Lot-by-Lot Acceptance-Sampling Procedures Ensuring Quality ...

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Lot-by-LotAcceptance-SamplingProcedures

Chapter7

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EnsuringQualityStandardsthroughAcceptance-Sampling

• InSeptember2008,itwasdiscoveredthatChina-baseddairyfarmersanddistributorshadbeenaddingmelamine(aplasticsmanufacturingbyproduct)tomilktofalselyinflateproteinreadings.Whyweretheyabletopassthetests?Whilethetestsensuredthatproteinlevelswere sufficient,theydidnotensurethatallingredientswereunadulterated.• Toensuretop-qualitymilkgoingforward,theChinesegovernmenttookthreeboldactions:issuedalistofbannedfoodadditives,overhauledtheindustrytomoveitawayfromlocalfarmersandtowardmassproduction,andincreasedtestingonbannedsubstances.• Itisimportanttonote,however,thatacceptance-samplingcanonlyconfirmqualitycharacteristicsofthoseitemsthataretested.Itcannotconfirmoverallquality.

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LearningObjectives1. Understandtheroleofacceptance-samplinginmodernqualitycontrolsystems2. Understandtheadvantages anddisadvantagesofsampling3. Understandthedifference betweenattributes andvariablessamplingplansandthe

majortypesofacceptance-samplingprocedures4. Knowhowsingle-,double-,andsequential-samplingplansareused5. Understandtheimportanceofrandom-sampling6. KnowhowtodeterminetheOCcurveforasingle-samplingplanforattributes7. Understandtheeffects ofthesamplingplanparametersonsamplingplan

performance8. Knowhowtodesignsingle-,double-,andsequential-samplingplansforattributes9. Knowhowrectifyinginspectionisused10. Understandthestructure anduseofMILSTD105Eanditsciviliancounterpartplans11. Understandthestructure anduseoftheDodge–Romig systemofsamplingplans12. Understandthestructure anduseofMILSTD414anditsciviliancounterpartplans

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Discussiontopics

TheAcceptance-SamplingProblem

Single-SamplingPlansforAttributes

Double-,Multiple-,andSequential-

Sampling

MilitaryStandard105E(ANSI/ASQCZ1.4,ISO2859)

TheDodge–RomigSamplingPlans

MILSTD414(ANSI/ASQCZ1.9)

ChainSampling Continuous-Sampling

Skip-LotSamplingPlans

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§ Acceptance-sampling isconcernedwith inspectionanddecisionmaking regarding products,oneoftheoldestaspectsofqualityassurance.

§ Inthe1930sand1940s,acceptance-samplingwasoneofthemajorcomponentsofthefieldofstatisticalqualitycontrolandwasusedprimarily forincomingorreceiving inspection.

§ Inmore recentyears, ithasbecometypical toworkwithsupplierstoimprove their processperformance throughtheuseofSPCanddesignedexperiments andnottorelyasmuchonacceptance-samplingasaprimary qualityassurance tool.

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§ Acompany receives a shipmentofproductfromasupplier.

§ Asample istaken fromthe lot,andsomequalitycharacteristicoftheunitsinthesample isinspected.

§ Onthebasisofthe informationinthissample, adecisionismaderegarding lotdisposition;usually toacceptor rejecta lot.Sometimes we refer tothisdecisionas lotsentencing.

§ Accepted lotsare putintoproduction;rejected lotsmaybereturned tothesupplierormaybesubjected tosomeother lotdispositionaction.

Thepurposeofacceptancesampling istodispositionor sentencelots

TypicalApplicationofAcceptance-Sampling

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AcceptanceSampling

Three aspectsofsamplingare important:

1. It isthepurposeofacceptance-sampling tosentence lots,nottoestimate the lotquality.

2. Acceptance-sampling plansdonotprovideanydirect formofqualitycontrol.It simplyacceptsand rejects lots.§ Evenifalllotsareofthesamequality,samplingwillacceptsomelotsand

rejectothers,theacceptedlotsbeingnobetterthantherejectedones.§ Processcontrolsareusedtocontrolandsystematicallyimprovequality,

butacceptancesamplingdoesnot.

3. Themosteffective useofacceptance-sampling isnotto“inspectquality intotheproduct,”butrather asanaudit tooltoensurethat theoutputofaprocessconformstorequirements.

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AcceptanceSampling

Generally, there are threeapproaches tolotsentencing:

1. Acceptwithnoinspection;

2. 100%inspectionand

3. Acceptance-sampling.

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AdvantagesofSampling

1. It isusuallylessexpensive, because there islessinspection.

2. There islesshandlingoftheproduct,andthusreduceddamage.

3. It isapplicable todestructive testing.

4. Fewer personnelare involved ininspectionactivities.

5. Itoftengreatly reduces theamountofinspectionerror.

6. The rejection ofentire lotsasopposedtothesimple returnofdefectives oftenprovidesastrongermotivation tothesupplierforquality improvements.

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DisadvantagesofSampling

1. There are risksofaccepting “bad”lotsandrejecting “good”lots.

2. Lessinformation isusuallygenerated abouttheproductortheprocessthatmanufactured theproduct.

3. Acceptance-sampling requiresplanninganddocumentationoftheacceptance-sampling procedure,whereas 100%inspectiondoesnot.

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Acceptance-sampling ismostlikely tobeuseful inthe followingsituations:

1. Whentestingisdestructive.

2. Whenthecostof100%inspectionisextremelyhigh.

3. When100%inspectionisnottechnologicallyfeasibleorwouldrequiresomuchcalendartimethatproductionschedulingwouldbeseriouslyimpacted.

4. Whentherearemanyitemstobeinspectedandtheinspectionerrorrateissufficientlyhighthat100%inspectionmightcauseahigherpercentageofdefectiveunitstobepassedthanwouldoccurwiththeuseofasamplingplan.

5. Whenthesupplierhasanexcellentqualityhistoryandsomereductionininspectionfrom100%isdesiredbutthesupplier’sprocesscapabilityissufficientlylowastomakenoinspectionanunsatisfactoryalternative.

6. Whentherearepotentiallyseriousproductliabilityrisksandalthoughthesupplier’sprocessissatisfactory,aprogramforcontinuouslymonitoringtheproductisnecessary.

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TypesofSamplingplans

§ There areanumberofdifferentways toclassifyacceptance-samplingplans.

§ Onemajorclassification isbyvariables andattributes.§ Variables,ofcourse,arequalitycharacteristicsthataremeasuredona

numericalscale.§ Attributesarequalitycharacteristicsthatareexpressedona“go,no-go”

basis.

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TypesofSampling

§ Single Sampling§ Selectnitemsatrandomfromthelot.If

therearecorfewerdefectivesinthesample,acceptthelot,andiftherearemorethancdefectiveitemsinthesample,rejectthelot.

§ DoubleSampling§ Followinganinitialsample,adecisionbased

ontheinformationinthatsampleismadeeitherto(1)acceptthelot,(2)rejectthelot,or(3)takeasecondsample.

§ Ifthesecondsampleistaken,theinformationfromboththefirstandsecondsampleiscombinedinordertoreachadecisionwhethertoacceptorrejectthelot

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TypesofSampling

§ Multiple-Sampling§ Morethantwosamplesmayberequiredin

ordertoreachadecisionregardingthedispositionofthelot

§ Sequential-Sampling§ Unitsareselectedfromthelotoneatatime,

andfollowinginspectionofeachunit,adecisionismadeeithertoacceptthelot,rejectthelot,orselectanotherunit

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LotFormation

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Random-Sampling

§ The random-sampling conceptisextremely important inacceptance sampling.

§ Unlessrandomsamplesareused,biaswillbe introduced.

§ The techniqueoftensuggested fordrawing arandomsample istofirstassignanumber toeach item inthe lot(serial orothercodenumbersorathree-digit randomnumber torepresent the length,width,anddepthinacontainercouldbeused).

§ Thennrandomnumbersare drawn,where the rangeofthesenumbers isfrom1tothemaximumnumber ofunitsinthelot.

§ Thissequence ofrandomnumbersdetermines whichunitsinthelotwill constitute thesample.

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Stratifyingalot

§ Sometimestheinspectormaystratifythelot.

§ Thisconsistsofdividingthelotintostrataorlayersandthensubdividingeachstrataintocubes.

§ Unitsarethenselectedfromwithineachcube.

§ Itdoesnotnecessarilyensurerandomsamples,atleastitensuresthatunitsareselectedfromalllocationsinthelot.

§ Ifjudgmentmethodsareusedtoselectthesample,thestatisticalbasisoftheacceptance-samplingprocedureislost.

