chapter 10 lot-by-lot acceptance sampling for attributes.ppt
TRANSCRIPT
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QualityControl
Chapter 9- Lot-by-Lot
Acceptance SamplingBy AttributesPowerPoint presentation to accompany
BesterfieldQuality Control, 8e
PowerPoints created by Rosida Coowar
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Learning 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 sample selection; and decisions
regarding rejected lots
Determine the OC Curve for a single sampling plan
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
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Learning Objectives
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
Upon completing this course, the students should be able to:
describe important concepts pertaining to quality, quality system and total quality management
apply the principles of probability and statistics in quality planning, control and improvement and
utilize appropriate statistical tools and techniques
undertake complete problem-solving cycle using appropriate quality improvement tools and
techniques
design a quality system and incorporate important quality elements necessary for its proper
functioning
implement the concept of total quality in engineering and non-engineering processes
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Fundamental AspectsAcceptance sampling has declined in importance due to SPC
Various acceptance sampling practices include statisticalacceptance sampling, fixed percentage, occasional random
checks, and 100% inspection
Acceptance Sampling is a form of inspection applied to lots or
batches of items before or after a process to judge conformanceto predetermined standards
Sampling plans are established by severity (critical, major and
minor)
A single sampling plan is defined by the lot size, N, sample size,
n, and the acceptance number, c
Performed where there is a consumer-supplier relationship
(consumer-supplier, two different plants, to departments, etc)
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Acceptance SamplingAcceptance Sampling is very useful when:
Large numbers of items must be processed in a short amount of
time
The cost of passing defectives is low
Fatigue/boredom is caused by inspecting large numbers of items
Destructive testing is required
Information concerning producers quality, such as and R, p
or c charts, and Cpkis not available
When automated inspection is not available
X
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Acceptance 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 inspection
100% inspection
Acceptance Sampling
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Acceptance SamplingAdvantages compared to 100% inspection:
Less expensive
Reduced handling damage
Upgrade inspection job from monotonous piece-by-piece decisions to
lot-by-lot decisions
Reduces the amount of inspection error Stronger motivation for supplier to quality improvement due to
rejection of entire lots
Disadvantages
Risk of accepting bad lots and rejecting good lots
Less information generated
Requires planning and documentation
No assurance that entire lot conforms to specifications
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Sampling Plans
Sampling Plans specify the lot size, sample size, number of
samples and acceptance/rejection criteria
Sampling plans involve:
Single samplingDouble sampling
Multiple sampling
Sequential sampling
Randomsample
Lot
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Sampling Plans
Single 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
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Single Sampling Plan
A single sampling plan is one where: A representative sample of n items is drawn from 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
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Double Sampling Plan
ADouble 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 theacceptance rejection criteria based on the total number ofdefective observed in both the first and second sample(n2,c2,r2)
If values are not given for r1 and r2, they are equal to c2+1
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First Randomsample
Lot
C1 r1
First sample inconclusive,
take second sampleReject LotAccept Lot
Compare number of defective found in the first random sample to C1
and r1 and make appropriate decision.
Double Sampling Plan
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C2
Reject LotAccept Lot
Compare the total number of defective in both lots to C2and make
the appropriate decision
Lot First Random sample
Second Random sample
Double Sampling Plan
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Double Sampling Plan
AMultiple Sampling Plan is similar to the double sampling planin that successive trials are made, each of which hasacceptance, rejection and inconclusive options
In sequential sampling plan, items are sampled and inspected
one after another. A cumulative record is maintained, and adecision is made to accept or not accept the lot as soon asthere is sufficient cumulative evidence
All four types of sampling plans can give the same results; thechance of a lot being accepted under a single sampling plan isthe same under appropriate double, multiple, and sequentialsampling plan
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Double Sampling Plan
Which Plan you choose depends on:
Cost and timeNumber of samples needed and number of items in
each sample
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Lot Formation
Lot formation can influence effectiveness of sampling plan
Considerations before inspection:
Lots should be homogeneous (same machine, same operator, sameinput material) otherwise sampling plan does not function properlyand it is difficult to take corrective action
Larger lots are more preferable than smaller lots to reduceinspection cost
Lots should be conformable to the materials-handling systems usedin both the vendor and consumer facilities (not to confuse betweenhomogeneous lot and packaging requirements)
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Random Sampling Samples should be representative of the entire lot with
each unit in the lot having equal likelihood of beingselected (random 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 islost Assign a number to each unit in the lot Use random numbers to determine which units to be sampled and
inspected (use Table of random numbers) Discard any number not appropriate
Many units already have serial numbers A unique number can be assigned indicating certain
location in the packing
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Random Sampling Liquid can be assumed to be homogeneous thus sample
taken anywhere Stratification may also be carried out during sampling
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Nonaccepted Lots A number of courses of actions can be taken:
1. Nonaccepted lot can be passed to production andnoncnonforming units sorted out by production personnel(unsatisfactory, defeats purpose of sampling, and slowsproduction)
2. Nonaccepted lot can be rectified at consumers plant by personnelfrom either the producers or consumers plant (consumer isaware of producers poor quality, transportation costs are saved,space must be provided at consumers plant)
3. Nonaccepted lot can be returned to producer for rectification(most appropriate course of action, results in long-run qualityimprovement, shipping costs paid both ways by producer,
producers employees are aware of poor quality
It is expected that nonaccepted lots will receipt 100%inspection and nonconforming units are discarded.
