Neutrosophic Cubic Einstein Geometric Aggregation Operators

NeutrosophicCubicEinsteinGeometricAggregation OperatorswithApplicationtoMulti-Criteria DecisionMakingMethod

MajidKhan 1,MuhammadGulistan 1,* ,NaveedYaqoob 2,MadadKhan 3 and FlorentinSmarandache 4

1

DepartmentofMathematicsandStatistics,HazaraUniversity,Mansehra21130,Pakistan; majid_swati@yahoo.com

2 DepartmentofMathematics,CollegeofScienceAl-Zulfi,MajmaahUniversity,Al-Zulfi11932,SaudiArabia; na.yaqoob@mu.edu.saornayaqoob@ymail.com

3 DepartmentofMathematics,COMSATSUniversityIaslamabad,AbbottabadCampus, Abbottabad22060,Pakistan;madadmath@yahoo.com

4 DepartmentofMathematics,UniversityofNewMexico,Albuquerque,NM87301,USA; fsmarandache@gmail.com

* Correspondence:gulistanmath@hu.edu.pkorgulistanm21@yahoo.com

Received:26December2018;Accepted:31January2019;Published:16February2019

Abstract: Neutrosophiccubicsets(NCs)areamoregeneralizedversionofneutrosophicsets(Ns) andintervalneutrosophicsets(INs).Neutrosophiccubicsetsarebetterplacedtoexpressconsistent, indeterminateandinconsistentinformation,whichprovidesabetterplatformtodealwithincomplete, inconsistentandvaguedata.Aggregationoperatorsplayakeyroleindailylife,andinrelation toscienceandengineeringproblems.InthispaperwedefinedthealgebraicandEinsteinsum, multiplicationandscalarmultiplication,scoreandaccuracyfunctions.Usingtheseoperations wedefinedgeometricaggregationoperatorsandEinsteingeometricaggregationoperators.First, wedefinedthealgebraicandEinsteinoperatorsofaddition,multiplicationandscalarmultiplication. Wedefinedscoreandaccuracyfunctiontocompareneutrosophiccubicvalues.Thenwedefinedthe neutrosophiccubicweightedgeometricoperator(NCWG),neutrosophiccubicorderedweighted geometricoperator(NCOWG),neutrosophiccubicEinsteinweightedgeometricoperator(NCEWG), andneutrosophiccubicEinsteinorderedweightedgeometricoperator(NCEOWG)overneutrosophic cubicsets.Amulti-criteriadecisionmakingmethodisdevelopedasanapplicationtotheseoperators. Thismethodisthenappliedtoadailylifeproblem.

Keywords: neutrosophiccubicweightedgeometricoperator(NCWG);neutrosophiccubicordered weightedgeometricoperator(NCOWG);neutrosophiccubicEinsteinweightedgeometricoperator (NCEWG);neutrosophiccubicEinsteinorderedweightedgeometricoperator(NCEOWG)

Symmetry 2019, 11,247;doi:10.3390/sym11020247 www.mdpi.com/journal/symmetry

1.Introduction

ThetheoryoffuzzysetswasintroducedbyZadeh[1].Soonafter,itattractedexpertsofsciences andengineeringduetoitspossibilisticbehavior.Theapplicabilityoffuzzysetsextendedittointerval valuedfuzzysets(IVFs)[2,3].In1986,K.Atnassovdevelopedthetheoryofintuitionisticfuzzysets[4], whichwerefurtherextendedtointervalvaluedintuitionisticfuzzysetsin1989[5].In2012,Y.B. Jungeneralizedtheideaoffuzzysetsandintuitionisticfuzzysetstoformcubicsets[6].Smarandache presentedhistheoryregardingtheinconsistentandindeterminatebehaviorofdatain1999,andnamed ittheneutrosophicset[7].Neutrosophicsetsconsistofthreecomponents:Truth,indeterminateand falsehood,whichprovidesamoregeneralplatformtodealwithvagueandinsufficientdata.In2005, Wangetal.[8]presentedtheideaofintervalvaluedneutrosophicsets.Intervalvaluedneutrosophic setsprovidearangetoexpertswhichmakesthemmorecomfortablewithmakingthechoice.Junetal. definedtheneutrosophiccubicset[9,10].Neutrosophiccubicsetsareageneralizationofneutrosophic setsandintervalneutrosophicsets.Theyenableustochoosebothintervalvaluesandsinglevalue membership.Thischaracteristicofneutrosophiccubicsetsenablesustodealwithuncertainandvague datamoreefficiently.

Decisionmakingisoneofthemostimportantfactorsinscienceandday-to-daylifeaswell. Aggregationoperatorsareanimperativepartofmoderndecisionmaking.Alackofdataorinformation makesitdifficultfordecisionmakerstotakeanappropriatedecision.Thisuncertainsituationcan beminimizedusingthevaguenatureneutrosophiccubicsetanditsextensions.Neutrosophiccubic set(NCs)areamoregeneralizedversionofneutrosophicsets(Ns)andintervalneutrosophicsets (INs).Neutrosophiccubicsetsarebetterplacedtoexpressconsistent,indeterminate,andinconsistent information,whichprovidesabetterplatformtodealwithincomplete,inconsistent,andvaguedata.

Aggregationoperatorshaveakeyroleindailylife,scienceandengineeringproblems.Zhanetal.[11] intheirworkapplicationsofneutrosophiccubicsetsinmulti-criteriadecisionmakingin2017. Banerjeeetal.[12]usedgreyrationalanalysisintheirworkGRAformultiattributedecisionmakingin neutrosophiccubicsetenvironmentin2017.LuandYe[13]definedcosinemeasureforneutrosophic cubicsetsformultipleattribtedecisionmakingin2017.Pramaniketal.[14]definedneutrosophiccubic MCGDMmethodbasedonsimilaritymeasurein2017.ShiandYe[15]definedDombiaggregation operatorsofneutrosophiccubicsetformultipleattributedeicisionmakingin2018.Baolinetal.[16] appliedEinsteinaggregationsonneutrosophicsetsinanovelgeneralizedsimplifiedneutrosophic numberEinsteinaggregationoperator2018.Alotofworkhasbeendoneandisbeingdonebydifferent researchersindecisionmakingusingneutrosophiccubicsets.

Inthispaper,wedefinealgebraicandEinsteinsum,multiplicationandscalarmultiplication, scoreandaccuracyfunctions.Usingtheseoperations,wedefinegeometricaggregationoperators andEinsteingeometricaggregationoperators.First,wedefinealgebraicandEinsteinoperatorsof addition,multiplicationandscalarmultiplication.Wethendefinescoreandaccuracyfunctionsto compareneutrosophiccubicvalues.Followingthis,weproposeaneutrosophiccubicorderedweighted geometricoperator(NCOWG),neutrosophiccubicEinsteinweightedgeometricoperator(NCEWG), andaneutrosophiccubicEinsteinorderedweightedgeometricoperator(NCEOWG)overneutrosophic cubicsets.Amulti-criteriadecisionmakingmethodisthendevelopedasanapplicationforthese operators.Thismethodisthenappliedtoadailylifeproblem.

