Chapter 15 – Acceptance Sampling (A...

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Islamic University of Gaza - Palestine

Chapter 15 – Acceptance Sampling (AS)


Learning Outcomes

After careful study of this chapter You should be able to:• Understand the role of Acceptance Sampling (AS) in modern

quality control systems,• Understand the advantages and disadvantages of AS,• Know the major types of AS procedures,• Know how single- and double- sampling plans are used,• Know ho to determine the Operating Characteristic Curve (OCC)

for a single sampling-plan for Attribute,• Understand the effects of sampling plan parameters on

sampling plan performance,

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What is AS?

• AS is a form of inspection that is used to determinewhether or not goods are coherent (consistent) with aset standard of quality.

Other definition:• AS is a method used to accept or reject product

based on a random sample of the product.


What is AS? (cont.)

• The purpose of acceptance sampling is to sentence lots(accept or reject) rather than to estimate the quality of alot.

• Acceptance sampling plans do not improve quality. Thenature of sampling is such that acceptance sampling willaccept some lots and reject others even though they areof the same quality.

• The most effective use of acceptance sampling is as anauditing tool to help ensure that the output of a processmeets requirements.

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History of AS

• Acceptance Sampling –a historically significant topic but less used today.

• Part of Ch. 11 on Inspection Methods.

• Still used in some applications today.


History and Status

• Used extensively in WW II.• Many Military-Spec plans developed (105-E, ANSI/ASQC

Z1.4-1993).

• Still popular as a defense procurement tool.– Very large lots, screening tool.– Low bid suppliers – no history.

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• When product testing is– destructive– expensive– time consuming

• When developing new products

• When dealing with new suppliers

• When a supplier’s product has had excellent quality in thepast

When is Acceptance Sampling useful?


When can acceptance sampling be applied?

• Sampling at the Input stage– Prevents goods that don’t meet standards from entering

into the process– This saves rework time and money

At any point in production• The output of one stage is the input of the next

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When can acceptance sampling be applied? (cont.)

• Sampling at the Output stage– Can reduce the risk of bad quality being passed on from the

process to a consumer– This can prevent the loss of prestige, customers, and money

• Sampling at the Process stage– Can help adjust the process and reduce the amount of poor

quality in production– Helps to determine the source of bad production and enables

return for reprocessing before any further costs may beincurred


Typical Application of Acceptance Sampling

• A vendor delivers a product to a manufacturing company– The product is a raw material used by the company

• A sample of the shipment is taken– Quality characteristics of the units in the sample are

inspected.• Based on the observations made, the decision is made to

either accept or reject the entire shipment

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• The decision to accept or reject the shipment is based onthe following set standards:– Lot size = N– Sample size = n– Acceptance number = c– Defective items = d

• If d <= c, accept lot• If d > c, reject lot

Typical Application of Acceptance Sampling


• Mr. Smith owns and operates a manufacturing plant.

– He receives a shipment of 1,000 sheets of glass.– Of the shipment, Mr. Smith chooses to sample 50 sheets.

• If more than 2 are defective, he is sending back the entireshipment to the supplier.

• Mr. Smith observes 5 defective sheets of glass.

What should Mr. Smith do in reference to the number ofdefective items observed????

Acceptance Sampling Plan

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Acceptance Sampling Plan

• Therefore, according to the set standards mentionedabove:

• Remember, if d > c, reject lot• Since c = 2, and d = 5…

– Mr. Smith shouldreject the lot of

1,000 sheets of glass

– N = 1,000– n = 50– c = 2– d = 5

– N = 1,000– n = 50– c = 2– d = 5


Advantages and Disadvantages of AS

In contrast to 100% inspection• AS is usually less expensive than,• AS is less handling of the product (less damage),• AS is applicable to destructive testing,• Fewer employees are involved in inspection activities,• AS reduce the amount of inspection error,

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Disadvantages of AS

• There are risks of accepting bad lots and rejecting goodlots,

• The process of the goods being inspected is not improved

• Less information is generated about products andprocesses,

• AS requires planning and documentation but 100%doesn't require.


