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2.830J / 6.780J / ESD.63J Control of Manufacturing Processes (SMA 6303)
Spring 2008
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Control of
Manufacturing Processes
Subject 2.830/6.780/ESD.63
Spring 2008
Lecture #8
Process Capability &
Alternative SPC Methods
March 4, 2008
Manufacturing
1
Agenda
• Control Chart Review
– hypothesis tests: α, β and n
– control charts: α, β, n, and average run length (ARL)
• Process Capability
• Advanced Control Chart Concepts
Manufacturing
2
Average Run Length
• How often will the data exceed the ±3σ limits
if Δμx = 0?
Prob(x > μ x + 3σ x ) + Pr ob(x < μ x − 3σ x )
= 3 / 1000
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
-4
Manufacturing
-3
−3σ
0
-2
-1
μ0
1
2
3
+3σ
4
3
Detecting Mean Shifts:
Chart Sensitivity
• Consider a real shift of Δμx:
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Sample Number
• How many samples before we can expect to
detect the shift on the xbar chart?
Manufacturing
4
Average Run Length
• How often will the data exceed the ±3σ limits
if Δμx = +1σ?
Prob(x > μ x + 2σ x ) + Prob(x < μ x − 4σ x )
= 0.023 + 0.001 = 24 / 1000
0.45
0.45
Actual
0.4
0.4
0.35
0.35
Assumed
Distribution
0.3
0.3
Distribution
0.25
0.25
-4
Manufacturing
−3σ
-4
-3
0.2
0.2
0.15
0.15
0.1
Δμ
0.1
0.05
0.05
0
-3
-2
-2
-1
μ
-1
0
pe
0
01
12
23
+3σ
34
4
5
Definition
• Average Run Length (arl): Number of runs (or
samples) before we can expect a limit to be
exceeded = 1/pe
– for Δμ = 0
– for Δμ = 1σ
arl = 3/1000 = 333 samples
arl = 24/1000 = 42 samples
Even with a mean shift as large as 1σ, it
could take 42 samples before we know it!!!
Manufacturing
6
Effect of Sample Size n on ARL
• Assume the same Δμ = 1σ
– Note that Δμ is an absolute value
• If we increase n, the Variance of xbar
σx
decreases:
σx =
n
• So our ± 3σ limits move closer together
Manufacturing
7
ARL Example
Original
Distribution
-4
−3σ
-3
0.450.45
0.45
0.40.4
0.4
0.350.35
0.35
0.30.3
0.3
0.250.25
0.25
0.20.2
0.2
0.150.15
0.15
0.10.1
Δμ 0.1
0.050.05
0.05
00
-2
-4
new limits
-3
-1
-4
-2
−3σ∗
-3
-1
μ
0-2
New
Distribution
pe
0
1-1
1
20
31
+3σ∗
422
3
34
+3σ
4
same absolute shift
As n increases pe increases so ARL decreases
Manufacturing
8
Another Use of the
Statistical Process Model:
The Manufacturing -Design Interface
• We now have an empirical model of the
process
0.45
0.4
0.35
0.3
How “good” is the
process?
0.25
0.2
0.15
0.1
0.05
Is it capable of producing
what we need?
Manufacturing
0
-4
-3
−3σ
-2
-1
0
μ
1
2
3
4
+3σ
9
Process Capability
• Assume Process is In-control
• Described fully by xbar and s
• Compare to Design Specifications
– Tolerances
– Quality Loss
Manufacturing
10
Design Specifications
• Tolerances: Upper and Lower Limits
Characteristic
Dimension
Target
Lower
x*
Specification
Limit
Upper
Specification
Limit
LSL
USL
Manufacturing
11
Design Specifications
• Quality Loss: Penalty for Any Deviation from
Target
QLF = L*(x-x*)2
How to
Calibrate?
