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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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