NPTEL_Acceptance Sampling (1)

Copyright Tapan P Bagchi 1

Acceptance Sampling

Accepting or rejecting a lot (of parts, components, etc.) based

on the inspection of a sampledrawn from it


Acceptance Sampling

● Accept/reject entire lot based on sample results● Created by Dodge and Romig during WWII● Not consistent with TQM of Zero Defects● Does not estimate the quality of the lot

A “Lot” of goods


What is acceptance sampling?

Lot Acceptance SamplingA SQC technique, where a random sample is taken from a lot, and upon the results of appraising the sample, the lot will either be rejected or acceptedA procedure for sentencing incoming batchesor lots of items without doing 100% inspectionThe most widely used sampling plans are given by Military Standard (MIL-STD-105E)


What is acceptance sampling?

PurposesDetermine the quality level of an incoming shipment or at the end of productionJudge whether quality level is within the level that has been predetermined

But! Acceptance sampling gives you no idea about the process that is producing those items!


Acceptance Sampling

Another area of quality control and improvementClosely connected with inspection and testing of productInspection can occur at many points in a process

Acceptance Sampling: the inspection and classification of a sample of nits selected at random from a larger batch or lot and ultimate decision about disposition of the lot –

Lot Disposition or Lot Sentencing

Acceptance Sampling: the inspection and classification of a sample of nits selected at random from a larger batch or lot and ultimate decision about disposition of the lot –

Lot Disposition or Lot Sentencing

After productionAfter production

Two common points of inspectionWhen parts are receivedWhen parts are received


Sampling PlansAcceptance sampling: Form of inspection applied to lots or batches of items before or after a process, to judge conformance with predetermined standards

Sampling plans: Plans that specify lot size, sample size, number of samples, and acceptance/rejection criteria

Single-sampling

Double-sampling

Multiple-sampling and Sequential sampling


Types of sampling plans

Sampling by attributes vs. sampling by variablesIncoming vs. outgoing inspectionRectifying vs. non-rectifying inspection

What is done with nonconforming items found during inspectionDefectives may be replaced by good items

Single, double, multiple and sequential plans


How acceptance sampling works

Attributes (“go no-go” inspection)

Defectives-product acceptability across range

Defects-number of defects per unit

Variable (continuous measurement)

Usually measured by mean and standard deviation


Why use acceptance sampling?

Can do either 100% inspection, or inspect a sample of a few items taken from the lotComplete inspection

Inspecting each item produced to see if each item meets the level desiredUsed when defective items would be very detrimental in some way


Sampling TermsAcceptance quality level (AQL): the percentage of defects at which consumers are willing to accept lots as “good”

Lot tolerance percent defective (LTPD or RQL): the upper limit on the percentage of defects that a consumer is willing to accept

Consumer’s risk: the probability that a lot contained defectives exceeding the LTPD will be accepted

Producer’s risk: the probability that a lot containing the acceptable quality level will be rejected


Why not 100% inspection?

Problems with 100% inspection

Generally very expensive

Can’t use when product must be destroyed to test it

Handling by inspectors can induce defects

Inspection may be very tedious so defective items may slip through even 100% inspection


A Lot-by-Lot Sampling Plan

N(Lot) n

Count Number

Conforming

Accept orReject Lot

Specify the plan (n, c) given N For a lot size N, determine

the sample size n, and the acceptance number c.

Reject lot if number of defects > c Specify course of action if lot is rejected


Quality Definitions

Acceptance quality level (AQL)The smallest percentage of defectives that will make the lot definitely acceptable. A quality level that is the base line requirement of the customer

RQL or Lot tolerance percent defective (LTPD)Quality level that is unacceptable to the customer


How acceptance sampling works

RememberYou are not measuring the quality of the lot, but, you are to sentence the lot to either reject or accept it

Sampling involves risks:Good product may be rejectedBad product may be acceptedBecause we inspect only a sample, not the whole lot!