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Guidelines forusingAcceptance-Sampling

§ Acceptance-Sampling Plan§ Statementofthesamplesizetobeusedandtheassociatedacceptanceor

rejectioncriteriaforsentencingindividuallots.

§ Sampling Scheme§ Setofproceduresconsistingofacceptance-samplingplansinwhichlot

sizes,samplesizes,andacceptanceorrejectioncriteriaalongwiththeamountof100%inspectionandsamplingarerelated.

§ Sampling System§ Aunifiedcollectionofoneormoreacceptance-samplingschemes.

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Acceptance-Samplingprocedures

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Discussiontopics




Sampling






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Single-Sampling plan

KeyVariables inSampling Plan

N= lotsize

n=number inspected

d=numberofdefectivesobserved

c=max#observed defects

Example

N=10000,n=89,c=2

§ Ifthenumberofobserveddefectives dislessthanorequaltoc=2,the lotwillbeaccepted

§ Ifthenumberofobserveddefectives disgreaterthanc=2,thelotwillberejected.

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TheOCCurve

§ Animportant measure oftheperformance ofanacceptance-samplingplan istheoperating-characteristic (OC)curve.

§ Thiscurveplotstheprobabilityofaccepting the lotversusthelotfractiondefective. Itshowstheprobability thatalotsubmittedwithacertain fractiondefective will beeither accepted orrejected.

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OCCurveExample

§ Theprobabilityofobservingexactlyddefectivesis

§ Theprobabilityofacceptanceissimplytheprobabilitythatdislessthanorequaltoc,or

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OCCurveExample

TheOCcurveshowsthediscriminatorypowerofthesamplingplan.

§ Forexample,inthesamplingplann=89,c=2,ifthelotsare2%defective,theprobabilityofacceptanceisapproximately0.74.

§ Thismeansthatif100lotsfromaprocessthatmanufactures2%defectiveproductaresubmittedtothissamplingplan,wewillexpecttoaccept74ofthelotsandreject26ofthem.

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Effectofn&conOCcurvesAsamplingplanthatdiscriminatedperfectlybetween goodandbadlotswouldhaveanOCcurvethatlookslikethefigurebelow.

TheOCcurvebecomesmoreliketheidealizedOCcurveshapeasthesamplesizeincreases.

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EffectofcontheOCcurve

§ Generally, changing theacceptance numberdoesnotdramatically change theslopeoftheOCcurve.

§ Astheacceptance number isdecreased, theOCcurve isshiftedtothe left.

§ Planswithsmaller valuesofcprovidediscriminationatlower levelsoflotfractiondefective thandoplanswithlarger valuesofc.

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Specificpoints ontheOCcurve

§ Frequently, thequalityengineer’s interest focusesoncertainpointsontheOCcurve.

§ Thesupplier isusuallyinterested inknowing§ whatleveloflotorprocessqualitywouldyieldahighprobabilityof

acceptanceor§ Converselywhatleveloflotorprocessqualitywillyieldalowprobability

ofacceptance?

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AcceptableQualityLevel(AQL)

§ AQL - Poorestlevel ofquality forthesupplier’s processthattheconsumerwouldconsider tobeacceptable asaprocessaverage.§ AQLisapropertyofthesupplier’smanufacturingprocess;itisnota

propertyofthesamplingplan.§ TheconsumerwilloftendesignthesamplingproceduresothattheOC

curvegivesahighprobabilityofacceptanceattheAQL.§ AQLisnotusuallyintendedtobeaspecificationontheproduct,norisita

targetvalueforthesupplier’sproductionprocess.

AQL issimplyastandardagainstwhichtojudgethe lots.

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LotTolerancePercentDefective(LTPD)

§ LTPD - Poorestlevel ofqualitythat theconsumer iswilling toaccept inan individual lot.Alsocalled rejectable quality level(RQL)andlimiting quality level (LQL).§ Thelottolerancepercentdefectiveisnotacharacteristicofthesampling

planbutisaleveloflotqualityspecifiedbytheconsumer.§ Itispossibletodesignacceptance-samplingplansthatgivespecified

probabilitiesofacceptanceattheLTPDpoint.

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Type-A&Type-BOCcurves

§ Type-BOCcurve§ IntheconstructionoftheType-BOCcurveitisassumedthatthesamples

comefromalargelotorthatweweresamplingfromastreamoflotsselectedatrandomfromaprocess.

§ Thebinomialdistributionistheexactprobabilitydistributionforcalculatingtheprobabilityoflotacceptance.

§ Type-AOCcurve§ Usedtocalculateprobabilitiesofacceptanceforanisolatedlotoffinite

size.§ SupposethatthelotsizeisN,thesamplesizeisn,andtheacceptance

numberisc.Theexactsamplingdistributionofthenumberofdefectiveitemsinthesampleisthehypergeometric distribution.

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ComparisonofType-A&Type-B OCcurves

§ The type-A OCcurvewillalways liebelow the type-BOCcurve.

§ However, thisdifference isonlysignificantwhen the lotsizeissmall relative tothesamplesize.

§ Unlessotherwise stated,alldiscussionofOCcurvesinthistext isintermsofthetype-BOCcurve.

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OtherAspectsofOCcurvebehavior

§ Twoapproaches todesigningsamplingplansthatareencountered inpracticehavecertain implicationsforthebehavioroftheOCcurve.

§ These approachesare§ theuseofsamplingplanswithzeroacceptancenumbers(c=0)and§ theuseofsamplesizesthatareafixedpercentageofthelotsize.

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§ PlanswithzeroacceptancenumbershaveOCcurvesthathaveaverydifferentshapethantheOCcurvesofsamplingplansforwhich.

§ Generally,samplingplanswithc=0haveOCcurvesthatareconvexthroughouttheirrange.

§ Asaresultofthisshape,theprobabilityofacceptancebeginstodropveryrapidly,evenforsmallvaluesofthelotfractiondefective.

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Example

§ IfAQL=1%,thisimpliesthatwewouldliketoacceptlotsthatare1%defectiveorbetter.§ Ifn=89,c=1isused,the

probabilityoflotacceptanceattheAQLisabout0.78.

§ Ifn=89,c=0isused,theprobabilityoflotacceptanceattheAQLisabout0.41.Nearly60%ofthelotsofAQLqualitywillberejected ifweuseanacceptancenumberofzero.

§ Analternativeapproachtousingzeroacceptancenumbersistousechain-samplingplans.

§ Undercertaincircumstances,chainsamplingworksconsiderablybetterthanacceptance-samplingplanswithc=0.

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§ Theprincipaldisadvantageofsamplingplansinwhichthesamplesizeisafixedpercentageofthelotsizeisthatthedifferentsamplesizesofferdifferentlevelsofprotection.

§ Althoughsamplingproceduressuchasthisonewereinwideusebeforethestatisticalprinciplesofacceptance-samplingweregenerallyknown,theirusehas(unfortunately)notentirelydisappeared.

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§ Acommonapproachtothedesignofanacceptance-samplingplanistorequirethattheOCcurvepassthroughtwodesignatedpoints.

§ Supposethatwewishtoconstructasamplingplansuchthattheprobabilityofacceptanceis1-α forlotswithfractiondefectivep1,andtheprobabilityofacceptanceisforlotswithfractiondefectivep2.

§ Assumingthatbinomialsampling(withtype-BOCcurves)isappropriate,weseethatthesamplesizenandacceptancenumbercarethesolutionto

ThetwosimultaneousequationsinEquation(7.3)arenonlinear,andthereisnosimple,directsolution.

DesigningaSingle-Sampling planwithaspecifiedOCCurve

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UsingaNomograph

• Thenomograph inFigure7.10canbeusedforsolvingtheseequations.• Twolinesaredrawnonthenomograph,oneconnectingp1 and1-α,andtheotherconnectingp2 andβ.• Theintersectionofthesetwolinesgivestheregionofthenomograph inwhichthedesiredsamplingplanislocated.

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UsingaNomographTo illustrate theuseofthenomograph, supposewewish toconstructa samplingplanforwhichp1 =0.01,α =0.05,p2 =0.06,andβ =0.10.Locatingtheintersectionofthe linesconnecting(p1 =0.01,1- α =0.95)and(p2 =0.06,β =0.10)onthenomograph indicates thattheplann=89,c=2isveryclosetopassingthroughthesetwopointsontheOCcurve.Obviously, sincenandcmustbeintegers,thisprocedurewillactually produceseveral plansthathaveOCcurves thatpassclosetothedesiredpoints.For instance, ifthefirstline isfollowedeither totheclinejustabove the intersectionpointortothecline justbelowit,andthealternatesamplesizesare read fromthechart, thiswillproduce twoplansthatpassalmostexactly throughthep1,1- α pointbutthatmaydeviate somewhat fromthep2,β point.Asimilarprocedure couldbefollowedwith thep2,β line.The resultoffollowingbothofthese lineswouldbe fourplansthatpassapproximately through thetwopointsspecifiedontheOCcurve.

