Transcript Document

Acceptance Sampling and
Statistical Process Control
1
Probability Review
Permutations (order matters)
•
Number of permutations of n objects taken x
at a time
n
Px 
n!
(n  x)!
2
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
3
P2 
3!
(3  2 )!

3!
 6
1!
3
Probability Review
Combinations (order does not matter)
•
Number of combinations of n objects taken x
at a time
n
C
x
n
n!

 
 x   x! (n  x)!


4
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
3
3!
3!
6
  


 3
2
2! (3  2 )!
2!1!
2
 
5
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”
n x
nx
P(x)    p (1  p)
; x  0,1,2,..., n
x
6
Probability Review
Binomial Distribution (cont.)
•
Example: What is the probability of 2
defectives in 4 items if p = 0.20?
4
P(2)    (0.20)
2
2
(0.80)
2
 0.1536
7
Probability Review
Binomial Distribution Exercises
8
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
P(r)
 D  N  D

r 

n  r



 N 

n 







9
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?
P(0)
 4   96 

0

 10 





 0.652
 100 


 10



10
Probability Review
Hypergeometric Distribution Exercises
11
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?
 10 
 (0.40)
P(0)  
0 
•
0
(0.60)
10
 0.665
We got 0.652 from the straight hypergeometric.
12
Probability Review
•
Poisson Distribution (to approximate Binomial)
•
•
Use when n20 and p0.05
=np (average number of defects)
P(x)

e
λ
λ
x
; x  0,1,2,...
x!
•
•
Example: What is the probability of finding 2 defects
in 4 items if p = 0.20?
 0.8
2
e
(0.8)
P(2) 
 0.1438
2!
We got 0.1536 with the straight Binomial.
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Acceptance Sampling
•
Definition: process of accepting or rejecting a
lot by inspecting a sample selected according to
a predetermined sampling plan
•
Notation:
N
n
c
p
=
=
=
=
batch size
sample size
acceptance number
proportion defective (known or
long-term average)
Pa = probability that a batch will be
accepted
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 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) = 
15
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.
16
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).
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OC Curves
•
Type A 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
18
Acceptance Sampling Plans
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Considerations:
•
-risk
•
-risk
•
Acceptable Quality Level (AQL)
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Lot Tolerance Percent Defective (LTPD)
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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)
20
Acceptance Sampling Plans
•
Single Sampling
•
•
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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
21
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)
22
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.
23
Average Outgoing Quality
Average Outgoing Quality (AOQ) = proportion
defective after inspection and rectification
•
AOQ curve relates outgoing quality to incoming
quality
AOQ 
(N  n)  p  Pa
N
•
Average Outgoing Quality Level (AOQL) = point
where outgoing quality is worst
•
Maximum AOQ over all possible values of p
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Average Outgoing Quality Level
(1 
AOQL
n
N

)
 y
n
(Table 1 provides values of y for c = 0,1,2,…,40)
For small sampling fractions,
AOQL

y
n
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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
n 
y
AOQL
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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
AOQL

y
n
•
Adjust c-value and n to manipulate the AOQL
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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
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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
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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
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Acceptance Sampling
Acceptance Sampling Plan Exercise
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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
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Other Sampling Plans
•
Chain 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
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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
34
Acceptance Sampling Trade-Off
•
Example: Desired AOQL = 5%,
Batch Size 1-800 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
35
Creating Work Units
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Homogeneity
•
•
•
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
36
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
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Sample Selection
Systematic Sampling
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Select a random start between 1 and 10
•
Select every xth item until n items are selected
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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
39
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
40
U.S. Census Bureau Applications
0 to 10,000 HIDs
AQL = 4%
Sample = 200 HIDs
Matched
Unmatched
Added
Acceptance Number = 14
Road
Water
Other
10,001 to 500,000 HIDs
AQL = 4%
Sample = 315 HIDs
Matched
Unmatched
Added
Acceptance Number = 21
Road
Water
Other
41
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
42
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
43
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
44
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
45
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
46
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
47
Control Charts
•
CL represents the process average
•
Upper and lower limits represent the
region the process moves in under
random variation
48
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
49
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)
50
Control Charts
A process is considered to be “in control” if
the process data move randomly between
the upper and lower limits
51
Control Charts
Time for Enumerating Blocks
14
UCL
Hours to Enumerate Block
12
10
8
CL
6
4
2
LCL
0
1
3
5
7
9
11
13
15
17
19
21
23
25
Day
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Control Charts
•
Examples:
•
Process: enumerator canvassing a block
•
Measure: hours to canvass the block
•
Is the process in statistical control?
53
Control Charts
•
The process data in time order move
randomly inside the control limits
•
Therefore, the process IS in statistical
control
54
Control Charts
Time for Enumerating Blocks
14
12
Hours to Enumerate
10
8
UCL
6
CL
4
2
LCL
0
1
3
5
7
9
11
13
15
17
19
21
23
25
Day
55
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
56
Control Charts
Time to Enumerate Blocks
25
Hours to Enumerate Block
20
15
UCL
10
CL
5
LCL
0
1
3
5
7
9
11
13
Day
15
17
19
21
23
25
57
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 retraining the lister
58
Control Charts
Time to Enumerate Blocks
14
UCL
12
Hours to Enumerate Blocks
10
8
CL
6
4
2
LCL
0
1
3
5
7
9
11
13
Day
15
17
19
21
23
25
59
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:
60
Control Charts
Time to Enumerate Blocks
14
UCL
12
Hours to Enumerate
10
8
CL
6
4
2
LCL
0
1
3
5
7
9
11
13
Day
15
17
19
21
23
25
61
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
62
Control Charts
Control Chart Exercises
63