Transcript Lot by lot acceptance sampling

IENG 486 - Lecture 18
Introduction to Acceptance Sampling,
Mil Std 105E
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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


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Requires MS Word for Nomograph
Requires MS Excel for AOQ
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Acceptance Sampling
Company receives shipment from
vendor
Sample taken from lot,
Quality characteristic inspected
Lot Sentencing:
Accept lot?
NO
YES
Use lot in
production
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Return lot
to vendor
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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

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Lots should be packaged to minimize shipping risks and make selection of
sample units easy
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Random Sampling

IMPORTANT:



Watch for Salting:


Units selected for inspection from lot must be chosen at random
Should be representative of all units in a lot
Vendor may put “good” units on top layer of lot knowing a lax inspector might
only sample from the top layer
Suggested technique:
1.
2.
3.
4.
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Assign a number to each unit, or use location of unit in lot
Generate / pick a random number for each unit / location in lot
Sort on the random number – reordering the lot / location pairs
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


If # of defectives > c = 2

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Accept lot
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:
 n i
n i
Pa  P  d  c      p 1  p 
i 0  i 
c
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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
Probability of Acceptance, Pa
1.0
0.8
0.6
Pa
0.4
0.2
0.0
0.00
n=89
c=2
0.02
0.04
0.06
0.08
0.10
Lot fraction defective, p

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
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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:
Probability of Acceptance, Pa
1.00
0.01
0.02
0.03
0.04
Lot fraction defective, p
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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
Probability of Acceptance, Pa
1.00
0.80
Pa
n=50, c=1
0.60
n=100, c=2
0.40
n=200, c=4
0.20
n=1000, c=20
0.00
0.00
0.02
0.04
0.06
0.08
0.10
Lot fraction defective, p
 Precision with which a sampling plan differentiates between good
and bad lots increases as the sample size increases
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Effect of c on OC Curve
Probability of Acceptance, Pa
1.0
0.8
Pa
n=89, c=2
0.6
0.4
0.2
0.0
0.00
n=89, c=1
n=89, c=0
0.02
0.04
0.06
0.08
0.10
Lot fraction defective, p
 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
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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
level


  P
  P  Lot is rejected 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

  P

 has unacceptable quality level 
  P  Lot accepted 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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p1 = AQL = .01
p - Axis
Greek - Axis
p2 = LTPD = .06
n = 120
 = .10
1 –  = 1 – .05 = .95
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




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Removed or
Reworked or
Returned to vendor or
Replaced with known good items
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Rectifying Inspection
Programs
Incoming Lots:
Fraction Defective
p0
Inspection
Activity
Rejected Lots:
100%
Inspected
Fraction
Defective = 0
Accepted
Lots
Fraction
Defective
p0
Outgoing Lots:
Fraction Defective
p1  p0
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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
AOQ 
Pa p  N  n 
N
 N - Lot size, n = # units in sample
 Assumes all known defective units replaced with good ones,
that is,


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If lot rejected, replace all bad units in lot
If lot accepted, just replace the bad units in sample
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Development of AOQ

If lot accepted:
Number defective units in lot:
 # units

fraction


remaining 
 p  N  n  


 defective  in lot




 Expected number of defective units:
 Lot
  # defective 
 Pa   p   N  n   Prob 


 accepted   units in lot 

Average fraction defective,
Average Outgoing Quality, AOQ:
AOQ 
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Pa p  N  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
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