2.Preliminaries

Thissectionconsistsoftwoparts:Notations,whichconsistsofnotationswiththeirdescriptions andsomepreviousdefinitions;andresults.Werecommendthereadertosee[1–3,6–9,16].

2.1.Notations

Thissectionconsistsofsomenotationswiththeirdescriptions,asshowninTable 1

Table1. Somenotationswiththeirdescriptions.

extreme

ψU is referredasupperfuzzyfunction.

valuedfuzzyset.

2.2.Pre-DefinedDefinitions

Thissectionconsistsofsomepredefineddefinitionsandresults.

Definition1[1]. Amapping ψ: U → [0,1] iscalledafuzzyset,and ψ(u) iscalledamembershipfunction, simplydenotedby ψ

Definition2[2,3]. Amapping Ψ : U → D[0,1] ,where D[0,1] istheinterval valueof [0,1],calledtheintervalvaluedfuzzyset(IVF).Forall u ∈ U Ψ(u)= ψL (u), ψU (u) |ψL (u), ψU (u) ∈ [0,1] and ψL (u) ≤ ψU (u) ismembershipdegreeof u in Ψ.Thisis simplydenotedby Ψ = ΨL , ΨU

Definition3[6]. Astructure C = u, Ψ(u), Ψ(u) |u ∈ U isacubicsetin U inwhich Ψ(u) isIVFin U, thatis, Ψ = ΨL , ΨU and Ψ isafuzzysetin U.Thiscanbesimplydenotedby C = Ψ, Ψ CU denotesthe collectionofcubicsetsinU.

Definition4[7]. Astructure N = {(TN (u), IN (u), FN (u))|u ∈ U} isaneutrosophicset(Ns),where {TN (u), IN (u), FN (u) ∈ [0,1]} arecalledtruth,indeterminacyandfalsityfunctions,respectively.Thiscanbe simplydenotedbyN = (TN , IN , FN ).

Definition5[8]. Anintervalneutrosophicset(INs)in U isastructure N = TN (u), IN (u), FN (u) |u ∈ U ,where TN (u), IN (u), FN (u) ∈ D[0,1] iscalledtruth, indeterminacyanfalsityfunctionin U,respectively.Thiscanbesimplydenotedby N = TN , IN , FN . Forconvenience,wedenoteN = TN , IN , FN byN = TN = TL N , TU N , IN = I L N , IU N , FN = FL N , FU N

Definition6[9]. AstructureN= u, TN (u), IN (u), FN (u), TN (u), IN (u), FN (u) |u ∈ U is neutrosophiccubicset(NCs)in U,inwhich TN = TL N , TU N , IN = I L N , IU N , FN = FL N , FU N isaninterval neutrosophicsetand (TN , IN , FN ) isaneutrosophicsetin U.Simplydenotedby N = TN , IN , FN , TN , IN , FN , [0,0] ≤ TN + IN + FN ≤ [3,3] and 0 ≤ TN + IN + FN ≤ 3. NU denotesthecollectionofneutrosophiccubic setsinU.SimplydenotedbyN = TN , IN , FN , TN , IN , FN .

Definition7[16]. Thet-operatorsarebasicallyunionandintersectionoperatorsinthetheoryoffuzzysets, whicharedenotedbyt-conorm (Γ∗) andt-norm (Γ),respectively.Theroleoft-operatorsisveryimportantin fuzzytheoryanditsapplications.

Definition8[16]. Γ∗ : [0,1] × [0,1] → [0,1] iscalledt-conormifitsatisfiesthefollowingaxioms:

Axiom1. Γ∗(1, u) = 1 and Γ∗(0, u) = 0;

Axiom2. Γ∗(u, v) = Γ∗(v, u) forallaandb; Axiom3. Γ∗(u, Γ∗(v, w)) = Γ∗(Γ∗(u, v), w) foralla,bandc; Axiom4. Ifu ≤ u andv ≤ v ,then Γ∗(u, v) ≤ Γ∗(u , v )

Mostknownt-conormsareasfollows:

1. Thedefaultt-conorm: Γ∗ max(u, v) = max(u, v).

2. Theboundedt-conorm: Γ∗ bounded (u, v) = min(1, u + v)

3. Thealgebraict-conorm: Γ∗ algebraic (u, v) = u + v uv

Definition9[16]. Γ : [0,1] × [0,1] → [0,1] iscalledt-normifitsatisfiesthefollowingaxioms:

Axiom5. Γ(1, u) = uand Γ(0, u) = 0; Axiom6. Γ(u, v) = Γ(v, u) forallaandb; Axiom7. Γ(u, Γ(v, w)) = Γ(Γ(u, v), w) foralla,bandc; Axiom8. Ifu ≤ u andv ≤ v , then Γ(u, v) ≤ Γ(u , v )

Mostwellknownt-normsareasfollows:

1. Thedefaultt-norm: Γmin(u, v) = min(u, v)

2. Theboundedt-norm: Γbounded (u, v) = max(0, u + v 1)

3. Thealgebraict-norm: Γalgebraic (u, v) = uv

If Γ∗(u, v), Γ(u, v) arecontinuousand Γ∗(u, u) > u, Γ(u, u) < u,then Γ∗ and Γ aresaidtobe Archimedest-conormandt-norm,respectively.Anypairofdualt-conorm(Γ∗)andt-norm(Γ)isused. Itisknownthatt-normsandt-conormsoperatorssatisfytheconditionofconjunctionanddisjunction operators,respectively.However,thealgebraicoperations,likealgebraicsumandproduct,arenot uniqueandmaycorrespondtounionandintersection.Thet-conormsandt-normsfamilieshave avastrange,whichcorrespondstounionsandintersections.Amongthese,theEinsteinsumand Einsteinproductaregoodchoicessincetheygivethesmoothapproximationlikealgebraicsumand algebraicproduct,respectively.Einsteinsum ⊕E andEinsteinproduct ⊗E areexamplesoft-conorm andt-norm,respectively:

Γ∗ E (u, v)= u + v 1 + uv ΓE (u, v)= uv 1 + (1 u)(1 v)

Groupdecisionmakingisanimportantaspectofdecisionmakingtheory.Weareoftenin situationsinwhichwehavetodealwithmorethenoneexpert,attributeandalternative.Motivated bysuchsituations,amulti-attributedecisionmakingmethodformorethenoneexpertisproposed onneutrosophiccubicaggregationoperators.Thiswholeworkconsistedofsixsections.InSection 3, wedefinesomealgebraicEinsteinoperationsandscoreandaccuracyfunctions,alongwithsome importantresultsandexamples.Onthebasisofthesedefinitionsandresults,wedefinegeometric andEinsteingeometricaggregationoperatorsonneutrosophiccubicsetsinSection 4.InSection 5, analgorithmisproposedbasedonneutrosophiccubicgeometricandEinsteingeometricaggregation operatorstodealwithmulti-attributedecisionmakingproblems.Inthefinalsection,anumerical examplefromdailylifeispresentedasanapplicationofthework.

3.OperationsonNeutrosophicCubicSets

Inthissection,weintroducesomenewoperationsonneutrosophiccubicsetswhicharefurther usedinthearticle.