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OCcurve

§ AlthoughanytwopointsontheOCcurvecouldbeusedtodefinethesamplingplan, itiscustomary inmany industriestouse theAQLandLTPDpointsforthispurpose.

§ Whenthe levelsoflotquality specifiedare p1=AQLandp2=LTPD,thecorrespondingpointsontheOCcurveare usuallyreferred toastheproducer’s riskpointandtheconsumer’s riskpoint,respectively.

§ Thus,α wouldbecalled theproducer’s riskandβ wouldbecalledtheconsumer’s risk.

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§ Acceptance-sampling programsusuallyrequire corrective actionwhen lotsare rejected.

§ Thus,arectifying inspectionprogramserves to“correct”lotquality.

RectifyingInspection

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HandlingRejectedLots

§ Rejected lotsmaybehandled inanumber ofways.

§ Thebestapproachistoreturn rejected lotstothesupplierandrequire ittoperformthescreeningand rework activities.

§ However, inmanysituations,because thecomponentsor rawmaterials are required inorder tomeet productionschedules,screeningand rework takeplaceat theconsumer level.

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Averageoutgoingquality(AOQ)

§ AOQ iswidely usedfortheevaluation ofa rectifyingsamplingplan.

§ TheAOQ isthequality inthe lotthat resultsfromtheapplicationofrectifying inspection.

§ It istheaverage valueoflotquality thatwouldbeobtainedoveralongsequence oflotsfromaprocesswithfractiondefective p.

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AOQ

§ Assumethatthe lotsizeisNandthatalldiscovereddefectivesare replaced withgoodunits.Then inlotsofsizeN,wehave§ nitemsinthesamplethat,afterinspection,containnodefectives,because

alldiscovereddefectivesarereplaced§ N-nitemsthat,ifthelotisrejected,alsocontainnodefectives§ N-nitemsthat,ifthelotisaccepted,containp(N-n)defectives

§ Thus,lotsintheoutgoingstageofinspectionhaveanexpectednumberofdefective unitsequal toPap(N-n),whichwemayexpressasanaverage fractiondefective, calledtheaverageoutgoingqualityor

• AslotsizeNbecomes large relativetothesamplesizen,

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AOQcurve§ Average outgoingqualitywillvaryasthe

fractiondefectiveoftheincominglotsvaries.Thecurve thatplotsaverageoutgoingqualityagainstincominglotqualityiscalledanAOQcurve.§ Whentheincomingqualityisverygood,

theaverage outgoingqualityisalsoverygoodwhereaswhentheincominglotqualityisverybad,mostofthelotsarerejected andscreened.

§ Thisleadstoaverygoodlevelofqualityintheoutgoinglots.

§ ThemaximumordinateontheAOQcurverepresents theworstpossibleaverage qualitythatwouldresultfromtherectifyinginspectionprogram,andthispointiscalledtheaverageoutgoingqualitylimit(AOQL).

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Example

§ theAOQLisseentobeapproximately0.0155.

§ Thatis,nomatterhowbadthefractiondefectiveisintheincominglots,theoutgoinglotswillneverhaveaworsequalitylevelontheaveragethan1.55%defective.

§ LetusemphasizethatthisAOQLisanaveragelevelofquality,acrossalargestreamoflots.

§ Itdoesnotgiveassurancethatanisolatedlotwillhavequalitynoworsethan1.55%defective.

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AverageTotalInspection (ATI)

§ Another importantmeasure relative torectifying inspectionisthetotalamountofinspection requiredbythesamplingprogram.

§ Ifthe lotscontainnodefective items,nolotswillbe rejected, andtheamountofinspectionper lotwillbe thesamplesizen.Iftheitemsarealldefective,every lotwillbesubmitted to100%inspection,andtheamountofinspectionper lotwillbe the lotsizeN.

§ Ifthe lotqualityis0<p<1, theaverage amountofinspectionperlotwill vary between thesamplesizenandthelotsizeN. Ifthelotisofqualitypandtheprobabilityoflotacceptance isPa,thentheaverage total inspection(ATI) perlotwillbe

ATI =n+ (1– Pa)(N– n)

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Example7.3:CalculatingATI

§ To illustrate theuseofEquation (7.6),considerExample 7.2withN=10,000,n=89,c=2,andp=0.01.Then, sincePa =0.9397,wehave

ATI =n+ (1- Pa)(N– n)=89+ (1– 0.9397)(10,000– 89)=687

§ Remember that thisisanaverage numberofunitsinspectedovermanylotswith fractiondefective p=0.01.

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ATICurve

§ SpecificationoftheAOQLisnotsufficienttodetermineauniquesamplingplan.

§ Therefore,itisrelativelycommonpracticetochoosethesamplingplanthathasaspecifiedAOQLand,inaddition,yieldsaminimumATIataparticularleveloflotquality.

§ Theleveloflotqualityusuallychosenisthemostlikelylevelofincominglotquality,whichisgenerallycalledtheprocessaverage.

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Proceduresforgeneratingrectifyinginspection plans

§ Theprocedure forgenerating theseplansisrelativelystraightforward andisillustrated inDuncan(1986).

§ Tables ofsamplingplansthatminimizeATI foragivenAOQLandaspecifiedprocessaverage phavebeendeveloped byDodgeandRomig.

§ It isalsopossibletodesigna rectifying inspectionprogramthatgivesa specified levelofprotectionat theLTPDpointandthatminimizes theaverage totalinspection foraspecifiedprocessaverage p.TheDodge–Romig sampling inspection tablesalsoprovide theseLTPDplans.

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Discussiontopics




Sampling






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Double-Sampling plans

§ Adouble-samplingplanisaprocedureinwhich,undercertaincircumstances,asecondsampleisrequiredbeforethelotcanbesentenced.

§ Double-SamplingPlanParametersn1 =samplesizeonthefirstsamplec1 =acceptancenumberofthefirstsamplen2 =samplesizeonsecondsamplec2 =acceptancenumberforbothsamples

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Double-Sampling plan

Advantages

§ Theprincipaladvantageofadouble-samplingplanwithrespecttosingle-samplingisthatitmayreducethetotalamountofrequiredinspection.

§ Furthermore,insomesituations,adouble-samplingplanhasthepsychologicaladvantageofgivingalotasecondchance.

Disadvantages

§ Unlesscurtailmentisusedonthesecondsample,undersomecircumstancesdouble-samplingmayrequiremoretotalinspectionthanwouldberequiredinasingle-samplingplanthatoffersthesameprotection.

§ Itisadministrativelymorecomplex,whichmayincreasetheopportunityfortheoccurrenceofinspectionerrors.

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Example7.4:IllustrationofaDouble-SamplingPlan

§ Supposen1 =50,c1 =1,n2 =100,andc2 =3.Thus,arandomsampleofn1 =50itemsisselected fromthelot,andthenumberofdefectives in thesample, d1,isobserved.

§ Ifd1 ≤ c1 =1,the lotisacceptedonthe firstsample. Ifd1 >c2 =3,the lotisrejected onthe firstsample.

§ Ifc1 <d1 ≤ c2,a secondrandomsample ofsizen2 =100isdrawnfromthe lot,and thenumberofdefectives inthissecondsample,d2,isobserved.

§ Nowthecombinednumberofobserved defectives fromboththefirstandsecondsample, d1 +d2,isusedtodetermine thelotsentence. Ifd1 +d2 ≤ c2 =3,the lotisaccepted. However, ifd1 +d2

>c2 =3,the lotisrejected. The operationofthisdouble-samplingplan isillustratedgraphically inFigure 7.14.

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Example7.4:IllustrationofaDouble-SamplingPlan

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Double-SamplingOCcurve

§ Adouble-samplingplanhasaprimaryOCcurvethatgivestheprobabilityofacceptanceasafunctionoflotorprocessquality.

§ ItalsohassupplementaryOCcurvesthatshowtheprobabilityoflotacceptanceandrejectiononthefirstsample.

§ TheOCcurvefortheprobabilityofrejectiononthefirstsampleissimplytheOCcurveforthesingle-samplingplann=n1andc=c2.