A resubmitted lot is not normally reinspected
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Statistical 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
1. Assume p02. Calculate np0value
3. Attain Pavalues from Poisson table using applicable cand np0values
4. Plot point (100p0, 100Pa)
5. Repeat steps 1, 2, 3 and 4 until a smooth curveobtained
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Operating Characteristic Curve
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 andinvestigating the properties of a Lot Acceptance SamplingPlan
There are two types of OC curves:
Type A
Gives the probability of acceptance of an individual lot comingfrom finite production
Type BGives the probability of acceptance for lots coming from a
continuous production
As n/N
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Statistical Aspects
OC Curve Properties
Acceptance sampling plans with similar properties can givedifferent OC curves
1. Sample size as a fixed percentage of lot size
2. Fixed sample size
3. As sample size increases, the curve becomes steeper4. As the acceptance number decreases, the curve becomes steeper
OC curve with n = 300 and c = 2 is steeper than that with n =50 and c = 0
Plans with c = 0 have the perception that nonconforming unitswill not be tolerated and should be used for criticalnonconformities
For major and minor nonconformities, c > 0 should beconsidered
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AnOperating Characteristic Curve (OCC) is a probability
curve for a sampling plan that shows the probabilities ofaccepting lots with various lot quality levels (% defectives).
0
0.1
0.20.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 .05 .10 .15 .20
Probability
ofaccepting
lot
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 Plans
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Consider a double sampling plan N = 2400, n1= 150, c1=
1, r1= 4, n2= 200, c2= 5, r2= 6 If there is one or fewer nonconforming unit on the first
sample, the lot is accepted
(Pa)I= P(1or less)IA second sample is taken only if there are 2 or 3
nonconforming units on the first sample
Then the lot can be accepted by obtaining:
Two NC on the first and 3 or less NC on second or
Three NC on first and two or less NC on second
(Pa)II= (P2)I(P3orless)II+ (P3)I(P2orless)II (Pa)combined= (Pa)I+ (Pa)II
Double Sampling Plan
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Double Sampling Plan
Inspect a sample
of 150 from lotof 2400
If 1 or lessNonconforming
units accept lots andstop
If 4 or moreNonconforming units
the lot is not acceptedand stop
If 2 or 3 nonconformingunits, inspect a second
sample of 200
If 5 or lessNonconforming units
On both samples,Accept the lot
If 6 or moreNonconforming units
On both samplesThe 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
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OCC for Double Sampling Plan
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OCC for a Multiple Sampling Plan
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Acceptable Quality Level (AQL)
Conflicting interest between consumer and producer when
acceptance sampling is used
Sampling carries risks of not accepting acceptable lots andof accepting unacceptable lots
Producers risk is the probability, for a given (n,c)
sampling plan, of rejecting a lot that has a defect levelequal to the AQL The producer suffers when this occurs,because a lot with acceptable quality was rejected. Thesymbol is commonly used for the Type I error and
typical values for range from 0.2 to 0.01 (typically 0.05) Pa= 1 - a
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Acceptance Quality Level (AQL) AQLacceptance quality limit
The AQL is a percent defective that is the base linerequirement for the quality of the producer's product. Theproducer would like to design a sampling plan such thatthere is a high probability of acceptinga lot that has a
defect level less than or equal to the AQL Consumers risk is the probability, for a given (n,c)
sampling plan, of accepting a lot with a defect level equalto the LTP or LQ. The consumer suffers when this occurs,because a lot with unacceptable quality was accepted.The symbol is commonly used for the Type II error andtypical values range from 0.2 to 0.01
LQ is limiting quality
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0.2
0.3
0.40.5
0.6
0.7
0.8
0.9
1
P
robability
ofac
cepting
lot
00.1
0 .05 .10 .15 .20 Lot quality (% defective)
AQL - percentage level of defects atwhich a customer is willing to accept
Acceptable Lot
Producers Risk = probability acceptable lot is rejected
OCC, AQL and Producers Risk
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0.2
0.3
0.40.5
0.6
0.7
0.8
0.9
1
P
robability
ofac
cepting
lot
00.1
0 .05 .10 .15 .20 Lot quality (% defective)
LTPD - upper limit on the percentage
of defectives that a customer is
willing to accept.
