TM 720 - Lecture 11
Acceptance Sampling Plans
4/13/2015
TM 720: Statistical Process Control
1
Assignment:

Reading:
•
•

Finish Chapter 14
•
•
Sections 14.1 – 14.2
Sections 14.4
Start Chapter 12
Assignment:
•
•
Download and complete Assign 08: Acceptance Sampling
•
•
Requires MS Word for Nomograph
Requires MS Excel for AOQ
Solutions for 8 will post on Thursday
4/13/2015
TM 720: Statistical Process Control
2
Acceptance Sampling
Company receives shipment from
vendor
Sample taken from lot,
Quality characteristic inspected
Lot Sentencing:
Accept lot?
NO
YES
Use lot in
production
4/13/2015
Return lot
to vendor
TM 720: Statistical Process Control
3
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.
4/13/2015
TM 720: Statistical Process Control
4
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.
4/13/2015
TM 720: Statistical Process Control
5
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
4/13/2015
TM 720: Statistical Process Control
6
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
4/13/2015
TM 720: Statistical Process Control
7
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
4/13/2015
TM 720: Statistical Process Control
8
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 minimized shipping risks and make
selection of sample units easy
4/13/2015
TM 720: Statistical Process Control
9
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.
2.
3.
4.
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
4/13/2015
TM 720: Statistical Process Control
10
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
4/13/2015
TM 720: Statistical Process Control
11
How to Compute the OC Curve
Probabilities

Assume that the lot size N is large (infinite)

d - # defectives ~ Binomial()
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
4/13/2015
TM 720: Statistical Process Control
12
Example

Lot fraction defective is p = 0.01,
n = 89 and c = 2. Find probability of accepting lot.
4/13/2015
TM 720: Statistical Process Control
13
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
4/13/2015
TM 720: Statistical Process Control
14
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
4/13/2015
TM 720: Statistical Process Control
15
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
4/13/2015
TM 720: Statistical Process Control
16
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.
4/13/2015
TM 720: Statistical Process Control
17
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

The precision with which a sampling plan differentiates
between good and bad lots increases as the sample size
increases
4/13/2015
TM 720: Statistical Process Control
18
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
4/13/2015
TM 720: Statistical Process Control
19
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 –  )
4/13/2015
TM 720: Statistical Process Control
20
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 
  P

 has an acceptable quality level

  P  Lot is rejected p  AQL 
4/13/2015
TM 720: Statistical Process Control
21
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 = LPTD = Lot Tolerance Percent Defective
 probability bad lot is accepted by the consumer when lot really has
a fraction defective  p2

That is,
 Lot accepted given that lot

  P

 has unacceptable quality level 
  P  Lot accepted p  LTPD 
4/13/2015
TM 720: Statistical Process Control
22
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]
4/13/2015
TM 720: Statistical Process Control
23
Use a Binomial Nomograph to Find Sampling Plan
(Figure 14-9, p. 658)


Draw two lines on nomograph
•
•
•
Line 1 connects p1 = AQL to (1- )
Line 2 connects p2 = LTPD to 
Pick n and c from intersection of lines
Example: Suppose
•
•
•
•
p1 = 0.01,
α = 0.05,
p2 = 0.06,
β = 0.10.
Find the acceptance sampling plan.
4/13/2015
TM 720: Statistical Process Control
24
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
4/13/2015
TM 720: Statistical Process Control
25
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
4/13/2015
TM 720: Statistical Process Control
26
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
4/13/2015
TM 720: Statistical Process Control
27
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,
•
•
If lot rejected, replace all bad units in lot
If lot accepted, just replace the bad units in sample
4/13/2015
TM 720: Statistical Process Control
28
Development of AOQ

If lot accepted:
Number defective units in lot:
 p N






Expected number of defective units:
 Pa

 n
 # units
 fraction  
 
  remaining
defective

 in lot



 Lot
  # defective 
p   N  n   Prob 


 accepted   units in lot 
Average fraction defective,
Average Outgoing Quality, AOQ:
4/13/2015
AOQ 
TM 720: Statistical Process Control
Pa p  N  n 
N
29
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.
4/13/2015
TM 720: Statistical Process Control
30
Military Standard 105E
(MIL STD 105E)
(ANSI/ASQC Z1.4, ISO 2859)

Most widely used acceptance sampling system for
attributes

MIL STD 105E is Acceptance Sampling System

•
collection of sampling schemes
Can be used with single, double or multiple sampling
plans
•
We will consider single sampling plans for this course
4/13/2015
TM 720: Statistical Process Control
31
Inspection Types


Normal Inspection
•
Tightened Inspection
•
•

Used at start of inspection activity
Instituted when vendor’s recent quality history has
deteriorated
Acceptance requirements for lots are more stringent
Reduced Inspection
•
•
Instituted when vendor’s recent quality history has been
exceptionally good
Sample size is usually smaller than under normal inspection
4/13/2015
TM 720: Statistical Process Control
32
Switching Rules
Start
AND conditions
- Production Steady
- 10 consecutive lots accepted
- Approved by responsible
authority
Reduced
2 out of 5 consecutive lots
rejected
Normal
Tightened
OR conditions
- Lot rejected
- Irregular production
- Lot meets neither accept
nor reject criteria
- Other conditions warrant
return to normal inspection
5 consecutive
lots accepted
10 consecutive lots remain
on tightened inspection
Discontinue
Inspection
4/13/2015
TM 720: Statistical Process Control
33
Procedure for
MIL STD 105E

STEP 1: Choose AQL
•
MIL STD 105E designed around Acceptable Quality
Level, AQL
• Recall that the Acceptable Quality Level, AQL, is
producer's largest acceptable fraction defective in
process
•
Typical AQL range:
• 0.01%  AQL  10%
• Specified by contract or authority responsible for
sampling
4/13/2015
TM 720: Statistical Process Control
34
Procedure for
MIL STD 105E

STEP 2: Choose inspection level
•
•
•
•
Level II
•
Designated as normal
Level I
•
•
Requires about one-half the amount of inspection as Level II
Use when less discrimination needed
Level III
•
•
Requires about twice as much
Use when more discrimination needed
Four special inspection levels used if very small samples
necessary
•
S-1, S-2, S-3, S-4
4/13/2015
TM 720: Statistical Process Control
35
Procedure for
MIL STD 105E



STEP 3–Determine lot size, N
•
Lot size most likely dictated by vendor
STEP 4: Find sample size code letter
•
•
From Table 14-4, p 675
Given lot size, N, and Inspection Level, use table to determine
sample size code letters
STEP 5: Determine appropriate type sampling plan
•
Decide if Single, Double or Multiple sampling plan is to be
used
4/13/2015
TM 720: Statistical Process Control
36
Procedure for
MIL STD 105E

STEP 6: Find Sample Size, n, and
Acceptance Level, c
• Given sample size letter code, use Master
Tables: 14-5, 14-6, and 14-7 on pp.676-678
• Find n and c for all three inspection types:
• Normal Inspection
• Tightened Inspection
• Reduced Inspection
4/13/2015
TM 720: Statistical Process Control
37
Example

Suppose product comes from vendor in lots of
size 2000 units. The acceptable quality level is
0.65%. Determine the MIL STD 105E
acceptance-sampling system.
4/13/2015
TM 720: Statistical Process Control
38
Questions & Issues
4/13/2015
TM 720: Statistical Process Control
39