Acceptance Sampling Flow Chart

Lot Receivedfrom Supplier

Random Sampleof MaterialSelected

Items Inspectedand Analyzed

ResultsCompared with

AcceptanceCriteria

Define andAnalyze

Sampling Plan

Accept the Lot Reject the Lot

Send Lot toInventory orProduction

DecideDisposition,

Return toSupplier

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Terminology

• Producer’s risk – risk associated with rejecting a lotof ‘good’ quality.

• Producers Risk– The risk associated with a producer rejecting a lot of

materials that actually have good quality• Also referred to

as a Type I Error

Acceptable Quality Level (AQL) – Numerical definition ofa ‘good’ lot, associated with the producer’s risk.

• Consumer’s risk – risk of accepting a ‘poor’ lot.• Limiting Quality Level - Numerical definition of a

‘poor’ lot, associated with the consumer’s risk.


Terminology

Consumer’s risk – risk of accepting a ‘poor’ lot.• Consumers Risk

– The risk associated with a consumer accepting a lot ofmaterials that actually have poor quality

• Also referred toas a Type II Error

• Limiting Quality Level - Numerical definition of a‘poor’ lot, associated with the consumer’s risk.

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Role of Producer and Consumer

Producer Consumer

Risk is a ‘good’ lot will berejected and sent back.

Risk is a ‘bad’ lotwill be accepted.

Take a SampleSize ‘n’,

Accept if ‘c’ or less.


Example: Sunshine Jeans

• Sunshine Jeans store receives a shipment of 300 pairs ofjeans from its warehouse.

• It is common practice for the store to sample 5% of the totalreceived.

• The acceptance number under any and all circumstancesfor Sunshine Jeans is 10.

• Of the 15 pairs of jeans observed, 2 were defective.

• What conclusion should the store manager come to basedon this information?

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• The store manager has just found out that the clerk whoinspected the samples made a huge mistake…

• The actual number of defective pairs of jeans sampled was12.

• What type of risk is involved with the error made atSunshine Jeans?

• How might this error affect the store and their customers?

Example: Sunshine Jeans (cont.)


Examples

• Producer’s risk is 5% for an AQL of 0.02.– Means batches that are 2% nonconforming are

good and prefer to reject these no more than 5% ofthe time.

• Consumer’s risk is 10% for an LQL of 0.08.– Means batches that are 8% nonconforming are bad

and prefer to accept these only 10% of the time.

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Operating Characteristic (OC) Curve

• The Operating Characteristic Curve is typically used torepresent the four parameters (Producers Risk,Consumers Risk, AQL and Lot Tolerance PercentDefective (LTPD) of the sampling plan as shown belowwhere the P on the x axis represents the percent defectivein the lot:



OC Curve

0

0.2

0.4

0.6

0.8

1

1.2

P

Pro

babi

lity

of A

ccep

tanc

e

LTPDAQL

Producers Risk

Consumers Risk

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• Note: if the sample is lessthan 20 units the binomialdistribution is used tobuild the OC Curveotherwise the Poissondistribution is used.

• Defines the performanceof a sampling plan.• Plots probability ofacceptance versusproportion nonconforming(p).

0

0.2

0.4

0.6

0.8

1

1.2

0 0.03 0.06 0.09 0.12 0.15

(p)P

ac

ce

pta

nc

e



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•As the lot percent nonconforming increases, the probabilityof acceptance decreases, just as you would expect.

•When the process percent nonconforming is below the prescribedlevel, 4.0% in this example, the probability of acceptance is 100%.

•For quality worse than this level, higher than 4%, theprobability of acceptance immediately drops to 0%. The dividingline between 100% and 0% acceptance is called the AcceptableQuality Level (AQL).


Actual OC Curves

• Are determined by sample size [n] and acceptancenumber [c].– Accept the lot if ‘c’ or fewer nonconforming are obtained,

reject if more.• OC assumes a binomial distribution (if lot size is 10x

sample size).• Calculate P-accept for range of incoming p levels.

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Considerations in Selecting a Sampling Plan

• Operating characteristics (OC) curve• Average outgoing quality (AOQ) curve (will not be discussed)



• An OC curve shows how well a particular sampling plan(n,c) discriminates between good and bad lots.

• The vertical axis is the probability of accepting a lot for aplan.

• The horizontal axis is the actual percent defective in anincoming lot.