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Example7.5:ComputationofanOCcurveforaDouble-Sampling Plan

§ Wenowillustrate thecomputationoftheOCcurve fortheplann1

=50,c1 =1,n2 =100,c2 =3.IfPa denotestheprobabilityofacceptanceonthecombinedsamples,andand denote theprobabilityofacceptance onthe firstandsecondsamples,respectively, then

§ PaI isjusttheprobability thatwe willobserved1 ≤ c1 =1

defectives outofarandomsampleofn1 =50items.Thus

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§ Ifp=0.05isthefractiondefective inthe incominglot,then

§ Toobtaintheprobabilityofacceptance onthesecondsample,wemustlistthenumberofways thesecondsample canbeobtained.Asecondsample isdrawnonlyifthereare twoorthreedefectives onthefirstsample—that is, ifc1<d1≤ c2.

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§ Thus,theprobabilityofacceptance onthesecondsample is

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§ Theprobabilityofacceptance ofa lotthathas fractiondefective p= 0.05istherefore

§ OtherpointsontheOCcurvearecalculated similarly. Rememberthat thesebinomialprobabilitiescanbecalculated usingMinitabonExcel.

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TheAverageSampleNumbercurve§ Insingle-sampling, thesizeofthesample inspected fromthe lot

isalwaysconstant,whereas indouble-sampling, thesizeofthesampleselected dependsonwhether ornotthesecondsample isnecessary.

§ Ageneral formula fortheaverage sample number indouble-sampling, ifweassumecomplete inspectionofthesecondsample, isASN=n1P1+(n1+n2)(1– P1)=n1+n2(1– P1)

§ where P1 istheprobabilityofmakingalot-dispositioning decisiononthefirstsample.ThisisP1=P{lot isacceptedonthefirstsample}+P{lot isrejectedonthefirstsample}

§ Iftheaboveequation isevaluated forvariousvaluesoflotfractiondefectivep, theplotofASNversuspiscalledanaveragesamplenumbercurve.

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Curtailment

§ Inpractice, inspectionofthesecondsample isusually terminatedandthe lotrejected assoonasthenumber ofobserveddefectiveitems inthecombinedsampleexceeds thesecondacceptancenumberc2.

§ Thisisreferred toascurtailment ofthesecondsample.

§ Theuseofcurtailed inspectionlowers theaverage samplenumber required indouble-sampling.

§ Ifcurtailed inspectionisusedinsingle-sampling oronthefirstsampleofdouble-sampling, theestimate oflotorprocessfalloutobtained fromthesedata isbiased.

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§ P(n1, j)- probabilityofobservingexactly jdefectives ina sampleofsizen1,

§ PL(n2,c2-j)- probability ofobservingc2 - jorfewer defectives inasampleofsizen2 ,and

§ PM(n2+1, c2-j+2) - probabilityofobservingc2-j+2defectives inasampleofsizen2 +1.

ASNcurve formulaforadouble-sampling planwithcurtailment on thesecond sample

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ASNcurve foradouble-sampling planwithcurtailmenton thesecond sample

§ theASNcurvefordouble-samplingwithoutcurtailmentonthesecondsampleisnotlowerthanthesamplesizeusedinsingle-samplingthroughouttheentirerangeoflotfractiondefective.

§ Iflotsareofverygoodquality,theywillusuallybeacceptedonthefirstsample,whereasiflotsareofverybadquality,theywillusuallyberejectedonthefirstsample.

§ ThisgivesanASNfordouble-samplingthatissmallerthanthesamplesizeusedinsingle-samplingforlotsthatareeitherverygoodorverybad.

§ However,iflotsareofintermediatequality,thesecondsamplewillberequiredinalargenumberofcasesbeforealotdispositiondecisioncanbemade.

§ Inthisrangeoflotquality,theASNperformanceofdouble-samplingisworsethansingle-sampling.

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Unlesscare isexercised toensure thatlotorprocessqualityisinthe rangewhere double-sampling ismosteffective,then theeconomicadvantages ofdouble-sampling relativetosingle-sampling maybe lost.

Ifcurtailment isusedonthesecondsample, theaveragesamplenumber curve fordouble-samplingalways liesbelowthesamplesizeusedinsingle-sampling.

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DesigningDouble-Sampling planswithspecified P1,1-α ,P2andβ.

§ It isoftennecessary tobeable todesignadouble-samplingplanthathasaspecifiedOCcurve.

§ Let (p1,1-α)and(p2,β)bethe twopointsofinterestontheOCcurve.

§ If,inaddition,we imposeanother relationshipontheparametersofthesamplingplan, thena simpleprocedurecanbeusedtoobtainsuchplans.

§ Themostcommonconstraintistorequire thatn2 isamultipleofn1.

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RectifyingInspection

§ Whenrectifying inspectionisperformed withdouble-sampling,theAOQcurve isgivenby

§ assuming thatalldefective itemsdiscovered,either insamplingor100%inspection,are replaced withgoodones.Theaveragetotal inspectioncurve isgivenby

§ Remember Pa =PIa +PII

a thatistheprobabilityoffinal lotacceptance andthat theacceptance probabilitiesdependonthelevel oflotorprocessqualityp.

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MultipleSamplingPlans

§ Thisplanwilloperateasfollows:If,atthecompletionofanystageofsampling,thenumberofdefectiveitemsislessthanorequaltotheacceptancenumber,thelotisaccepted.

§ If,duringanystage,thenumberofdefective itemsequalsorexceedstherejectionnumber,thelotisrejected;otherwisethenextsampleistaken.

§ Themultiple-samplingprocedurecontinuesuntilthefifthsampleistaken,atwhichtimealotdispositiondecisionmustbemade.

§ Thefirstsampleisusuallyinspected100%,althoughsubsequentsamplesareusuallysubjecttocurtailment.

7 – 68

Multiple-Sampling plans

§ TheconstructionofOCcurvesformultiple-samplingisastraightforwardextensionoftheapproachusedindouble-sampling.

§ Similarly,itisalsopossibletocomputetheaveragesamplenumbercurveofmultiple-samplingplans.

§ Onemayalsodesignamultiple-samplingplanforspecifiedvaluesofp1,1,p2,and.

§ Foranextensivediscussionofthesetechniques,seeDuncan(1986).

§ Theprincipaladvantageofmultiple-samplingplansisthatthesamplesrequiredateachstageareusuallysmallerthanthoseinsingleordouble-sampling;thus,someeconomicefficiencyisconnectedwiththeuseoftheprocedure.

§ However,multiple-samplingismuchmorecomplextoadminister.

18

7 – 69

SequentialSamplingplans

§ Insequential-sampling, we takeasequence ofsamples fromthelotandallowthenumberofsamples tobedetermined entirelybythe resultsofthesamplingprocess.

§ Inpractice, sequential samplingcantheoretically continueindefinitely, until thelot isinspected100%.

§ Inpractice, sequential-sampling plansareusually truncatedafterthenumber inspected isequal tothree times thenumber thatwouldhavebeen inspectedusingacorrespondingsingle-samplingplan.

§ Ifthesamplesizeselected ateachstage isgreater thanone, theprocessisusuallycalled groupsequential-sampling.

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Item-by-ItemSequential-Sampling

§ Ifthesamplesizeinspectedateachstage isone, theprocedure isusuallycalled item-by-itemsequential-sampling.

§ Item-by-item sequential-sampling isbasedonthesequentialprobability ratio test(SPRT),developed byWald (1947).

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§ Thecumulativeobservednumberofdefectivesisplottedonthechart.

§ Foreachpoint,theabscissaisthetotalnumberofitemsselecteduptothattime,andtheordinateisthetotalnumberofobserveddefectives.

§ Iftheplottedpointsstaywithintheboundariesoftheacceptanceandrejectionlines,anothersamplemustbedrawn.

§ Assoonasapointfallsonorabovetheupperline,thelotisrejected.

§ Whenasamplepointfallsonorbelowthelowerline,thelotisaccepted.

7 – 72


§ Theequations forthe twolimit lines forspecifiedvaluesofp1,1-α,p2,andβ are

where

19

7 – 73


§ Insteadofusingagraph todetermine the lotdisposition,thesequential-sampling plancanbedisplayed ina table.

§ Theentries inthe tableare foundbysubstitutingvaluesofnintotheequationsfortheacceptance and rejection linesandcalculatingacceptance and rejection numbers.

§ Acceptance andrejection numbersmustbe integers, sotheacceptance number isthenextinteger less thanorequal toXA,andtherejectionnumber isthenext integer greater thanorequal toXR.

7 – 74

Example

Forn=45,

§ XA =-1.22+0.028n=-1.22+0.028(45)=0.04(accept)

§ XR =1.57+0.028n=1.57+0.028(45)=2.83(reject)

§ theacceptance number is0andthe rejectionnumber is3.

