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DesignofExperiments(DOE)

Thesubjectmattercontainedhereiniscoveredbyacopyrightownedby:

FORDMOTORCOMPANY

CORPORATEQUALITYDEVELOPMENTCENTER

DEARBORN,MI

Copyright?2001FordMotorCompany

Thisdocumentcontainsinformationthatmaybeproprietary.ThecontentsofthisdocumentmaynotbeduplicatedbyanymeanswithoutthewrittenpermissionofFordMotorCompany.

Allrightsreserved

TableofContents

TOC\h\z\t"p-head1,1,p-head2,2"

Introduction

3

Agenda

3

PurposeoftheCourse

5

Goal

5

Objectives

5

DesignofExperiments(DOE)History

7

DOEandtheSTAEngineer

7

DefinitionofDOE

7

UsesofDOE

9

Planning

10

StepsforExperimentalDesign

10

DOEPlanAnalysis

13

CommonFailures

13

InfluencesonControlFactors

13

EvaluationProcedures

15

StepsoftheDOE

17

ChecklistforDOE

19

QuestionstoAsktheSupplieraboutthePlan

21

Addingup(Pool)StandardDeviations

22

StepsforSimpleAnalysis

24

FactorialExperiments

25

BasicConcept

25

InteractionEffects

29

FractionalFactorials

30

UseoftheResults

32

APracticalAidforExperimenters

33

SignificanceofFactorEffects

35

Activity:TypicalThree-factortwo-levelexperiment

36

FractionalFactorialDesign

37

DOEMethods

39

ClassicalVersusTaguchiMethods

39

Taguchi’sLossFunction

41

Optimalcostofquality

42

TaguchiMethod

44

TaguchiCake-BakingExample

45

TipsfromTaguchi

47

Signal-to-NoiseRatio

49

ControllingNoise

51

InnerandOuterArrays

52

SignaltoNoiseforMaximumorMinimum

54

ConfirmingtheExperiment

56

EvolutionaryOperation(EvOp)

58

ANOVAMethods

60

AssumptionsforApplyingANOVA

60

AnalysisofVariance

62

CalculationsforANOVATable

64

CaseStudies

69

CaseStudy1—TheMismatchedMuffler

70

CaseStudy2

71

Summary

74

AdditionalResources

76

SubjectMatterExpert(SME)

76

AdditionalTraining

76

References

76

ReviewofAnswerstoPre-TestandPost-Test

77

Introduction

WelcometotheDesignofExperiments(DOE)course,partoftheCorporateQualityDevelopmentCentercurriculum.ThisisoneofthecoursesintheQualityToolsmodule,whichisdesignedtoprovideSTAEngineerswithpracticalknowledgeofthetoolsrequiredtosuccessfullyaccomplishtheirjobs.

Agenda

Introduction

DesignofExperiment(DOE)History

Planning

DOEPlanAnalysis

FactorialExperiments

DOEMethods

CaseStudies

Summary

AdditionalResources

PurposeoftheCourse

Thiscoursepresentsstatisticalconceptsneededtodesign,conduct,analyze,andinterpretmulti-factorexperiments,whichareusedinfactorscreening,characterizingandoptimizingofprocesses.

Goal

ThegoalofthiscourseistoprovideSTAEngineerswiththeknowledgetoreviewaSupplier’sDOEtodetermineifithasbeensetup,performed,implementedandanalyzedcorrectly.

Objectives

Uponcompletionofthiscourse,theparticipantwillbeableto:

DefinetheroleoftheSTAEngineerinrelationtoDOEandprocessimprovement

DefinethepurposeofDOEandapplicationtypes(ClassicalandTaguchi)

IdentifycriteriaforconductingaDOE

ExplainthebasicstepsofconductingaDOE

RecognizeappropriateandinappropriateoutcomesandprocessesofaSupplierDOE(casestudies)

IdentifycommonfailuresthatanSTAEngineermayencounterwhilereviewingaDOE

IdentifystrategicquestionsthatshouldbeaskedwhenreviewingaDOE

ExplaintherelationshipbetweenDOEandtherestofthequalitytools

IdentifyadditionalresourcesavailabletoassistwiththeconductoranalysisofaDOE

DesignofExperiments(DOE)History

ThehistoryofDOEgoesbacktothe1920s,whenitwasusedinagriculture.TodayitisawidelyexpectedengineeringtoolusedatFordandbyitsSuppliers.