• For a given sampling plan, points for the OC curve can bedeveloped using the Poisson probability distribution

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OC Curve (continued)

• Management may want to:– Specify the performance of the sampling procedure by

identifying two points on the graph:• AQL and• LTPD and

– Then find the combination of n and c that provides a curvethat passes through both points


Operating Characteristics (OC) Curves

• OC curves are graphs which showthe probability of accepting a lotgiven various proportions ofdefects in the lot

• X-axis shows % of items that aredefective in a lot- “lot quality”

• Y-axis shows the probability orchance of accepting a lot

• As proportion of defects increases,the chance of accepting lotdecreases

• Example: 90% chance of acceptinga lot with 5% defectives; 10%chance of accepting a lot with 24%defectives

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.10.10

.20.20

.30.30

.40.40

.50.50

.60.60

.70.70

.80.80

.90.90Pr

obab

ility

of A

ccep

ting

the

Lot

Prob

abili

ty o

f Acc

eptin

g th

e Lo

t

0 5 10 15 20 250 5 10 15 20 25

1.001.00

% Defectivesin Lots

AQL = 3%LTPD = 15%

Consumer’s Risk ( ) = 8.74%

Producer’s Risk ( ) = 3.67%

n = 15, c = 0


Constructing an (oc) Curve

• For a specified single sampling plan, the OC curvemay be constructed using a binomial distributionif n is small relative to the lot size– p -- true fraction nonconforming– n -- sample size– c -- acceptance number

• We know thatknk )p1(p

k)!-(nk!n!=)DEFECTIVESP(k

c

0k

knka )p1(p

)!kn(!k!n)ck(PP:So

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• Suppose we have a sampling plan defined by thefollowing parameters:– n = 100– c = 2– What is the probability of accepting a lot with 0.5%

defectives?

c

0k

knka )p1(p

)!kn(!k!n)ck(PP

2

0k

k100k )005.01()005.0()!k100(!k

!100

Constructing an (oc) Curve (cont.)


2

0k

k100k )005.01()005.0()!k100(!k

!100

21002

11001

01000

)005.01()005.0()!2100(!2

!100

)005.01()005.0()!1100(!1

!100

)005.01()005.0()!0100(!0

!100

9859.00757.03044.06058.0

Constructing an (oc) Curve (cont.)

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AQL, LTPD, Consumer’s Risk (α) & Producer’s Risk (β)

• AQL is the small % of defects thatconsumers are willing to accept;order of 1-2%

• LTPD is the upper limit of thepercentage of defective itemsconsumers are willing to tolerate

• Consumer’s Risk (α) is the chanceof accepting a lot that contains agreater number of defects than theLTPD limit; Type II error

• Producer’s risk (β) is the chance alot containing an acceptablequality level will be rejected; Type Ierror


Example: Constructing an OC Curve

• Lets develop an OC curve for asampling plan in which a sample of5 items is drawn from lots of N=1000items

• The accept /reject criteria are set upin such a way that we accept a lot ifno more that one defect (c=1) isfound

• Using Table 6-2 and the rowcorresponding to n=5 and x=1

• Note that we have a 99.74% chanceof accepting a lot with 5% defectsand a 73.73% chance with 20%defects

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Example: Constructing an OC Curve

– Example 1: Given a producers risk of .05 and anAQL of .015 determine a sampling plan

• c = 1: n = .355/.015 ~ 24• c = 4: n = 1.97/.015 ~ 131

– Example 2: Given a consumers risk of .1 and aLTPD of .08 determine a sampling plan

• c = 0: n = 2.334/.08 ~ 29• c = 5: n = 9.274/.08 ~ 116


Three alternatives for specifying sampling plans

• Producers Risk and AQL specified• Consumers Risk and LTPD specified• All four parameters specified

• For the first two cases we simply choose theacceptance number and divide the appropriatecolumn by the associated parameter to get the samplesize.

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Three alternatives for specifying sampling plans (cont.)