Acceptance Sampling


Three approaches for sentencing a lot

• when 100% inspection is too costly• Trusted supplier with potential risk

1. Single sampling1. Single sampling

2. Rectified sampling 2. Rectified sampling

3. Double/multiple sampling3. Double/multiple sampling

• Inspect every item in the lot, then remove the defective units by discarding, reworking, or returning to suppliers• For critical process. Defective input may result high failure cost• From new supplier

• For low cost or low impact material on the subsequent process• From trusted supplier


Types of Sampling Plan

Sampling Plan for VariablesSampling Plan for VariablesSampling Plan for AttributesSampling Plan for Attributes

1. Single-sampling plan2. Double-sampling plan3. Multiple-sampling plan4. Sequential sampling plan

Standard sample size

Smaller sample size on average


Advantages and Disadvantages of Sampling

SSP’s AdvantagesLess expensive because of less inspection

Works with single sample Protects both producer and consumer

Rejection on entire lot motivates quality improvement for suppliers

DisadvantagesRisk of accepting a lot of poor quality Risk of rejecting a lot of acceptable qualityRequires planning and documentationRequire extensive study on customer’s requirement


Lot Formation

1. Lots should be such that …produced on the same machines, by same operators, from common raw materials, at approximately the same timeperiod

2. Larger lots are better than smaller lotsThese are more representative of overall quality

3. Lots should be conformable to the material handling systems and personnel


Random SamplingInspected units should be selected at randomInspected units should represent all items in the lotPotential bad stories:

Sampled from the front, or top of pile Did not randomizeDid not stratify the lot and did not sample from each stratum


The Single Sampling Plan

The most common and easiest plan to use but not most efficient in terms of average number of samples neededOne sample drawn from the lot and 100% inspectedSingle sampling plan

N = lot sizen = sample size (randomized)c = acceptance numberd = number of defective items in sample

Rule: If d ≤ c, accept lot; else reject the lot


d ≤ c ?

Reject lot

YesAccept lot

Do 100% inspection

Return lot to supplier

Inspect all items in the sample

Defectives found = d

No

Take a randomized sample of size n

from the lot N

The Single Sampling procedure


Single-Sampling Plans for Attributes

A lot of size N has been submitted for inspectionSample size nAcceptance number of defective cLot sentencing is based on one sample of size n

Example:N = 10,000

n= 89c = 2

Example:N = 10,000

n= 89c = 2

N = 10,000n = 89

Sample exactly n parts and inspect every part


Producer’s & Consumer’s Risksdue to mistaken sentencing of lot

TYPE I ERROR = P(reject good lot)α or Producer’s risk

5% is common

TYPE II ERROR = P(accept bad lot)β or Consumer’s risk10% is typical value


Acceptance sampling contd.

Producer’s riskRisk associated with a lot of acceptable quality rejected

Alpha α= Prob (committing Type I error)

= P(rejecting lot at AQL quality level)

= producers risk


Acceptance sampling contd.

Consumer’s riskReceive shipment, assume good quality, actually bad quality

Beta β= Prob (committing Type II error)= Prob (accepting a lot at RQL quality level) = consumers risk

The Operating Characteristic (OC) curve for a sampling plan quantifies these risks


Take a randomized sample of size n from

the lot of unknown quality p

The Single Sampling procedure



d ≤ c ?Yes

Reject lot

Accept lot

No


Do 100% inspection


The Operating Characteristic or OC

Curve

The OC curve indicates a sampling plan’s ability to

discriminate between good and bad lots


What is the Operating Characteristic (OC) Curve?

It is a graph of the % defective (p) in a lot or batch vs. the probability that the sampling plan will accept the lotShows probability of lot acceptance Pa as function of lot quality level (p)

It is based on the sampling planCurve indicates discriminating power of the planAids in selection of plans that are effective in reducing risk Helps to keep the high cost of inspection down


OC Definitions on the CurvePr

obab

ility

of A

ccep

ting

Lot

Lot Quality (Fraction Defective)

100%

75%

50%

25%

.03 .06 .09

α = 0.1090%

β = 0.10

AQ

L

RQ

LIndifferent

Good Bad


OC Curve TermsAcceptable Quality Level (AQL)

Percentage of defective items a customer is willing to accept from you (a property of mfg. process)

Lot Tolerance Percent Defective (LTPD)Upper limit on the percentage of defects a customer is willing to accept ( a property of the consumer)

Average Outgoing Quality (AOQ)Average of rejected lots and accepted lots

Average Outgoing Quality Limit (AOQL)Maximum AOQ for a range of fractions defective


The Perfect OC CurvePr

obab

ility

of A

ccep

ting

Lot


100%

75%

50%

25%

.03 .06 .09

This curve distinguishes perfectly between good and bad lots.