§ Note thatthe lotcannotbeaccepted untilatleast44unitshavebeen tested.

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Example7.6DevelopingaSequentialSamplingPlan

§ Supposewe wishtofinda sequential-sampling plan forwhichp1

=0.01,α =0.05,p2 =0.06,andβ =0.10.Thus,

§ Therefore the limit linesare XA =-1.22+0.028n(accept)andXR =1.57+0.028n(reject)

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TheOCCurveandASNCurveforSequential Sampling

§ TheOCcurve forsequential-sampling canbeeasilyobtained. Twopointsonthecurveare (p1,1-α)and(p2,β).Athirdpoint,near themiddleofthecurve, isp=sandPa =h2/(h1 +h2).

§ Theaverage samplenumber takenundersequential-sampling is

20

7 – 77

RectifyingInspectionforSequential-Sampling

§ Theaverage outgoingquality (AOQ)forsequential-sampling isgivenapproximately by

§ Theaverage total inspectionisalsoeasilyobtained. Note thattheamountofsampling isA/CwhenalotisacceptedandNwhen itisrejected. Therefore, theaverage total inspectionis

7 – 78

Discussiontopics




Sampling






7 – 79

DescriptionoftheStandard

§ StandardsamplingproceduresforinspectionbyattributesweredevelopedduringWorldWarII.

§ MILSTD105Eisthemostwidelyusedacceptance-samplingsystemforattributesintheworldtoday.

§ Theoriginalversionofthestandard,MILSTD105A,wasissuedin1950.Sincethen,therehavebeenfourrevisions;thelatestversion,MILSTD105E,wasissuedin1989.

§ MILSTD105Eisacollectionofsamplingschemes;therefore,itisanacceptancesamplingsystem.

§ Thereisaderivativecivilianstandard,ANSI/ASQCZ1.4,whichisquitesimilartothemilitarystandard.

§ ThestandardwasalsoadoptedbytheInternationalOrganizationforStandardizationasISO2859.

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§ Thestandardprovidesforthreetypesofsampling:single-sampling,double-sampling,andmultiple-sampling.

§ Foreachtypeofsamplingplan,aprovisionismadeforeithernormalinspection,tightenedinspection,orreducedinspection.

§ TheprimaryfocalpointofMILSTD105Eistheacceptablequalitylevel(AQL).ThestandardisindexedwithrespecttoaseriesofAQLs.

§ Whenthestandardisusedforpercentdefectiveplans,theAQLsrangefrom0.10%to10%.Fordefectsperunitsplans,thereareanadditionaltenAQLsrunningupto1,000defectsper100units.

§ ItshouldbenotedthatforthesmallerAQLlevels,thesamesamplingplancanbeusedtocontroleitherafractiondefectiveoranumberofdefectsperunit.

§ TheAQLsarearrangedinaprogression,eachAQLbeingapproximately1.585timestheprecedingone.

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7 – 81

InspectionLevels

§ Thesample sizeusedinMILSTD105Eisdetermined bythe lotsizeandby thechoiceofinspectionlevel.

§ Three general levels ofinspectionareprovided.§ LevelIIisdesignatedasnormal.§ LevelIrequiresabouthalftheamountofinspectionthatLevelIIdoesand

maybeusedwhenlessdiscriminationisneeded.§ LevelIIIrequiresabouttwiceasmuchinspectionasLevelIIandshouldbe

usedwhenmorediscriminationisneeded.

§ There arealso fourspecial inspectionlevels, S-1,S-2,S-3,andS-4.Thespecial inspection levelsusevery small samplesandshouldbeemployedonlywhen thesmall samplesizesarenecessary andwhengreater sampling riskscanormustbetolerated.

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§ Fora specifiedAQLandinspectionlevel andagiven lot size,MILSTD105Eprovidesanormal samplingplanthat istobeusedaslongas thesupplier isproducingtheproductatAQLqualityorbetter.

§ Italsoprovidesaprocedure forswitching totightened andreduced inspectionwhenever there isanindication that thesupplier’squalityhaschanged.

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Switchingrules

7 – 84

SwitchingRules

22

7 – 85

ProcedureAstep-by-stepprocedure forusingMILSTD105Eisasfollows:

1. ChoosetheAQL.

2. Choosethe inspectionlevel.

3. Determine thelot size.

4. Findtheappropriate samplesizecode letter fromTable 7.4.

5. Determine theappropriate typeofsamplingplan touse(single,double,multiple).

6. Enter theappropriate table tofindthe typeofplantobeused.

7. Determine thecorrespondingnormaland reduced inspectionplanstobeusedwhenrequired.

7 – 86

SamplesizecodelettersforMILSTD105E

7 – 87 7 – 88

23

7 – 89 7 – 90

Example7.7UseofMILSTD105C

Supposethataproductissubmitted inlotsofsizeN=2,000.Theacceptablequalitylevel is0.65%.Wewilluse thestandard togenerate normal, tightened, and reducedsingle-sampling plansforthissituation.Forlotsofsize2,000undergeneral inspection level II,Table 7.4indicates thattheappropriate samplesizecode letter isK.Therefore, fromTable 7.5,forsingle-sampling plansundernormalinspection, thenormal inspectionplan isn=125,c=2.Table7.6indicatesthatthecorrespondingtightened inspectionplanisn=125,c=1.Note thatinswitchingfromnormaltotightenedinspection, thesample sizeremains thesame, buttheacceptancenumber isreducedby1.Thisgeneral strategy isusedthroughoutMILSTD105Eforatransition totightened inspection.If thenormalinspectionacceptance number is1,2,or3,theacceptance numberforthecorrespondingtightened inspectionplan isreducedby1.

7 – 91

Example7.7UseofMILSTD105C

Ifthenormal inspectionacceptance number is5,7,10,or14,thereduction inacceptance number fortightened inspectionis2.Foranormalacceptance numberof21,the reduction is3.Table 7.7indicates thatunder reduced inspection, thesamplesizeforthisexample wouldben=50,theacceptancenumberwouldbec=1,andtherejectionnumberwouldbe r=3.Thus,iftwodefectiveswere encountered, the lotwouldbeaccepted, butthenext lotwouldbeinspectedundernormal inspection.

Inexamining the tables,note thatifavertical arrow isencountered,the firstsamplingplanaboveorbelowthearrowshouldbeused.When thisoccurs,thesample sizecodeletter andthesample sizechange. Forexample, ifasingle-sampling plan isindexedbyanAQLof1.5%andasample sizecodeletter ofF, thecode letter changes toGandthesamplesizechangesfrom20to32.

7 – 92

Discussion

§ MILSTD105EpresentstheOCcurvesforsingle-samplingplans.Thesearealltype-BOCcurves.

§ TheOCcurvesforthematchingdouble- andmultiple-samplingplansareroughlycomparablewiththoseforthecorrespondingsingle-samplingplans.

§ TheOCcurvespresentedinthestandardarefortheinitiial-samplingplanonly.

§ TheyarenottheOCcurvesfortheoverallinspectionprogram,includingshiftstoandfromtightenedorreducedinspection.

24

7 – 93

MILSTD105ESummary

1. MILSTD105EisAQL-oriented.

2. Thesample sizesselected foruseinMILSTD105Eare 2,3,5,8,13,20,32,50,80,125,200,315,500,800,1250,and2000.Thus,notall samplesizesarepossible.

3. Thesample sizesinMILSTD105Eare related tothelot sizes.

4. Theswitching rules fromnormal totightened inspectionandfromtightened tonormal inspectionarealsosubject tosomecriticism.

5. Aflagrant andcommonabuseofMILSTD105Eisfailure tousetheswitching rulesatall.

7 – 94

Comparisonbetween thecivilianandmilitarystandard

Acivilianstandard,ANSI/ASQCZ1.4orISO2859,isthecounterpartofMILSTD105E.Itwasadoptedin1981&differsfromMILSTD105Einthefollowing5ways:1. Theterminology“nonconformity,”“nonconformance,”and“percent

nonconforming” isused.2. Theswitchingruleswerechangedslightlytoprovideanoptionforreduced

inspectionwithouttheuseoflimitnumbers.3. Severaltablesthatshowmeasuresofschemeperformance(includingthe

switchingrules)wereintroduced.SomeoftheseperformancemeasuresincludeAOQL,limitingqualityforwhichPa 0.10andPa 0.05,ASN,andoperating-characteristiccurves.

4. Asectionwasaddeddescribingproperuseofindividualsamplingplanswhenextractedfromthesystem.

5. Afigureillustratingtheswitchingruleswasadded.Theserevisionsmodernizetheterminologyandemphasizethesystemconceptofthecivilianstandard.Alltables,numbers,andproceduresusedinMILSTD

105EareretainedinANSI/ASQCZ1.4andISO2859.

