > M3 > M1

x = 3, y = 12, γ = 3 S(R1)= 1.8285, S(R2)= 2.2592, S(R3)= 1.8958, S(R4)= 2.2557; M2 > M4 > M3 > M1

Table4. Rankingordersofdecisionresultusingdifferentvaluesfor x and y forINWDPGBM.

ParameterValuesINWDPGBMOperatorRankingOrders

x = 1, y = 0, γ = 3

S(R1)= 1.5032, S(R2)= 1.7934, S(R3)= 1.5136, S(R4)= 1.8037; M4 > M2 > M3 > M1

x = 1, y = 5, γ = 3 S(R1)= 1.8220, S(R2)= 2.2256, S(R3)= 1.8717, S(R4)= 2.1140; M2 > M4 > M3 > M1

x = 3, y = 7, γ = 3 S(R1)= 1.8539, S(R2)= 2.2686, S(R3)= 1.9094, S(R4)= 2.1584; M2 > M4 > M3 > M1

x = 5, y = 10, γ = 3 S(R1)= 1.8583, S(R2)= 2.2745, S(R3)= 1.9146, S(R4)= 2.1647; M2 > M4 > M3 > M1

x = 1, y = 10, γ = 3 S(R1)= 1.7814, S(R2)= 2.1710, S(R3)= 1.8248, S(R4)= 2.0632; M2 > M4 > M3 > M1

x = 10, y = 4, γ = 3

x = 3, y = 12, γ = 3

S(R1)= 1.8087, S(R2)= 2.2259, S(R3)= 1.8656, S(R4)= 2.2164; M2 > M4 > M3 > M1

S(R1)= 1.8671, S(R2)= 2.2866, S(R3)= 1.9254, S(R4)= 2.1781; M4 > M2 > M3 > M1

Symmetry 2018, 10,459 20of32

FromTables 3 and 4,wecanobservethatwhenthevaluesoftheparameterincrease,thescore valuesobtainedusingINWDPBMdecrease.WhileusingtheINWDPGBMoperator,thescorevalues increasebutthebestchoiceis M2 for x = y ≥ 1.

FromTable 5,wecanseethatdifferentrankingordersareobtainedfordifferentvaluesof γ.When γ = 0.5 and γ = 2,thebestchoiceis M4 bytheINWPBMoperator;whenweusetheINWPGBM operator,itis M2.Similarly,forothervaluesof γ > 2,thebestchoiceis M2 whiletheworstis M1

Table5. Rankingordersofdecisionresultusingdifferentvaluesfor γ

ParameterValuesINWDPBMOperatorINWDPGBMOperatorRankingOrders

x = 1, y = 1, γ = 0.5

x = 1, y = 1, γ = 2

x = 1, y = 1, γ = 4

x = 1, y = 1, γ = 7

x = 1, y = 1, γ = 10

x = 1, y = 1, γ = 15

x = 1, y = 1, γ = 20

S(R1)= 1.6662, S(R2)= 2.1025, S(R3)= 1.7606, S(R4)= 2.1972;

S(R1)= 1.7783, S(R2)= 2.2015, S(R3)= 1.8408, S(R4)= 2.2091;

S(R1)= 1.8229, S(R2)= 2.2363, S(R3)= 1.8803, S(R4)= 2.2219;

S(R1)= 1.8375, S(R2)= 2.2455, S(R3)= 1.9037, S(R4)= 2.2315;

S(R1)= 1.8418, S(R2)= 2.2477, S(R3)= 1.9160, S(R4)= 2.2365;

S(R1)= 1.8447, S(R2)= 2.2488, S(R3)= 1.9270, S(R4)= 2.2409;

S(R1)= 1.8460, S(R2)= 2.2492, S(R3)= 1.9328, S(R4)= 2.2432;

