1. Introduction to Variability
Acceptance Sampling

2. IENG 486 Statistical Quality & Process Control
Acceptance Sampling
(c) , D.H. Jensen, R.C. Wurl

3. Three Important Aspects of Acceptance Sampling
IENG 486 Statistical Quality & Process Control
Three Important Aspects of Acceptance Sampling
Purpose is to sentence lots, not to estimate lot quality
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.
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.
(c) , D.H. Jensen, R.C. Wurl

4. Three Approaches to Lot Sentencing
IENG 486 Statistical Quality & Process Control
Three Approaches to Lot Sentencing
Accept with no inspection
100% inspection – inspect every item in the lot, remove all defectives Defectives – returned to vendor, reworked, replaced or discarded
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.
(c) , D.H. Jensen, R.C. Wurl

5. Acceptance Sampling Used When:
IENG 486 Statistical Quality & Process Control
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
(c) , D.H. Jensen, R.C. Wurl

6. Advantages of Acceptance Sampling over 100% Inspection
IENG 486 Statistical Quality & Process Control
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
(c) , D.H. Jensen, R.C. Wurl

7. Disadvantages of Acceptance Sampling (vs 100% Inspection)
IENG 486 Statistical Quality & Process Control
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 – 
Truth
Lot Good Lot Bad
Belief
Lot Good
Lot Bad
No Error
b Error
a Error
No Error
(c) , D.H. Jensen, R.C. Wurl

8. Disadvantages of Acceptance Sampling (vs 100% Inspection)
IENG 486 Statistical Quality & Process Control
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
(c) , D.H. Jensen, R.C. Wurl

9. TM 720: Statistical Process Control
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
(c) D.H. Jensen & R.C. Wurl

10. IENG 486 Statistical Quality & Process Control
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:
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
(c) , D.H. Jensen, R.C. Wurl

11. Single Sampling Plans for Attributes
IENG 486 Statistical Quality & Process Control
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
(c) , D.H. Jensen, R.C. Wurl

12. How to Compute the OC Curve Probabilities
IENG 486 Statistical Quality & Process Control
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:
(c) , D.H. Jensen, R.C. Wurl

13. IENG 486 Statistical Quality & Process Control
Example
Lot fraction defective is p = 0.005, n = 89 and c = 2. Find probability of accepting lot.
(c) , D.H. Jensen, R.C. Wurl

14. IENG 486 Statistical Quality & Process Control
Example
Lot fraction defective is p = 0.01, n = 89 and c = 2. Find probability of accepting lot.
(c) , D.H. Jensen, R.C. Wurl

15. IENG 486 Statistical Quality & Process Control
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
(c) , D.H. Jensen, R.C. Wurl

16. IENG 486 Statistical Quality & Process Control
OC Curve
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
(c) , D.H. Jensen, R.C. Wurl

17. IENG 486 Statistical Quality & Process Control
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:
(c) , D.H. Jensen, R.C. Wurl

18. IENG 486 Statistical Quality & Process Control
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.
(c) , D.H. Jensen, R.C. Wurl

19. IENG 486 Statistical Quality & Process Control
Effect of n on OC Curve
Precision with which a sampling plan differentiates between good and bad lots increases as the sample size increases
(c) , D.H. Jensen, R.C. Wurl

20. IENG 486 Statistical Quality & Process Control
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
(c) , D.H. Jensen, R.C. Wurl

21. Producer and Consumer Risks in Acceptance Sampling
IENG 486 Statistical Quality & Process Control
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 – a )
and
a bad lot will be accepted (Consumer’s Risk – b )
(c) , D.H. Jensen, R.C. Wurl

22. IENG 486 Statistical Quality & Process Control
Producer’s Risk - a
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 a 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,
(c) , D.H. Jensen, R.C. Wurl

23. Producer’s Risk - a
n=89, c=2, p=AQL=0.01, find producer risk a

24. IENG 486 Statistical Quality & Process Control
Consumer’s Risk - b
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
b is the probability a bad lot is accepted by the consumer when the lot really has a fraction defective  p2
That is,  
(c) , D.H. Jensen, R.C. Wurl

