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Acce

ptan

ceS

ampl

ing

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Acc

eptanceSam

pling

Accept/rejectentirelotbasedons

ampleresults

Created

byDodgean

dRomigdur

ingWWII

NotconsistentwithTQMofZeroD

efects

Doesno

testimatethequalityofth

elot

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Whatisaccep

tancesampling?

LotAccepta

nceSampling

ASQC

technique,wherearandomsam

pleis

takenfromalot,a

nduponth

eresultso

f

appraisingthesample,thelo

twille

itherbe

rejecte

doraccepted

Aprocedureforsentencingincomingbatches

orlots

ofitemswithoutdoing

100%inspection

Themostwidelyusedsamplingplansa

re

givenb

yMilitaryS

tandard(M

IL-STD-10

5E)

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Whatisaccep

tancesampling?

Purposes

Determ

inethequalitylevelofan

incoming

shipmentorattheendofproduction

Judge

whetherqua

lityleveliswithintheleve

l

thatha

sbeenpredetermined

But!Acceptance

sampling

givesyou

noidea

aboutthe

process

thatis

producingthoseitems!

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Typesofsamplingplans

Samplingbyattributesvs.

sampling

by

variables

Incomingvs.outgoinginsp

ection

Rectifyingvs.non-rectifyin

ginspection

Whatisdonewithnonconformin

gitemsfoun

d

during

inspection

Defectivesmaybe

replacedbygooditems

Single,

double,m

ultiplean

dsequen

tial

plans

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How

accept

ancesampling

w

orks

Attributes(gono-go

inspectio

n)

Defectives-produ

ctacceptabilityacrossrange

Defects-numberofdefectsp

erunit

Variable

(continuou

smeasurement)

Usuallymeasuredbymean

andstanda

rd

deviation

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W

hyus

eacceptance

sampling?

Candoeither100%

inspection,orinspect

a

sampleofafewitem

stakenfromthelot

Complete

inspection

Inspectingeachitemproducedtoseeif

each

itemmeetsthelev

eldesired

Usedw

hendefectiveitemsw

ouldbeve

ry

detrime

ntalinsom

eway

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Whynot100%ins

pection

?

Problemsw

ith100%inspection

Verye

xpensive

Cantusewhenp

roductmustbedestroyedto

test

Handlingbyinspectorscaninducedefects

Inspec

tionmustbeverytedioussodefective

itemsdonotslipthroughins

pection

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TheSingle

SamplingPlan

Themostcommonan

deasiestpla

ntousebut

not

mostefficientinterms

ofaveragenumberofsa

mples

needed

Singlesa

mplingplan

N=lotsize

n=samplesize(ra

ndomized)

c=acceptancenum

ber

d=numberofdefectiveitemsin

sample

Rule:Ifdc,accept

lot;elserejectthelot

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Producers&Consum

ersRis

ks

duetomista

kensentencing

TYPEIERROR

=P(rejectgoodlot)

orP

roducersrisk

5%iscommon

TYPEIIERROR=P(acceptbadlo

t)

orC

onsumers

risk

10%istypicalva

lue

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Q

uality

Definitions

Acceptancequalitylevel(AQL)

Thesm

allestpercentageofdefectivesth

atwill

maketh

elotdefinitelyacceptable.Aquality

levelthatisthebaselinerequirementof

the

customer

RQLorLottolerance

percentdefective(LT

PD)

Quality

levelthatisunacceptabletothe

customer

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Accep

tancesamplingcont

d.

Producersrisk

Riskas

sociatedwithalotofacceptablequality

rejected

Alpha

=Prob(c

ommittingTy

peIerror)

=P(rejec

tinglotatAQ

Lqualitylev

el)

=producersrisk

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Tak

earandomized

sampleofsizenfrom

thelotof

unk

nownqualityp

T

heSing

le

Samplin

g

p

rocedure

Inspectallitemsinthe

sample

Defectivesfound=d

d

c

? No

Yes

Rejectlot

Accept

lot

Returnlot

tosupplier

Do100%

ins

pection

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OperatingChara

cteristic

(OC)Cu

rve

Itisagraphofthe%d

efective(p)inalotorbatchvs.

theproba

bilitythatthe

samplingplanwillacceptthelot

Showsprobabilityoflo

tacceptancePaasfunctionof

lotquality

level(p)