6.3.ComparingwiththeOtherMethods

S(R1)= 1.7870, S(R2)= 2.2347, S(R3)= 1.9103, S(R4)= 2.1812; M4 > M2 > M3 > M1 M2 > M4 > M3 > M1

S(R1)= 1.8491, S(R2)= 2.2786, S(R3)= 1.9213, S(R4)= 2.1799; M4 > M2 > M3 > M1 M2 > M4 > M3 > M1

S(R1)= 1.8740, S(R2)= 2.2856, S(R3)= 1.9275, S(R4)= 2.1751; M2 > M4 > M3 > M1 M2 > M4 > M3 > M1

S(R1)= 1.8747, S(R2)= 2.2763, S(R3)= 1.9331, S(R4)= 2.1669; M2 > M4 > M3 > M1 M2 > M4 > M3 > M1

S(R1)= 1.8701, S(R2)= 2.2698, S(R3)= 1.9373, S(R4)= 2.1622; M2 > M4 > M3 > M1 M2 > M4 > M3 > M1

S(R1)= 1.8642, S(R2)= 2.2637, S(R3)= 1.9414, S(R4)= 2.1582; M2 > M4 > M3 > M1 M2 > M4 > M3 > M1

S(R1)= 1.8608, S(R2)= 2.2604, S(R3)= 1.9435, S(R4)= 2.1562; M2 > M4 > M3 > M1 M2 > M4 > M3 > M1

Toillustratetheadvantagesandeffectivenessofthedevelopedmethodinthisarticle,wesolve theaboveexamplebyfourexistingMADMmethods,includingINweightedaveragingoperator,IN weightedgeometricoperator[12],thesimilaritymeasuredeﬁnedbyYe[15],Muirheadmeanoperators developedbyLiuetal.[42],INpoweraggregationoperatordevelopedbyLiuetal.[37].

FromTable 6,wecanseethattherankingordersarethesameastheonesproducedbytheexisting aggregationoperatorswhentheparametervalues x = 1, y = 0, γ = 3,buttherankingordersare differentwhentheinterrelationshipamongattributesareconsidered.Thatiswhythedeveloped methodbasedontheproposedaggregationoperatorsismoreﬂexibleduetheparameterandpractical asitcanconsidertheinterrelationshipamonginputarguments.

Table6. Rankingorderofthealternativesusingdifferentaggregationoperators.

AggregationOperatorParameterScoreValuesRankingOrder

INWAoperator[12]No

S(R1)= 1.8430, S(R2)= 2.2497, S(R3)= 1.9151, S(R4)= 2.2788; M4 > M2 > M3 > M1

INWGAoperator[12]No S(R1)= 1.7286, S(R2)= 2.0991, S(R3)= 1.7751, S(R4)= 2.1608; M4 > M2 > M3 > M1

Similaritymeasure Hammingdistance[15] No D1(R∗ , R1)= 0.7948, D1(R∗ , R2)= 0.9581, D1(R∗ , R3)= 0.8805, D1(R∗ , R4)= 0.9725; M4 > M2 > M3 > M1

Generalizedpower Aggregationoperator[37] Yes λ = 1 S(R1)= 1.8460, S(R2)= 2.2543, S(R3)= 1.9163,S(R4)= 2.2799; M4 > M2 > M3 > M1

INWMMoperator[42] Yes P(1,1,1) S(R1)= 1.8054, S(R2)= 2.2321, S(R3)= 1.9172,S(R4)= 2.2773; M4 > M2 > M3 > M1

INWDMMoperator[42] Yes P(1,1,1) S(R1)= 1.6260, S(R2)= 1.9202, S(R3)= 1.7061,S(R4)= 2.0798; M4 > M2 > M3 > M1

ProposedINWDPBM

x = 1, y = 0, γ = 3 Yes S(R1)= 1.9319, S(R2)= 2.4172, S(R3)= 2.0936, S(R4)= 2.4222; M4 > M2 > M3 > M1 ProposedINWDPGBM

x = 1, y = 0, γ = 3 Yes S(R1)= 1.5032, S(R2)= 1.7934, S(R3)= 1.5136, S(R4)= 1.8037; M4 > M2 > M3 > M1

INWDPBMoperatorinthis article Yes x = y = 1, γ = 3

S(R1)= 1.8087, S(R2)= 2.2259, S(R3)= 1.8656, S(R4)= 2.2164; M2 > M4 > M3 > M1

INWDPBMoperatorinthis article Yes x = y = 1, γ = 3 S(R1)= 1.8671, S(R2)= 2.2866, S(R3)= 1.9254, S(R4)= 2.1781; M2 > M4 > M3 > M1

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Fromtheabovecomparativeanalysis,wecanknowtheproposedmethodhasthefollowing advantages,thatis,itcanconsidertheinterrelationshipamongtheinputargumentsandcanrelieve theeffectoftheawkwarddatabyPWVatthesametime,anditcanpermitmorepreciserankingorder thantheexistingmethods.TheproposedmethodcantaketheadvantagesofPAoperatorandBM operatorconcurrently,thesefactorsmakesitalittlecomplexincalculations.