7 – 95

Discussiontopics




Sampling






7 – 96

TheDodge–Romig SamplingPlans

§ H.F.DodgeandH.G.Romig developed a setofsamplinginspectiontables forlot-by-lotinspectionofproductbyattributesusingtwo typesofsamplingplans:§ plansforlottolerancepercentdefective(LTPD)protectionand§ plansthatprovideaspecifiedaverageoutgoingqualitylimit(AOQL).

§ Foreachoftheseapproaches tosamplingplandesign, there aretables forsingle- anddouble-sampling.

§ Sampling plansthatemphasizeLTPDprotection, suchastheDodge–Romig plans,areoftenpreferred toAQL-orientedsamplingplans, suchasthoseinMILSTD105E,particularly forcritical componentsandparts.

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§ TheDodge–Romig AOQLplansaredesignedsothattheaveragetotalinspectionforagivenAOQLandaspecifiedprocessaveragepwillbeminimized.

§ Similarly,theLTPDplansaredesignedsothattheaveragetotalinspectionisaminimum.

§ ThismakestheDodge–Romigplansveryusefulforin-plantinspectionofsemifinishedproduct.

§ TheDodge–Romig plansapplyonlytoprogramsthatsubmitrejectedlotsto100%inspection.

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AOQLplans

§ TheDodge–Romig (1959)tablesgiveAOQLsamplingplansforAOQLvaluesof0.1%,0.25%,0.5%,0.75%,1%,1.5%,2%,2.5%,3%,4%,5%,7%,and10%.

§ ForeachoftheseAOQLvalues, sixclassesofvalues fortheprocessaverage arespecified.

§ Anexample oftheDodge–Romig samplingplansisshowninTable7.8.

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AOQLplans

7 – 100

AOQLplanExample

§ SupposethatweareinspectingLSImemoryelementsforapersonalcomputerandthattheelementsareshippedinlotsofsizeN=5,000.

§ Thesupplier’sprocessaveragefalloutis1%nonconforming.

§ Wewishtofindasingle-samplingplanwithanAOQL3%.

§ FromTable7.8,wefindthattheplanisn=65&c=3

§ Table7.8alsoindicatesthattheLTPDforthissamplingplanis10.3%.

§ ThisisthepointontheOCcurveforwhichPa=0.10.

§ Therefore,thesamplingplann=65,c=3givesanAOQLof3%nonconformingandprovidesassurancethat90%ofincominglotsthatareasbadas10.3%defectivewillberejected.

26

7 – 101

Examplecontd.

§ Assuming thatincomingquality isequal totheprocessaverageandthat theprobabilityoflotacceptance at thislevelofquality isPa =0.9957,we findthattheaverage totalinspectionforthisplanis

§ Thus,we will inspectapproximately 86units,ontheaverage, inorder tosentencea lot.

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LTPDplans

§ TheDodge–Romig LTPDtablesaredesignedsothat theprobabilityoflotacceptance at theLTPDis0.1.

§ Tables areprovided forLTPDvaluesof0.5%,1%,2%,3%,4%,5%,7%,and10%.

§ Table 7.9foranLTPDof1%isrepresentative oftheseDodge–Romig tables.

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LTPDplans

7 – 104

Example

§ SupposethatLSImemory elements forapersonal computerareshippedfromthesupplier inlotsofsizeN=5000.

§ Thesupplier’s processaverage fallout is0.25%nonconforming,andwe wishtouseasingle-sampling planwithanLTPDof1%.

§ From inspectionofTable 7.9,thesamplingplan thatshouldbeusedisn=770,c=4

§ Ifwe assumethatrejectedlotsarescreened100%andthatdefective itemsare replaced withgoodones, theAOQL forthisplan isapproximately 0.28%.

27

7 – 105

Discussiontopics




Sampling






7 – 106

GeneralDescriptionoftheStandard

§ MILSTD414isalot-by-lotacceptance-sampling plan forvariables. The standardwas introducedin1957.

§ The focalpointofthisstandard istheacceptable quality level(AQL),whichranges from0.04%to15%.

§ There are fivegeneral levelsofinspection,andlevel IV isdesignated as“normal.”

§ Aswith theattributes standard,MILSTD105E,samplesizecodelettersare used,butthesamecode letter doesnotimply thesamesample sizeinbothstandards.

§ All thesamplingplansandprocedures inthestandardassumethat thequalitycharacteristic ofinterest isnormally distributed.

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OrganizationofMILSTD414

7 – 108

Procedure1§ Take a randomsample ofnitemsfromthelotandcomputethe

statistic

§ Note thatZLSL intheaboveequationsimplyexpressesthedistancebetween thesample average and the lowerspecification limit instandarddeviation units.

§ The larger thevalueofZLSL,the fartherthesampleaverage isfromthe lower specification limit,andconsequently, thesmalleristhe lotfractiondefective p.

§ Ifthere isacriticalvalue ofpofinterestthatshouldnotbeexceededwithstatedprobability,we cantranslate thisvalueofpintoacriticaldistance— say,k—forZLSL.

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7 – 109

Procedure1(contd.)

§ Thus,ifZLSL ≥k,wewouldacceptthelotbecausethesample dataimply that thelotmean issufficiently farabove theLSLtoensurethat thelot fractionnonconformingissatisfactory.

§ However, ifZLSL <k, themean istooclosetotheLSL,andthe lotshouldberejected.

7 – 110

Procedure2

§ Take a randomsample ofnitemsfromthelotandcomputeZLSLusingtheequationbelow.

§ UseZLSL toestimate thefractiondefective ofthelotorprocessasthearea underthestandardnormalcurvebelowZLSL.

§ (Actually, usingQLSL=ZLSL√(n/(n-1))asa standardnormalvariable isslightlybetter, because itgives abetter estimate ofp.)Letp̂betheestimateofpsoobtained.Iftheestimateexceeds aspecifiedmaximum valueM,reject the lot;otherwise, accept it.

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MILSTD414(ANSI/ASQC Z1.9)

§ SpecificationLimits§ Inthecaseofsingle-specificationlimits,eitherProcedure1orProcedure2

maybeused.§ Iftherearedouble-specificationlimits,thenProcedure2mustbeused.

§ LotVariability§ Iftheprocessorlotvariabilityisknownandstable,thevariabilityknown

plansarethemosteconomicallyefficient.§ Whenlotorprocessvariabilityisunknown,eitherthestandarddeviationor

therangeofthesamplemaybeusedinoperatingthesamplingplan.

§ The range methodrequiresa larger samplesize,andwedonotgenerally recommend itsuse.

7 – 112

MILSTD414(ANSI/ASQC Z1.9)

§ MILSTD414isdividedintofoursections.

§ SectionA isageneral descriptionofthesampling plans,includingdefinitions, samplesizecode letters,andOCcurvesforthevarioussamplingplans.

§ SectionB ofthestandardgivesvariables samplingplansbasedonthesamplestandarddeviation forthecaseinwhichtheprocessorlotvariability isunknown.

§ SectionC presentsvariables samplingplansbasedonthesamplerange method.

§ SectionD givesvariables samplingplansforthecasewhere theprocessstandarddeviation isknown.

29

7 – 113

Example7.8UsingMILSTD414

Considera soft-drinkbottlerwhoispurchasingbottlesfromasupplier.The lower specification limitonburstingstrengthis225psi.Supposethat theAQLat thisspecification limit is1%.Suppose thatbottlesare shippedinlotsofsize100,000.Findavariables samplingplan thatusesProcedure1fromMILSTD414.Assumethatthe lotstandarddeviation isunknown.

7 – 114

Example7.8UsingMILSTD414

Solution

FromTable 7.10,ifweuse inspectionlevel IV, thesamplesizecodeletter isO.FromTable 7.11we findthat samplesizecodeletter Oimpliesa samplesizeofn100.

Foranacceptable quality level of1%,onnormal inspection, thevalueofk is2.00.Iftightenedinspectionisemployed, theappropriate valueofkis2.14.

Notethatnormalandtightened inspectionuse thesame tables.TheAQLvalues fornormal inspectionare indexedat the topofthetable,andtheAQLvalues fortightened inspectionare indexed fromthebottomofthe table.

7 – 115

UseoftheTables

7 – 116

30

7 – 117

UseoftheTables

§ MILSTD414containsaprovisionforashifttotightened orreduced inspectionwhen thisiswarranted.

§ Theprocessaverage isusedasthebasisfordetermining whensuchashiftismade.

§ Usually, theprocessaverage iscomputedusinginformation fromthepreceding tenlots.