3.1.AlgebraicAddition,MultiplicationandScalarMultiplication

Weintroducethealgebraicaddition,multiplication,andscalarmultiplicationonneutrosophic cubicsets(NCs).Animportantresultofexponentialmultliplictionisestablishedonthebasisofthese defintions,whichprovidesthebasistodefineneutrosophiccubicgeometricaggregationoperators.

Definition10. Thesumoftwoneutrosophiccubicsets(NCs), A = TA, IA, FA, TA, IA, FA ,where TA =



Definition11. Theproductbetweentwoneutrosophiccubicsets(NCs), A = TA, IA, FA, TA, IA, FA , where TA = TL A, TU A , IA = I L A, IU A , FA = FL A, FU A and B = TB, IB, FB, TB, IB, FB ,where TB = TL B , TU B , IB = I L B , IU B , FB = FL B , FU B isdefinedas A ⊗ B =      TL A TL B , TU A TU B , I L A I L B , IU A IU B , FL A + FL B FL A FL B , FU A + FU B FU A FU B , TA + TB TA TB, IA + IB IA IB, FA FB

  

Thefollowingresultisestablishedtodealwiththeexponentialmultiplicationonneutrosophic cubicvalues.Thisresultenablesustodefinegeometricaggregationoperatorsalongsomeimportant resultsonneutrosophiccubicsets. Theorem1. Let A = TA, IA, FA, TA, IA, FA ,where TA = TL A, TU A , IA = I L A, IU A , FA = FL A, FU A , bea neutrosophiccubicvalue,thentheexponentialoperationcanbedefinedby Ak = 

(TL A )k , (TU A )k , (I L A )k , (IU A )k , 1 1 FL A k,1 1 FU A k , 1 (1 TA )k,1 (1 IA )k , (FA )k 

whereAk = A ⊗ A⊗,... ⊗ A(k times),andAk isaneutrosophiccubicvalueforeverypositivevalueofk.

    





 

(TL A ), (TU A ) , (I L A ), (IU A ) , FL A, FU A , TA, IA, FA



 (TL A )m+1 , (TU A )m+1 , (I L A )m+1 , (IU A )m+1 , 1 1 FL A m + FL A FL A + 1 FL A m FL A,1 1 FU A m + FU A FU A + 1 FU A m FU A , 1 (1 TA )m + TA TA + (1 TA )m TA,1 (1 IA )m + IA IA + (1 IA )m IA, (FA )m+1



    



(TL A )m+1 , (TU A )m+1 , (I L A )m+1 , (IU A )m+1 , 1 1 FL A

1 FU A

1

(FA



      





Definition13. TheEinsteinunionbetweentwoneutrosophiccubicsets(NCs), A = TA, IA, FA, TA, IA, FA where TA = TL A, TU A , IA = I L A, IU A , FA = FL A, FU A ,and B = TB, IB, FB, TB, IB, FB where TB = TL B , TU B , IB = I L B , IU B , FB = FL B , FU B isdefinedas

A ∨ B = Γ TA, TB , Γ IA, IB , Γ∗ FA, FB , Γ∗{TA, TB }, Γ∗{IA, IB }, Γ{FA, FB }

Definition14. TheEinsteinintersectionbetweentwoneutrosophiccubicsets(NCS), A = TA, IA, FA, TA, IA, FA ,where TA = TL A, TU A , IA = I L A, IU A , FA = FL A, FU A and B = TB, IB, FB, TB, IB, FB , where TB = TL B , TU B , IB = I L B , IU B , FB = FL B , FU B isdefinedas

A ∧ B = Γ∗ TA, TB , Γ∗ IA, IB , Γ FA, FB , Γ{TA, TB }, Γ{IA, IB }, Γ∗{FA, FB } .

OnthebasisofEinsteinunionandintersectiontheEinsteinsumandproductisdefinedover neutrosophiccubicvalues.

Definition15. TheEinsteinsumbetweentwoneutrosophiccubicsets(NCS), A = TA, IA, FA, TA, IA, FA , where TA = TL A, TU A , IA = I L A, IU A , FA = FL A, FU A and B = TB, IB, FB, TB, IB, FB , where TB = TL B , TU B , IB = I L B , IU B , FB = FL B , FU B isdefinedas

L A +TL B 1+TL A TL B , TU A +TU B 1+TU A TU B , I L A +I L B 1+I L A I L B , IU A +IU B 1+IU A IU B , FL A FL B 1+(1 FL A )(1 FL B ) , FU A FU B 1+(1 FU A )(1 FU B ) TA TB 1+(1 TA )(1 TB ) , IA IB 1+(1 IA )(1 IB ) , FA +FB 1+FA FB

 

Definition16. TheEinsteinproductbetweentwoneutrosophiccubicsets(NCS), A = TA, IA, FA, TA, IA, FA ,where TA = TL A, TU A , IA = I L A, IU A , FA = FL A, FU A and B = TB, IB, FB, TB, IB, FB , where TB = TL B , TU B , IB = I L B , IU B , FB = FL B , FU B isdefinedas

TL A TL B 1+(1 TL A )(1 TL B ) , TU A TU B 1+(1 TU A )(1 TU B ) , I L A I L B 1+(1 I L A )(1 I L B ) , IU A IU B 1+(1 IU A )

(1+TL A )k (1 TL A )k (1+TL A )k +(1 TL A )k , (1+TU A )k (1 TU A )k (1+TU A )k +(1 TU A )k , (1+I L A )k (1 I L A )k (1+I L A )k +(1 I L A )k , (1+IU A )k (1 IU A )k (1+IU A )k +(1 IU A )k , 2(FL A )k (2 FL A )k +(FL A )k , 2(FU A )k (2 FU A )k +(FU A )k , 2(TA )k (2 TA )k +(TA )k , 2(IA )k (2 IA )k +(IA )k , (1+FA )k (1 FA )k (1+FA )k +(1 FA )k

Afterdefiningthescalarmultiplicationovertheneutrosophiccubicset,weestablishedthe followingresult,whichdealswiththeEinsteinexponentialmultiplicationonneutrosophiccubic values.ThisresultenabledustodefineEinsteingeometricaggregationoperatorsalongwithsome importantresultsonneutrosophiccubicsets.

Theorem2. Let A = TA, IA, FA, TA, IA, FA ,where TA = TL A, TU A , IA = I L A, IU A , FA = FL A, FU A ,bea neutrosophiccubicvalue,thentheexponentialoperationdefinedby

2(TL A )k 2( TL A )k +(TL A )k , 2(TU A )k (2 TU A )k +(TU A )k , 2(I L A )k (2 I L A )k +(I L A )k , 2(IU A )k (2 IU A )k +(IU A )k , (1+FL A )k (1 FL A )k (1+FL A )k +(1 FL A )k , (1+FU A )k (1 FU A )k (1+FU A )k +(1 FU A )k , (1+TA )k (1 TA )k (1+TA )k +(1 TA )k , (1+IA )k (1 IA )k (1+IA )k +(1 IA )k , 2(FA )k (2 FA )k +(FA )k

where AEk = A ⊗E A ⊗E ... ⊗E A(k times),moreover AEk isaneutrosophiccubicvalueforeverypositive valueofk.





