1. Acceptance Sampling and Statistical Process Control

2. Probability Review Permutations (order matters)
Number of permutations of n objects taken x at a time

3. Probability Review Permutations (cont.)
Example: Number of permutations of 3 letters (A, B, C) taken 2 at a time
(A,B,C)  AB, BA, AC, CA, BC, CB

4. Probability Review Combinations (order does not matter)
Number of combinations of n objects taken x at a time

5. Probability Review (cont.)
Combinations
Example: Number of combinations of 3 letters (A, B, C) taken 2 at a time
(A, B, C) = AB, AC, BC

6. Probability Review Binomial Distribution
Sum of a series of independent, identically distributed Bernoulli random variables
Probability of x defectives in n items
p = probability of “success” (usually determined from long-term process average)
1-p = probability of “failure”

7. Probability Review Binomial Distribution (cont.)
Example: What is the probability of 2 defectives in 4 items if p = 0.20?

8. Probability Review
Binomial Distribution Exercises

9. Probability Review Hypergeometric Distribution
Random sample of size n selected from N items, where D of the N items are defective
Probability of finding r defectives in a sample of size n from a lot of size N

10. Probability Review Hypergeometric Distribution (cont.)
Example: What is the probability of finding 0 defects in 10 items taken from a lot of size 100 containing 4 defects?

11. Probability Review
Hypergeometric Distribution Exercises

12. Probability Review Binomial Approximation of Hypergeometric
Use when n/N  0.10
P ~ D/N
Example: What is the probability of finding 0 defects in 10 items selected from a lot of size 100 containing 40 defects?
We got from the straight hypergeometric.

13. Probability Review Poisson Distribution (to approximate Binomial)
Use when n20 and p0.05
=np (average number of defects)
Example: What is the probability of finding 2 defects in 4 items if p = 0.20?
We got with the straight Binomial.

14. Acceptance Sampling
Definition: process of accepting or rejecting a lot by inspecting a sample selected according to a predetermined sampling plan
Notation: N = batch size
n = sample size
c = acceptance number
p = proportion defective (known or long-term average)
Pa = probability that a batch will be accepted

15.  and  Risks
Type I Error: rejecting an acceptable lot (a.k.a. producer’s risk)
P(Type I Error) = 
Type II Error: accepting an unacceptable lot (a.k.a. consumer’s risk)
P(Type II Error) = 

16. Operating Characteristic (OC) Curves
OC Curves characterize acceptance sampling plans.
OC Curves are complete plotting of Pa for a lot at all possible values of p.

17. OC Curves
Steepness of OC curves indicates the power of the acceptance sampling plans to distinguish “good” lots from “bad” lots.
Power = 1-P(accepting lot | p)
= P(rejecting lot | p)
For large values of p (i.e., large number of defects), we want the Power to be large (i.e., close to 1).

18. OC Curves Type A OC Curve Type B OC Curve
Uses known value for p (i.e., lot composition is known)
Can use hypergeometric distribution
Type B OC Curve
Uses process average for p
Can use binomial distribution

19. Acceptance Sampling Plans
Considerations:
-risk
-risk
Acceptable Quality Level (AQL)
Lot Tolerance Percent Defective (LTPD)

20. Acceptance Sampling Plans
Acceptable Quality Level (AQL)
Maximum percent defective that is acceptable
 = P(rejecting lot | p = AQL)
Corresponds to higher Pa (left-hand side of OC Curve)
Lot Tolerance Percent Defective (LTPD)
Worst quality that is acceptable (accepted with low probability)
 = P(accepting lot | p = LTPD)
Corresponds to lower Pa (right-hand side of OC Curve)

21. Acceptance Sampling Plans
Single Sampling
Take a single sample from the lot for inspection
Quality of sampled work determines lot decision
Double Sampling
One small sample from the lot for inspection
If quality of first sample is acceptable, that sample determines lot decision
If quality of first sample is unacceptable or not clear, select a second small sample to make lot decision

22. Acceptance Sampling Plans
Single-Sampling vs. Double-Sampling Plans
Either plan can satisfy AQL and LTPD requirements
Double sampling often results in smaller total sample sizes
Double sampling can increase cost if second sample is required often
Psychological advantage to double sampling (second chance)

23. Inspection Inspection involves verifying the quality of a work unit.
Most inspection includes rectification of errors found.
Rectification:
100% of defective work units are repaired or replaced
Rejected lots are 100% verified and rectified
Verifiers should have higher level of understanding to establish confidence in the inspection/rectification process.

24. Average Outgoing Quality
Average Outgoing Quality (AOQ) = proportion defective after inspection and rectification
AOQ curve relates outgoing quality to incoming quality
Average Outgoing Quality Level (AOQL) = point where outgoing quality is worst
Maximum AOQ over all possible values of p

25. Average Outgoing Quality Level
(Table 1 provides values of y for c = 0,1,2,…,40)
For small sampling fractions,

26. Average Outgoing Quality Level
Determining sample size for desired outgoing quality (AOQL):
Given desired AOQL
Given desired acceptable number of defects, c
Given y-value from Table 1 for desired c

27. Average Outgoing Quality Level
Determining AOQL for known sample size and desired acceptable number of rejects, c:
Given sample size, n
Given desired c-value
Given y-value from Table 1 for desired c-value
Adjust c-value and n to manipulate the AOQL

28. Additional Acceptance Sampling Terminology
Average Sample Number (ASN)
Average number of sample units inspected to reach lot decision
In single-sampling plan: ASN = n
In double-sampling plan: n1  ASN  n1 + n2

29. Additional Acceptance Sampling Terminology
Average Total Inspection (ATI):
Average number of units inspected per lot
In single-sampling plan: n  ATI  N
ATI = n + (1-Pa)(N-n)
Better incoming quality  less inspection