§ Fulldetailsoftheswitchingproceduresaredescribed in thestandardand ina technicalmemorandum onMILSTD414,publishedbytheUnitedStatesDepartment oftheNavy, BureauofOrdnance.

7 – 118

UseoftheTables§ FractionDefective§ EstimationofthefractiondefectiveisrequiredinusingProcedure2ofMILSTD

414.§ Itisalsorequiredinimplementingtheswitchingrulesbetweennormal,

tightened,andreducedinspection.§ Inthestandard,threetablesareprovidedforestimatingthefractiondefective.

§ StandardDeviation§ WhenstartingtouseMILSTD414,onecanchoosebetweentheknown

standarddeviationandunknownstandarddeviationprocedures.§ ItisagoodideatomaintaineitheranRorschartontheresultsofeachlotso

thatsomeinformationonthestateofstatisticalcontrolofthescatterinthemanufacturingprocesscanbecollected.

§ Whenaknownsplanisused,itisalsonecessarytomaintainacontrolchartoneitherRorsasacontinuouscheckontheassumptionofstableandknownprocessvariability.

7 – 119

UseoftheTables

§ Mixedvariables/attributes acceptance-sampling plans§ MILSTD414containsaspecialprocedureforapplicationofmixed

variables/attributesacceptance-samplingplans.§ Ifthelotdoesnotmeettheacceptabilitycriterionofthevariablesplan,an

attributessinglesamplingplan,usingtightenedinspectionandthesameAQL,isobtainedfromMILSTD105E.

§ Alotcanbeacceptedbyeitheroftheplansinsequencebutmustberejectedbyboththevariablesandattributesplan.

7 – 120

Discussion

§ In1980, theAmericanNationalStandardsInstituteandtheAmericanSocietyforQualityControl releasedanupdatedcivilianversionofMILSTD414knownasANSI/ASQCZ1.9.MILSTD414wasoriginallystructuredtogiveprotection essentiallyequivalenttothatprovidedbyMILSTD105A(1950).

§ WhenMILSTD105Dwasadoptedin1963, thisnewstandardcontainedsubstantiallyrevisedtablesandprocedures thatledtodifferences inprotection betweenitandMILSTD414.

§ Consequently, itisnotpossibletomove directlyfromanattributessamplingplaninthecurrentMIL STD105Etoacorresponding variablesplaninMILSTD414iftheassurance ofcontinued protectionisdesired forcertainlotsizesand AQLs.

§ TheciviliancounterpartofMILSTD414,ANSI/ASQCZ1.9,restoresthisoriginalmatch.Thatis,ANSI/ASQCZ1.9isdirectlycompatiblewithMILSTD105E(and itsequivalentciviliancounterpartANSI/ASQCZ1.4).

31

7 – 121

Discussion

§ Thisequivalence wasobtainedbyincorporating the followingrevisionsinANSI/ASQC Z1.9:

1. Lotsizerangeswere adjusted tocorrespondtoMILSTD105D.

2. Thecode lettersassigned tothevariouslotsizeranges werearranged tomake protectionequal tothatofMILSTD105E.

3. AQLsof0.04,0.065,and15were deleted.

4. Theoriginal inspectionlevels I, II, III, IV,andVwere relabeled S3,S4,I, II, III, respectively.

5. Theoriginal switchingruleswere replaced bythoseofMILSTD105E,withslightrevisions.

7 – 122

ChangesmadetotheStandard§ Inaddition, tomodernizing terminology, thewordnonconformity

wassubstitutedfordefect,nonconformancewassubstitutedfordefective, andpercentnonconformingwassubstitutedforpercentdefective.

§ Theoperating-characteristic curveswere recomputed andreplotted, andanumberofeditorial changeswere made tothedescriptive material ofthestandardtomatchMILSTD105Eascloselyaspossible.

§ Finally, anappendixwasincludedshowing thematchbetweenANSI/ASQC Z1.9,MILSTD105E,andthecorrespondingcivilianversionANSIZ1.4.

§ Thisappendixalsoprovidedselected percentage pointsfromtheOCcurvesofthesestandardsandtheir differences.

7 – 123

Discussion§ Asofthiswriting, theDepartment ofDefensehasnotofficially

adoptedANSI/ASQC Z1.9andcontinuestouseMILSTD414.

§ Theprincipaladvantage oftheANSI/ASQC Z1.9standardisthat itispossibletostartinspectionbyusinganattributes samplingscheme fromMILSTD105EorANSI/ASQC Z1.4, collect sufficientinformation tousevariables inspection,andthenswitchtothevariables scheme, whilemaintaining thesameAQL-code lettercombination.

§ AsinMILSTD414,ANSI/ASQC Z1.9assumes thatthequalitycharacteristic isnormallydistributed.

§ Iftheassumptionofnormality isbadlyviolated, either a specialvariables samplingproceduremustbedeveloped orwe mustreturn toattributes inspection.

7 – 124

Discussiontopics




Sampling






32

7 – 125

ChainSampling

§ Forsituationsinwhichtestingisdestructiveorveryexpensive,samplingplanswithsmallsamplesizesareusuallyselected.

§ Thesesmallsamplesizeplansoftenhaveacceptancenumbersof0.

§ Planswithzeroacceptancenumbersareoftenundesirable,however,inthattheirOCcurvesareconvexthroughout.

§ Figures7.6and7.8presentOCcurvesofsamplingplansthathaveacceptancenumbersof0andacceptancenumbersthataregreaterthan0.

§ Dodge(1955)suggestedanalternateprocedure,knownaschainsampling,thatmightbeasubstituteforordinarysingle-samplingplanswithzeroacceptancenumbersincertaincircumstances.

7 – 126

ChainSampling

§ Chainsampling plansmakeuseofthecumulative resultsofseveral preceding lots.Thegeneral procedure isasfollows:1. Foreachlot,selectthesampleofsizenandobservethenumberof

defectives.2. Ifthesamplehaszerodefectives,acceptthelot;ifthesamplehastwoor

moredefectives,rejectthelot;andifthesamplehasonedefective,acceptthelotprovidedtherehavebeennodefectivesinthepreviousilots.

§ Thus,forachainsampling plangivenbyn=5,i =3,alotwouldbeaccepted iftherewere nodefectives inthesampleoffive,or ifthere were onedefective inthesampleoffiveandnodefectiveshadbeen observed inthesamples fromtheprevious three lots.

§ Thistypeofplan isknownasaChSP-1plan.

7 – 127

ChainSampling

§ TheeffectofchainsamplingistoaltertheshapeoftheOCcurveneartheoriginsothatithasamoredesirableshape.

§ Figure7.21showsOCcurvesforChSP-1planswithn=5,c=0,andi =1,2,3,and5.

§ Thecurvefori =1isdotted,anditisnotapreferredchoice.

§ Inpractice,valuesofi usuallyvarybetween3and5,sincetheOCcurvesofsuchplansapproximatethesingle-samplingplanOCcurve.

7 – 128

ChainSampling§ ThepointsontheOCcurveofaChSP-1planare givenbythe

equation

§ where P(0,n)andP(1,n)aretheprobabilitiesofobtainingzeroandonedefectives, respectively, outofarandomsampleofsizen.To illustratethecomputations, considertheChSP-1planwithn=5,c=0,andi =3.Forp=0.10,wehave

33

7 – 129

Conditions forChain SamplingTheproperuseofchainsampling requires that the followingconditionsbemet:

1. Thelotshouldbeoneofaseriesinacontinuingstreamoflots,fromaprocessinwhichthereisrepetitiveproductionunderthesameconditions,andinwhichthelotsofproductsareofferedforacceptanceinsubstantiallytheorderofproduction.

2. Lotsshouldusuallybeexpectedtobeofessentiallythesamequality.

3. Thesamplingagencyshouldhavenoreasontobelievethatthecurrentlotisofpoorerqualitythanthoseimmediatelypreceding.

4. Thereshouldbeagoodrecordofqualityperformanceonthepartofthesupplier.

5. Thesamplingagencymusthaveconfidencethatthesupplierwillnottakeadvantageofitsgoodrecordandoccasionallysendabadlotwhensuchalotwouldhavethebestchanceofacceptance.

7 – 130

Discussiontopics




Sampling






7 – 131

Continuous-Sampling

§ All thesamplingplansdiscussedpreviouslyare lot-by-lotplans.

§ However, manymanufacturingoperations,particularly complexassemblyprocesses,donotresultin thenatural formationoflots.

§ Whenproductioniscontinuous,twoapproachesmaybeusedtoformlots.

§ The firstprocedureallows theaccumulationofproductionatgivenpointsintheassemblyprocess.

