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04/19/23 IENG 486 Statistical Quality & Process Control 1
IENG 486 - Lecture 18
Introduction to Acceptance Sampling,
Mil Std 105E
04/19/23 IENG 486 Statistical Quality & Process Control 2
Assignment
Reading: Chapter 9
Sections 9.1 – 9.1.5: pp. 399 - 410 Sections 9.2 – 9.2.4: pp. 419 - 425 Sections 9.3: pp. 428 - 430
Homework: Due 03 DEC CH 9 Textbook Problems:
1a, 17, 26 Hint: Use Excel!
Last Assignment: Download and complete Last Assign: Acceptance Sampling
Requires MS Word for Nomograph Requires MS Excel for AOQ
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Acceptance Sampling
Company receives shipment fromvendor
Sample taken from lot,Quality characteristic inspected
Lot Sentencing:Accept lot?
YES
Return lotto vendor
NO
Use lot inproduction
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Three Important Aspects of Acceptance Sampling
1. Purpose is to sentence lots, not to estimate lot quality
2. Acceptance sampling does not provide any direct form of quality control. It simply rejects or accepts lots. Process controls are used to control and systematically improve quality, but acceptance sampling is not.
3. Most effective use of acceptance sampling is not to “inspect quality into the product,” but rather as audit tool to insure that output of process conforms to requirements.
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Three Approaches to Lot Sentencing
1. Accept with no inspection
2. 100% inspection – inspect every item in the lot, remove all defectives
Defectives – returned to vendor, reworked, replaced or discarded
3. Acceptance sampling – sample is taken from lot, a quality characteristic is inspected; then on the basis of information in sample, a decision is made regarding lot disposition.
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Acceptance Sampling Used When:
Testing is destructive 100% inspection is not technologically feasible 100% inspection error rate results in higher percentage of
defectives being passed than is inherent to product Cost of 100% inspection extremely high Vender has excellent quality history so reduction from 100% is
desired but not high enough to eliminate inspection altogether Potential for serious product liability risks; program for
continuously monitoring product required
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Advantages of Acceptance Sampling over 100% Inspection
Less expensive because there is less sampling Less handling of product hence reduced damage Applicable to destructive testing Fewer personnel are involved in inspection activities Greatly reduces amount of inspection error Rejection of entire lots as opposed to return of defectives
provides stronger motivation to vendor for quality improvements
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Disadvantages of Acceptance Sampling (vs 100% Inspection)
Always a risk of accepting “bad” lots and rejecting “good” lots Producer’s Risk: chance of rejecting a “good” lot – Consumer’s Risk: chance of accepting a “bad” lot –
Less information is generated about the product or the process that manufactured the product
Requires planning and documentation of the procedure – 100% inspection does not
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Lot Formation
Lots should be homogeneous Units in a lot should be produced by the same:
machines, operators, from common raw materials, approximately same time
If lots are not homogeneous – acceptance-sampling scheme may not function effectively and make it difficult to eliminate the source of defective products.
Larger lots preferred to smaller ones – more economically efficient
Lots should conform to the materials-handling systems in both the vendor and consumer facilities
Lots should be packaged to minimize shipping risks and make selection of sample units easy
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Random Sampling
IMPORTANT: Units selected for inspection from lot must be chosen at random Should be representative of all units in a lot
Watch for Salting: Vendor may put “good” units on top layer of lot knowing a lax inspector might
only sample from the top layer
Suggested technique:1. Assign a number to each unit, or use location of unit in lot2. Generate / pick a random number for each unit / location in lot3. Sort on the random number – reordering the lot / location pairs4. Select first (or last) n items to make sample
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Single Sampling Plans for Attributes
Quality characteristic is an attribute, i.e., conforming or nonconforming
N - Lot size n - sample size c - acceptance number
Ex. Consider N = 10,000 with sampling plan n = 89 and c = 2 From lot of size N = 10,000 Draw sample of size n = 89 If # of defectives c = 2
Accept lot If # of defectives > c = 2
Reject lot
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How to Compute the OC Curve Probabilities
Assume that the lot size N is large (infinite)
d - # defectives ~ Binomial(p,n)where
p - fraction defective items in lot n - sample size
Probability of acceptance:
0
P 1c
n iia
i
nP d c p p
i
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Example
Lot fraction defective is p = 0.01, n = 89 and c = 2. Find probability of accepting lot.
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OC Curve
Performance measure of acceptance-sampling plan displays discriminatory power of sampling plan
Plot of: Pa vs. p Pa = P[Accepting Lot] p = lot fraction defective
p = fraction defective in lot Pa = P[Accepting Lot]
0.005 0.9897
0.010 0.9397
0.015 0.8502
0.020 0.7366
0.025 0.6153
0.030 0.4985
0.035 0.3936
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OC curve displays the probability that a lot submitted with a certain fraction defective will be either accepted or rejected given the current sampling plan
Probability of Acceptance, Pa
0.00.20.40.60.81.0
0.00 0.02 0.04 0.06 0.08 0.10
Lot fraction defective, p
Pa
n=89c=2
OC Curve
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Ideal OC Curve
Suppose the lot quality is considered bad if p = 0.01 or more A sampling plan that discriminated perfectly between good
and bad lots would have an OC curve like:
1.00
0.040.01 0.02 0.03
Lot fraction defective, p
Probability of Acceptance, Pa
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Ideal OC Curve
In theory it is obtainable by 100% inspection IF inspection were error free.