Itisbasedonthesamplingplan

Curveind

icatesdiscrim

inatingpow

eroftheplan

Aidsinse

lectionofpla

nsthataree

ffectiveinreducing

risk

Helpstokeepthehigh

costofinspectiondown

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OperatingCharacteristicCu

rve

AQL

LTPD

=0.10

=0.05

Probabilityofacceptance,Pa

{0.60

0.40

0.20

0.020.04

0.06

0.08

0.10

0.12

0.14

0.16

0.180.20

0.80{

Propo

rtiondefective

p

1.00

OCcu

rveforn

and

c

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OC

Curve

Calcu

lation

TheWays

ofCalculatin

gOCCurves

Binomialdistribution

Hype

rgeometricd

istribution

Pa=P(rdefectivesfoundin

asampleof

n)

Poissonformula

P(r)=((np)re-np)/r!

Larsonnomogram

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OC

CurveCalcula

tionby

P

oisson

distribution

APoisson

formulacan

beused

P(r)=((np)re-np)/r!

=Prob(exac

tlyrdefectivesinn)

Poissonis

alimit

Limitationsofusing

Poisson

nN/10totalbatch

Little

faithinPois

sonprobabilitycalculationwhenn

isqu

itesmallandpquitelarge.

ForPoisson,Pa=P(r

c)

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p

Forus,Pa=P(r

c)

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OCCurveCalculationbyBinom

ial

Distribution

Notethat

wecannotalwaysusethe

binomial

distributionbecause

Binomialsarebasedonconstantprobabilities

N

isnotinfinite

p

changesas

itemsaredra

wnfromthe

lot

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OCCurvebyBinomialForm

ula

.12

.115

.11

.162

.10

.223

.09

.300

.08

.394

.07

.502

.06

.620

.05

.739

.04

.845

.03

.930

.02

.980

.01

.998

Pd

Pa

Usingthisformulawithn

=52and

c=3andp=.01,.02,...,.1

2wefind

datavaluesasshownon

theright.

Thisgivenstheplotshow

nbelow.

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TheIdea

lOCC

urve

Idealcurve

wouldbe

perfectlype

rpendicular

from0to100%fora

fractiondef

ective=AQL

Itwillaccep

teverylotwith

pAQLan

drejectever

y

lotwithp>

AQL

p

A

QL

1.0

0.0P

a

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Properties

ofOCCurves

Theaccep

tancenumbercandsam

plesizenaremost

important

factorsindefiningtheOC

curve

Decreasin

gtheacceptancenumberispreferred

over

increasing

samplesize

ThelargerthesamplesizethesteeperistheOC

curve

(i.e.,itbec

omesmorediscriminatingbetweengo

odand

badlots)

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Properties

ofOCCurves

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AOQ

Curve

0.015

AOQL

Average

Outgoing

Quality

0.010

0.005

0.10

0

.09

0.01

0.02

0.03

0.04

0.050.06

0.07

0.08

AQL

LTPD

(Incoming)PercentD

efective

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DoubleSa

mpling

Plans

Takesm

allinitials

ample

If#de

fectivesupperlimit,

reject

If#de

fectivesbetweenlimits,takesecond

sample

Accepto

rrejectlotbasedon2samples

Lessinspectionth

aninsing

le-sampling

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MultipleSampling

Plans

Advanta

ge:Usess

mallersam

plesizes

Takeinitialsample

If#d

efectives

upperlimit,reject

If#d

efectivesb

etweenlim

its,re-sample

Continuesampling

untilaccep

torrejectlot

basedonallsampledata

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SequentialSam

pling

Theultim

ateexten

sionofm

ultiple

sampling

Itemsareselected

fromalo

toneata

time

Afterinspectionof

eachsam

pleadecision

ismadetoaccept

thelot,re

jectthelo

t,or

toselect

anotheritem

InSkipLotSamplingonlyafractionofthe

lotssubm

ittedare

inspected

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Choos

ingAS

amplin

gMeth

od

Anecon

omicdec

ision

Singles

amplingp

lans

high

sampling

costs

Double/Multiples

amplingp

lans

lowsamplingcosts

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Tak

earandomized

sampleofsizenfrom

thelotof

unk

nownqualityp

De

signingThe

SingleSampling

plan

Inspectallitemsinthe

sample

Defectivesfound=d

d

c

? No

Yes

Rejectlot

Accept

lot

Returnlot

tosupplier

Do100%

ins

pection

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Poiss

ondistributionfo

rDefects

Poissonp

arameter:

=np

P(r)=(np

)re-np/r!=Prob(exactlyrd

efectivesinn)

Thisformulamaybeu

sedtoformu

lateequation

s

involvingAQL,RQL,andtogive

n(n,c).

