09 lot-by-lot acceptance sampling for attributes

Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved

QualityQuality ControlControl

Chapter 9- Lot-by-Lot Chapter 9- Lot-by-Lot Acceptance SamplingAcceptance Sampling

By AttributesBy Attributes

Chapter 9- Lot-by-Lot Chapter 9- Lot-by-Lot Acceptance SamplingAcceptance Sampling

By AttributesBy AttributesPowerPoint presentation to accompanyPowerPoint presentation to accompany

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OutlineOutline

Fundamental Aspects Statistical Aspects Sampling Plan Design


Learning ObjectivesLearning Objectives

When you have completed this chapter you should be able to:

Know the advantages and disadvantages of sampling; the types of sampling plans and selection factors; criteria for formation of lots; criteria for sampling selection; and decisions regarding rejected lots.

Determine the OC Curve for a single sampling plan.


Learning Objectives-cont’d.Learning Objectives-cont’d.


Determine the equations needed to graph the OC Curve for a double sampling plan.

Know the properties of OC Curves. Know the consumer-producer relationships of

risk, AQL, and LQ.


Learning Objectives-cont’d.Learning Objectives-cont’d.


Determine the AOQ curve and the AOQL for a single sampling plan.

Determine single sampling plans for stipulated producers risk and for stipulated consumers risk.


Fundamental AspectsFundamental Aspects

Acceptance Sampling is a form of inspection applied to lots or batches of items before or after a process to judge conformance to predetermined standards.


Acceptance SamplingAcceptance Sampling

Acceptance Sampling is very useful when: Large numbers of items must be Large numbers of items must be

processed in a short amount of time.processed in a short amount of time. The cost of “passing defectives” is low.The cost of “passing defectives” is low. Fatigue/boredom is caused by inspecting Fatigue/boredom is caused by inspecting

large numbers of items.large numbers of items.


Acceptance SamplingAcceptance Sampling

Acceptance Sampling is very useful when:Destructive testing is required


Acceptance SamplingAcceptance Sampling Three important aspects of sampling:

Involves random sampling of the entire lot

Accept and Reject Lots (does not improve the quality) “Lot Sentencing”

Audit Tool Three approaches to “lot sentencing”:

Accept with no inspection100% inspectionAcceptance Sampling


Acceptance SamplingAcceptance Sampling Advantages

Less expensiveReduced damageReduces the amount of inspection

error Disadvantages

Risk of accepting “bad” lots and rejecting “good” lots

Less information generatedRequires planning and

documentation


Sampling PlansSampling Plans

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

Single sampling Double sampling Multiple sampling

Random sample

Lot


Sampling PlansSampling PlansSingle Sampling Plan

N = lot size

n = sample size

C=acceptance number

If c or less non-conforming units are found in the sample, the lot is accepted, else it is rejected.


Single Sampling PlanSingle Sampling PlanA single sampling plan is one where:

A representative sample of n items is A representative sample of n items is drawn drawn from a lot size of N itemsfrom a lot size of N items

Each item in the sample is examined and classified as good/defective

If the number of defective exceeds a specified rejection number (c) the whole lot is rejected; otherwise the whole lot is accepted


Double Sampling PlanDouble Sampling Plan

A Double Sampling Plan allows to take a second sample if the results of the original sample are inconclusive. Specifies the lot size, size of the initial sample, the accept/reject/inconclusive criteria for the initial sample (N, n1, c1 (Ac), r1(Re)) Specifies the size of the second sample and the acceptance rejection criteria based on the total number of defective observed in both the first and second sample (n2,c2,r2)


First Random sample

Lot

C1 r1

First sample inconclusive, take second sample

Reject LotAccept Lot

Compare number of defective found in the first random sample to Compare number of defective found in the first random sample to C1C1 and and r1 r1 and make appropriate decision.and make appropriate decision.



C2

Reject LotAccept Lot

Compare the total number of defective in both lots to Compare the total number of defective in both lots to C2C2 and make and make the appropriate decisionthe appropriate decision

Lot First Random sample

Second Random sample



Double Sampling PlanDouble Sampling PlanA Multiple Sampling Plan is similar to the double sampling plan in that successive trials are made, each of which has acceptance, rejection and inconclusive options.

