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Thustherearemultiplereasonsforconsideringtheeffectofmeasurementcostonproductprofitability,

beginning with the global objective of improving competitive posture,just discussed. The effect of

measurementcostbecomesyetmore intenselyfocusedwhen,as isoftenthecase,measurementsare

alsousedtocontrolthemanufacturingprocess. Processcontrolishighlydesirableasitnotonlyreduces

thenumberofmeasurements (andhence costs),but itproactivelyadjusts theprocessand thus can

reducethenumberofoutofspecificationcomponents.Costsarenowleveragedupasafewprocess

control measurements may affect several hundred manufactured components. Additionally, anerroneousmeasurementsystemnowhastheopportunitytomisadjustthemanufacturingparameters,

creatingoutofspecificationcomponents,andthentopassthemontothecustomer. Atalowerlevel,

anunderstandingofmeasurement cost canbeapplied tooptimize the contributionofmeasurement

resources to the financial bottom line, tojustify and predict the benefits ofmeasurement resource

expendituresandtofocusavailablemetrologyresourcestobestpromoteprofitability.

Inrecentyearstherehavebeensignsofashiftawayfromthetraditionalviewofproductmetrology,

andtowardanunderstandingofthevaluemeasurementcanbringtoamanufacturedproduct. Most

noticeably,thisconcepthasgainedtractioninsemiconductormanufacturingand,tosomeextent,in

chemicalmeasurement. However,ithasbeenourobservation,basedonourconsultingandsoftware

activitiesintheareaofCMMmeasurementuncertaintyevaluation,thatinthefieldofdimensional

measurement,and3Dmetrologyinparticular,theconventionalperspectiveremainsdominant. While

therearesignsofashiftamonglargemanufacturingoperations,certainlyinthearenaofmediumto

smallbusinessestherehasbeennogroundswellofchange.

Itappearstheremightbeseveralreasonsthatcoordinatemeasurementhaslaggedotherareasof

metrologyinthisregard. CMMsarecomplexsystems. Theiroutputissubjecttotheeffectsofmultiple,

sometimesinterrelatedinfluences. ThustheuncertaintyofaparticularCMMmeasurementresultis

technicallydifficultandpossiblyexpensivetoevaluate. TheaverageCMMpractitionermaybe

inadequatelyequippedtoperformthisevaluationanddefenditsresults. Perhapsmostproblematicis

thefact

that

measurement

cost

analysis

is

acombined

technical/business

exercise.

Metrologists

may

lacksufficientlydetailedinsighttotheeconomicimplicationsoftheirmeasurements,whilebusiness

managersmayfindthetechnicalaspectsdaunting.

Ourpurposehereistodiscussbothtechnicalandbusinessdecisionmakingcomponentsofthisissueas

itappliestocomplexmeasurementsystems,CMMsinparticular,withsomegeneralitybutinsufficient

detailtohelpprepareCMMmetrologiststocommunicateconcerningtheirdataanditseconomic

impactwithothersinthemanufacturingenterprise. Weareobligedtomakeclearattheoutsetthat,in

ordertomakemeaningfuljudgmentsoftheeconomicsofCMMmeasurementuncertainty,itistask

specificmeasurementuncertainty1thatwemusthave;moregenericexpressionsofCMMmeasurement

1TaskspecificmeasurementuncertaintyistheuncertaintyapplicabletoevaluationofaspecificGD&Tparameter

ofaspecificpartfeature,underparticularconditionsofmanufactureandmeasurement;forexample,The

uncertaintyofthediameterofthemaximuminscribedcylinderthatwilljustfitinsidethisnominally3inch

diameterhole,measuredwiththisparticularCMM,underthesespecificconditions,is0.0008inchesat95%

confidence.

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uncertaintywillnotdo. Inwhatfollows,thediscussionwillcoverinturntwodistinctphasesoftheCMM

costanalysisprocess:riskanalysis,whichisapurelytechnicalexercise,andprofitabilitycalculation,

whichinmanyofitsaspectsisabusinessexercise.

RecentTechnicalActivitiesTherehavebeenrecentdevelopmentsintwoareasrelevanttoCMMmeasurementcostanalysis. Oneoftheseistheavailabilityofcommercialsoftwarepackagesfordevelopmentoftaskspecificuncertainty

statementsforCMMmeasurements. Onesuchproductwasdescribedinanearlierpaperinthissession

[1]. Theotheristheappearanceofanumberofnationalandinternationalstandardsaddressingthe

someofthetechnicalaspectsofmeasurementuncertaintyevaluation[214]. Whilethesepublications

dealwithmanyofthetechnicalaspectsofmeasurementcostanalysisandtwoofthem[7,8]touchon

theroleofproductacceptancedecisionrulesasbusinessissues,therehasbeennodetaileddiscussion

oftheeconomicimpactsofCMMmeasurementuncertainty.