DOEandtheSTAEngineer

TheroleofanSTAEngineeristounderstandDOEinordertomakesoundjudgmentswhendealingwithSuppliers.TheSTAEngineerwillneedtorecognizeifaSupplierhastheabilitytosetup,perform,implement,andanalyzetheimprovementprocesscorrectly.

DefinitionofDOE

DOEisatotalplanofactionaimedatobtainingknowledgeaboutagivenprocesstoimproveitortosolveaproblem.Theobjectiveofadesignedexperimentistoobtainmoreinformationwithlessexpenditureofresourcesthancanbeobtainedbytraditional(onefactoratatime)techniques.

DOEwaspioneeredbyR.A.Fisher,anagriculturalscientist,inEnglandinthe1920s.Heusedthetechniquetostudytheeffectontheoutcomeofmultiplevariablessimultaneously.Fisherwantedtofindouthowmuchrain,water,fertilizer,sunshine,etc.wereneededtoproducethebestcrop.

PESTICIDE

B2

B1

2"

4"

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FERTILIZER

A2

UsesofDOE

DesignofExperimentscanplayakeyroleinunderstandingandimprovingthereliabilityofFord’svehicles.

Experimentationcanbeusedto:

Modeldegradationoffunctioninvehiclesystems

Identifyfactorsthatsignificantlyimprovesystemlifeordegradationrate

Modelmultivariatefunctionalrelationshipsthatcanbeusedforoptimizationstudies

DOEatFordwill:

Reduceimperfectionsinparts

Reducecosts

Reduceguesswork

Reducelosttime

Improvecustomerrelations

ImproverelationswithSuppliers

Improveproductivity

ClassicalDOEprovidesapredictiveequation.

TaguchiDOEquicklysolvesproblems.

Planning

StepsforExperimentalDesign

Statetheproblem(s):Usequalitymeasurestoclearlyindicatethelevelofqualityorloss.ThismaycomefromtheGlobal8Danalysis.Theproblemstatementshouldaddressthefollowing:

Whatdataexiststhatcharacterizestheproblemasitoccurs

Howtheproblemisobserved

Whentheproblemoccurs

Howseveretheproblemis

Wheretheproblemoccurs

Statetheobjectiveoftheexperiment:Thisstatementshouldaddressthescopeoftheexperimentandshouldbebasedon:

Theproblemstatement

Competitivebenchmarkinformationconcerningtheproblem

Customerinformationconcerningtheproblem

StartDate_________ EndDate______________

Selectthequalitycharacteristicsandmeasurementsystems:Thecharacteristics(responses,dependentvariables,oroutputvariables)shouldberelatedtocustomerneedsandexpectations.Thechartbelowcapturestheresponse,thetype,andtheanticipatedrangethathelpstodeterminethemethodofmeasurement

Response

Type

AnticipatedRange

MeasurementMethod/Accuracy

StepsforExperimentalDesign,continued

Selectthefactorsthatmayinfluencetheselectedqualitycharacteristics:Processflowdiagrams,cause/effectdiagrams,specifications,statisticalprocesscontrolchartresultsaresomesourcesforthisinformationandmaybecapturedinachartsimilartotheonebelow.

Factor

Type

ControllableorNoise

RangeofInterest

Levels

AnticipatedInteractionswith

HowMeasured

1.

2.

3.

4.

Determinethenumberofresourcestobeusedintheexperiment:Considerthedesirednumber,thecostperresource,timeperexperimentaltrialandthemaximumallowablenumberofresources.

Determinewhichdesigntypesandanalysisstrategiesareappropriate:Discussadvantagesanddisadvantagesofeach.

Selectthebestdesigntypeandanalysisstrategytosuittheneeds.

Determineifalltherunscanberandomizedandwhichfactorsaremostdifficulttorandomize.

Conducttheexperimentandrecordthedata:Monitorboththeeventsforaccuracy.

Analyzethedata,drawconclusions,makepredictions,anddoconfirmatorytests.

Assessresults,makedecisions,anddocumentresults:Evaluatenewstateofqualityandcomparewithlevelpriortoimprovementeffort.

DOEPlanAnalysis

Whenanalyzingtheplan,itisimportanttounderstandthecommonfailuresandinfluencesofcontrolfactorsinordertoverifythattheplanhasaccountedforthesefactors.