• Producers Risk and AQL specified• Consumers Risk and LTPD specified

– Example 1: Given a producers risk of .05 and an AQL of.015 determine a sampling plan

• c = 1: n = .355/.015 ~ 24• c = 4: n = 1.97/.015 ~ 131


• c = 0: n = 2.334/.08 ~ 29• c = 5: n = 9.274/.08 ~ 116

• Note: you can check these with the OC curve postedon my web site


Three alternatives for specifying sampling plans

• Producers Risk and AQL specified• Consumers Risk and LTPD specified

– Example 1: Given a producers risk of .05 and an AQL of.015 determine a sampling plan

• c = 1: n = .355/.015 ~ 24• c = 4: n = 1.97/.015 ~ 131


• c = 0: n = 2.334/.08 ~ 29• c = 5: n = 9.274/.08 ~ 116

• Note: you can check these with the OC curve postedon my web site

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All four parameters specified

• When all four parameters are specified we must first finda value close to the ratio LTPD/AQL in the table. Thenvalues of n and c are found.

• Example: Given producers risk of .05, consumers risk of.1, LTPD 4.5%, and AQL of 1% find a sampling plan.– Since 4.5/1 = 4.5 is between c= 3 and c = 4. Using the

n(AQL) column the sample sizes suggested are 137 and 197respectively. Note using this column will ensure aproducers risk of .05. Using the n(LTPD) column will ensurea consumers risk of .1


Designing Sampling Plans

Excerpt From a Sampling Plan Table with Producers Risk = 0.05 and Consumers Risk = 0.10C LTPD/AQL n(AQL) n(LTPD)0 44.89 0.052 2.3341 10.946 .355 3.8862 6.509 .818 5.3243 4.89 1.366 6.684 4.057 1.97 7.9925 3.549 2.613 9.2746 3.206 3.286 10.5357 2.957 3.981 11.7728 2.768 4.695 12.9969 2.618 5.426 14.205

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Sample Calculations

• Binomial table only goes up to n=20.• Approximate Binomial by Poisson, u=np.• Calculate p(2 or less). This is P-accept.

• Example n=50, p=0.03, u=1.5. P(x≤2)=0.809.

• Vary p from 0 to 0.15.


0

0.2

0.4

0.6

0.8

1

1.2

0 0.03 0.06 0.09 0.12 0.15(p)

P a

ccep

tan

ce

Single sampling plan n = 50, c= 2

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Producer and Consumer Risk

• Assume AQL (acceptable quality level) is 0.01. Then P-accept = .986.

• Producer’s Risk is 1-0.986 = 0.014.

• Assume LQL(limiting quality level) is 0.11. Then Paccept= 0.076.

• Consumer’s Risk is 0.076.


Designing Plan Performance

• Vary n and c to obtain different OC curves.

• Single and multiple sampling.

• Refer to standard published sampling plans.

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Double Sampling Plan

Application of double sampling requires that a first sample of sizen1 is taken at random from the (large) lot.

The number of defectives is then counted and compared to thefirst sample's acceptance number a1 and rejection number r1.

Denote the number of defectives in sample 1 by d1 and in sample2 by d2, then:

– If d1<= a1, the lot is accepted.If d1 >= r1, the lot is rejected.If a1 < d1 < r1, a second sample is taken.


If a second sample of size n2 is taken, the number ofdefectives, d2, is counted.

The total number of defectives is D2 = d1 + d2.

Now this is compared to the acceptance number a2 and therejection number r2 of sample 2.

In double sampling, r2 = a2 + 1 to ensure a decision on the sample.

– If D2 <= a2, the lot is accepted.If D2 >= r2, the lot is rejected.

Double Sampling Plan (cont.)

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Vary n and c

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15p

P ac

cept

n=50 c=2n=80 c=2n=50 c=1n=100 c=2


50020

11

0.9984120.9170290.5590930.1854920.0352850.0042970.0003682.39E-051.24E-065.31E-081.95E-096.29E-11

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

0 0.05 0.1 0.15

P A

ccep

t

p

n = 50, c = 2

n = 100 , c = 4

n = 200 , c = 8

n = 500, c= 20

Vary n and c (cont.)

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Summary

• Acceptance Sampling– a form of inspection that is used to determine whether or not

goods are coherent with a set standard of quality– It does not improve the process that the products are derived

from,• Uses of Acceptance Sampling• Risks

– Producers Risk– Consumers Risk

• Stages of Inspection– Input Stage– Process Stage– Output Stage


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Chapter 15 – Acceptance Sampling (A...

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