CMMRiskAnalysisThegeneraltopicofriskanalysisinmeasurementhasbeencoveredin[10];thereaderisreferredthere

forbasicinformation. Here,wewilldescribethegeneralprocessandwillgiveamoredetailed

descriptionofthepropertiesofCMMmeasurementsthatmaketheirtreatmentsomewhatunique. As

theseuniqueaspectsarementioned,suggestedapproachesfordealingwiththemwillbeprovided.

Amanufactureditemthatissubjecttosomespecificationmaybeinoneoftwoconditions: Itmay

conformtothespecification(C)oritmayfailtoconform(C). Iftheitemissubjectedtomeasurement

therearetwopossibleoutcomes:itmaypass( P )orfail( F )theinspection. Thus,therearefour

possibleoutcomes

of

aproduct

inspection:

The

item

may

conform

to

specification

and

pass

inspection

( PC)oritmaynotconformtospecificationandfailinspection( FC),bothofwhicharedesired

outcomes,resultinginacceptanceofagoodpartorrejectionofabadpart. Inthepresenceof

measurementerrortwootherconditionsmayberealized: Thepartmay,infact,notconformto

specificationbutpassinspection( PC). Thisisanundesirableoutcome,resultinginacceptanceofan

outoftoleranceitem. Alternatively,apartthatconformstospecificationmayfailtheinspection( FC).

This,too,isundesirablesinceitleadstorejectionofagooditem. Theprobabilitiesoftheselasttwo

eventsarecalled,respectively,theconsumersandproducersrisksinreferencetothepartywho

usuallybearsthecostoftheerror. Therelationshipsoftheserisksareoftenrepresentedina

contingencytable(Figure1).

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Calculationoftheconform/nonconformprobabilitiesrequiresknowledgeoftheproductionprocess

probabilitydensity. Theconformanceprobabilityisgivenby

( | ) ( | ) , ( | ) 1 ( | )C

p C I p x I dx p C I p C I= =

where,forexample, ( | )p C I denotestheprobabilityofconformancegivenanypriorinformation,I ,

( | )p x I istheprocessprobabilitydensityandtheintegralisovertheacceptanceregion.

Thenthe

consumers

risk

is

( ) ( ) ( )0 0 0| | |Cx R x R

R p PCx I dx p PC xI p x I dx

= = i

wheretherangeofintegrationisoverallvaluesofxthatareoutsidetheacceptancezone,

( )0|p PC xI istheprobabilitythatacharacteristicknowntobenonconformingneverthelessproduces

Figure1.ContingencyTableforProductMeasurement

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ameasurementresultwithintheacceptancezone,seeFigure2,and ( )0|p x I isthepriorprobability

densityforthemeasurand. Similarly,theproducersriskisgivenby

( ) ( )0 0| |Px C

R p FC xI p x I dx

= i

wherenowtheintegrationisovertheacceptancezone. Figure2isanillustrationoftheconsumersand

producersrisksforthecaseofaGaussianerrordistribution.

Figure2. Consumer'sandProducer'sRisks

AssumptionofGaussianformfortheproductionandmeasurementprobabilitydensitiesisacommon

practice,andofferstheadvantageofcomputationaltractability;thisistreatedindetailin[10]. Such

assumptionscanproverashinthecaseofCMMmetrologyandcanleadtosubstantialerrors. Infact,

normaldistributionsoftenturnouttobeasmuchtheexceptionastherule.

ProductionProbabilityDensityTherearetwoissuesinapplyingprobabilitydensitiesinmeasurementriskanalysis. Thefirstisto

determinetheinherentprocessvariability. Thisistypicallydonebymeasuringalargesampleofthe

productandplottingahistogramoftheresults. Anynonrepeatabilityofthemeasurementsystemwill

besuperimposedontheproductionvariation. Ideallythecontributionofmeasurementvariabilitywill

bemadenegligiblebyusingameasurementsystemwithsufficientlysmallrepeatability. Alternatively,

measurementrepeatabilitycanbedeterminedbyrepeatedlymeasuringastableartifactandlookingat

thedistributionoftheresults. Iftherepeatabilityofthemeasurementsystemsodeterminedis

characterizedbyastandarddeviation,m ,andthetotalstandarddeviationofameasurementis T ,

theprocessstandarddeviationis2 2

p T m = . Itis,ofcourse,prudenttoperiodicallyverifythat

theprocessremainsstable.

