Statistical Process Control. Take periodic samples from a process Plot the sample points on a control chart Determine if the process is within limits Correct the process before defects occur. Common Causes. Assignable Causes. Average. Grams. (a) Mean. Assignable Causes. Average. Grams.
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Common Causes
Assignable Causes
Average
Grams
(a) Mean
Assignable Causes
Average
Grams
(b) Spread
Assignable Causes
Average
Grams
(c) Shape
Control Chart Examples
UCL
Nominal
Variations
LCL
Sample number
Control Charts
Assignable causes likely
UCL
Nominal
LCL
1 2 3
Samples
Mean
-3 -2 -1 +1 +2 +3
68.26%
95.44%
99.74%
The Normal
Distribution
= Standard deviation
Type I error:
Probability of searching for
a cause when none exists
UCL
Process
average
LCL
(a) Three-sigma limits
Type I error:
Probability of searching for
a cause when none exists
UCL
Process
average
LCL
(b) Two-sigma limits
Type II error:
Probability of concluding
that nothing has changed
UCL
Shift in process
average
Process
average
LCL
(a) Three-sigma limits
Type II error:
Probability of concluding
that nothing has changed
UCL
Shift in process
average
Process
average
LCL
(b) Two-sigma limits
Constructing a Control Chart
Types of Data
Control Charts For Attributes
Based on the binomial distribution
UCL = p + 3 p(1-p) /n
= 0.10 + 3 0.10 (1-0.10) /100
= 0.190
Proportion
SampleDefect Defective
1 6 .06
2 0 .00
3 4 .04
. . .
. . .
20 18 .18
200
100 jeans in each sample
total defectives
total sample observations
200
20 (100)
p =
=
= 0.10
0.2
0.18
0.16
0.14
0.12
0.1
Proportion defective
0.08
0.06
0.04
0.02
0
Sample number
0
2
4
6
8
10
12
14
16
18
20
.
.
Based on the Poisson distribution
Count # of defects per roll in 15 rolls of denim fabric.
SampleDefects
1 12
2 8
3 16
. .
. .
15 15
190
24
21
18
15
.
12
Number of defects
9
6
3
0
0
2
4
6
8
10
12
14
Sample number
Slip Ring Diameter (cm)
Sample 12345 X R
15.025.014.944.994.96 4.98 0.08
25.015.035.074.954.96 5.00 0.12
34.995.004.934.924.99 4.97 0.08
45.034.915.014.984.89 4.96 0.14
54.954.925.035.055.01 4.99 0.13
64.975.065.064.965.03 5.01 0.10
75.055.015.104.964.99 5.02 0.14
85.095.105.004.995.08 5.05 0.11
95.145.104.995.085.09 5.08 0.15
105.014.985.085.074.99 5.030.10
50.09 1.15
3s Control Chart Factors
Sample size X-chartR-chart
nA2D3D4
21.8803.27
31.0202.57
40.7302.28
50.5802.11
60.4802.00
70.420.081.92
80.370.141.86
Process Capability
Process Capability
Natural
control
limits
Natural
control
limits
Design
specs
PROCESS
PROCESS
Process can meet specifications
Process cannot meet specifications
Natural
control
limits
Design
specs
PROCESS
Process capability exceeds specifications
Acceptance Sampling
Outline
Some Sampling Plans
A Single Sampling Plan
Consider a single sampling plan with n = 10 and c = 2
A Single Sampling Plan
Acceptable fraction defective in a lot
Maximum fraction defective accepted in a lot
Type I error = P(reject a good lot)
Type II error = P(accept a bad lot)
{
0.80
0.60
0.40
0.20
{
0.20
0.18
0.02
0.04
0.06
0.08
0.10
0.12
0.14
0.16
Operating Characteristic Curve
1.00
1-a = 0.05
OC curve for n and c
Probability of acceptance, Pa
b = 0.10
Percent defective
AQL
LTPD
AQL = 0.1
LTPD = 0.3
= 0.05
= 0.10
Average Outgoing Quality
Average Outgoing Quality
AOQ Curve
0.015
AOQL
0.010
Average
Outgoing
Quality
0.005
0.10
0.09
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
AQL
LTPD
(Incoming) Percent Defective
Reading and Exercises