ThescorevaluesandrankingordersbythesemethodsareshowninTable 6

7.Conclusions

ThePBMoperatorcantaketheadvantageofPAoperator,whichcaneliminatetheimpactof awkwarddatagivenbythepredisposedDMs,andBMoperator,whichcanconsiderthecorrelation betweentwoattributes.TheDombioperationsofTNandTCNproposedbyDombihavetheedge ofgoodflexibilitywithgeneralparameter.Inthisarticle,wecombinedPBMwithDombioperation andproposedsomeaggregationoperatorstoaggregateINNs.Firstly,wedefinedsomeoperational lawsforINSsbasedonDombiTNandTCNanddiscussedsomepropertiesoftheseoperations. Secondly,weextendedPBMoperatorbasedonDombioperationstointroduceINDPBMoperator, INWDPBMoperator,INDPGBMoperator,INWDPGBMoperatoranddiscussedsomepropertiesof theseaggregationoperators.Thedevelopedaggregationoperatorshavetheedgethattheycantake thecorrelationamongtheattributesbyBMoperator,andcanalsoremovetheeffectofawkwarddata byPAoperatoratthesameandduetogeneralparameter,sotheyaremoreflexibleintheaggregation process.Further,wedevelopedanovelMADMmethodbasedondevelopedaggregationoperators todealwithintervalneutrosophicinformation.Finally,anillustrativeexampleisusedtoshowthe effectivenessandpracticalityoftheproposedMADMmethodandcomparisonweremadewiththe existingmethods.TheproposedaggregationoperatorsareveryusefultosolveMADMproblems.

Infutureresearch,weshalldeﬁnesomedistinctaggregationoperatorsforSVHFSs,INHFSs, doublevaluedneutrosophicsetsandsoonbasedonDombioperationsandapplythemtoMAGDM andMADM.

AuthorContributions: Q.K.andP.L.visualizedandworktogethertoachievethiswork.Q.K.andP.L.wrotethe paper.T.M.,F.S.andK.U.offeredplentyofadviceenhancingthereadabilityofwork.Allauthorsapprovedthe publicationwork.

Funding: NationalNaturalScienceFoundationofChina:71771140,71471172,SpecialFundsofTaishanScholars ProjectofShandongProvince:ts201511045.

Acknowledgments: We,theauthorsofthisprojectaredeeplygratefulforthevaluablesuggestionsandcomments providedbytherespectedreviewerstoimprovethequalityandvalueofthisresearch.

Nopotentialconﬂictofinterestwasreportedbyauthors.

Symmetry 2018, 10,459 22of32

AppendixA.BasicConceptofPBMOperator DeﬁnitionA1. Forpositiverealnumbers ℘h (h = 1,2,..., l) andx, y > 0 theaggregationmapping[54] PBMx,y (℘1, ℘2,..., ℘l ) =          1 l2 l l ∑ i = 1, j = 1 i = j           l(T(℘i ) + 1) l ∑ o=1 (T(℘o ) + 1) ℘i      x ×      l(T(℘i ) + 1) l ∑ o=1 (T(℘o ) + 1) ℘j      y               1 x+y (A1) issaidtobepowerBonferronimean(PBM)meanoperator.

ConﬂictsofInterest:

ThereforeaccordingtotheDeﬁnition6,wehave in ≤ INDPBM in1, in2,..., inl

Inasimilarway,theotherpartcanbeproved.Thatis in ≤ INDPBM in1, in2,..., inm ≤ in + .Hence in ≤ INDPBM in1, in2,..., inm ≤ in + References

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Therefore according to the Definition (6), we have   12,,...,. linINDPBMininin 

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Symmetry 2018, 10,459 30of32

          1 22 ,1 1111111111 111 L i LLL ij ij iji iji LLL iji llxyllx FLin xyxyFLFLFL lll FLFLFL                                                                                     1 ,1 1 L L ij ij L j y FL FL l FL                                                                                      1 22 ,1 1111111111 1111 l U UUU ij iji ij ii UUjj U ii llxyllxy in xyxy llll                                                                                 1 ,1 U U ij ij U j                                                     



  

Thentherearethefollowingscores

                       Then there are the following scores

222 LULULU TRTRIDIDFLFL 

11. 222 LULULU TRTRIDIDFLFL

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