Thesecondissueischoiceofanappropriateandconvenientrepresentationoftheprobabilitydensity.

Productiondensitiesfortwosidedmeasurements,e.g.size,lengthorangle,canoftenbereasonably

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representedasGaussians. Onesidedmeasurements,e.g.form,positionororientation,represent

instancesinwhichvaluesnearzeroarenotfrequentlyobservedandtheGaussiandistributionis

thereforeinappropriate,sinceitwillassignfiniteprobabilitiestovaluesthatarephysicallyimpossible. It

isconceivablethatdistributionsofinherentlyonesidedquantitiescouldbemodeledbyaGaussian

distributiontruncatedatzero,butitismorepracticaltouseadistributionmodelthatnaturallygoesto

zero.The

beta

distribution

is

one

such

that

works

well;

it

is

treated

in

Appendix

IIof

[10]

and

illustrated

inFigure3.

Figure3. GammaProbabilityDensityforaOnesidedMeasurement

MeasurementProbabilityDensityWhileCMMssharemanysimilaritieswithotherdimensionalmeasurementtechnologies,similar

complexitiesarisewithregardtothemeasurementprobabilitydensity,dueprimarilytothemultitudeof

variablesthatcaninfluencethemeasurementresultandthecomplexitiesoftheirinteractions[1].

WhileapproximatelyGaussianmeasurementdensitiesaresometimesobserved,thevarietyof

distributiontypesistypicallygreatanddifficulttopredictfromintuitionoranalysis. Afewobserved

measurementdistributionsarepresentedasexamplesinfigures47,whichwerechosenfortheir

relativesimplicity

and

represent

cases

where

asingle

error

source

was

known

to

predominate.

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Figure4. MeasurementDistributionforaLengthmeasurement

Thisisanobservedmeasurementdistributionforalengthmeasurementmadeunderconditionswhere

thegeometricerrorsoftheCMMwereknowntopredominate. Inthisinstance,aGaussiandistribution

mightbeareasonableapproximation.

Figure5. MeasurementDistributionwithTemperatureVariation

Here.InFigure5,themajorinfluencewastemperaturevariation,usingacontrollerthatkeptthe

temperaturebetweentwofixedlimits. Thisdistributionmightbereasonablywellmodeledasuniform.

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Figure6. DiameterMeasurementwith3lobeFormError

Figure6showsthesizemeasurementdistributionproducedbyinteractionoftheprobingpatternwitha

3lobeformerroronacylindricalhole. ItexhibitstheclassicalUshapeexpectedfromthissource.

Finally,Figure7showsthemeasurementdensityforaflatnessmeasurement,madeunderthesame

conditionsasthelengthmeasurementofFigure4.

Figure7.

Flatness

Measurement

with

predominantly

Random

Errors

Evidently,properrepresentationofthemeasurementprobabilitydensityisarathermorecomplex

problemthanmodelingtheproductiondensity. Assuggestedintheprevioussection,measurement

variabilitycanbedeterminedbyrepeatedmeasurementofastableartifact. Alternatively,

measurementvariabilitycanbeestimatedbysimulationtechniquessuchasthosedescribedin[1]. The

effortrequiredisconsiderablylessandthisis,infact,howthedistributionsinFigures47were

determined.

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Giventhevarietyofshapesthatcanbeobservedformeasurementdistributions,analysisrequiresa

versatilerepresentation. Whitehousehassuggestedthebetadistributionprovidesasuitably

comprehensivecapability[15]. Thebetaprobabilitydensityisgivenby

( )( )

( ) ( )( )111

, , 1

,

baf a b y y y

a b

=

where

( ) ( ) ( )( )

111

0

, 1ba

a b y y dy=

and a and b areadjustableparameters. Theabilityofthisdistributiontorepresentmanyofthe

observedmeasurementdistributionsisillustratedinFigure8.

Anotherworkablealternativeistousethehistogramdatadirectly,performinganumericalintegration

overtheappropriateinterval.