AEm =

U



isaneutrosophiccubicvalue.Thenweprove AEk+1 isneutrosophiccubicvalue. Consider,

2(TL A )m (2 TL A )m +(TL A )m , 2(TU A )m (2 TU A )m +(TU A )m , 2(I L A )m (2 I L A )m +(I L A )m , 2(IU A )m (2 IU A )m +(IU A )m , (1+FL A )m (1 FL A )m (1+FL A )m +(1 FL A )m , (1+FU A )m (1 FU A )m (1+FU A )m +(1 FU A )m , (1+TA )m (1 TA )m (1+TA )m +(1 TA )m , (1+IA )m (1 IA )m (1+IA )m +(1 IA )m , 2(FA )m (2 FA )m +(FA )m

1

,

      

4(I L A )m+1 (2 I L A )m +(I L A )m ((2 I L A )+I L A ) 1+ 1 2(I L A )m (2 I L A )m +(I L A )m 1 2I L A (2 I L A )+I L A

,

4(IU A )m+1 (2 IU A )m +(IU A )m ((2 IU A )+IU A ) 1+ 1 2(IU A )m (2 IU A )m +(IU A )m 1 2IU A (2 IU A )+IU A

    ,     (1+FLA )m (1 FLA )m (1+FLA )m +(1 FLA )m + (1+FLA ) (1 FLA ) (1+FLA )+(1 FLA ) 1+ (1+FLA )m (1 FLA )m (1+FLA )m +(1 FLA )m (1+FLA ) (1 FLA ) (1+FLA )+(1 FLA )

(1+FU A )m (1 FU A )m (1+FU A )m +(1 FU A )m + (1+FU A ) (1 FU A ) (1+FU A )+(1 FU A ) 1+ (1+FU A )m (1 FU A )m (1+FU A )m +(1 FU A )m (1+FU A ) (1 FU A ) (1+FU A )+(1 FU A )



4(TLA )m+1 (2 TLA )m +(TLA )m ((2 TLA )+TLA ) 1

+

(2 TLA )m +(TLA )m 2(TLA )m (2 TLA )m +(TLA )m (2 TLA )+TLA 2TLA (2 TLA )+TLA

,

,

    , (1+TA )m (1 TA )m (1+TA )m +(1 TA )m + (1+TA ) (1 TA ) (1+TA )+(1 TA ) 1+ (1+TA )m (1 TA )m (1+TA )m +(1 TA )m (1+TA ) (1 TA ) (1+TA )+(1 TA ) , (1+IA )m (1 IA )m (1+IA )m +(1 IA )m + (1+IA ) (1 IA ) (1+IA )+(1 IA ) 1+ (1+IA )m (1 IA )m (1+IA )m +(1 IA )m (1+IA ) (1 IA ) (1+IA )+(1 IA ) , 4(FA )m+1 ((2 FA )m +(FA )m )((2 FA )+FA ) 1+ 1 2(FA )m (2 FA )m +(FA )m 1 2FA (2 FA )+FA           

4(TU A )m+1 (2 TU A )m +(TU A )m ((2 TU A )+TU A ) 1+ (2 TU A )m +(TU A )m 2(TU A )m (2 TU A )m +(TU A )m (2 TU A )+TU A 2TU A (2 TU A )+TU A

4(IU A )m+1 (2 IU A )m +(IU A )m ((2 IU A )+IU A ) 1+ (2 IU A )m +(IU A )m 2(IU A )m (2 IU A )m +(IU A )m (2 IU A )+IU A 2IU A (2 IU A )+IU A

    ,                                                 

(1+FLA )m (1 FLA )m ((1+FLA )+(1 FLA ))+ (1+FLA )m +(1 FLA )m ((1+FLA ) (1 FLA )) ((1+FLA )m +(1 FLA )m )((1+FLA )+(1 FLA )) (1+FLA )m +(1 FLA )m ((1+FLA )+(1 FLA ))+(1+FLA )m+1 (1+FLA )m (1 FLA ) (1 FLA )m (1+FLA )+(1 FLA )m+1 ((1+FLA )m +(1 FLA )m )((1+FLA )+(1 FLA )) , (1+FU A )m (1 FU A )m ((1+FU A )+(1 FU A ))+ (1+FU A )m +(1 FU A )m ((1+FU A ) (1 FU A )) ((1+FU A )m +(1 FU A )m )((1+FU A )+(1 FU A )) (1+FU A )m +(1 FU A )m ((1+FU A )+(1 FU A ))+(1+FU A )m+1 (1+FU A )m (1 FU A ) (1 FU A )m (1+FU A )+(1 FU A )m+1 ((1+FU A )m +(1 FU A )m )((1+FU A )+(1 FU A ))

    ,

      

   

                                               

4(I L A )m+1 (2 I L A )m +(I L A )m ((2 I L A )+I L A ) (2 I L A )m +(I L A )m ((2 I L A )+I L A ) + (2 I L A )m (I L A )m ((2 I L A ) I L A ) (2 I L A )m +(I L A )m ((2 I L A )+I L A ) , 4(IU A )m+1 (2 IU A )m +(IU A )m ((2 IU A )+IU A ) (2 IU A )m +(IU A )m ((2 IU A )+IU A ) + (2 IU A )m (IU A )m ((2 IU A ) IU A ) (2 IU A )m +(IU A )m ((2 IU A )+IU A )

, 1 + FU A m+1 + 1 + FU A m 1 FU A 1 FU A m 1 + FU A 1 FU A m+1 + 1 + FU A m+1 1 + FU A m 1 FU A + 1 FU A m 1 + FU A 1 FU A m+1 1 + FU A m+1 + 1 + FU A m 1 FU A 1 FU A m 1 + FU A + 1 FU A m+1 + 1 + FU A m+1 1 + FU A m 1 FU A + 1 FU A m 1 + FU A + 1 FU A m+1

(1 + TA )m+1 + (1 + TA )m (1 TA ) (1 TA )m (1 + TA ) (1 TA )m+1+ (1 + TA )m+1 (1 + TA )m (1 TA ) + (1 TA )m (1 + TA ) (1 TA )m+1 (1 + TA )m+1 + (1 + TA )m (1 TA ) (1 TA )m (1 + TA ) + (1 TA )m+1+ (1 + TA )m+1 (1 + TA )m (1 TA ) + (1 TA )m (1 + TA ) + (1 TA )m+1

, (1 + IA )m+1 + (1 + IA )m (1 IA ) (1 IA )m (1 + IA ) (1 IA )m+1+ (1 + IA )m+1 (1 + IA )m (1 IA ) + (1 IA )m (1 + IA ) (1 IA )m+1 (1 + IA )m+1 + (1 + IA )m (1 IA ) (1 IA )m (1 + IA ) + (1 IA )m+1+ (1 + IA )m+1 (1 + IA )m (1 IA ) + (1 IA )m (1 + IA ) + (1 IA )m+1

1 +I Lm A (2 I L A )+ (2 I L A )m+1 I L A (2 I L A )m +(I L A )m+1 (I L A )m (2 I L A ) , 4(IU A )m+1 (2 IU A )m+1 +IU A (2 IU A )m +(IU A )m+1 +IUm A (2 IU A )+ (2 IU A )m+1 IU A (2 IU A )m +(IU