§ Thesecondprocedurearbitrarily marksoffagivensegment ofproductionasa“lot.”

7 – 132

Continuous-Sampling

§ FirstProcedure§ Thefirstprocedureallowstheaccumulationofproductionatgivenpoints

intheassemblyprocess.§ Thishasthedisadvantageofcreatingin-processinventoryatvariouspoints,

whichrequiresadditionalspace,mayconstituteasafetyhazard,andisagenerallyinefficientapproachtomanaginganassemblyline.

§ SecondProcedure§ Thedisadvantageofthisapproachisthatifalotisultimatelyrejectedand

100%inspectionofthelotissubsequentlyrequired,itmaybenecessarytorecallproductsfrommanufacturingoperationsthatarefurtherdownstream.

§ Thismayrequiredisassemblyoratleastpartialdestructionofsemifinisheditems.

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7 – 133

Continuous-Sampling

§ Continuous-sampling plansconsistofalternating sequences ofsampling inspectionandscreening (100%inspection).

§ Theplansusuallybeginwith100%inspection,andwhenastatednumberofunitsisfoundtobefreeofdefects (thenumber ofunitsi isusuallycalled theclearancenumber), samplinginspectionisinstituted.

§ Sampling inspectioncontinuesuntila specifiednumberofdefective unitsisfound,atwhich time100%inspectionisresumed.

§ Continuous-samplingplansare rectifying inspectionplans,inthatthequalityoftheproductisimprovedby thepartial screening.

7 – 134

CSP-1

§ Continuous-samplingplanswere firstproposedbyHaroldF.Dodge(1943).

§ Dodge’s initialplan iscalledCSP-1.

§ ACSP-1planhasanoverallAOQL.

§ Thevalue oftheAOQLdependsonthevaluesoftheclearance number i andthesamplingfractionf.

7 – 135

CSP-1

§ Table 7.12presentsvariousvaluesofi andfforCSP-1thatwillleadtoastipulatedAOQL.

§ Note inthe table thatanAOQLof0.79%couldbeobtainedusinga samplingplanwith i =59andf=1/3,orwith i =113andf=1/7.

§ Thechoiceofi andfisusuallybasedonpracticalconsiderationsinthemanufacturingprocess.

§ Asageneral rule,however, itisnotagoodidea tochoosevaluesoffsmaller than1/200becausetheprotectionagainstbadquality inacontinuousrunofproductionthenbecomesverypoor.

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CSP-1

§ Theaverage numberofunitsinspected ina100%screeningsequence following theoccurrenceofadefect isequal to

where q=1- p,andpisthefractiondefectiveproducedwhentheprocessisoperating incontrol.

§ Theaverage numberofunitspassedunder thesamplinginspectionprocedure beforeadefective unitisfoundis

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CSP-1

§ Theaverage fractionoftotalmanufactured unitsinspectedinthelongrun is

§ Theaverage fractionofmanufactured unitspassedunder thesamplingprocedure is

§ WhenPa isplottedasafunctionofp,weobtainanoperatingcharacteristiccurve foracontinuous-samplingplan.

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OCCurves forCSP-1

§ Graphsofoperating-characteristic curvesforseveral valuesoffandi forCSP-1plansareshowninFigure 7.23.

§ Note thatformoderate-to-smallvaluesoff,i hasmuchmoreeffectontheshapeofthecurve thanf.

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OtherContinuous-Sampling Plans

§ DodgeandTorrey (1951)proposedCSP-2andCSP-3.

§ CSP-2§ 100%inspectionwillnotbereinstatedwhenproductionisundersampling

inspectionuntiltwodefectivesampleunitshavebeenfoundwithinaspaceofKsampleunitsofeachother.

§ ChooseKequaltotheclearancenumberi.§ CSP-2plansareindexedbyspecificAOQLsthatmaybeobtainedby

differentcombinationsofi andf.

§ CSP-3§ Designedtogiveadditionalprotectionagainstspottyproduction.§ Requiresthatafteradefectiveunithasbeenfoundinsamplinginspection,

theimmediatelyfollowingfourunitsshouldbeinspected.Ifanyofthesefourunitsisdefective,100%inspectionisimmediatelyreinstituted.Ifnodefectivesarefound,theplancontinuesasunderCSP-2.

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MILSTD1235C

§ Anothercommonobjection tocontinuous-samplingplansistheabrupt transitionbetween sampling inspectionand100%inspection. Lieberman andSolomon(1955)havedesignedmultilevel continuoussamplingplanstoovercome thisobjection.

§ Muchofthework oncontinuous-samplingplanshasbeenincorporated intoMILSTD1235C.

§ Thestandardprovidesforfivedifferent typesofcontinuous-samplingplans.Tables toassisttheanalyst indesigningsamplingplansare presented inthestandard.§ CSP-1andCSP-2areapartofMILSTD1235C.§ Inaddition,therearetwoothersingle-levelcontinuous-sampling

procedures,CSP-FandCSP-V.§ ThefifthplaninthestandardisCSP-T,amultilevelcontinuous-samplingplan.

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MILSTD105E

§ MILSTD105E,whichdoesfocusontheAQL, isdesigned formanufacturing situationsinwhichlotting isanaturalaspectofproduction,andprovidesa setofdecisionrules forsentencinglotssothatcertain AQLprotection isobtained.

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Discussiontopics




Sampling






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Skip-Lot SamplingPlans

§ Thissectiondescribes thedevelopment andevaluation ofasystemoflot-by-lotinspectionplans inwhichaprovisionismadeforinspectingonlysome fractionofthesubmitted lots.

§ These plansareknownasskip-lot sampling plans.

§ Dodge (1956)initiallypresentedskip-lotsamplingplansasanextensionofCSP-typecontinuous-samplingplans.

§ Theversionofskip-lotsampling initiallyproposedbyDodgerequired asingledetermination oranalysis toascertain the lot’sacceptability orunacceptability.

§ These plansarecalledSkSP-1.

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§ Skip lotsamplingplansdesignated SkSP-2followthenext logicalstep; thatis,each lottobesentenced issampledaccording toaparticular attribute lotinspectionplan.Perry (1973)givesagooddiscussionoftheseplans.

§ Askip-lot samplingplanoftypeSkSP-2usesa specified lotinspectionplancalled the“reference-sampling plan,” togetherwith the following rules:1. Beginwithnormalinspection,usingthereferenceplan.Atthisstageof

operation,everylotisinspected.2. Wheni consecutivelotsareacceptedonnormalinspection,switchto

skippinginspection.Inskippinginspection,afractionfofthelotsisinspected.

3. Whenalotisrejectedonskippinginspection,returntonormalinspection.

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§ Theparameters fandi aretheparametersoftheskip-lotsamplingplanSkSP-2,where, i isapositiveinteger, andfliesintheinterval0<f<1.

§ LetPdenotetheprobabilityofacceptanceofalotfromthereference-sampling plan.Then, Pa(f,i)istheprobabilityofacceptance fortheskip-lotsamplingplanSkSP-2,where

§ It canbeshownthatforf2<f1,agivenvalue oftheclearancenumber i,andaspecifiedreference-samplingplan,

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Skip-Lot SamplingPlans§ Furthermore, fori<j,a fixedvalue off,andagiven reference-

samplingplan,

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§ Avery importantpropertyofa skip-lotsamplingplanistheaverage amountofinspection required.

§ Theaverage samplenumber ofaskip-lotsamplingplan is

§ where Fistheaverage fractionofsubmittedlotsthataresampledandASN(R)istheaveragesamplenumberofthereference-samplingplan.It canbeshownthat

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Example

§ Considera reference-samplingplanofn=20andc=1.Since theaverage samplenumber fora single-samplingplan isASN=n,wehaveASN(SkSP)=n(F)

§ Figure 7.26presentstheASNcurve forthe reference-samplingplann=20,c=1andtheskip-lotsamplingplansshowninthefigure.

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§ Wenotethat forsmallvaluesofincominglot fractiondefective,the reductionsinaverage samplenumber areverysubstantial fortheskip-lotsamplingplansevaluated.

§ Ifthe incoming lotquality isverygood,consistentlyclosetozerofractionnonconforming,say, thenasmallvalue off,perhaps1/4or1/5, couldbeused.Ifincomingquality isslightlyworse, thenanappropriate value offmightbe1/2.

§ Skip-lot samplingplansareaneffective acceptance-samplingprocedureandmaybeusefulasasystemofreduced inspection.

§ Theyseem toworkbestwhen thesupplier’s processesare inastateofstatisticalcontrolandwhen theprocesscapability isadequate toensurevirtually defect-free production.

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End- Chapter7

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