Figure8. ExamplesoftheBetaProbabilityDistribution

Insummary,theconsumersandproducersrisksdependontheproductionandmeasurement

probabilitydensities,andtheacceptancelimits. Knowingandunderstandingtheserisksprovide

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solutionstothetechnicalquestionsabouttheconsequencesofbaddecisions. Theyareofvalueinand

ofthemselvesifthetechnicalrequirementsofanacceptanceprocessareknownanditisnecessaryto

determineifaspecificCMMisuptothemeasurementchoreathand. Theyare,aswell,aprerequisite

todealingwithbusinessconcernsaboutproductprofitability.

CMMCostAnalysisInordertounderstandtheeconomicimplicationsofmeasurementdecisionsandtooptimizethisfacet

ofthemanufacturingoperationitisrequiredtoreducethetechnicalrisksofmeasurementdecisionsto

theirimpactonproductprofitability. TypeIandTypeIImeasurementerrorswillbothincurcosts.

Rejectionofapartthatis,infact,good(TypeIerror)resultsinlossoftheentireamountinvestedin

productionofthatitem. Thesecostsshouldbereadilyaccessibleandwellknown. Theywillinclude,for

example,thecostsofmanufactureandinspectionaswellasofmaterials,scrapdisposaland,perhaps,

unnecessaryrework. Someofthecostsarisingfromacceptanceofapartthatis,infact,bad(TypeII

error)areequallyaccessible. Examplesmightbewarrantyandfailureanalysiscosts. Othercost

consequencesof

aType

IIerror

may

be

more

difficult

to

quantify

but

can

easily

exceed

the

readily

documentedcosts. Examplesincludedamagedcustomerperceptionofproductquality,lossoffuture

salesandcostoflawsuits.

LossFunctionsInordertoquantifythecostofincorrectdecisionsaboutproductconformance,itisnecessaryto

associatedeviationofthequalitycharacteristicfromtheidealwithaneconomicloss. Thisassociationis

generallyexpressedasalossfunction. Realisticassessmentofproductioneconomicsanddevelopment

ofoptimummeasurementstrategieswilldependcriticallyonchoiceofanappropriatelossfunction.

SomepossiblechoicesareshowninFigure9forthecaseofabilateraltolerancewhere,generally,the

nominalvalue

of

the

quality

characteristic

is

optimum.

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Figure9. QualityLossFunctions;BilateralToleranceor"NominaltheBest"

Theclassicallossfunctionhasoftenbeenastepfunction,whichstatesthatcostsdonotdependonthe

actualvalueofthequalitycharacteristicaslongasitiswithinspecification. Althoughthestepfunction

hasbeen

widely

used

in

the

past

and

is

mathematically

convenient,

it

is

now

generally

acknowledged

thatthequadraticlossfunction,

( ) ( )2

2

IIT

kL x x T=

originallyproposedbyTaguchi[16],isabetterexpressionofthecostconsequencesoflesstangible

factorssuchasthecustomersqualityperception. Here,II

k isthecostofaTypeIIerror, isthe

tolerancewidthandT isthenominalortargetvalue. Inthismodel,anydeviationfromthenominal

valueincursacostpenalty. TheTaguchilossfunctionisalsorecognizedtoembodysomedegreeof

unrealism,in

that

increases

without

limit.

More

realistically,

there

will

generally

be

adegree

of

nonconformancebeyondwhichthefulllosswillhavebeenrealized. Thisrealitycouldbe

accommodatedbychoosingadeviationatwhichthelossfunctionundergoesatransitionfroma

quadraticexpressiontoaconstantvalueofmaximumloss. This,too,presentssomedifficultyinthatthe

suddentransitiontoconstantlossseemsunrealisticandhasledtoproposalofavarietyoflossfunctions

basedoninvertedprobabilitydistributions. Theearliestoftheseistheinvertednormalloss,originally

suggestedbySpiring[17]andhavingtheform

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( )( )

2

21 exp

2S

x TL x K

=

where KisthemaximumlossduetoaTypeIIerror,T isthetargetvalueand isashapeparameter

suchthat

if

/ 4 = thelossatT willbeveryclose(within0.9997of) K.

OneoftheinterestingandproblematicfeaturesofCMMevaluationsofGD&Tparametersisthat,

contrarytothemoregeneralqualityenvironment,wherethemajorityofqualitycharacteristicsareof

thebilateraltoleranceornominalthebesttype,manyGD&Tparametersarelessthebetteror

singlesidedvariety. Analogouslossfunctionscanbegeneratedforthesinglesidedtolerancecase,

Figure10,ascanmorecomplicatedfunctionstodealwithinstanceswheredeviationsinonedirection

shouldbepenalizeddifferentlythanthoseintheother,seeforexampleFigure11. Unfortunatelyfor

theCMMmetrologists,theliteratureonthiscategoryoflossfunctionsisrelativelysparse.