What would allow you to achieve a curve like this?


OC Curves


Prob

abili

ty o

f Acc

eptin

g Lo

t100%

75%

50%

25%

.03 .06 .09

OC Curves come in various shapes depending on the sample size and risk of α and β errors

This curve is more discriminating

This curve is less discriminating


OC Curve and its interpretation

The oc-curve enables us to evaluate the probability of acceptance (Pa) for any true lot quality level-on a what-if basis. This way, one can design sampling plans that perform the way one wants. The OC curve is Interpreted as follows:

If the lot quality is 0.093 fraction defective, then Pa, is 0.05. If the lot quality is 0.018 fraction defective, then Pa, is 0.95.


Operating Characteristic Curve

AQL LTPD

β = 0.10

α = 0.05Pr

obab

ility

of

acce

pta

nce

, P a

{

0.60

0.40

0.20

0.02 0.04 0.06 0.08 0.10 0.12 0.14 0.16 0.18 0.20

0.80

{

Proportion defective p

1.00

OC curve for n and c


OC Curve helps visualize producer’s and consumer’s points

Type I and Type II decision errors correspond logically to the two decision points on the oc-curve. Type I error -- Wrongful RejectionA type I error is associated with the producer's point -- to reject when the true value of the quality characteristic is AQL. The risk of rejecting an AQL lot is the producer'srisk (α = alpha risk)Type II error -- Wrongful acceptance

A Type II error is to accept when the true value of the quality characteristic is RQL -- at the consumer's point. The risk of accepting a lot, if it is an RQL lot, is the consumer's risk (ß = beta risk).


Decision Criteria

0

1.00

Prob

abili

ty o

f acc

eptin

g lo

t

Lot quality (fraction defective)

“Good” “Bad”

Ideal

Not verydiscriminating


Consumer’s and Producer’s Risk

00.10.20.30.40.50.60.70.80.9

1

0 .05 .10 .15 .20 .25

Prob

abili

ty o

f acc

eptin

g lo

t

Lot quality (fraction defective)

α = .10

β = .10

“Good”

AQL

“Bad”Indifferent

LTPD


Example: QC Curve for n = 10, c = 1Example: QC Curve for n = 10, c = 1

00.10.20.30.40.50.60.70.80.9

1

0 .10 .20 .30 .40 .50

Prob

abili

ty o

f acc

epta

nce

Fraction defective in lot

.9139

.7361

.5443

.3758

.2440.1493

.0860


Types of OC Curves

Type AGives the probability of acceptance for an individual lot coming from finite production

Type BGive the probability of acceptance for lots coming from a continuous process or infinite size lot


OC Curve Calculation

The Ways of Calculating OC Curves

Binomial distribution (Type B)

Hypergeometric distribution (Type A)

Pa = P(r defectives found in a sample of n)Poisson formula

P(r) = ( (np)r e-np)/ r!Larson nomogram


OC Curve Calculation by Poisson distribution

A Poisson formula can be usedP(r) = ((np)r e-np) /r! = Prob(exactly r defectives in n)

Poisson is a limit Limitations of using Poisson

n ≤ N/10 total batch Little faith in Poisson probability calculation when n is quite small and p quite large.

For Poisson, Pa = P(r ≤ c)

Copyright Tapan P Bagchi 43p

For us, Pa = P(r ≤ c)


OC Curve Calculation by Binomial Distribution

Note that we cannot always use the binomial distributionbecause

Binomials are based on constant probabilities

N may not be infinitep changes as items are drawn from the lot


OC Curve by Binomial FormulaPa p (%)

.998 .01

.980 .02

.930 .03

.845 .04

.739 .05

.620 .06

.502 .07

.394 .08

.300 .09

.223 .10

.162 .11

.115 .12

Using this formula with n = 52 and c=3 and p = .01, .02, ...,.12 we find data values as shown on the right. This givens the plot shown below.


