Outline
2.008
1. Optimization
2. Statistical Process Control
3. In-Process Control
Process Control
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Variation: Common and
Special Causes
What is quality?
Pieces vary from each other:
Size
Size
Size
Size
But they form a pattern that, if stable, is called a distribution:
Size
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Common and Special Causes
(cont’d)
Size
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Common and Special Causes
(cont’d)
If only common causes of variation are present,
the output of a process forms a distribution that is
stable over time and is predictable:
Distributions can differ in…
Location
Size
Shape
Spread
Prediction
Size
Size
Size
…or any combination of these
Tim
e
Size
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1
Common and Special Causes
(cont’d)
Variation: Run Chart
Specified diameter = 0.490” +/-0.001”
If special causes of variation are present, the
process output is not stable over time and is not
predictable:
0.4915
Diameter of part
0.4910
Prediction
0.4905
0.4900
0.4895
0.4890
0.4885
Tim
0.4880
e
0
10
20
Size
Process
change
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30
40
50
Run number
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Process Control
8
Statistical Process Control
1. Detect disturbances
(special causes)
In Control
(Special causes eliminated)
Tim
e
2. Take corrective actions
Out of Control
(Special causes present)
Size
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Central Limit Theorem
Shewhart Control Chart
• A large number of independent
events have a continuous probability
density function that is normal in
shape.
Average (of 10 samples ) Diameter
1.010
• Averaging more samples increases
the precision of the estimate of the
average.
Upper control limit
1.005
1.000
Lower control limit
0.995
0.990
0
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11
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10
20
30
40
50
Run number
60
70
80
90
100
12
2
Sampling and Histogram
Creation
Sampling and Histogram
Creation (cont’d)
20
10
1
9
2
10
8
6
10
4
0
9
1
2
3
4
5
6
7
Outcome
8
9
8
Taking ∞ random
samples, the
resulting histogram
would look like this
4
6
2
10
3
7
5
4
6
2
3
4
5
6
Outcome
7
8
9
13
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Probability distribution
0.5
0
25
20
15
10
5
0
1
2
3
4
1
2
3
4
5
6
7
Outcome
8
9
10
5
Take 10 random samples,
calculate their average, and
repeat ∞ times, the resulting
histogram would approach
the continuous distribution
shown
5
6
7
8
9
10
14
Normal Distribution
Underlying distribution
1.5
1
30
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Distribution resulting from
averaging of 2 random values
Underlying D istribution
35
Take 10 random samples,
calculate their average, and
repeat 100 times, the
resulting histogram would
resemble
10
Uniform Distributions
1.5
40
3
7
1
Probability distr ibution
1
2
Frequency
8
Frequency
Taking 100 random
samples, the
resulting histogram
would look like this
Wheel of Fortune: Equal
probability of outcome 1-10,
P=0.1
14
12
Frequency
18
16
Wheel of Fortune: Equal
probability of outcome 1-10,
P=0.1
Distribution resulting from the
average of 2 random values
1
0.5
0
0.5
0
1
Distribution resulting from the
average of 3 random values
0
0.5
0
0
1
Distribution resulting from the
average of 10 random values
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0
0
15
Precision
Not precise
0
Distribution resulting from the
average of 3 random values
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
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0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Distribution resulting from the
average of 10 random values
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
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Shewhart Control Chart
Precise
• Upper Control Limit (UCL), Lower
Control Limit (LCL)
Not accurate
• Subgroup size (5 < n < 20)
Accurate
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3
Shewhart Control Chart (cont’d)
Shewhart Control Chart (cont’d)
Control chart based on 100 samples
Control chart based on average of 10 samples, note the step
change that occurs at run 51
1.010
Upper control limit
Average (of 10 samples ) Diameter
Average (of 10 samples ) Diameter
1.010
1.005
1.000
Lower control limit
0.995
1.005
Upper control limit
1.000
Lower control limit
0.995
Step disturbance
Step disturbance
0.990
0.990
10
0
20
30
40
50
Run number
60
70
80
90
100
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10
0
19
20
30
40
50
Run number
60
70
80
90
100
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Control Charts
20
Setting the Limits
Idea: Points outside the limits will signal that something is wrong-
Upper control limit
an assignable cause. We want limits set so that assignable causes
Process average
Lower control limit
are highlighted, but few random causes are highlighted
accidentally.