Wecanu

sePoissontablestoappr

oximatelyso

lve

theseequ

ations.Pois

soncanapproximatebinomial

probabilitiesifnislarg

eandpsma

ll.

Q.Ifwesam

ple50items

fromalarge

lot,whatisthe

probabilitythat2ared

efectiveifthedefectrate

(p)=

.02?Whatistheprobabilitythatno

morethan3

defectsarefoundoutofthe50?

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Samp

lingPlan

DesignbyBinomia

l

Dis

tribution

Binomia

ldistributio

n:

P(xdefe

ctivesinn)=[n!/(x!(n-x))!]px(1-p

)n-x

Recalln!/(x!(n-x))!

=waystochoosexin

n

Q

.If4samp

les(items)

arechosenfroma

populationwithad

efectrate=

.1,whatisthe

probabilitythat

a)exactly1outof4is

defective?

b)atm

ost1outof4is

defective?

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Solvingfor(n,c)

Todesigna

singlesamplingplanweneedtwopoints.

Typicallythe

searep1=AQL,p2=LTPDand

,

arethe

Producer'sR

isk(TypeIe

rror)andCo

nsumer'sRis

k(Type

IIerror),respectively.By

binomialformulas,nand

care

thesolution

to

Thesetwos

imultaneous

equationsar

enonlinears

othere

isnosimple,directsolution.TheLarsonnomogram

can

helpushere

.

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TheLarson

Nomogr

am

Appliestos

ingle

samplingplan

Basedonb

inomial

distribution

Uses1-=PaatAQL

=PaatRQL

Canproduc

eOC

curve

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De

finitionsandT

erms

Reference:NISTEngineeringStatis

ticsHandbook

Acceptable

QualityLev

el(AQL):Th

eAQLisap

ercent

defectivethatisthebase

linerequirementforthequality

oftheprodu

cer'sproduct.Theproduc

erwouldlike

to

designasamplingplans

uchthatthereisahigh

probabilityo

facceptinga

lotthathas

adefectleve

lless

thanorequaltotheAQL

.

LotToleran

cePercentDefective(LTPD)a

lsoca

lle

d

RQL(Rejec

tionQuality

Level):The

LTPDisa

designatedhighdefectle

velthatwou

ldbeunacce

ptable

totheconsu

mer.Theconsumerwouldlikethesampling

plantohave

alowproba

bilityofacceptingalotwitha

defectlevelashighasth

eLTPD.

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TypeIError(Producer'sRisk):This

istheproba

bility,

foragiven(n,c)samplin

gplan,ofrejectingalotthathas

adefectlevelequaltoth

eAQL.Theproducersuffers

whenthisoccurs,becausealotwithacceptablequality

wasrejected.Thesymbol

iscomm

onlyusedforthe

TypeIerror

andtypicalv

aluesfor

rangefrom0.2to

0.01.

TypeIIErro

r(Consume

r'sRisk):Thisistheprobability,

foragiven(n,c)samplin

gplan,ofac

ceptingalot

witha

defectlevelequaltothe

LTPD.Thec

onsumersuffers

whenthisoccurs,becausealotwithunacceptable

qualitywas

accepted.Th

esymbol

iscommonly

used

fortheType

IIerrorandtypicalvaluesrangefrom

0.2to

0.01.

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OperatingC

haracteristic(OC)Curv

e:Thiscurve

plotsthepro

babilityofac

ceptingthelot(Y-axis)ve

rsus

thelotfractionorpercentdefectives(X-axis).

TheOCcurveistheprim

arytoolford

isplayingand

investigating

thepropertiesofasamp

lingplan.

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AverageOutgoingQua

lity(AOQ):Acommon

procedure,

whensamplingandtestin

gisnon-

destructive,isto100%inspectreject

edlotsandreplace

alldefective

swithgoodunits.Inthis

case,allreje

cted

lotsaremadeperfectan

dtheonlyde

fectsleftare

those

inlotsthatwereaccepte

d.