+1 (1 TA )m+1 )

((1+TA )m+1 +(1 TA )m+1 ) ,

4(TL A )m+1 2 (2 TL A )m+1 +(TL A )m+1 , 4(TU A )m+1 2 (2 TU A )m+1 +(TU A )m+1 , 4(I L A )m+1 2 (2 I L A )m+1 +(I L A )m+1 , 4(IU A )m+1 2 (2 IU A )m+1 +(IU A )m+1 , (1+FL A )m+1 (1 FL A )m+1 (1+FL A )m+1 +(1 FL A )m+1 , (1+FU A )m+1 (1 FU A )m+1 (1+FU A )m+1 +(1 FU A )m+1 , (1+TA )m+1 (1 TA )m+1 (1+TA )m+1+(1 TA )m+1 , (1+IA )m+1 (1 IA )m+1 (1+IA )m+1+(1 IA )m+1 , 4(FA )m+1 2 (2 FA )m+1+(FA )m+1

2(TL A )m+1 (2 TL A )m+1 +(TL A )m+1 , 2(TU A )m+1 (2 TU A )m+1 +(TU A )m+1 , 2(I L A )m+1 (2 I L A )m+1 +(I L A )m+1 , 2(IU A )m+1 (2 IU A )m+1 +(IU A )m+1 , (1+FL A )m+1 (1 FL A )m+1 (1+FL A )m+1 +(1 FL A )m+1 , (1+FU A )m+1 (1 FU A )m+1 (1+FU A )m+1 +(1 FU A )m+1 , (1+TA )m+1 (1 TA )m+1 (1+TA )m+1+(1 TA )m+1 , (1+IA )m+1 (1 IA )m+1 (1+IA )m+1+(1 IA )m+1 , 2(FA )m+1 (2 FA )m+1+(FA )m+1

Whichshowsthat k = m + 1isaneutrosophiccubicvalue.

3.3.ScoreandAccuracyFunctionofNeutrosophicCubicSet

Forthecomparisonoftwoneutrosophicvalues,thescoreandaccuracyfunctionaredefined. Thescorefunctionisusedtocomparetwoneutrosophiccubicvalues;sometimesthescoreof twoneutrosophiccubicvaluesbecomesequal,althoughtheyhavedifferentcomponentsoftruth, indeterminancyandfalsityfunctions.Thissituationcanbeovercomebythehelpofanaccuracy function.Thefollowingdefinition,alongwithexamples,providesabetterviewofunderstandingto thereader.

Definition18. Let N = TN , IN , FN , TN , IN , FN ,where TN = TL N , TU N , IN = I L N , IU N , FN = FL N , FU N , beaneutrosophiccubicvalueandwedefinethescorefunctionas

S(N) = TL N FL N + TU N FU N + TN FN

Sometimesthesituationarisesthatthescoreoftwoneutrosophiccubicvaluesareequal.Insucha situation,acomparisonismadeonthebasisofanaccuracyfunction.

Definition19. Let N = TN , IN , FN , TN , IN , FN ,where TN = TL N , TU N , IN = I L N , IU N , FN = FL N , FU N , beaneutrosophiccubicvalue,theaccuracyfunctionisdefinedas

H(u)= 1 9 TL N + I L N + FL N + TU N + IU N + FU N + TN + IN + FN

Thefollowingdefinitionisaccomplishedforthecomparisonrelationoftheneutrosophic cubicvalues.

Definition20. Let N1 and N2 betwoneutrosophiccubicvalues,where SN1 and SN2 arescoresand HN1 and HN2 areaccuracyfunctionsofN1 andN2,respectively.

1. IfSN1 > SN2 ⇒ N1 > N2

2. IfSN1 = SN2 andHN1 > HN2 ⇒ N1 > N2 HN1 = HN2 ⇒ N1 = N2

Example1. Let N1 = ([0.5,0.9][0.6,0.9][0.1,0.4],0.3,0.4,0.4) and N2 = ([0.2,0.8][0.5,0.9][0.4,0.8],0.4,0.45,0.8) betwoneutrosophicsets. Then

SN1 = 0.8, andSN2 = 0.6

SN1 > SN2 ⇒ N1 > N2

Inthefollowingexamplethescorefuntionsareequal,soaccuracyfunctionsareusedtocompare neutrosophiccubicvalues.

Example2. Let N1 = ([0.4,0.9][0.5,0.8][0.1,0.7],0.4,0.5,0.8) and N2 = ([0.4,0.6][0.5,0.9][0.6,0.7],0.7,0.5,0.3) betwoneutrosophicsets.

SN1 = 0.1, SN2 = 0.1

SN1 = SN2 ⇒ N1 = N2 HN1 = 0.566, HN2 = 0.577 HN1 < HN2 ⇒ N1 < N2

4.NeutrosophicCubicGeometricandEinsteinGeometricAggregationOperators

Inthissection,weintroducetheconceptofneutrosophiccubicgeometricaggregationoperators andneutrosophiccubicEinsteingeometricaggregationoperators.

Thissectionconsistsoftwosub-sections:InSection 4.1,theneutrosophiccubicgeometric aggregationoperatorsaredefinedonthebasisofSection 3.1;andinSection 4.2,theneutrosophiccubic EinsteingeometricaggregationoperatorsaredefinedonthebasisofSection 3.2

4.1.NeutrosophicCubicWeightedGeometricAggregationOperator

WedefineneutrosophiccubicgeometricaggregationoperatorsusingSection 3.1 Definition21. Wedefinetheneutrosophiccubicweightedgeometricoperator(NCWG)as NCWG : Rm → RdefinedbyNCWGw (N1, N2,..., Nm )= m ⊗ j=1 N wj j

InNCWG,theneutrosophiccubicvaluesarefirstweightedthenaggregated.

Definition22. Wedefinetheneutrosophiccubicorderedweightedgeometricoperator(NCOWG)as

NCOWG : Rm → RdefinedbyNCOWGw (N1, N2,..., Nm )= m ⊗ j=1 N wj (γ)j where N(γ)j aredescendingorderedneutrosophiccubicvalues,andtheweight W =(w1, w2, ... , wm )T of correspondingneutrosophiccubicvaluesNj (j = 1,2,3,..., m) issuchthateachwj ∈ [0,1] and m ∑ wj j=1 = 1.

InNCOWG,theneutrosophiccubicvaluesarefirstarrangedindecendingorder,weightedand thenaggregated.