The Ideal OC Curve

p AQL

1.0

0.0

Pa● Ideal curve would be perfectly perpendicular from 0 to 100% for a fraction defective = AQL

● It will accept every lot with p ≤ AQL and reject every lot with p > AQL


Guidelines for choosing Producer’s and Consumer’s decision points (AQL and RQL)To choose the Producer's Point (AQL) in practice

Lots at the producer's point quality level (AQL) should be accepted most of the time. Define AQL accordingly. Take into account historical quality levels and the consequence of returning a lot to the producer at AQL quality level. Choose the producers risk of rejecting a lot that is of AQL quality. Typical: α = 0.05.

To choose the Consumer's Point (RQL) in practiceLots at the consumer's point quality level (RQL) should be rejected most of the time. Take into account the economic consequence of the consumer’s accepting a lot at RQL quality level and define RQL accordingly. Choose the consumers risk of accepting a lot that is of RQL quality. Typical: β = 0.05.


The OC Curve by binomial formula

Operating characteristic (OC) curve

Curve plots the probability of acceptingthe lot (Pa) versus the lot fraction defective (p)


The OC Curve for Single SP


Ideal OC curve

Theoretically , it can be achieved by 100% inspection of the lotDue to sampling, ideal OC curve cannot be perfectly achieved.If sample size (n) is large, the OC curve shape will approach the ideal OC curve


Properties of OC Curves

The acceptance number c and sample size n are most important factors in defining the OC curveDecreasing the acceptance number (c) is preferred over increasing sample size (n)The larger the sample size n the steeper is the OC curve (i.e., it becomes more discriminating between good and bad lots)


Effects of nn on OC curves

As sample size (n) increases,increases, the OC curve shape will approach the idealOC curve ~ more discriminating

Note that c is kept in proportional to n

movement of OC


Effects of cc on OC curves

Increasing acceptance number (c) does not significantly change the shape of OC

But Producer’s risk (β) increases

movement of OC


OC curveAcceptable quality level (AQL)

The poorest quality level for the supplier’s process that a consumer would consider to be acceptable A property of the supplier’s manufacturing process, not a property of the sampling plan

Lot tolerance percent defective (LTPD)The protection obtained for individual lots of poor qualityAlso called rejectable quality level (RQL)LTPD is a level of lot quality specified by consumer, not a characteristic of the sampling planSampling plans can be designed to have specified performance at the AQL and the LTPD points


Type-A and Type-B OC Curves


Poisson distribution for Defects

Poisson parameter: λ = npP(r) = (np)r e-np/r! = Prob(exactly r defectives in n)

This formula may be used to formulate equations involving AQL,RQL, α and β to given (n, c).

We can use Poisson tables to approximately solve these equations. Poisson can approximate binomial probabilities if n is large and p small.

Q. If we sample 50 items from a large lot, what is the probability that 2 are defective if the defect rate (p) = .02? What is the probability that no more than 3 defects are found out of the 50?


Hypergeometric DistributionHypergeometric formula:

r defectives in sample size n when M defectives are in N.This distribution is used when sampling from a small population. It is used when the lot size is not significantly greater than the sample size. (Can’t assume here each new part picked is unaffected by the earlier samples drawn).

Q. A lot of 20 tires contains 5 defective ones (i.e., p = 0.25). If an inspector randomly samples 4 items, what is the probability of 3 defective ones?

⎟⎟⎠

⎞⎜⎜⎝

⎛

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−−

=

Nn

Mr

MNrn

rP )(


Properties of OC Curves for finite size lots


Properties of OC Curves

If the acceptance level c is changed, the shape of the curve will change. All curves permit the same fraction of sample to be nonconforming.