CommonFailures

ThecommonfailuresthatoccurwhenconductingaDOEare:

Dataiscollectedwhenthereisonlyonevariable.

Supplieroftenleavesouttheinteraction.

Supplierhasnotidentifiedrecentchangesintheprocess.

InfluencesonControlFactors

Temperature

Differentoperators

Humidity

Locationofplant

Environmentalfactors

Lackofconsistency

Differentsamplesize

EvaluationProcedures

Definepreciselytheproceduresforrunningtheexperiment,indicatingwhichfactorscanbeeasilychangedfromoneruntothenext.

Getinformationregardingpastdataandrepeatability.

Determinedesirabilityandopportunitiesforrunningtheexperimentinstages.

Listrelationshipbetweentheindependentvariableandresponsevariable.

StepsoftheDOE

Tasks

TaskAids

Who

Stateproblem(s)

QualityFunctionDeployment,testfailures,warrantyitems,scrapitems,ParetoAnalysis

Productand/orprocessexperts

Stateobjective(s)

Customerrequirements,competitivebenchmarks

Selectqualitycharacteristic(s)&measurementsystem(s)

Gagerepeatability&reproducibilityanalysis

Selectfactorsandinteractions;determinecontrolandnoisefactors

Fishbonediagram,flowcharts,SPCcharts

Selectlevels

Specificationlimits,operationallimits

Selectorthogonalarray(s)

OAselectiontablesD-1,D-2;blankOAs

Assignfactors&interactionstoorthogonalarray(s)

AssignmenttablesD-3,D-4;interactiontables;OAmodificationrules

DOEexpert

Conducttests

Computersoftware,trialdatasheets,randomizationplan,partserializationplan,materiallogisticsplan

Product,process,andDOEexperts

Analyzeandinterpretdata

Observationmethod,columneffectsmethod,ANOVA,computersoftware,plotting,ranking(magnitude&timeorder)

DOEexpert

ConductConfirmationTest

Estimatesofthemeanconfirmationexperimentflowchart

Product,process,andDOEexperts

ChecklistforDOE

Identifytheproperpeopletobeinvolvedintheprocess/productteamandtheleaderoftheinvestigationteam.

Describeinmeasurabletermstheproblem—howthepresentsituationdiffersfromwhatisdesired.

Obtainagreementfromthoseinvolvedon:

Scopeoftheinvestigation

Otherconstraintssuchastimeorresources

Obtainagreementonthegoaloftheinvestigation.

DetermineifstagingforDOEisappropriateorifotherresearch,suchasSPC,shouldbeaccomplishedfirst.

Usebrainstormingandproblem-solvingtoolstodeterminewhatfactorsmaybeimportantandwhichofthemcouldinteract.Totalagreementisrequiredtoeliminateany.

Choosearesponsethatrelatestotheunderlyingcauseandnotthesymptomandismeasurable,ifpossible.

Determinethetestproceduretobeusedandassessrepeatabilityandreproducibilityifapplicable.

Determinewhichofthefactorsarecontrollableandwhicharenot.

Determinethelevelstobetestedforeachfactor(experimentboldly).

Chooseordeveloptheappropriateexperimentaldesign.

Obtainfinalagreementfromallinvolvedpartiesonthe:

Goal

Approach

Allocationofroles

Experimentaldesign

Testprocedure

Timingoftheworkplan

Arrangetostageappropriateproduct,machinery,andtestingfacilities.

Monitortheexperimenttoensureproperproceduresarefollowed.

Usetheappropriatetechniquestoanalyzethedata.

Prepareasummaryoftheexperimentwithgraphicalportrayalofconclusionsandrecommendations.

QuestionstoAsktheSupplieraboutthePlan

Hastheproblembeenadequatelydefinedbasedonreportedeffects?

Wasthe8Dprocessfollowedtodetermineinterimactionsandidentifyprocesselementscontributingtotheproblem?

Issufficientstatisticaldata/evidenceavailabletonarrowdownvariablestothesignificantfew?Whatstatisticaltoolswereused?

IsthereanestablishedprocedureforapplyingDOEtechniquesandevaluatingtestresults?

Hasadequatescreeningbeenperformedtodetermineprocessalternativesagainst“must”criteriaand/oreliminateunacceptablealternatives?