x*=target
Manufacturing
12
Use of Tolerances: Process Capability
• Define Process using a Normal Distribution
• Superimpose x*, LSL and USL
• Evaluate Expected Performance
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
LSL
-4
-3
−3σ
Manufacturing
0.05
x* 0
-2
-1
μ0
1
USL
2
3
+3σ
4
13
Process Capability
• Definitions
(USL − LSL)
tolerance range
Cp =
=
6σ
99.97% confidence range
• Compares ranges only
• No effect of a mean shift
Manufacturing
14
Process Capability: Cpk
C pk
⎛ (USL − μ ) (LSL − μ) ⎞
= min ⎝
,
⎠
3σ
3σ
= Minimum of the normalized deviation from
the mean
•
Compares effect of offsets
Manufacturing
15
Cp = 1; Cpk = 1
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-4
Manufacturing
-3
-2
-1
0
1
2
3
4
16
Cp = 1; Cpk = 0
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-4
-3
-2
Manufacturing
-1
0
1
2
3
4
17
Cp = 2; Cpk = 1
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-4
Manufacturing
-3
-2
-1
0
1
2
3
4
18
Cp = 2; Cpk = 2
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-4
Manufacturing
-3
-2
-1
0
1
2
3
4
19
Effect of Changes
• In Design Specs
• In Process Mean
• In Process Variance
• What are good values of Cp and Cpk?
Manufacturing
20
Cpk Table
Manufacturing
Cpk
z
P<LS or
P>USL
1
3
1E-03
1.33
4
3E-05
1.67
5
3E-07
2
6
1E-09
21
The “6 Sigma” problem
P(x > 6σ) = 18.8x10-10
Cp=2
0.45
Cpk=2
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
-4
LSL
−3σ∗
-3
-2
0
-1
0
1
+3σ∗
2
3
4
USL
6σ
Manufacturing
22
The 6 σ problem: Mean Shifts
P(x>4σ) =
Cp=2
31.6x10-6
0.45
Cpk=4/3
0.4
Even with a mean shift of 2σ
we have only 32 ppm out of spec
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
-4
-3
-2
-1
0
1
LSL
2
3
4
USL
4σ
Manufacturing
23
Capability from the Quality Loss Function
QLF = L(x) =k*(x-x*)2
x*
Given L(x) and p(x) what is E{L(x)}?
Manufacturing
24
Expected Quality Loss
E{L(x)} = E[k(x − x*)
2
]
= k [E(x ) − 2 E(xx*) + E(x * )]
2
2
= kσ + k( μx − x*)
2
x
Penalizes
Variation
Manufacturing
2
Penalizes
Deviation
25
Process Capability
•
•
•
•
•
The reality (the process statistics)
The requirements (the design specs)
Cp - a measure of variance vs. tolerance
Cpk - a measure of variance from target
Expected Loss - an overall measure of
goodness
Manufacturing
26
Xbar Chart Recap
• xbar - S (or R) charts
– plot of sequential sample statistics
– compare to assumptions
• normal
• stationary
• Interpretation
– hypothesis tests on μ and σ
– confidence intervals
– “randomness”
• Application
– Real-time decision making
Manufacturing
27
Real-Time
1.8
1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
1
3
5
7
9
11
13
15
17
19
21
23
25
27
29
31
33
35
37
39
41
43
45
47
49
Sample Number
Manufacturing
28
Beyond Xbar
• Good Points
– Simple and “transparent”
– Enforces Assumptions
• Normality (via Central Limit)
• Independent (via long sampling times)
• Limitations
– n>1 to get Xbar and S
– ARL is typically large
• Not very sensitive to small changes
– Slow time response
Manufacturing
29
Beyond Xbar
• What if n=1?
– Have a Lot of Data
– Want Fast Response to Changes
• How to Compute Control Chart Statistics?
– Running Chart and Running Variance?
– Running Average and Running Variance?
– Running Average with Forgetting Factor
• How to Increase Sensitivity to Small, Persistent Mean
Shift?