Figure10. LossFunctionsforaUnilateralTolerance

Figure11. NonsymmetricLossFunctionExample

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Choiceoftheappropriatelossfunctionforaparticularproductisgenerallybeyondtheexpertiseand

knowledgebaseoftheCMMmetrologist;heretheneedforcollaborationafinancialanalystfamiliar

withthecustomerbaseandsalesenvironmentisindicated. Themeasurementspecialistwillalmost

certainlyfindfinancialandsalespersonnellackinginunderstandingofthebusinessofmeasurement. A

mutualeducationalexchangewillalmostcertainlyberequired.

ProfitCalculationMethodsofmeasurementcostcomputationcanvaryconsiderablyincomplexityandsophistication. At

oneextreme,elaborateandhighlydetailedmodelshavebeendevelopedforanalyzingthecostto

benefittradeoffsofsemiconductormeasurement;forexample,see[18]. Ontheotherhand,

significantlysimplermodelsareusefulinmostinstances,especiallyforapplicationtoestablished

metrologyfacilitieswherethereisalonghistoryofmeasurementcostdata. Suchasimplemodelis

usedintheexampleprofitcalculationsthatfollow. Specifically,weusehereamodelderivedfromone

originallyproposedbyWilliamsandHawkins[19,20]todealwithGaussianerrordistributionsandthe

Taguchilossfunction,andexpandedbyustocovertherangeoferrordistributionsandlossfunctions

describedabove.

Intheiroriginaltreatment,WilliamsandHawkinsdevelopedanexpressionfortheperunitprofitunder

conditionsofmeasurementerror:

( )( ) ( )222UnitProfit IIs Ik

k T P F k

= +

whereSk isthesellingpriceperunit, Ik isthecostofmanufacturingandtestingoneproductunit(also

equaltothecostofaTypeIerror),II

k istheestimatedcostofaTypeIIerror,T isthenominalortarget

value, isthetolerancebandwidth, and arethemeanandstandarddeviation,respectively,of

thequalitycharacteristicasseenbythecustomer(thatis,afteracceptance)and ( )P F istheprobability

ofaunitpassingitsinspectiontest. Thisexpressionincludestheeffectsofsaleprice,TypeIandTypeII

errors,andcustomerperceptionofquality. Analogousexpressionshavebeendevelopedforotherloss

functionsanddistributions.

MultipleparametersAcomplicationuniquetocoordinatemeasurement,ascomparedtomostotherdimensional

measurementtechnologies,arisesfromthefactthatitisusualformultiplecharacteristicstobe

evaluatedonthesamemeasurementapparatusinasingleinspectionoperation. Itismostusefulto

endupwithasinglelossparameterthatexpressestheneteconomiceffectofallproductnonidealities,

whichleadstotheconsiderationofhowtomostappropriatelycombinethelossesarisingfromthe

variousdimensionalrequirements. Ithasbeensuggestedthatthelossfunctionsfortheindividual

parameterscanbeaddedtoformanetmultivariatelossfunction[21],butthisseemsintuitively

extremeforacasewherethecasewhereacommonmeasurementapparatus,acommonenvironment

andinsomecasesagainstcommondatumsandcommonproductionmachinerymakelikelysome

degreeofcorrelationamongthevariousGD&Tparameterevaluations. Forthisreason,wepreferthe

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recommendationofWilliamsandHawkins[20]thatthelossfunctionsbecombinedastheirrootsumof

squares,

( )( )2

T i

i

L E L=

whichtendstolessentheseverityofthepenaltyformeasurementsysteminducedcorrelation. Finally,

itshouldberecognizedthatthedevelopmentofmultivariatelossfunctionsforcorrelatedcharacteristics

remainsanactivefieldofresearchandthatnewdevelopmentsaretobeexpected.

ExamplesInthissectionwepresentafewspecificexamplesoftheeffectsofvariousmeasurementvariableson

productprofitability,asillustrationsofthevalueofperforminglosscalculations.