0
10
20
30
40
50
60
70
80
90
100
Run number
1.
2.
3.
Convention for Control Charts:
Collection:
Gather data and plot on chart.
•
Control:
Calculate control limits from process data, using simple formulae.
•
Identify special causes of variation; take local actions to correct.
•
Capability:
•
Quantify common cause variation; take action on the system.
• Upper control limit (UCL) = x + 3σsg
• Lower control limit (LCL) = x - 3σsg
(Where σsg represents the standard deviation of a subgroup of samples)
These three phases are repeated for continuing process improvement.
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Setting the Limits (cont’d)
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Benefits of Control Charts
Properly used, control charts can:
Convention for Control Charts (cont’d):
• Be used by operators for ongoing control of a process
• Help the process perform consistently, predictably, for quality and
cost
σsub group = σsg ≠ σprocess
σsubgroup = σprocess/√n
• Allow the process to achieve:
– Higher quality
UCL = x + 3σprocess/√n
– Lower unit cost
– Higher effective capacity
LCL = x - 3σprocess/√n
• Provide a common language for discussing process performance
As n increases, the UCL and LCL move closer to the center line,
• Distinguish special from common causes of variation; as a guide
to local or management action
making the control chart more sensitive to shifts in the mean.
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4
Process Capability
Process Capability Index
In Control and Capable
(Variation from common cause reduced)
1. Cp = Range/6σ
Lower specification limit
Upper specification limit
Tim
2. Cpk
e
In Control but not Capable
(Variation from common causes excessive)
Size
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Process Capability
• Take an example with:
Mean = .738
Standard deviation, σ = .0725
UCL = 0.900
LCL = 0.500
• Normalizing the specifications:
UCL − x
σ
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Process Capability (cont’d)
LCL
0.500
UCL
0.900
x
0.738
• Using the tables of areas under the Normal Curve, the
proportions out of specification would be:
PUCL= 0.0129
σ
ZLCL
ZUCL
-3.28
2.23
PLCL = 0.0005
Ptotal = 0.0134
0.900 − 0.738
= 2.23
0.0725
LCL − x 0.500 − 0.738
Z LCL =
=
= −3.28
0.0725
σ
ZMIN = 2.23 (on an absolute basis)
Z UCL =
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• The Capability Index would be:
CPK = Zmin/3 = 2.23 / 3 = 0.74
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Process Capability (cont’d)
Process Capability (cont’d)
• If this process could be adjusted toward the center of the
specification, the proportion of parts falling beyond either
or both specification limits might be reduced, even with no
change in Standard deviation.
• For example, if we confirmed with control charts a new
mean = 0.700, then:
• The proportions out of specification would be:
Z UCL =
Z LCL =
UCL − xnew
σ
LCL − xnew
σ
Zmin = 2.76
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0.900 − 0.700
= 2.76
0.0725
0.500 − 0.700
=
= −2.76
0.0725
=
LCL
0.500
PUCL= 0.0029
PLCL = 0.0029
Ptotal = 0.0058
UCL
0.900
x
0.700
• The Capability Index would be:
CPK = Zmin/3 = 2.76 / 3 = 0.92
σ
ZLCL
ZUCL
-2.76
2.76
29
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5
Improving Process Capability
• To improve the chronic performance of the
process, concentrate on the common causes
that affect all periods. These will usually require
management action on the system to correct.
• Chart and analyze the revised process:
– Confirm the effectiveness of the system by continued
monitoring of the Control Chart
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