25. Consumer’s Risk - b
n89, c=2, p=LTPD=0.05, find consumer risk b

26. Designing a Single-Sampling Plan with a Specified OC Curve
TM 720: Statistical Process Control
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 – a = P[Lot is accepted | p = AQL]
β = P[Lot is accepted | p = LTPD]
(c) D.H. Jensen & R.C. Wurl

27. Use a Binomial Nomograph to Find Sampling Plan (Figure 15-9, p. 661)
TM 720: Statistical Process Control
Use a Binomial Nomograph to Find Sampling Plan (Figure 15-9, p. 661)
Draw two lines on nomograph
Line 1 connects p1 = AQL to (1- a)
Line 2 connects p2 = LTPD to b
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.
(c) D.H. Jensen & R.C. Wurl

28. P1 = AQL = .01
P - Axis
Greek - Axis
n = 140
P2 = LTPD = .05
 = .10
1 –  = 1 – .05 = .95
c = 3
Take a sample of size 140.
Accept lot if defectives ≤ 3.
Otherwise, reject entire lot!

29. Check – Producer Risk

30. Check – Consumer Risk

31. Rectifying Inspection Programs
TM 720: Statistical Process Control
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
(c) D.H. Jensen & R.C. Wurl

32. Rectifying Inspection Programs
TM 720: Statistical Process Control
Rectifying Inspection Programs
(c) D.H. Jensen & R.C. Wurl

33. Where to Use Rectifying Inspection
TM 720: Statistical Process Control
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
(c) D.H. Jensen & R.C. Wurl

34. Average Outgoing Quality: AOQ
TM 720: Statistical Process Control
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
(c) D.H. Jensen & R.C. Wurl

35. TM 720: Statistical Process Control
Development of AOQ
If lot accepted: Number defective units in lot:
Expected number of defective units:
Average fraction defective, Average Outgoing Quality, AOQ:
(c) D.H. Jensen & R.C. Wurl

36. TM 720: Statistical Process Control
Example for AOQ
Suppose N = 10,000, n = 89, c = 2, and incoming lot quality is p = Find the average outgoing lot quality.
(c) D.H. Jensen & R.C. Wurl

37. Military Standard 105E (MIL STD 105E) (ANSI/ASQC Z1.4, ISO 2859)
TM 720: Statistical Process Control
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
(c) D.H. Jensen & R.C. Wurl

38. TM 720: Statistical Process Control
Inspection Types
Normal Inspection
Used at start of inspection activity
Tightened Inspection
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
(c) D.H. Jensen & R.C. Wurl

39. TM 720: Statistical Process Control
Switching Rules
(c) D.H. Jensen & R.C. Wurl

40. TM 720: Statistical Process Control
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
(c) D.H. Jensen & R.C. Wurl

41. TM 720: Statistical Process Control
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
(c) D.H. Jensen & R.C. Wurl

42. TM 720: Statistical Process Control
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
(c) D.H. Jensen & R.C. Wurl

43. TM 720: Statistical Process Control
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
Find n and c for all three inspection types:
Normal Inspection
Tightened Inspection
Reduced Inspection
(c) D.H. Jensen & R.C. Wurl

44. TM 720: Statistical Process Control
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.
(c) D.H. Jensen & R.C. Wurl

45. Normal Insp. Level
Lot Size
= 2000

46. AQL
Plan K
Sample 125 units.
Ac = 2, accept if defects ≤ 2.
Re = 3, reject entire lot if defects ≥ 3.

47. AQL
Plan K
Sample 125 units
Ac = 1, accept if defects ≤ 1.
Re = 2, reject entire lot if defects ≥ 2.

48. AQL
Plan K
Sample 50 units
Ac = 1, accept if defects ≤ 1.
Re = 3, reject entire lot if defects ≥ 3.