1 FL Nj w1 ,1 1 FU Nj w1 , 1 1 TNj w1 ,1 1 INj w1 , FNj w1 wj , 2 ∏ j=1 FU Nj

(TL Nj )w1 , (TU Nj )w1 , (I L Nj )w1 , (IU Nj )w1 , 2 ∏ j=1 (TL Nj )wj , 2 ∏ j=1 (TU Nj )wj , 2 ∏ j=1 (I L Nj )wj , 2 ∏ j=1 (IU Nj )wj , 1 2 ∏ j=1 1 FL Nj

m w2 ,1 1 FU Nj

wj , m ∏ j=1 IU Nj w2 ,1 1 INj

wj 1 m ∏ j=1 (1 FL Nj )wj ,1 m ∏ j=1 (1 FU Nj )wj 1 m ∏ j=1 (1 TNj ) wj ,1 m ∏ j=1 (1 INj ) wj , m ∏ j=1 FNj w2 , (FNj )w2

wj       =

wheretheweightW =(w1, w2,..., wm )T ofNj (j = 1,2,3,..., m) suchthatwj ∈ [0,1] and m ∑ wj j=1 = 1 Proof. Bymathematicalinductionfor m = 2,using 2 ⊗ j=1 N wj j = Nw1 1 ⊗ Nw2 2               

(TL Nj )w2 , (TU Nj )w2 , (I L Nj )w2 , (IU N wj , 1 2 ∏ j=1 1 TNj

wj ,1 2 ∏ j=1 1 INj

wj , 2 ∏ j=1 FNj

wj

              

wj I L Nm+1

wm+1 , n ∏ j=1 IU Nj

wj IU Nm+1

1 n ∏ j=1 (1 FL Nj )wj + 1 (1 FL Nm+1 )wm+1 1 n ∏ j=1 (1 FL Nj )wj 1 (1 FL Nm+1 )wm+1 , 1 n ∏ j=1 (1 FU Nj )wj + 1 (1 FU Nm+1 )wm+1

n ∏ j=1 (1 FU Nj )wj 1 (1 FU Nm+1 )wm+1 ,

n ∏ j=1 1 TNj wj + 1 1 TNm+1 wm+1

n ∏ j=1 1 TNj wj 1 1 TN

INj

j=1

∏ j=1 1 INj

j

1 FNj

Nm

j=1 (1 FL Nj )wj 1 + n ∏ j=1 (1 FL Nj )wj +(1 FL Nm+1 )wm+1 n ∏ j=1 (1 FL Nj )wj (1 FL Nm+1 )wm+1 , 2 n+1 ∏ j=1 (1 FU Nj )wj 1 + n+1 ∏ j=1 (1 FU Nj )wj +(1 FU Nm+1 )wm+1 n ∏ j=1 (1 FU Nj )wj (1 FU Nm+1 )wm+1

       

,

1

J

4.2.NeutrosophicCubicEinsteinWeightedGeometricAggregationOperator

WedefineneutrosophiccubicEinsteingeometricaggregationoperatorsusingSection 3.2 Definition23. TheneutrosophiccubicEinsteinweightedgeometricoperator(NCEWA)isdefinedas NCEWG : Rm → R,definedbyNCEWGw (N1, N2,..., Nm )= m ⊗ j=1 NE j wj

where, W =(w1, w2, , wm )T istheweightof Nj (j = 1,2,3, , m),suchthat wj ∈ [0,1] and m ∑ wj j=1 = 1.

Thatis,firstalltheneutrosophicvaluesareweightedthenaggregatedusingEinsteinoperations. Definition24. OrderneutrosophiccubicEinsteinweightedgeometricoperator(NCEOWG)isdefinedas NCEOWG : Rm → RbyNCEOWGw (N1, N2,..., Nm )= m ⊗ j=1 BE j wj where Bj isthe jth largest, W =(w1, w2, , wm )T istheweightof Nj (j = 1,2,3, , m),suchthat wj ∈ [0,1] and m ∑ wj j=1 = 1

j

+FU Nj wj m ∏ j=1 1 FU Nj wj m ∏ j=1 1+FU Nj wj + m ∏ j=1 1 FU Nj wj

,

NCEWG(Nj                                       where W =(w1, w2, ... , wm )T istheweightvectorof Nj (j = 1,2,3, ... , m),suchthat wj ∈ [0,1] and m ∑ wj j=1 = 1.

Proof. Weusemathematicalinductiontoprovethisresult,for m = 2,usingdefinition(Einsteinsum andEinsteinscalarmultiplication).

2 TL N1 w1 2 TL N1 w1 +TL N1 , 2 TU N1 w1 2 TU N1 w1 +TU N1 , 2 I L N1 w1 2 I L N1 w1 +I L N1 , 2 IU N1 w1 2 IU N1 w1 +IU N1 , (1+FL N1 )w1 (1 FL N1 )w1 (1+FL N1 )w1 +(1 FL N1 )w1 , (1+FU N1 )w1 (1 FU N1 )w1 (1+FU N1 )w1 +(1 FU N1 )w1 , (1+TN1 )w1 (1 TN1 )w1 (1+TN1 )w1 +(1 TN1 )w1 , (1+IN1 )w1 (1 IN1 )w1 (1+IN1 )w1 +(1 IN1 )w1 , 2(FN1 )w1 (2 FN1 )w1 +FN1

2

 

2 2 ∏ j=1 TL Nj wj



2 ∏ j=1 1+FU Nj wj 2 ∏ j=1 1 FU Nj wj 2 ∏ j=1 1+FU Nj wj + 2 ∏ j=1 1 FU Nj wj 

2 ∏ j=1 1+INj wj 2 ∏ j=1 1 INj wj 2 ∏ j=1 1+INj wj + 2 ∏ j=1 1 INj wj , 2 2 ∏ j=1 FNj wj 2 ∏ j=1 2 FNj wj + 2 ∏ j=1 FNj wj

 

2 TL N2 w2 2 TL N2 w2 +TL N2 , 2 TU N2 w2 2 TU N2 w2 +TU N2 , 2 I L N2 w2 2 I L N2 w2 +I L N2 , 2 IU N2 w2 2 IU N2 w2 +IU N2 , (1+FL N2 )w2 (1 FL N2 )w2 (1+FL N2 )w2 +(1 FL N2 )w2 , (1+FU N2 )w2 (1 FU N2 )w2 (1+FU N2 )w2 +(1 FU N2 )w2 , (1+TN2 )w2 (1 TN2 )w2 (1+TN2 )w2 +(1 TN2 )w2 , (1+IN2 )w2 (1 IN2 )w2 (1+IN2 )w2 +(1 IN2 )w2 , 2(FN2 )w2 (2 FN2 )w2 +FN2                                  

n ∏ j=1 1+FU Nj wj n ∏ j=1 1 FU Nj wj n ∏ j=1 1+FU Nj wj + n ∏ j=1 1 FU Nj wj   , n ∏ j=1 1+TNj wj n ∏ j=1 1 TNj wj n ∏ j 1 1+TNj wj + n ∏ j 1 1 TNj wj ,

n ∏ j=1 1+INj wj n ∏ j=1 1 INj wj n ∏ j 1 1+INj wj + n ∏ j 1 1 INj wj , 2 n ∏ j=1 FNj wj n ∏ j=1 2 FNj wj + n ∏ j=1 FNj wj

as NE n+1 wn+1 =                      

TLNn+1 wn+1

                      so n ⊗ j 1 NE j wj ⊗E NE m+1 wm+1 =                                          2 n ∏ j=1 TLNj wj n ∏ j 1 2 TLNj wj + n ∏ j 1 TLNj wj , 2 n ∏ j=1 TUNj wj n ∏ j 1 2 TUNj wj + n ∏ j 1 TUNj wj     ,     2 n ∏ j=1 I L Nj wj n ∏ j 1 2 I L Nj wj + n ∏ j 1 I L Nj wj , 2 n ∏ j=1 IUNj wj n ∏ j 1 2 IUNj wj + n ∏ j 1 IUNj wj     ,     n ∏ j=1 1+FLNj wj n ∏ j=1 1 FLNj wj n ∏ j 1 1+FLNj wj + n ∏ j 1 1 FLNj wj , n ∏ j=1 1+FUNj wj n ∏ j=1 1 FUNj wj n ∏ j 1 1+FUNj wj + n ∏ j 1 1 FUNj wj     , n ∏ j 1 1+TNj wj n ∏ j 1 1 TNj wj n ∏ j 1 1+TNj wj + n ∏ j 1 1 TNj wj , n ∏ j 1 1+INj wj n ∏ j 1 1 INj wj n ∏ j 1 1+INj wj + n ∏ j 1 1 INj wj , 2 n ∏ j=1 FNj wj n ∏ j 1 2 FNj wj + n ∏ j 1 FNj wj                                      ⊕E                               2

