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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
BesterfieldBesterfieldQuality Control, 8eQuality Control, 8e
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Quality Control, 8eQuality Control, 8e
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Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
OutlineOutline
Fundamental Aspects Statistical Aspects Sampling Plan Design
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
Learning Objectives-cont’d.Learning Objectives-cont’d.
When you have completed this chapter you should be able to:
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
Learning Objectives-cont’d.Learning Objectives-cont’d.
When you have completed this chapter you should be able to:
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
Acceptance SamplingAcceptance Sampling
Acceptance Sampling is very useful when:Destructive testing is required
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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)
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Double Sampling PlanDouble Sampling Plan
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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 Plan
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
OCC for Double Sampling PlanOCC for Double Sampling Plan
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
Double Sampling PlanDouble 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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
OCC for a Multiple Sampling PlanOCC for a Multiple Sampling Plan
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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. .
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
AOQ and Acceptance SamplingAOQ and Acceptance Sampling
ProducerProducerN=3000N=3000
n=89n=89c=2c=2
ConsumerConsumer
15 lots2% nonconforming
11 lots2% nonconforming
4 lots2% nonconforming
4 lots0% nonconforming
Figure 9-15 How acceptance Sampling works
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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.
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
Sampling Plan DesignSampling Plan Design
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
Sampling Plan DesignSampling Plan Design
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
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
Sampling Plan DesignSampling Plan Design
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 .
Besterfield: Quality Control, 8th ed.. © 2009 Pearson Education, Upper Saddle River, NJ 07458.All rights reserved
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