Obviously, ideal OC curve is unobtainable in practice
But, ideal OC curve can be approached by increasing sample size, n.
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Effect of n on OC Curve
Precision with which a sampling plan differentiates between good and bad lots increases as the sample size increases
Probability of Acceptance, Pa
0.00
0.20
0.40
0.60
0.80
1.00
0.00 0.02 0.04 0.06 0.08 0.10
Lot fraction defective, p
Pan=50, c=1
n=100, c=2
n=200, c=4
n=1000, c=20
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Effect of c on OC Curve
Changing acceptance number, c, does not dramatically change slope of OC curve.
Plans with smaller values of c provide discrimination at lower levels of lot fraction defective
Probability of Acceptance, Pa
0.0
0.2
0.4
0.6
0.8
1.0
0.00 0.02 0.04 0.06 0.08 0.10
Lot fraction defective, p
Pa
n=89, c=2
n=89, c=1
n=89, c=0
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Producer and Consumer Risks in Acceptance Sampling
Because we take only a sub-sample from a lot, there is a risk that:
a good lot will be rejected (Producer’s Risk – )
and a bad lot will be accepted
(Consumer’s Risk – )
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Producer’s Risk -
Producer wants as many lots accepted by consumer as possible so Producer “makes sure” the process produces a level of fraction defective
equal to or less than:
p1 = AQL = Acceptable Quality Level
is the probability that a good lot will be rejected by the consumer even though the lot really has a fraction defective p1
That is,
Lot is rejected given that process
has an acceptable quality levelP
Lot is rejectedP p AQL
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Consumer’s Risk -
Consumer wants to make sure that no bad lots are accepted Consumer says, “I will not accept a lot if percent defective is greater
than or equal to p2”
p2 = LTPD = Lot Tolerance Percent Defective
is the probability a bad lot is accepted by the consumer when the lot really has a fraction defective p2
That is,
Lot accepted given that lot
has unacceptable quality levelP
Lot acceptedP p LTPD
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Designing a Single-Sampling Plan with a Specified OC Curve
Use a chart called a Binomial Nomograph to design plan
Specify: p1 = AQL (Acceptable Quality Level)
p2 = LTPD (Lot Tolerance Percent Defective)
1 – = P[Lot is accepted | p = AQL]
β = P[Lot is accepted | p = LTPD]
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Use a Binomial Nomograph to Find Sampling Plan
(Figure 15-9, p. 643)
Draw two lines on nomograph Line 1 connects p1 = AQL to (1- ) Line 2 connects p2 = LTPD to Pick n and c from the intersection of the lines
Example: Suppose p1 = 0.01, α = 0.05, p2 = 0.06, β = 0.10.
Find the acceptance sampling plan.
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Greek - Axis
p - Axisp1 = AQL = .01
1 – = 1 – .05 = .95
p2 = LTPD = .06
= .10
n = 120
c = 3
Take a sample of size 120.
Accept lot if defectives ≤ 3.
Otherwise, reject entire lot!
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Rectifying Inspection Programs
Acceptance sampling programs usually require corrective action when lots are rejected, that is,
Screening rejected lots Screening means doing 100% inspection on lot
In screening, defective items are Removed or Reworked or Returned to vendor or Replaced with known good items
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Rectifying Inspection Programs
InspectionActivity
Rejected Lots: 100%
Inspected
AcceptedLots
FractionDefective
Incoming Lots:Fraction Defective
FractionDefective = 0
Outgoing Lots:Fraction Defective
0p
0p
1 0p p
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Where to Use Rectifying Inspection
Used when manufacturer wishes to know average level of quality that is likely to result at given stage of manufacturing
Example stages: Receiving inspection In-process inspection of semi-finished goods Final inspection of finished goods
Objective: give assurance regarding average quality of material used in next stage of manufacturing operations
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Average Outgoing Quality: AOQ
Quality that results from application of rectifying inspection Average value obtained over long sequence of lots from process with
fraction defective p
N - Lot size, n = # units in sample Assumes all known defective units replaced with good ones,
that is, If lot rejected, replace all bad units in lot If lot accepted, just replace the bad units in sample
aP p N nAOQ
N
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Development of AOQ
If lot accepted:Number defective units in lot:
Expected number of defective units:
Average fraction defective,Average Outgoing Quality, AOQ:
# units
fraction remaining
defectivein lot
p N n
Lot # defectiveProb
accepted units in lotaP p N n
aP p N nAOQ
N
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Example for AOQ
Suppose N = 10,000, n = 89, c = 2, and incoming lot quality is p = 0.01. Find the average outgoing lot quality.
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Questions & Issues