Which Plan you choose depends on: Cost and time Number of samples needed and number of items in each sample


Lot FormationLot FormationConsiderations before inspection:

Lots should be homogeneousLarger lots are more preferable than

smaller lotsLots should be conformable to the

materials-handling systems used in both the vendor and consumer facilities


Random SamplingRandom Sampling

Units selected for inspection should be chosen at random

If random samples are not used, bias can be introduced

If judgment methods are used to select the sample, the statistical basis of the acceptance-sampling procedure is lost


Statistical AspectsStatistical AspectsThe Operating Characteristic

Curve: Measures the performance of an

acceptance sampling plan Plots the probability of accepting the

lot versus the lot fraction defective Shows the probability that a lot

submitted with a certain fraction defective will be either accepted or rejected


Acceptable Quality Level Acceptable Quality Level (AQL)(AQL)

The AQL is a percent defective that is the base line requirement for the quality of the producer's product. The producer would like 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 Lot Tolerance Percent DefectiveDefective

The Lot Tolerance Percent Defective The Lot Tolerance Percent Defective LTPD or LQ is a designated high LTPD or LQ is a designated high defect level that would be defect level that would be unacceptable to the consumer. The unacceptable to the consumer. The consumer would like the sampling consumer would like the sampling plan to have aplan to have a low probability of low probability of acceptingaccepting a lot with a defect level as a lot with a defect level as high as the LTPD. high as the LTPD.


Type I Error (Producer’s Risk)Type I Error (Producer’s Risk) This is the probability, for a given

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


Type II Error (Consumer’s Type II Error (Consumer’s Risk)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.


Operating Characteristic Operating Characteristic CurveCurve

This curve plots the probability of accepting the lot (Y-axis) versus the lot fraction or percent defectives (X-axis). The OC curve is the primary tool for displaying and investigating the properties of a Lot Acceptance Sampling Plan.


OC CurvesOC Curves

There are two types of OC curves:Type A

Gives the probability of acceptance of an individual lot coming from finite production

Type BGives the probability of

acceptance for lots coming from a continuous production


An Operating Characteristic Curve (OCC) is a probability curve for a sampling plan that shows the probabilities of accepting lots with various lot quality levels (% defectives).

00.1

0.20.3

0.40.5

0.60.7

0.80.9

1

0 .05 .10 .15 .20

Prob

abilit

y of

acc

eptin

g lo

t

Lot quality (% defective)

Under this sampling plan, if the lot has 3% defective . the probability of accepting the lot is 90% . the probability of rejecting the lot is 10%

If the lot has 20% defective . it has a small probability (5%) of being accepted . the probability of rejecting the lot is 95%

0.7

0.8

OCCs for Single Sampling OCCs for Single Sampling PlansPlans


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prob

abilit

y of

acc

eptin

g lo

t

0

0.1

0 .05 .10 .15 .20 Lot quality (% defective)

AQL - percentage level of defects at which a customer is willing to accept

“Acceptable Lot”

Producer’s Risk = probability acceptable lot is rejected

OCC, AQL and Producer’s RiskOCC, AQL and Producer’s Risk


0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Prob

abilit

y of

acc

eptin

g lo

t

0

0.1

0 .05 .10 .15 .20 Lot quality (% defective)

LTPD - upper limit on the percentage LTPD - upper limit on the percentage of defectives that a customer is of defectives that a customer is willing to accept.willing to accept.