 2 n+1 ∏ j=1 TL Nj wj n+1 ∏ j=1 2 TL Nj wj + n+1 ∏ j=1 TL Nj wj , 2 n+1 ∏ j=1 TU Nj wj n+1 ∏ j=1 2 TU Nj wj + n+1 ∏ j=1 TU Nj wj   ,    2 n+1 ∏ j=1 I L Nj wj n+1 ∏ j 1 2 I L Nj wj + n+1 ∏ j 1 I L Nj wj , 2 n+1 ∏ j=1 IU Nj wj n+1 ∏ j 1 2 IU Nj wj + n+1 ∏ j 1 IU Nj wj   ,   

n+1 ∏ j 1 1+INj wj n+1 ∏ j 1 1 INj wj n+1 ∏ j=1 1+INj wj + n+1 ∏ j=1 1 INj wj , 2 n+1 ∏ j=1 FNj wj n+1 ∏ j=1 2 FNj wj + n+1 ∏ j=1 FNj wj

n+1 ∏ j=1 1+FU Nj wj n+1 ∏ j=1 1 FU Nj wj n+1 ∏ j 1 1+FU Nj wj + n+1 ∏ j 1 1 FU Nj wj                                   



soresultholdsforallvaluesof m Theorem7. LetNj = TNj , INj , FNj , TNj , INj , FNj ,where TNj = TL Nj , TU Nj , INj = I L Nj , IU Nj , FNj = FL Nj , FU Nj , (j = 1,2, , m) isacollectionofneutrosophiccubicvaluesand W =(w1, w2,..., wm )T isaweightvectorofNj (j = 1,2,3,..., m),withwj ∈ [0,1] and m ∑ wj j=1 = 1.

1. Idempotency:Ifforall Nj = TNj , INj , FNj , TNj , INj , FNj ,where TNj = TL Nj , TU Nj , INj = I L Nj , IU Nj , FNj = FL Nj , FU Nj , (j = 1,2, , m) areequal,thatis, Nj = N forallk,then NCEWGw (N1, N2,..., Nm )= N

2. Monotonicity: Let Bj = TBj , IBj , FBj , TBj , IBj , FBj , where TBj = TL Bj , TU Bj , IBj = I L Bj , IU Bj , FBj = FL Bj , FU Bj (j = 1,2, , m) bethecollectionofcubicvalues.If SB (u) ≥ SN (u) andBj (u) ≥ Nj (u) thenNCWGw (N1, N2,..., Nm ) ≤ NCWGw (B1, B2,..., Bm )

3. Boundary: N ≤ NCWGw {(N1)T , (N2)T ,..., (Nm )T } ≤ N+,where

N = min j TL Nj ,min j I L Nj ,1 max j FL Nj ,min j TNj ,min j INj ,1 max j FL Nj , N+ = max j TU Nj ,max j IU Nj ,1 min j FU Nj ,max j TNj ,max j INj ,1 min j FNj

Proof. FollowedbyTheorem2.

Theorem8. Let Nj = TNj , INj , FNj , TNj , INj , FNj ,where TNj = TL Nj , TU Nj , INj = I L Nj , IU Nj , FNj = FL Nj , FU Nj , (j = 1,2, , m) beacollectionofneutrosophiccubicvaluesand W =(w1, w2,..., wm )T isaweightvectoroftheNCOWA,withwj ∈ [0,1] and m ∑ wj j=1 = 1

1. Ifw=(1,0,...,0)T ,thenNCEOWG (N1, N2,..., Nm )= maxNj

2. Ifw(0,0,...,1)T ,thenNCEOWG (N1, N2,..., Nm )= minNj

3. Ifwj = 1, wj = 0,andj = j,thenNCEOWG (N1, N2,..., Nm )= Nj whereNj isthejthlargestof (N1, N2,..., Nm )

Proof. FollowedbyTheorem3.

5.AnApplicationofNeutrosophiccubicGeometricandEinsteinGeometricAggregation OperatortoGroupDecisionMakingProblems

Groupdecisionmakingisanimportantfactorofdecisionmakingtheory.Weareoftenina situationwithmorethenoneexpert,attributeandalternativetodealwith.Motivatedbysuch situations,amulti-attributedecisionmakingmethodformorethenoneexpertisproposedin thissection.

Inthissection,wedevelopanalgorithmforgroupdecisionmakingproblemsusingthegeometric andEinsteingeometricaggregations(NCWGandNCEWG)undertheneutrosophiccubicenvironment. Algorithm. Let F = {F1, F2,..., Fn } bethesetofnalternatives, H = {H1, H2,..., Hm } bethem attributessubjecttotheircorrespondingweight W = {w1, w2,..., wm } suchthat wj ∈ [0,1] and m ∑ j=1 wj = 1, and D = {D1, D2,...Dr } betherdecisionmakerswiththeircorrespondingweight V = {v1, v2,..., vr }. suchthatvj ∈ [0,1] and r ∑ j=1 vj = 1Themethodhasthefollowingsteps:

Step1. First,weconstructneutrosophiccubicdecisionmatricesforeachdecisionmaker D(s) = N(s) ij n×m (s = 1,2,..., r).

Step2. Alldecisionmatricesareaggregatedtoasinglematrixconsistingof m attributes,byNCWGand NCEWGcorrespondingtotheweightassignedtothedecisionmaker.

Step3. ByusingaggregationoperatorslikeNCWGandNCEWG,thedecisionmatrixisaggregatedbythe weightassignedtothemattributes.

Step4. The n alternativesarerankedaccordingtotheirscoresandarrangedindescendingordertoselect thealternativewithhighestscore.