AOQ'srefertothelong

term

defectlevelforthiscombinedLASPand100%

inspectionofrejectedlotsprocess.If

alllotscome

in

withadefectlevelofexactlyp,andtheOCcurve

forthe

chosen(n,c

)LASPindic

atesaproba

bilityp

aof

acceptings

uchalot,overthelongru

ntheAOQc

an

easilybesh

owntobe:

whereNisthelotsize.

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AverageSampleNumber(ASN):Fo

rasinglesampling

plan(n,c)weknoweach

andeverylo

thasasamp

leof

sizentaken

andinspecte

dortested.Fordouble,m

ultiple

andsequentialplans,the

amountofsamplingvaries

dependingo

nthenumberofdefectso

bserved.Forany

givendouble

,multipleor

sequentialplan,alongtermASN

canbecalcu

latedassumingalllotsco

meinwitha

defect

levelofp.A

plotoftheASN,versusth

eincomingd

efect

levelp,desc

ribesthesam

plingefficiencyofagivenlot

samplingscheme.

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TheM

IL-STD

-105E

approa

ch

AQueryfroma

Practitioner:SelectingAQL(ac

ceptablequalitylevels)

I'dlikesomeguidanceonselectin

ganacceptablequalitylevelandinspection

levelswhenusin

gsamplingproce

duresandtables

.Forexample,w

henIuse

MIL-STD-105E,howdoItodecidewhenIshoulduseGI,GIIorS2,S4?

--ConfusedinColumb

us,

Ohio

W.EdwardsDem

ingobservedthatthemainpurpo

seofMIL-STD-105wasto

beatthevendoroverthehead.

"Youcannotimprovethequalityintheprocessstre

amusingthisap

proach,"

cautionsDonWh

eeler,authorofUnderstandingStatisticalProcess

Control

(SPCPress,199

2)."Neithercanyousuccessfullyfilteroutthebadstuff.

Abouttheonlyplacethatthisprocedurewillhelpis

intryingtodetermine

whichbatcheshavealreadybeen

screenedandwhichbatchesare

raw,

unscreened,run-of-the-millbadstufffromyoursup

plier.Itaughtthe

se

techniquesforye

arsbuthaverep

entedofthiserro

rinjudgment.Th

eonly

appropriatelevelsofinspectionareallornone.Anythingelseisjustplaying

roulettewiththeproduct."

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AQLAccepta

nceSamplingb

y

AttributesbyMILST

D105E

Determin

elotsizeNa

ndAQLforthetaskatha

nd

Decideth

etypeofsamplingsing

le,double,etc.

Decideth

estateofinspection(e.g.normal)

Decideth

etypeofins

pectionlevel(usuallyII)

LookatT

ableKforsa

mplesizes

Lookatthesamplingplanstables(e.g.TableIIA)

Readn,AcandRenumbers

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MIL-S

TD-105E

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How

/When

wouldyouuse

Ac

ceptanceSampling?

Advantagesofaccepta

ncesampling

Lesshandlingdama

ges

Fewerinspectorsto

putonpayro

ll

100%inspectioncostsaretohig

h

100%testingwould

taketolong

Acceptanc

esamplinghassomedis

advantages

Riskincludedincha

nceofbadlotacceptanc

eand

goodlo

trejection

Sample

takenprovideslessinformationthan

100%

inspection

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Summary

Thereare

manybas

ictermsyo

uneedtoknow

tobeabletounderstandaccep

tancesampling

SPC,Acceptalot,R

ejectalot,C

ompleteInsp

ection,

AQL,LT

PD,SamplingPlans,Pro

ducersRisk,

Consum

ersRisk,Alpha,Beta,Defect,Defectives,

Attributes,Variables,ASN,ATI.

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Usef

ullink

s

ww.bioss.sari.ac.uk/sm

art/unix/mseqacc/slides/fra

mes.htm

Accep

tanceSamplin

gOverviewT

extandAudio

http://iew3.technion.a

c.il/sqconline/milstd105.htm

l

Online

calculatorfor

acceptances

amplingplans

ww.stats.uwo.ca/courses/ss316b/2002/accept_02red.pdf

Acceptancesamplin

gmathematicalbackground

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