– Integrate the Error
Manufacturing
30
Chart Design:
n=1 Designs - Running Averages
• Sensitivity: Ability to detect small changes
(e.g. mean shifts)
• Time Response: Ability to Catch Changes
Quickly
• Noise Rejection?: Higher Variance
Manufacturing
31
Xbar “Filtering”
1.4
1.2
1
0.8
0.6
Run Data
Xbar n=4
0.4
0.2
0
1
6
11
16
21
26
31
36
41
46
51
56
61
66
71
76
81
86
91
96
-0.2
-0.4Manufacturing
32
Filtering
• Reduced Peaks
• Hides intermediate data
• Reduces the “frequency content” of the output
Manufacturing
33
Independence and Correlation
• Independence: Current output does not
depend on prior
• Correlation: Measure of Independence
– e.g. auto correlation function
Rxx (τ ) = E[x(t)x(t + τ )]
Manufacturing
34
Correlation
Rxx (τ ) = E[x(t)x(t + τ )]
For a linear 1st order system
For an uncorrelated
τ~ 1 sec:
process
1
0.8
0.6
0.4
0.2
0
-4
Manufacturing
-3
Tmin
-2
-1
0
1
2
3
Tmax
4
35
Sampling: Frequency and
Distribution of Samples
1
1
0.8
0.8
0.8
0.6
0.6
0.6
0.4
0.4
0.4
0.2
0.2
0.2
1
0
-4
-3
Tmin
-2
0
0
-3
-2
012341 2 31 4
0-1
-4-1-3 -2 -1-4
T
Tmin Tm Tmax
2
3
Tmax
4
SAMPLE TIME
Manufacturing
36
Correlation and Sampling
Correlated
Samples
Uncorrelated
Samples
Correlation
Time (e.g.)
• Taking samples beyond correlation
time guarantees independence
Manufacturing
37
Sampling and Averaging
• Sampling Frequency Affects
– Time Response
– Correlation
• Averaging
– Filters Data
– Slows Response
Manufacturing
38
Alternative Charts: Running Averages
• More averages/Data
• Can use run data alone and
average for S only
• Can use to improve resolution
of mean shift
j+n
xRj
n measurements
at sample j
Manufacturing
SR j 2
1
= ∑ xi
n i=j
Running Average
1 j +n
2
=
(x
−
x
)
∑
Rj Running Variance
n − 1 i= j i
39
Specific Case: Weighted Averages
y j = a1 x j −1 + a2 x j −2 + a3 x j −3 + ...
• How should we weight measurements??
– All equally? (as with Running Average)
– Based on how recent?
• e.g. Most recent are more relevant than less
recent?
Manufacturing
40
Consider an Exponential Weighted Average
0.25
Define a weighting function
Wt − i = r (1 − r )
0.2
i
0.15
Exponential Weights
0.1
0.05
0
1
Manufacturing
2
3
4
5
6
7
8
9
10
11
12 13
14
15
16 17
18
19
20
41
Exponentially Weighted Moving Average: (EWMA)
Ai = rxi + (1 − r)Ai −1
Recursive EWMA
⎛ σ x2 ⎞ ⎛ r ⎞
2t
σA = ⎜
(
)
1
−
1
−
r
⎟⎝
⎝ n ⎠ 2 − r⎠
[
]
σA =
UCL, LCL = x ± 3σ A
Manufacturing
time
σx ⎛ r ⎞
2
n ⎝ 2 − r⎠
for large t
42
Effect of r on σ multiplier
0.9
0.8
plot of (r/(2-r)) vs. r
0.7
0.6
0.5
0.4
wider control limits
0.3
0.2
0.1
0
0
Manufacturing
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
r
43
SO WHAT?
• The variance will be less than with xbar,
σA =
σx
n
⎛ r ⎞
⎝ 2 − r⎠
= σx
⎛ r ⎞
⎝ 2 − r⎠
• n=1 case is valid
• If r=1 we have “unfiltered” data
– Run data stays run data
– Sequential averages remain
• If r<<1 we get long weighting and long delays
– “Stronger” filter; longer response time
Manufacturing
44
EWMA vs. Xbar
1
0.9
r=0.3
0.8
Δμ = 0.5 σ
0.7
xbar
EWMA
UCL EWMA
LCL EWMA
grand mean
UCL
LCL
0.6
0.5
0.4
0.3
0.2
0.1
0
0
Manufacturing
50
100
150
200
250
300
45
Mean Shift Sensitivity
EWMA and Xbar comparison
1.2
xbar
1
EWMA
UCL
EWMA
0.8
LCL
EWMA
0.6
Grand
Mean
3/6/03
UCL
0.4
LCL
Mean shift = .5 σ
0.2
n=5
r=0.1
0
1
5
Manufacturing
9
13
17
21
25
29
33
37
41
45
49
46
Effect of r
1
0.9
xbar
0.8
EWMA
0.7
UCL
EWMA
0.6
LCL
EWMA
0.5
Grand
Mean
0.4
UCL
0.3
LCL
0.2
0.1
r=0.3
0
1
3
Manufacturing
5
7
9 11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
47
Small Mean Shifts
• What if Δμx is small wrt σx ?