Istheresufficientstatisticalevidenceforverifyinglevelsofprocessstabilityandcapabilityovertherecentpast?

Haveanyprocesschanges,asdefinedinthePPAPmanual,beenmadeintherecentpastincludingsubcontractorprocesses?

Hasacriteriamatrixbeencompletedwithweightingof“desirablecriteria”(allpotentialimprovements)inselectionofresponsevariables?

Note:ResponsevariablesmustbecustomerCriticaltoQuality(CTQ)characteristicsinaddressinggivenproblems.

Indefiningtheexperimentalprogram,whatcriteriawasusedinselectingtheindependentfactorsaffectingtheresponsevariable?

Wasappropriatescreeningdonetoeliminatefactorsnotjudgedashavingamajorimpactonresponsevariable?

Tofurtherrefinethelistoffactors,weretraditionaltestingmethods,suchasTrialandError,SpecialBatchRuns,PilotRuns,andPlannedComparisons,usedtoidentifythe2-3factorshavingthebiggestimpactontheresponsevariable?

Was/istheexperimentdesignedtotesttheimpactofthefactorstogetherratherthantestingonefactoratatime?

Wasanobjectivemeasurementsystemusingqualifiedtestorgagingmethodsselectedtomeasureresponses?

Addingup(Pool)StandardDeviations

Firstsquarethestandarddeviations(=thevariance)

Multiplyeachvariance(S2)byn-1(toweightthemproperly).

Thenaddthemanddividebythesumofthetwo(n-1)’s

Hereistheformula:

StepsforSimpleAnalysis

Checkdataforaccuracy.

Conductvisualanalysisofthedata.

Calculatetheaverageateachlevelforeachcolumn.

Calculatetheeffectsandhalf-effectsforeachcolumn.

PlottheaveragesfromStep#3.

GenerateaParetodiagramoftheabsolutevalueofeachhalf-effectfromStep#4.

Determinethe“importance”ofeachhalf-effect.*Whenappropriate,constructinteractionplotsfor“important”interactions.

Generateapredictionequationusing“important”half-effects.

Baseduponyourobjective,selectthebestsettingsforimportantfactors/interactions.

UsingthepredictionequationgeneratedinStep#8,predicttheresponse.Usethisvalueasatargetforverificationruns.

* IfusingsoftwaresuchasDOEKISS,usestatisticalsignificancetodetermineimportanthalf-effects(coefficients).

AdaptedfromtheAirAcademyAssociation

FactorialExperiments

BasicConcept

ThefirstfactorialexperimentsweredoneinEngland,beforeWWII,byR.A.Fisher.Thesewereagriculturalexperiments.Thepurposewastoseehowvariousfactorsaffectedcropgrowthbyapplying“treatments”to“blocks”ofland.

Example1:

Ablockoflandisdividedintofourparts.Twotypesoffertilizer(A1andA2)areappliedfromeasttowest,andtwotypesofpesticide(B1andB2)areappliedfromnorthtosouth.

PESTICIDE

B2

B1

2"

4"

6"

8"

A2

A1

FERTILIZER

Thenumbersinsidethesquaresrepresenttheaveragegrowthforeachsquareofland.

Evenwithoutfurtheranalysis,itseemsobviousthatfertilizerA2andpesticideB2wouldbegoodchoices.

Example1:(continued)

Todeterminetheeffectofeachfactor,calculatetheaverageforlevels1and2ofeachfactor,andthenfindthedifference.

FERTILIZER

A1

A2

B1

B2

PESTICIDE

AVE.

EFFECT

AVE.

EFFECT

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InteractionEffects

Althoughtheeffectis4inchesandthepesticideeffectis2inches,acheckisneededtoseewhethertheseeffectsareadditive.Itispossiblethatwhenthebestfertilizerandthebestpesticidearecombined,theresultwillbe0inchesor12inchesinsteadof6inches.

Averagingdiagonallydoesthecheckforadditiveeffects.

Ifthefactoreffectsareadditive,thendiagonalaveragesarejustestimatesofthegrandaverageandtheinteractioneffectsshouldberelativelysmall.

Ifthereisalackofadditivity,itiscalledan“interaction.”