MeasurementUncertaintyandGuardBandsItiscommonpracticetoguardbandmeasurementstoreducetheincidenceandcostofmeasurement

errors. Mostcommonly,itisthecostofTypeIIerrorsthatisofmajorconcern;hencetheacceptance

zoneisreducedbysomeamount(knownasstringentacceptance)andanincreasedprobabilityof

rejectingagoodpartisacceptedintheinterestofnotexposingthecustomertobadparts. Itisusefulto

know,foragivenmeasurementuncertainty,theguardbandchoicethatwillmaximizeprofit. Inthis

example,weareconsideringthemeasurementofa100mmdiametershaftwithtolerancelimitsof1

mm. Theproductionprocessiscentered;thatisitproducespartswithameansizeof100mm. The

productionstandarddeviationis0.33mm. Themeasurementprocessisunbiased. Thesellingpriceof

onepartis$30,thecostofproducingapartis$7.50,andthecostofshippingabadpartistakentobe

$300. Thisistypicalofwhatmightberegardedasacriticalorhighconsequencepart,wheretheratio

ofTypeIIerrorcosttosellingpriceisoftenfoundtobe 10.

Guardbandingisrequiredinordertoachieveprofitability,withtheoptimumguardbandmultiplierbeing

ontheorderof0.65. Thereisasignificanteffectofmeasurementuncertainty,withtheprofit

approachingzeroatthehighendoftherangestudied.

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Figure12.EffectofMeasurementUncertaintyandGuardbandingonProfit

MeasurementBiasMeasurementbiasisproperlyconsideredtobeafactortobediscoveredandeliminated,howeverits

completeremoval

is

not

always

possible

and

the

effect

of

bias

on

profitability

can

be

significant.

Here

is

anexample,usingthesamegeneralconditionsasthepreviousonethatillustratesthepotential

magnitudeofbiasdrivenlossofprofit. Themeasurementstandarddeviationwastakenheretobe0.1

mmandaguardbandclosetotheoptimumindicatedintheearlierexamplewasused. Anundiscovered

oruncorrectedmeasurementbiasofthesamemagnitudeasthemeasurementuncertaintycancausea

profitdecrementofalmost7%.

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Figure13. EffectofMeasurementBias

MeasurementStrategyAsregardsmeasurementpointplacementstrategywithCMMs,thegeneralwisdomismoreisbetter.

Thisexample

illustrates

that

this

is

not

guaranteed

to

be

the

case.

The

feature

of

interest

is,

again,

cylindrical. Figure14showstheeffectonperunitprofitassamplingdensityisincreased,assumingthe

featureformerrorisrandomoverthefeaturesurface. Thesamplingpatternusedconsistedoftaking

halfthetotalnumberofpointsoneachoftwocircularsectionsneareachendofthefeature,anot

unreasonablestrategyiftheformerrorisknowntoberandom. TheupperplotofFigure14showsthat

theuncertaintyofthediametermeasurementdecreasessteadilywithincreasingpointdensityas

conventionalwisdomsuggests.

Figure15illustratestheresultsofthesamesetofmeasurementstrategieswhentheformerrorhasa3

lobeshapewiththesameamplitudeasthemaximumrandomexcursion. Theuncertaintythenadopts

anoscillatory

behavior

that

is

reflected,

although

not

to

an

extreme

degree

once

the

sampling

density

risesto7pointsperlevel,intheperunitprofit.

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Figure14. EffectofSamplingStrategyonProfit,RandomFormError

Figure15. EffectofSamplingStrategyonProfit,3lobeFormError

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AvailableToolsTwocommerciallyavailablesoftwarepackagesforCMMmeasurementuncertaintyevaluationare

knowntotheauthorsofthisreport;onewasdescribedinanearlierpaperinthissession[1]. Itwas

usedtosupplythemeasurementuncertaintyevaluationsusedinthispaper. Theotherisavailableasan

addontotheCMMcontrolanddataanalysissoftwareprovidedbysomespecificCMMvendors. The

nextreleaseofthefirstmentionedofthesepackageswillmakeavailableatoolkittocomputeproduct

profitabilityandoptimumguardbandselections.

Someoftheprofitabilitycalculationspresentedinthispapercanbeperformedwithspreadsheet

programs. Mostofthem,however,requirenumericalintegrationofprobabilitydensitieswhichare

difficulttoperforminspreadsheetapplications. Commercialsoftwareintendedfornumerical

computation(forexampleMATLAB2orMathematica3)ismuchbettersuitedforthecomputationof

measurementcostsandprofitability. ApackageofMATLABfunctionsdevelopedbyuswasusedto

preparethedataforthispaper.

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