6.Application

MobilecompaniesplayavitalroleinPakistan’sstockmarket.Theperformanceofthesecompanies affectsresourcesofcapitalmarketandhavebecomeacommonconcernofshareholders,government authorities,creditorsandotherstakeholders.Inthisexample,aninvestorcompanywantstoinvest hiscapitallevyinlistedcompanies.Theyacquiretwotypesofexperts:Attorneyandmarketmaker. Theattorneyisacquiredtolookatthelegalmattersandthemarketmakerisaquiredtoprovidehis expertiseincapitalmarketmatters.Dataarecollectedonthebasisofstockmarketanalysisandgrowth indifferentareas.Letthelistedmobilecompaniesbe (x1) Zong, (x2) Jazz, (x3) Telenorand (x4) Ufone,whichhavehigherratiosofearningsthantheothersavailableinthemarket,fromthethree alternativesof (A1) stockmarkettrends, (A2) policydirectionsand (A3) theannualperformance. Thetwoexpertsevaluatedthemobilecompanies xj, j = 1,2,3,4 withrespecttothecorresponding attributes (Ai, i = 1,2,3),andproposedtheirdecisionmakingmatricesconsistingofneutrosophic cubicvaluesinEquation(1)andEquation(2).TheEquation(3)representsthesinglematrixasthe aggregationofEqutiona1andEquation(2)byNCWGorNCEWG.TheEquation(4)isobtainedby applyingNCWGorNCEWGonattributes.Thedecisionmatricesareaggregatedtoasingledecision matrix.Attheendwerankthealternativesaccordingtotheirscoretogetthedesirablealternative(s).

Step1.WeconstructthedecisionmakermatricesinEquations(1)and(2).

Equation(1): Decisionmakingmatrixforthefirstexpert(attorney) Da is

X1

A1 A2

[0.2,0.6], [0.4,0.6], [0.5,0.8],0.7,0.4,0.3

X2 [0.3,0.5], [0.6,0.9], [0.3,0.6],0.3,0.6,0.7

X3 [0.6,0.9], [0.2,0.7], [0.4,0.9],0.5,0.5,0.6

X4 [0.4,0.8], [0.5,0.9], [0.3,0.8],0.5,0.8,0.5

[0.1,0.4], [0.5,0.8], [0.4,0.8],0.6,0.7,0.5

A3

[0.4,0.6], [0.2,0.7], [0.5,0.9],0.4,0.5,0.3

[0.5,0.9], [0.1,0.3], [0.4,0.8],0.8,0.3,0.6 [0.2,0.7], [0.1,0.6], [0.4,0.7],0.5,0.4,0.7

[0.2,0.6], [0.7,0.3], [0.3,0.8],0.4,0.6,0.5

[0.5,0.9], [0.7,0.9], [0.1,0.5],0.5,0.6,0.4

[0.2,0.7], [0.4,0.9], [0.5,0.7],0.6,0.4,0.5 [0.3,0.5], [0.5,0.9], [0.7,0.3],0.3,0.3,0.8

Equation(2): Decisionmakingmatrixforthesecondexpert(marketmaker) Dm is

A1 A2 A3

X1 [0.3,0.6], [0.2,0.6], [0.2,0.6],0.8,0.7,0.2

X2 [0.2,0.5], [0.6,0.9], [0.7,0.3],0.4,0.8,0.7

X3 [0.5,0.9], [0.2,0.6], [0.3,0.8],0.7,0.7,0.8

X4 [0.3,0.5], [0.3,0.9], [0.2,0.5],0.6,0.5,0.4

[0.3,0.8], [0.4,0.8], [0.3,0.8],0.6,0.7,0.4 [0.2,0.7], [0.2,0.6], [0.3,0.8],0.5,0.3,0.5

[0.4,0.9], [0.1,0.4], [0.5,0.8],0.6,0.5,0.7 [0.4,0.9], [0.1,0.4], [0.5,0.8],0.6,0.5,0.7

[0.2,0.5], [0.2,0.7], [0.5,0.8],0.6,0.7,0.2 [0.3,0.5], [0.3,0.9], [0.2,0.5],0.6,0.5,0.4

[0.4,0.7], [0.2,0.8], [0.7,0.3],0.6,0.7,0.7 [0.2,0.6], [0.5,0.9], [0.2,0.8],0.4,0.4,0.8

Step2. Let W = (0.4,0.6)T , thenthesinglematrixcorrespondingtoweight W byuseofNCWG operatoris

Equation(3): Thesingledecisionmatrix.

A1 A2 A3

0.2551,0.6000

0.2885,0.6732

0.3371,0.6968

[0.1933,0.6062], [0.4430,0.8001], [0.3418,0.8680], 0.6000,0.7000,0.4772

[0.2352,0.5577], [0.6000,0.9000], [0.3000,0.6634], 0.3618,0.7360,0.7000

[0.2000,0.5378], [0.2352,0.7000], [0.4279,0.8000], 0.5295,0.6634,0.2885

[0.2638,0.6581], [0.1999,0.6381], [0.3881,0.8484], 0.4621,0.3881,0.2223

[0.5253,0.8670], [0.1515,0.6000], [0.4621,0.8448], 0.3371,0.4621,0.3301

[0.3680,0.6325], [0.4210,0.9000], [0.1614,0.5000], 0.5626,0.5426,0.4000

0.3465,0.6325

0.2416,0.7449

0.2639,0.8385

0.3881,0.7000], 0.6000,0.6041,0.6118

0.5000,0.9000

0.2416,0.7647

[0.2375,0.6195], [0.2885,0.7916], [0.3567,0.8146], 0.6315,0.5757,0.2851

0.4426,0.7657], [0.2165,0.5915], [0.5382,0.7804], 0.4827,0.5729,0.5282

[0.3500,0.6616], [0.3335,0.8142], [0.3131,0.7498], 0.5791,0.6133,0.4439

[0.3327,0.6774], [0.3630,0.7787], [0.2888,0.7396], 0.4906,0.5359,0.5692

(4) Step4. Usingthescorefunctionwerankthealternativesas: S(X1) = 0.0321, S(X2) = 0.0548, S(X3) = 0.0839and S(X4) = 0.0969, X3 > X2 > X1 > X4 Themostdesirablealternativeis X3.

7.Conclusions

Dealingwithreallifeproblems,decisionmakersencounterincompleteandvaguedata. Thecharacteristicsofneutrosophiccubicsetsenablesdecisionmakerstodealwithsuchasituation. Consequently,foreachsituationwedefinedthealgebraicandEinsteinsum,productandscalar multiplication.Itisoftendifficulttocomparetwoormoreneutrosophiccubicvalues.Thescoreand accuracyfunctionsaredefinedtocomparetheneutrosophiccubicvaluesvalues.Usingtheseoperations wedefinedneutrosophiccubicgeometric,neutrosophiccubicweightedgeometric,neutrosophiccubic Einsteingeometric,andneutrosophiccubicEinsteinweightedgeometricaggregationoperatorswith someusefulproperties.Inthenextsection,amulti-criteriadecisionmakingalgorithmwasconstructed. Inthelastsection,adailylifeproblemwassolvedusingmulti-criteriadecisionmakingmethod (MCDM).Thispaperisbasedonsomebasicdefinitionsandaggregationoperators,whichcanbe furtherextendedtonewhorizons,likeneutrosophiccubichybridgeometricandneutrosophiccubic Einsteinhybridgeometricaggregationoperators.

AuthorContributions: Allauthorscontributedequally.

Acknowledgments: The3rdauthorwouldliketothanktheDeanshipofScientificResearchatMajmaahUniversity forsupportingthisworkunderProjectNumber1440-52.

ConflictsofInterest: Theauthorsdeclarenoconflictofinterest.

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