FERTILIZER

A1

A2

B1

B2

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EFFECT

AB

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FractionalFactorials

Sofartherearesixaveragesfromonlyfoursetsofmeasurements.Thistremendousefficiencycanbeincreasedfurtherifwehappentoknowfromexperiencethatthefactorsareadditive.

Example2:

Assumethatthewateringschedulewillbeadditivewithfertilizerandpesticide.

Athirdfactorcanbeaddedbyputtingdifferentwateringschedulesonthediagonalsquares.

C1

C2

PESTICIDE

B2

B1

A2

A1

FERTILIZER

Theadditionalfactormightchangetheaverages,butifthehypothesisaboutadditivityiscorrect,thedifferenceintheaverages(effect)willstaythesameforfactorsAandB.

Thisiscalledafractionalfactorialbecausethereisnotenoughdatatoseparatealltheinteractionsthataretheoreticallypossible.

UseoftheResults

Thefactoreffectsascomputedrepresentthedifferencebetweentheresponseatthehighandlowlevelsofthefactors.Ifthefactoreffectisdividedbythedifferencebetweenthehighandlowlevelsofthefactors,theresultswillbethechangeintheresponseforacodedunitchangeinthefactor.

Themodelunderlyingthetwo-levelfactorialiswrittenintermsofcodedfactors.Themodelis:

+++…++

higherorderinteractions

Scaling

-1+σ+1

wherepredictedresponse

,

Notethatthecoefficientsofthemodelarehalfofthecorrespondingfactoreffects,sincethecodedlevels(+1)and(-1)differbytwounits.Intheexample,theequationis:

Someworkersprefertoomitthetermswithinsignificanteffects.

SignificanceofFactorEffects

Ifacomputedfactoreffectislarger(inabsolutevalue)thanthe“minimumsignificantfactoreffect,”theexperimentercansafelyconcludethatthetrueeffect?isanonzero.Theminimumsignificantfactoreffect[MIN]isderivedfromanappropriatet-testofsignificance.Theformulais:

where

[MIN] =minimumsignificantfactoreffect

t =thevalueofstudent’s“t”atthedesiredprobabilitylevelforthenumberofdegreesoffreedomintheestimates

m = thenumberof+signsinthecolumn(forfactoreffectcolumnsandforthemeancolumn)

k =thenumberofreplicatesofeachtrial

s = pooledstandarddeviationofasingleresponseobservation

Intheexample,theestimateofthestandarddeviationis4.4with11degreesoffreedom.Thus,theminimumsignificanteffectis:

Activity:TypicalThree-factortwo-levelexperiment

FractionalFactorialDesign

DesignandAnalysisWorksheet

*Averageofallresponsesincolumn(Y)

**Ignoringthreefactorhigherinteractions

Average+:AverageofallYvaluesassociatedwithapositivecoefficient(+)inarespectivecolumn

Average-:AverageofallYvaluesassociatedwithanegativecoefficient(-)inarespectivecolumn

LocationEffect[Difference]:(Average+)-(Average-)

VarianceEffects[Ratio]:LargerAverage/SmallerAverage

ProportionEffect[Difference]:(Average+)-(Average-)

DOEMethods

ClassicalVersusTaguchiMethods

Classicalmethodologyemphasizes:

Usingsequentialexperimentationtomodelprocessbehavioranddevelopempiricalprocessmodels(includingmodelingtheeffectof“noise”factors)

Predictingfutureprocessbehavior,includingoptimalsettingsfromempiricalmodels

Investigatingandisolatefactorsaffectingmeanandvariationindependently

Selectingexperimentaldesignfromconsiderationofthetrade-offsinrunningafractionofafullfactorialdesign

Forexamplea28-4designinvestigatestheeffectsof8factorsin16runs,andthetrade-offsareknownbeforerunningtheexperiment.Additionalexperimentationmayberequiredtoclearlyidentifytheeffectsofinteractions.