“Rectifying” Inspection

When the lot is rejected, it requires corrective action: 100% inspection and replacement of defectives by good parts

Rectified

Passed


Average Outgoing Quality (AOQ)

Expected proportion of defective items passed to customer

Average outgoing quality limit (AOQL) isThe “maximum” point on AOQ curve

NnNpPinspectionrectifyingwithAOQ a )( −

=


Average outgoing quality -AOQ

Given the lot size of N, sample size of n, and fraction defective of pn items in the sample that, after inspection, contain no defectives, because all discovered defectives are replacedN – n items that, if the lot is rejected, also contain no defectivesN – n items that, if the lot is accepted p(N-n) defectives

If N >> n, then

AOQ = Pa p


Average Quality

Average outgoing quality (AOQ): Average of inspected lots (100%) and uninspected lots

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛ −×= NnNpPaAOQ

Pa = Probability of accepting lotp = Fraction defectiveN = Lot sizen = Sample size


AOQLimit (AOQL) Estimation

0 00.05 0.0460.1 0.074

0.15 0.0820.2 0.075

0.25 0.0610.3 0.045

0.35 0.030.4 0.019

0

0.02

0.04

0.06

0.08

0.1

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4

Approximate AOQL = .082

AO

Q (F

ract

ion

defe

ctiv

e ou

t)

Incoming fraction defective


AOQ Curve

0.015

0.010

0.005

0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

AOQLAverageOutgoingQuality

(Incoming) Percent Defective

AQL LTPD


Average outgoing quality -AOQ


Double Sampling Plans

Take small initial sampleIf # defectives < lower limit, accept

If # defectives > upper limit, reject

If # defectives between limits, take second sample

Accept or reject lot based on 2 samples

Less inspection than in single-sampling


Multiple Sampling Plans

Advantage: Uses smaller sample sizes

Take initial sample

If # defectives < lower limit, accept

If # defectives > upper limit, reject

If # defectives between limits, re-sample

Continue sampling until accept or reject lot based on all sample data


Sequential Sampling

The ultimate extension of multiple samplingItems are selected from a lot one at a timeAfter inspection of each sample a decision is made to accept the lot, reject the lot, or to select another item

In Skip Lot Sampling only a fraction of the lots submitted are inspected


Sequential Sampling PlansSequential Analysis is the technique by which we build up our sample one item at a time, and after inspecting each item, ask ourselves: "Can we be sure enough to accept or reject this batchon the information so far collected?“Seq SP offers Its value is in enabling reliable conclusions to be wrung from a minimum of data. This was deemed sufficient to require that it be classified "Restricted " within the meaning of the Espionage Act during the war of 1939-45.Used for lot-by-lot acceptance plans for applications needing the minimum possible average sample numbers.The SPR type sequential sampling plan is also known as Columbia sequential sampling because the methodology was developed at Columbia University.


Example of Sequential Decision Diagram

Results of sequential sampling data is plotted on the diagram sequentially as it is collected. The accept or reject decision is made when sample data plots on or over an acceptance line or rejection line. The stair step pattern is caused by the discrete nature of the sample size and the number of defectives in the n-table upon which this diagram is based.


Sequential Sampling: Acceptance and Rejection Limit lines


Equations for the Limit Lines

The equations for the two limit lines are functions of the parameters p1, , p2, and .

where


OC-Curve for Count Data in Sequential Sampling—similar to other sampling plans


Choosing A Sampling Method:Single, Double or Sequential?

It is an economic decisionSingle sampling plans

high sampling costs

Double/Multiple sampling planslow sampling costs

More complex to implement


Designing a Sampling Plan

The plan must protect both the consumer (the buyer) and the

supplier (the seller)


d ≤ c ?

Reject lot

YesAccept lot

Do 100% inspection




No

Take a randomized sample of size n from

the lot of unknown quality p

Designing The Single Sampling plan


Sampling Plan Design by Binomial Distribution

Binomial distribution:

P(x defectives in n) = [n!/(x!(n-x))!]px(1- p)n-x

Recall n!/(x!(n-x))! = ways to choose x in n

Q. If 4 samples (items) are chosen from a population with a defect rate = .1, what is the probability that a) exactly 1 out of 4 is defective?

b) at most 1 out of 4 is defective?