Consumers Risk = probability unacceptable is accepted
Unacceptable Lot
OCC, LTPD and Consumers Risk
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Average Outgoing Quality (AOQ) A common procedure, when sampling and testing is non-
destructive, is to 100% inspect rejected lots and replaceall defectives with good units. In this case, all rejected lotsare made perfect and the only defects left are those inlots that were accepted
The procedure is called rectifying inspection AOQ = (100p0)(Pa)
Rectifying inspection results in nonaccepted lots havingAOQ better than incoming quality
The limit of percent nonconforming is average outgoingquality limit (AOQL). A plot of theAOQ(Y-axis) versus theincoming lot p(X-axis) will start at 0 for p= 0, and returnto 0 for p= 1 (where every lot is 100% inspected and
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Average Outgoing Quality (AOQ) rectified). In between, it will rise to a maximum. This
maximum, which is the worst possible long termAOQ, iscalled the Average Outgoing Quality LevelAOQL.
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TheAverage Outgoing Quality (AOQ)is the average of
rejected lots (100% inspection) and accepted lots ( asample of items inspected)
Typically the term (N-n)/N is very close to 1; therefore, theequation most often used is:
ac
ac
N - nAOQ = P * p( ) where
N
P = Probability of accepting a lot
p = Fraction defective
n = sample size
N = Lot size
Average Outgoing Quality
AOQ = P *p where
P = Probability of accepting a lot
p = Fraction defective
ac
ac
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AOQ and Acceptance Sampling
ProducerN=3000
n=89c=2
Consumer
15 lots2% nonconforming
11 lots2% nonconforming
4 lots2% nonconforming
4 lots0% nonconforming
Figure 9-15 How acceptance Sampling works
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AOQ and Acceptance Sampling
Total Number NumberNonconforming
11 lots-
2% Nonconforming
11(3000)=33,000 33,000(0.02)=660
4 lots-
0% Nonconforming
4(3000)(0.98)=11,760 0
44,760 660
Percent Nonconforming (AOQ) = 660/44,760 X
100 =1.47%
Figure 9-15 contd.
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Average Sample Number (ASN) For a single sampling (n,c) we know each and every lot has
a sample of size ntaken and inspected or tested
For double, multiple and sequential plans, the amount ofsampling varies depending on the number of defectsobserved
For double sampling plan,
ASN = n1+ n2(1-P1)Where P1is the probability of a decision on the first sample
ASN curves are a valuable tool for justifying double or
multiple sampling when inspection costs are great becauseof inspection time, equipment costs, or equipmentavailability
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Average Total Inspection (ATI) When rejected lots are 100% inspected, it is easy to
calculate theATIif lots come consistently with a defectlevel of p.
ATI is the amount inspected by both the consumer andproducer
It assumes that rectified lots will receive 100% inspection For a lot acceptance sampling plan (LASP) with (n,c) and a
probability paof accepting a lot with defect levelp, wehave:
ATI=n + (1 - Pa) (N - n)
where Nis the lot size
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Sampling Plan Design Suppose is known and the AQL is also known then :
Sampling plan with stipulated producers risk (a, AQL) Sampling plan with stipulated consumers risk
Sampling plan with stipulated producers and consumers risk
can be designed
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Sampling Plan Design Stipulated Producers Risk
a= 0.05 AQL = 1.2% Pa=0.95 p0.95= 0.012
Assume values for c, find np0.95for this c value, calculate n
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Sampling Plan Design Stipulated Consumers Risk
= 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
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Sampling Plan Design Stipulated Producers and Consumers risk
= 0.10 = 0.10AQL=0.9 LQ= 7.8
Find the ratio of P0.10/P0.95. From table 9-4 C is between 1and 2. Find n for c =1 and n for c =2 .
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Sampling Plan Design Have 4 plans
Select plan based on: Lowest sampling size
Greatest sampling size
Plan exactly meets consumers stipulation and is as close
as possible to producers stipulation Plan exactly meets producers stipulation and is as close
as possible to consumers stipulation
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Sampling Plan Design
An assembly plant plans a single sampling plan such that itslots of products that are no worse than 2% nonconformingis accepted at least 99% of the time. Recommend thepossible plans. Select a plan with higher consumersprotection. Draw the OC curve for the selected plan using 7
points. Table of Poisson distribution and np values forcorresponding c, a, and bvalues are given.
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Note: x is c while lt is np0
Table of Poisson distribution
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Table of np vlaues with corresponding c, a, and bvalues
THE END
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THE END
IQ + EQ + SQ = TQ
We have indeed created man in the best of molds. (At-Tin: 4)
Taqwim: mold, symmetry, form, nature, constitution