Taguchimethodologyemphasizes:

Robustdesign-searchingforthesetofconditionstoachieveoptimumbehavior

Minimizingthelossfunction-andeconomiclossduetorunningatuncontrolledconditions(noise)

Selectingexperimentaldesignfromexaminationoflineargraphs,whichallowsinvestigationofdesiredinteractioneffects,basedonprocessknowledge

Taguchi’sLossFunction

Taguchi’slossfunctionisatheoreticalquadraticrelationship.Taguchicontends,asproductcharacteristicsdeviatefromthenormalaim,lossesincreaseaccordingtoaparabolicfunction.Considerthefollowingdiagrams:

LSLUSLLSLUSL

All All All

TaguchiConcept

Target

TraditionalConcept

L

Bad Good Bad

Formula:L=K(Y–T)2

Where;

L=lossindollars T=targetvalue(normalaim)

K=costcoefficient Y=actualqualityvalue

Althoughtheequationcannotbeproven,itemphasizesthepointthataconsistentproductminimizesthetotalloss.Merelyattemptingtoproduceaproductwithinspecificationsdoesn’tpreventloss.Taguchifurtherdefinesqualityasthelossinflictedonsocietyaftertheshipmentofaproduct.

Example: Thespecificationsforaproductare6and14,withatargetof10.If20%oftheproductisproducedatexactly8,20%ontargetand60%atexactly11,whatisthelossfunction?

Solution: L=.2K(8-10)2+.6K(11-10)2

L=.8K+.6

L=1.4K

Optimalcostofquality

HistoricalView CurrentView

TaguchiMethod

Taguchi’smaincontributionishisconceptofrobustness.

Whendevelopingadesign,oraprocess,twotypesoffactorsareconsidered:

Controllablefactors(orDesignfactors)—InnerArray

Noisefactors(orEnvironmentalfactors)—OuterArray

Controllablefactorsarecanbesetandmaintained.

Noisefactorsareimpossible,difficult,ortooexpensivetocontrol.

TaguchiusesthestatisticcalledSignaltoNoiseRatio(S/N).Theprimarypurposeistomaximizeperformancewhileminimizingvariation.

TherearethreetypesofS/Nratios:

Maximizingtheresponse:

Minimizingtheresponse:

Targetresponse(yatoptimumvalueandminimizeS2)

TaguchiCake-BakingExample

TipsfromTaguchi

Thegoalofengineeringexperimentsistoeconomicallyimprovereal-worldproductsandprocesses,notscientificknowledge.

Useaconsistentmethodthat’sversatileenoughtoworkalmostanytime(orthogonalarrays).

Useasystemtoadaptthemethodtotheproblem,notviceversa(lineargraphs).

Studylotsofvariables,includingthe“noise”variablesthatthedesigndoesn’tcontrol.

Studyalimitednumberofinteractions,selectedbyengineeringknowledgeandexperience.

Chooseparametersthatminimizevariation,andalsomovethedesigntowardthetarget.

Predicttheresults.

Runaconfirmingexperimenttocheckreal-worldreproducibility.

Becarefulwhatyouoptimize.Selectaqualitycharacteristicthat’spracticalandinfluencesmarketshare.

Beforewecanbegin,wemustbeabletodefinequalityinmeasurableterms.

Signal-to-NoiseRatio

Tobesureofchoosingthebestfactorlevels,considerboththemeanresponselevelandthevariation.Toaccomplishthismoreeasily,itisusefultohaveasinglestatisticthatcorrelateswellwithquality,asmeasuredbythelossfunction.

Signal-to-noiseratio()iscommonlyusedintheelectronicsindustry.IthasbeenadaptedbyTaguchiforevaluatingexperiments.

Tomakethenumbersadditive,theyareusuallyconvertedtodecibels(d?).

Taguchihasdefinedawholegroupofstatisticshecallssignal-to-noiseratios.Ahighsignal-to-noiseratioforanexperimentalconditionmeanstherewasalargechangeinthedesireddirectionand/orareductioninvariation,i.e.,ahighsignal-to-noiseratiomeansbetterquality.

ControllingNoise

Noiseisthevariationoffactorsthataren’tnormallycontrolledbytheprocessorproduct.

OuterNoise

Temperature,humidity,customer,etc.

InnerNoise

Deteriorationwithtime,etc.

ProcessNoise

Betweenproduct,etc.

Robust

Notsensitivetonoise.Arobustdesignshowslittlefunctionalvariation,regardlessofnoise.

InnerandOuterArrays

Ifsomenoisefactorscanbetemporarilycontrolledduringtheexperiment,an“outerarray”issetuptodothis.Thepurposeistoensurethatthenoiseisproperlydistributedamongtheruns.Signal-to-noiseratioscanautomaticallyexploitnonlinearitiesandin