• But it is “persistent”
• How could we detect?
– ARL for xbar would be too large
Manufacturing
48
Another Approach: Cumulative Sums
• Add up deviations from mean
– A Discrete Time Integrator
j
C j = ∑ (x i − x)
i=1
• Since E{x-μ}=0 this sum should stay near zero
• Any bias in x will show as a trend
Manufacturing
49
Mean Shift Sensitivity: CUSUM
8
7
t
Ci = ∑ (x i − x )
6
i =1
5
4
Mean shift = 1σ
3
Slope cause by
mean shift Δμ
2
1
49
47
45
43
41
39
37
35
33
31
29
27
25
23
21
19
17
15
13
11
9
7
5
3
1
0
-1
Manufacturing
50
Control Limits for CUSUM
• Significance of Slope Changes?
– Detecting Mean Shifts
• Use of v-mask
– Slope Test with Deadband
Upper decision line
⎛1 − β⎞
d = ln⎜
⎟
δ ⎝ α ⎠
2
δ=
θ
Δx
σx
⎛ Δx ⎞
θ = tan ⎜ ⎟
⎝ 2k ⎠
−1
d
Lower decision line
Manufacturing
where
k = horizontal scale
factor for plot
51
Use of Mask
8
7
6
5
θ=tan-1(Δμ/2k)
k=4:1; Δμ=0.25 (1σ)
tan(θ) = 0.5 as plotted
4
3
2
1
49
47
45
43
41
39
37
35
33
31
29
27
25
23
21
19
17
15
13
11
9
7
5
3
1
0
-1
Manufacturing
52
An Alternative
• Define the Normalized Statistic
Zi =
Xi − μ x
σx
• And the CUSUM statistic
Which has an
expected mean of
0 and variance of 1
t
Si =
∑Z
i =1
t
i
Which has an
expected mean of
0 and variance of 1
Chart with Centerline =0 and Limits = ±3
Manufacturing
53
Example for Mean Shift = 1σ
5
Normalized CUSUM
4
3
2
Mean Shift = 1
47
45
43
41
39
37
35
33
31
29
27
25
23
21
19
17
15
13
11
9
7
5
3
0
1
1
σ
-1
Manufacturing
54
Tabular CUSUM
• Create Threshold Variables:
Ci = max[0, xi − ( μ 0 + K ) + Ci −1 ] Accumulates
+
+
Ci = max[0,( μ 0 − K ) − xi +
−
deviations
Ci −1 ] from the
mean
−
K= threshold or slack value for
accumulation
Δμ
K=
2
typical
Δμ = mean shift to detect
H : alarm level (typically 5σ)
Manufacturing
55
Threshold Plot
6
μ
0.495
σ
0.170
k=δμ/2
0.049
h=5σ
0.848
5
C+ C-
4
C+
CH threshold
3
2
1
0
1
3
Manufacturing
5
7
9
11 13 15 17 19 21 23 25 27 29 31 33 35 37 39 41 43 45 47 49
56
Alternative Charts Summary
• Noisy Data Need Some Filtering
• Sampling Strategy Can Guarantee
Independence
• Linear Discrete Filters have Been Proposed
– EWMA
– Running Integrator
• Choice Depends on Nature of Process
Manufacturing
57
Summary of SPC
• Consider Process a Random Process
– Can never predict precise value
• Model with P(x) or p(x)
– Assume p(x,t) = p(x)
• Shewhart Hypothesis
– In-control = purely random output
• Normal, independent stationary
• “The best you can do!”
– Not in-control
• Non-random behavior
• Source can be found and eliminated
Manufacturing
58
The SPC Hypothesis
0.9
0.8
0.7
Process
0.6
Y
0.5
0.4
0.3
0.2
0.1
0
1
2
3
4
5
6
7
8
9
10
11
Sample Number
12
13
14
15
16
17
18
19
20
In-Control
...
...
p(y)
Not
In-Control
Manufacturing
59