Designing a single-sampling plan with a specified OC curve

To construct sampling plan such that

Probability of acceptance for lots with fraction defective p1

Probability of acceptance for lots with fraction defective p2

Two simultaneous conditions that the (n, c) plan should meet


Solving for (n, c)

To design a single sampling plan we need two points. Typically these are p1 = AQL, p2 = LTPD and , are the Producer's Risk (Type I error) and Consumer's Risk (Type II error), respectively. By binomial formulas, n and c are the solution to

These two simultaneous equations are nonlinear so there is no simple, direct solution. The Larson nomogram can help us here.


SSP Design by Larson Nomogram

● Applies to single sampling plan

● Based on binomial distribution

● Uses1-α = Pa at AQL

β = Pa at RQL● Can produce OC

curve


Two simultaneous conditions that the (n, c) plan will meet

Designing a single sampling plan by Larson’s Binomial Nomogram

Inputs are AQL, α, RQL and β.Outputs are n and c


The MIL-STD-105E approachA Query from a Practitioner: Selecting AQL (acceptable quality levels)“I'd like some guidance on selecting an acceptable quality level and inspection levels when using sampling procedures and tables. For example, when I use MIL-STD-105E, how do I to decide when I should use GI, GII or S2, S4?”

-- Confused in Columbus, Ohio

W. Edwards Deming observed that the main purpose of MIL-STD-105 was to beat the vendor over the head.

"You cannot improve the quality in the process stream using this approach," cautions Don Wheeler, author of Understanding Statistical Process Control (SPC Press, 1992). "Neither can you successfully filter out the bad stuff. About the only place that this procedure will help is in trying to determine which batches have already been screened and which batches are raw, unscreened, run-of-the-mill bad stuff from your supplier. I taught these techniques for years but have repented of this error in judgment. The only appropriate levels of inspection are all or none. Anything else is just playing roulette with the product."


MIL-STD-105E

Original version (MIL STD 105A) issued in 1950 as tables; Last version (MIL STD 105E) in 1989; ISO adopted it as ISO 2859Plan covers sampling by attributes for given lot size (N) and acceptable quality level (AQL).Prescribes sample size n, acceptance number c, and rejection number rStandard included three types of inspection—normal, tightened and reduced and gives switching rulesPlans assure producer’s risk (α) of 0.01 – 0.1. The only way to control the consumer’s risk (β) is to change inspection level


Military Standard 105ESampling procedure for inspection by attributes developed duringWorld War II and is the most widely used acceptance-sampling system for attributes in the world todayA collection of sampling schemes; therefore an acceptance-sampling systemProvides for three types of sampling: single, double, and multiplePrimary focal point is the acceptable quality level (AQL)

Different AQLs may be designated for different types of defects: critical, major, and minorGenerally specified in contract or by authority responsible for sampling

Sample size is determined by lot size and by choice of inspection level


AQL Acceptance Sampling by Attributes by MILSTD 105E

Determine lot size N and AQL for the task at handDecide the type of sampling—single, double, etc.Decide the state of inspection (e.g. normal)Decide the type of inspection level (usually II)Look at Table K for sample sizesLook at the sampling plans tables (e.g. Table IIA)Read n, Ac and Re numbers


Military Standard 105E Procedure1. Choose the AQL2. Choose the inspection level3. Determine the lot size4. Fine the appropriate sample size code letter (from

the MIL-STD 105E Table)5. Determine the appropriate type of sampling plan to

use (single)6. Enter the appropriate table to find the type of plan to

be used7. Determine the corresponding normal and reduced

inspection plans to be used when required


MIL STD 105E Inspection levels and Sample Size codes


OC Curves under different Inspection Conditions


MIL STD 105E “Normal” Sample Size n and defective limit c


The OC curve of a MIL STD 105E Sampling Plan


MIL-STD-105E Normal Inspection


MIL-STD-105E Reduced Inspection


MIL-STD-105E Tightened Inspection


Military Standard 105ESampling rules for switching between normal, tightened and reduced inspection


OC Curves under different MIL Inspection Conditions


Discussion of Military Standard

Several points about MIL STD 105E should be emphasized:MIL STD 105E is AQL-oriented, no concern for βNot all possible sample sizes are possible (2,3,5,8,13,20,32,50,etc.)Sample sizes are related to lot sizesSwitching rules are subject to criticism for both misswitchingbetween inspection plans and discontinuation even though there has been no actual quality deteriorationMIL STD 105E has been replaced by MIL STD 1916 by the US Department of Defense as of October 2008 http://assist.daps.dla.mil/

http://assist.daps.dla.mil/


How/When would you use Acceptance Sampling?