30. Additional Acceptance Sampling Terminology
Average Fraction Inspected (AFI):
Average fraction of units inspected per lot
AFI = ATI/N
In single-sampling plan: n/N  AFI  1

31. Acceptance Sampling
Acceptance Sampling Plan Exercise

32. Other Sampling Plans Continuous Sampling
Not possible to use lots/batches
Level of inspection depends on perceived quality level
Continuous Sampling Plan 1 (CSP1)
Start at 100% inspection
After i consecutive non-defectives, go to sample inspection
Go back to 100% inspection when a defective is found

33. Other Sampling Plans Chain Sampling Skip-Lot Sampling
Like standard sampling plans with c=0
However, allow 1 defect in a lot if previous i lots were defective-free
Useful for small lots where c=0 is required
Skip-Lot Sampling
Lot-based continuous sampling
Lots inspected 100% until i lots are defective-free, then go to sample inspection of lots

34. Acceptance Sampling Trade-Off
Sampling Fraction vs. Acceptance Criteria
Assuming a set quality level (AOQL), we can make choices regarding inspection rates and acceptance criteria
To decrease inspection rate, we must tighten acceptance criteria
To allow more defects, we must increase sample size

35. Acceptance Sampling Trade-Off
Example: Desired AOQL = 5%, Batch Size items
25% inspection  accept if 6 or fewer defects in a sample of size 60
10% inspection  accept if 5 or fewer defects in a sample of size 60
See Tables 1 and 2

36. Creating Work Units Homogeneity Batch Size Alike items within a batch
Helps to keep samples representative
Batch Size
Rule-of-thumb: ½ to 1 day of work
Very small batch  frequent QC, more paperwork
Very large batch  delayed feedback, higher rework risk

37. Sample Selection
Important for sample selection to be completely random
Random Number Table
Number batch items 1 through N
Select random point in random number table
Use next n numbers in the table as the batch items to select

38. Sample Selection Systematic Sampling
Select a random start between 1 and 10
Select every xth item until n items are selected

39. U.S. Census Bureau Applications
Address Canvassing
Three random starts within listing pages
One random start provides start for check of total listings
Two random starts provide start for check of added housing units and deleted housing units
No errors allowed
QA form example from upcoming 2004 Census Test

40. U.S. Census Bureau Applications
Review of Map Improvement Files
Select map “features” for QC review
Acceptance Sampling plan
Global AQL requirement
Sample size dependent on “batch” size
Acceptance number dependent on sample size
Uses stratified sampling

41. U.S. Census Bureau Applications
Sample = 200 HIDs
Acceptance Number = 14
0 to 10,000 HIDs
AQL = 4%
Matched
Unmatched
Added
Road
Water
Other
Sample = 315 HIDs
Acceptance Number = 21
10,001 to 500,000 HIDs
AQL = 4%
Matched
Unmatched
Added
Road
Water
Other

42. Process Control Allows us to plan quality into our processes
Spend fewer resources on inspection and rework
Observe processes, collect samples, measure quality, determine if the process produces acceptable results

43. Process Control Stresses prevention over inspection
Traditional approach to QC:
Most resources spent on inspection and rework
Process Control approach to QC:
Most resources spent on prevention with relatively little spent on inspection and rework
Lower overall cost

44. Sources of Variation Random Sources
Cause of variation is common or unassignable
Process is still “in control”
Difficult or impossible to eliminate
Requires modification to the process itself

45. Sources of Variation Non-Random Sources
Cause of variation is special and assignable
Could be difficult to eliminate
Causes process to be “out of control”
Can address the specific cause of the variation

46. Sources of Variation Diagnosing non-random variation
“2-out-of-3” – if two out of three consecutive points are out of control
“4-out-of-5” – if four out of five consecutive points are out of control
“7 successive” – if seven consecutive points are on one side of the process average

47. Control Charts Graphical device for assessing statistical control
Plots data from a process in time order
Three reference lines:
Upper Limit
Center Line (CL)
Lower Limit

48. Control Charts CL represents the process average
Upper and lower limits represent the region the process moves in under random variation

49. Control Charts
Control chart limits can be Control Limits or Specification Limits
Control Limits: based on quality capability of the process
Control limits traditionally set at 3 standard deviations away from the CL
Specification Limits: based on quality goal

50. Control Charts
Example: Chemical process with assumed concentration of 3%, desired concentration  3.2% and  2.8%
Specification Limits: LSL = 2.8, USL = 3.2
Control Limits: LCL = 2.5, UCL = 3.5 (based on true capability of process)

51. Control Charts
A process is considered to be “in control” if the process data move randomly between the upper and lower limits

52. Control Charts

53. Control Charts Examples: Process: enumerator canvassing a block
Measure: hours to canvass the block
Is the process in statistical control?

54. Control Charts
The process data in time order move randomly inside the control limits
Therefore, the process IS in statistical control

55. Control Charts

56. Control Charts Some points above the UCL
Therefore, process IS NOT in statistical control
Might be hard-to-enumerate blocks (special cause)
Field supervisor might want to send a more experienced lister to canvass those blocks

57. Control Charts

58. Control Charts Process has definite upward trend
Therefore, process IS NOT in statistical control
Perhaps lister has forgotten proper techniques (special cause)
Field supervisor might want to consider re-training the lister

59. Control Charts

60. Control Charts Sometimes, patterns are hard to see
This example seems to show a random distribution of points
However, look what happens when we connect the points:

61. Control Charts

62. Control Charts The points are all inside the control limits
So, the process IS in statistical control
However, there is a definite cyclical pattern
These types of patterns are not random and should be investigated

63. Control Charts
Control Chart Exercises