Consumer’s Risk = probability unacceptable is acceptedConsumer’s Risk = probability unacceptable is accepted

“Unacceptable Lot”

OCC, LTPD and Consumer’s RiskOCC, LTPD and Consumer’s Risk


OCC for Double Sampling PlanOCC for Double Sampling Plan



Inspect a sample Inspect a sample of 150 from lot of 150 from lot

of 2400of 2400

If 1 or less If 1 or less NonconformingNonconforming

units accept lots andunits accept lots andstopstop

If 4 or more If 4 or more Nonconforming units Nonconforming units the lot is not acceptedthe lot is not accepted

and stopand stopIf 2 or 3 nonconformingIf 2 or 3 nonconforming units, inspect a secondunits, inspect a second

sample of 200sample of 200

If 5 or lessIf 5 or lessNonconforming unitsNonconforming units

On both samples, On both samples, Accept the lotAccept the lot

If 6 or more If 6 or more Nonconforming unitsNonconforming units

On both samplesOn both samplesThe lot is not acceptedThe lot is not accepted

Figure 9-5 Graphical description of the double sampling plan: N=2400,n1=150,c1=1r1=4, n2=200, c2=5, and r2=6


OCC for a Multiple Sampling PlanOCC for a Multiple Sampling Plan


Average Outgoing Quality Average Outgoing Quality (AOQ)(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. .


The Average Outgoing Quality (AOQ) is the average of rejected lots (100% inspection) and accepted lots ( a sample of items inspected))

ac

ac

N - nAOQ = P * p( ) where

NP = Probability of accepting a lot

p = Fraction defective

n = sample size

N = Lot size

Average Outgoing QualityAverage Outgoing Quality


AOQ and Acceptance SamplingAOQ and Acceptance Sampling

ProducerProducerN=3000N=3000

n=89n=89c=2c=2

ConsumerConsumer

15 lots2% nonconforming




Figure 9-15 How acceptance Sampling works


AOQ and Acceptance AOQ and Acceptance SamplingSampling

Total NumberTotal Number Number Number NonconformiNonconformi

ngng11 lots-11 lots-

2% 2% NonconformingNonconforming

11(3000)=33,00011(3000)=33,000 33,000(0.02)=66033,000(0.02)=660

4 lots-4 lots-

0% 0% NonconformingNonconforming

4(3000)4(3000)(0.98)=11,760(0.98)=11,760

00

44,76044,760 660660

Percent Nonconforming (AOQ) = Percent Nonconforming (AOQ) = 660/44,760 X 100 =1.47%660/44,760 X 100 =1.47%

Figure 9-15 cont’d.


Typically the term (N-n)/N is very close to 1; therefore, the equation most often used is:

AOQ = P * p where

P = Probability of accepting a lot

p = Fraction defective

ac

ac

Average Quality of Inspected Average Quality of Inspected LotsLots


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 Average Outgoing Quality Level AOQL.

Average Outgoing Quality LevelAverage Outgoing Quality Level


Average Total Inspection (ATI)

When rejected lots are 100% inspected, it is easy to calculate the ATI if lots come consistently with a defect level of p. For a LASP (n,c) with a probability pa of accepting a lot with defect level p, we have:ATI = n + (1 - pa) (N - n) where N is the lot size.


Average Sample Number Average Sample Number (ASN)(ASN)

For a single sampling (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.


Average Sample Number Average Sample Number (ASN)(ASN)

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 plan scheme. ASN = n1 + n2 (1 – P1) for a double sampling plan.


Sampling Plan DesignSampling Plan Design

Suppose α is known and the AQL is also known then : Sampling plan with stipulated

producer’s risk Sampling plan with stipulated

consumer’s risk Sampling plan with stipulated

producer’s and consumer’s risk

can be designed.



Stipulated Producer’s Risk α = 0.05 AQL = 1.2% Pa=0.95 P0.95= 0.012

Assume values for C, find np0.95 for this c value, calculate n



Stipulated Consumer’s Risk β = 0.10 LQ = 6.0% Pa=0.10 P0.10= 0.060

Assume values for C, find np0.95 for this c value, calculate n



Stipulated Producer’s and Consumer’s risk α = 0.10 β = 0.10 AQL=0.9 LQ= 7.8

Find the ratio of P0.10/P0.95. From table 9-4 C is between 1 and 2. Find n for c =1 and n for c =2 .


Sampling Plan DesignSampling Plan Design Have 4 plans. Select plan based on:

Lowest sampling size Greatest sampling size Plan exactly meets consumer’s

stipulation and is as close as possible to producer’s stipulation

Plan exactly meets producer’s stipulation and is as close as possible to consumer’s stipulation

09 lot-by-lot acceptance sampling for attributes

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