Advantages of acceptance samplingLess handling damagesFewer inspectors to put on payroll100% inspection costs are to high100% testing would take to long

Acceptance sampling has some disadvantagesRisk included in chance of bad lot “acceptance” and good lot “rejection”Sample taken provides less information than 100% inspection


Summary

Many basic terms you need to know to be able to understand acceptance sampling

SPC, Accept a lot, Reject a lot, Complete Inspection, AQL, LTPD, Sampling Plans, Producer’s Risk, Consumer’s Risk, Alpha, Beta, Defect, Defectives, Attributes, Variables, ASN, ATI.


Some Key Definitions and TermsReference: NIST Engineering Statistics Handbook

Acceptable Quality Level (AQL): A percent defective that is the base line requirement for the quality of the producer's product. The producer would like the buyer to design a sampling plan such that there is a high probability of accepting a lot that has a defect level less than or equal to the AQL.

Lot Tolerance Percent Defective (LTPD) also calledRQL (Rejection Quality Level): A designated high defect level that would be unacceptable to the consumer. The consumer would like the sampling plan to have alow probability of accepting a lot with a defect level as high as the LTPD.


Type I Error (Producer's Risk): The probability for a given (n, c) sampling plan of rejecting a lot that has a defect level equal to AQL. The producer suffers when this occurs, because α lot with acceptable quality was rejected. The symbol a is commonly used for the Type I error and typical values for α to range from 0.2 to 0.01.

Type II Error (Consumer's Risk): This is the probability, for a given (n, c) sampling plan, of accepting a lot with a defect level equal to the LTPD. The consumer suffers when this occurs, because a lot with unacceptable quality was accepted. The symbol β is commonly used for the Type II error and typical values range from 0.2 to 0.01.


Average Outgoing Quality (AOQ):A common procedure, when sampling and testing is non-destructive, is to 100% inspect rejected lots and replace all defectives with good units. In this case, all rejected lots are made perfect and the only defects left are those in lots that were accepted. AOQ's refer to the long term defect level for this combined LASP and 100% inspection of rejected lots process. If all lots come in with a defect level of exactly p, and the OC curve for the chosen (n, c) LASP indicates a probability Pa of accepting such a lot, over the long run the AOQ can easily be shown to be:

where N is the lot size.


Average Outgoing Quality Level (AOQL): A plot of the AOQ (Y-axis) versus the incoming lot p (X-axis) will start at 0 for p = 0, and return to 0 for p = 1 (where every lot is 100% inspected and rectified). In between, it will rise to a maximum. This maximum, which is the worst possible long term AOQ, is called the AOQL.

Average Total Inspection (ATI): When rejected lots are 100% inspected (rectifying inspection), it is easy to calculate the ATI if lots come consistently with a defect level of p. For a sampling plan (n, c) with a probability Paof accepting a lot with defect level p, we have

ATI = n + (1 - Pa) (N - n)where N is the lot size.


Average Sample Number (ASN):For a single sampling plan (n, c) we know each and every lot has a sample of size n taken and inspected or tested.

For double, multiple and sequential plans, the amount of sampling varies depending on the number of defects observed.

For any given double, multiple or sequential plan, a long term ASN can be calculated assuming all lots come in with a defect level of p.

A plot of the ASN, versus the incoming defect level p, describes the sampling efficiency of a given lot sampling scheme.


Useful links

http://www.bioss.sari.ac.uk/smart/unix/mseqacc/slides/frames.htmAcceptance Sampling Overview Text and Audio

http://iew3.technion.ac.il/sqconline/milstd105.htmlOnline calculator for acceptance sampling plans

http://www.stats.uwo.ca/courses/ss316b/2002/accept_02red.pdfAcceptance sampling mathematical background

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