Lecture 9: Robust Design
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Robust Design
A New Definition of Quality.
The Signal-to-Noise Ratio.
Orthogonal Arrays.
Lecture 9: Robust Design
Spanos
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Robust Design
A New Definition of Quality.
The Signal-to-Noise Ratio.
Orthogonal Arrays.
Lecture 9: Robust Design
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The Taguchi Philosophy
Quality is related to the total loss to society due to
functional and environmental variance of a given product
Quality is related to the total loss to society due to
functional and environmental variance of a given product
Taguchi's method focuses on Robust Design through use of:
• S/N Ratio to quantify quality
• Orthogonal Arrays to investigate quality
Taguchi starts with a new definition of Quality:
Lecture 9: Robust Design
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Meeting the specs vs. hitting the target
better quality
worse quality
m
m-5
m+5
Lecture 9: Robust Design
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Quadratic Loss Function:
L(y) = k (y - m)
2
Fig 2.3 pp 18 from
Quality Engineering Using Robust Design
by Madhav S. Phadke
Prentice Hall 1989
Lecture 9: Robust Design
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Quadratic Loss Function on Normal Distribution
Average quality loss due to µ and
σ:
Fig 2.5 pp 26
E(Q) = k [(µ-m) +
σ ]
2
2
Lecture 9: Robust Design
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Exploiting non-linearity:
Fig 2.6 pp 28
Lecture 9: Robust Design
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Parameters are classified according to function:
Fig 2.7 pp 30
Lecture 9: Robust Design
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Orthogonal Arrays
b = (X
T
X)
-1
X
T
y
V(b) = (X
T
X)
-1
σ
2
During Regression Analysis, an orthogonal arrangement of
the experiment gave us independent model parameter
estimates:
Orthogonal arrays have the same objective:
For every two columns all possible factor combinations
occur equal times.
L
4
(2
3
) L
9
(3
4
) L
12
(2
11
) L
18
(2
1
x 3
7
)
Lecture 9: Robust Design
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Simple CVD experiment for defect reduction
max n = -10 log (MSQ def)
10
Lecture 9: Robust Design
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Simple CVD experiment for defect reduction (cont)
Using the L
9
orthogonal array:
Lecture 9: Robust Design
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Estimation of Factor Effects (ANOM)
m = 1
9
η
1
+
η
2
+
η
3
+...+
η
9
m
A
1
= 1
3
η
1
+
η
2
+
η
3
m
A
2
= 1
3
η
4
+
η
5
+
η
6
m
A
3
= 1
3
η
7
+
η
8
+
η
9
...
m
B
2
= 1
3
η
2
+
η
5
+
η
8
...
m
D
3
= 1
3
η
3
+
η
4
+
η
8
η A
i
,B
j
,C
k
,D
l
= μ+α
i
+β
j
+γ
k
+δ
l
+e
α
i
= 0
Σ
β
i
= 0
Σ
γ
i
= 0
Σ
δ
i
= 0
Σ
Lecture 9: Robust Design
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Analysis of CVD defect reduction experiment
Fig 3.1 pp 46
Tab 3.4 pp 55
Lecture 9: Robust Design
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ANOVA for CVD defect reduction experiment
Grand total sum of squares:
η
i
2
= 19,425 (dB)
2
Σ
i=1
9
Total sum of squares:
η
i
-m
2
= 3,800 (dB)
2
Σ
i=1
9
Sum of squares due to mean:
m
2
= 15,625 (dB)
2
Σ
i=1
9
Sum of squares due to error:
e
i
2
= ??? (dB)
2
Σ
i=1
9
Sum of squares due to A: 3
m
A
i
-m
2
= 2,450 (dB)
2
Σ
i=1
3
Lecture 9: Robust Design
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ANOVA for CVD defect reduction experiment (cont)
Lecture 9: Robust Design
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Estimation of Error Variance
The experimental error is estimated from the ANOVA
residuals.
It is then used to estimate the error of the effects and to
determine their significance at the 5% level.
Lecture 9: Robust Design
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Confirmation Experiment
Once the optimum choice has been made, it is tested by
performing a confirmation run.
This run is used to "validate" the model as well as confirm
the improvements in the process.
Variance of prediction (for the model)
σ
pred
2
=
σ
e
2
n
0
+
σ
e
2
n
r
This gives us +/-2
σ limits on the confirmation experiment.
1
n
0
σ
e
2
= 1
n +
1
n
A
1
- 1
n +
1
n
B
1
- 1
n σ
e
2
Lecture 9: Robust Design
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The additive model
Fig 3.3 pp 63, enlarged 120%
Since we assumed additive model, we must make sure
that there are no interactions:
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Example: Large CVD experiment.
Objectives:
a) reduce defects n = -10 log (MSQ Def)
b) maximize S/N of rate n'= 10 log (µ /
σ )
c) adjust poly thickness to a 3600 Å target.
10
10
2
2
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Choosing the Control Factors
Tab 4.6-7 pp 88-90
Lecture 9: Robust Design
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Using the L
18
orthogonal array...
Tab 4.3 pp 78, enlarged 120%
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Data summary for large CVD experiment:
Tab 4.5 pp 85, enlarged 120%
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Data analysis for large CVD experiment (cont)
Fig 4.5 pp 86, enlarged 120%
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ANOVA table for large CVD experiment:
η
Lecture 9: Robust Design
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ANOVA table for large CVD experiment :
η’
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ANOVA tables for large CVD experiment:
η’’
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Combined Prediction Using the Additive Model
Tab 4.6-7 pp 88-90
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Verification for large CVD experiment
Tab 4.10-11 pp 92
for further reading: Quality Engineering Using Robust Design by Madhav S. Phadke
Prentice Hall 1989
Lecture 9: Robust Design
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“Inner” and “Outer” Arrays
•
Often one want to improve performance based on some “control”
factors, in the presence of some “noise” factors.
•
Two arrays are involved: the inner array explores the “control”
factors, and the entire experiment is repeated across an array of the
noise factors.
•
Inner arrays are typically orthogonal designs
•
Outer arrays are typically small, 2-level fractional factorial designs.
Lecture 9: Robust Design
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Why use S/N Ratios?
• They lead to an optimum through a monotonic function.
• They help improve additivity of the effects.
⎟
⎠
⎞
⎜
⎝
⎛
−
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
−
=
∑
∑
i
i
STB
i
i
LTB
Y
n
N
S
Y
n
N
S
s
Y
N
S
2
2
2
2
1
log
10
1
1
log
10
log
10
Nominal is best:
Larger is better:
Smaller is better:
Lecture 9: Robust Design
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Taguchi vs. RSM
Taguchi
RSM
Small number of runs
Explicit control of Interactions
Engineering Intuition
Statistical Intuition
“Complete” package
Training Issues
Additive Models
More General Models
Orthogonal Arrays
Fractional Factorials
A “Philosophy”
A Tool
Lecture 9: Robust Design
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Design of Experiments
Comparison of Treatments
Blocking and Randomization
Reference Distributions
ANOVA
MANOVA
Factorial Designs
Two Level Factorials
Blocking
Fractional Factorials
Regression Analysis
Robust Design
Analysis
Modeling