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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 #14
Aliasing and Higher Order Models
April 3, 2008
Manufacturing
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1
Outline
• Last Time
– Full Factorial Models
– Experimental Design
• Blocks and Confounding
• Single Replicate Designs
• Today
–
–
–
–
Fractional Factorial Designs
Aliasing Patterns
Implications for Model Construction
Process Optimization using DOE
Manufacturing
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2
Fractional Factorial Experiments
• What if we do less than full factorial 2k?
• Example: run < 23 experiments for 3 inputs
– From regression model for 3 inputs:
y = β 0 + β1 x1 + β 2 x2 + β12 x1 x2 + β 3 x 3
+ β13 x1 x 3 + β 23 x2 x 3 + β123z x1 x2 x 3 + ε
– We will not be able to find all 8 coefficients
Manufacturing
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23-1 Experiment
• Consider doing 4 experiments instead of 8; e.g.:
x1 x2 x1 x2
1 −1 −1 +1
2 +1 −1
−1
3 −1 +1
4 +1 +1
−1
+1
Manufacturing
• This is a 22 array
• Could also be for 3
inputs if we define
x3= x1x2
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23-1 Experiment
1
2
3
4
x1
−1
+1
−1
+1
x2
x3
−1
−1
+1
+1
+1
−1
−1
+1
But now we can only define
4 coefficients in the model:
e.g.:
)
y = β 0 + β1 x1 + β 2 x2 + β 3 x 3
i.e. no interaction terms
Manufacturing
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23-1 Experiment
Or we could choose other terms:
)
y = β 0 + β1 x1 + β 2 x2 + β13 x1 x 3
or:
)
y = β 0 + β1 x1 + β12 x1 x2 + β 3 x 3
or:
…
Manufacturing
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Confounding / Aliasing
• We actually have the following:
)
y=β
0
+ β '1 z1 + β '2 z2 + β '3 z3
• where the z variable represent sums of the various input
terms, e.g.
z1 = x1 x2 + x 3
z2 = x1 + x2 x 3 L
• where the specific choice of the experimental array
determines what these sums are
Manufacturing
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Confounding / Aliasing
23 Array: (Our X matrix)
Test
I
A
B
AB
C
AC
BC
ABC
(1)
1
-1
-1
1
-1
1
1
-1
a
1
1
-1
-1
-1
-1
1
1
b
1
-1
1
-1
-1
1
-1
1
ab
1
1
1
1
-1
-1
-1
-1
c
1
-1
-1
1
1
-1
-1
1
ac
1
1
-1
-1
1
1
-1
-1
bc
1
-1
1
-1
1
-1
1
-1
abc
1
1
1
1
1
1
1
1
Manufacturing
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Confounding / Aliasing
Consider upper half:
Test
I
A
B
AB
C
AC
BC
ABC
(1)
1
-1
-1
1
-1
1
1
-1
a
1
1
-1
-1
-1
-1
1
1
b
1
-1
1
-1
-1
1
-1
1
ab
1
1
1
1
-1
-1
-1
-1
c
1
-1
-1
1
1
-1
-1
1
ac
1
1
-1
-1
1
1
-1
-1
bc
1
-1
1
-1
1
-1
1
-1
abc
1
1
1
1
1
1
1
1
Look at columns for C - no change at all! or C = -I
Also AC = -A and BC = -B, and ABC = -AB
Manufacturing
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Confounding / Aliasing
Test
I
A
B
AB
C
AC
BC
ABC
(1)
1
-1
-1
1
-1
1
1
-1
a
1
1
-1
-1
-1
-1
1
1
b
1
-1
1
-1
-1
1
-1
1
ab
1
1
1
1
-1
-1
-1
-1
c
1
-1
-1
1
1
-1
-1
1
ac
1
1
-1
-1
1
1
-1
-1
bc
1
-1
1
-1
1
-1
1
-1
abc
1
1
1
1
1
1
1
1
ContrastA=[ -(1)+a-b+ab]
AC is an alias of A
ContrastAC=[ (1)-a+b-ab]
Note that alias of A =A*(-C)
Defining Relation I = -C
Manufacturing
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Choice of Design?
• Aliases
– Must have one of the pair assumed negligible
(“sparsity of effects”)
• Balance/Orthogonality
– Sufficient excitation of inputs
– Enable short-cut estimation of model effects and
model coefficients
Manufacturing
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Balance and Orthogonality
Test
I
A
B
AB
C
AC
BC
ABC
(1)
1
-1
-1
1
-1
1
1
-1
a
1
1
-1
-1
-1
-1
1
1
b
1
-1
1
-1
-1
1
-1
1
ab
1
1
1
1
-1
-1
-1
-1
c
1
-1
-1
1
1
-1
-1
1
ac
1
1
-1
-1
1
1
-1
-1
bc
1
-1
1
-1
1
-1
1
-1
abc
1
1
1
1
1
1
1
1
Note: All columns have equal number of + and - signs (Balance)
Sum of product of any two columns = 0 (Orthogonality)
-All combinations occur the same number of times
Manufacturing
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Balance/Orthogonality in 23-1
Test
I
A
B
C
AB
AC
BC
ABC
1
1
-1
-1
-1
1
1
1
-1
a
1
1
-1
-1
-1
-1
1
1
b
1
-1
1
-1
-1
1
-1
1
c
1
1
1
-1
1
-1
-1
-1
ab
1
-1
-1
1
1
-1
-1
1
ac
1
1
-1
1
-1
1
-1
-1
bc
1
-1
1
1
-1
-1
1
-1
abc
1
1
1
1
1
1
1
1
• A and B are balanced; C is not
• A, B and C are orthogonal
Manufacturing
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Better Subset – Balanced/Orthogonal
Test
I
A
B
C
AB
AC
BC
ABC
1
1
-1
-1
-1
1
1
1
-1
a
1
1
-1
-1
-1
-1
1
1
b
1
-1
1
-1
-1
1
-1
1
c
1
1
1
-1
1
-1
-1
-1
ab
1
-1
-1
1
1
-1
-1
1
ac
1
1
-1
1
-1
1
-1
-1
bc
1
-1
1
1
-1
-1
1
-1
abc
1
1
1
1
1
1
1
1
With this array:
- balance for A, B, C
- all but ABC are orthogonal
e.g. aliases of A:
A*ABC=A*I
A*A = I
BC aliased with A
Aliases:
A BC
B AC
C AB
- defining relation I=ABC
I ABC
Manufacturing
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Design Resolution
• Resolution III
– No main aliases with other main effects
– Main - interaction aliases
• Resolution IV
– No alias between main effects and 2 factor effects,
but others exist
• Resolution V
– No main and no 2 factor aliases ….
Manufacturing
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Design Resolution
Test
I
A
B
C
AB
AC
BC
ABC
1
1
-1
-1
-1
1
1
1
-1
a
1
1
-1
-1
-1
-1
1
1
b
1
-1
1
-1
-1
1
-1
1
c
1
1
1
-1
1
-1
-1
-1
ab
1
-1
-1
1
1
-1
-1
1
ac
1
1
-1
1
-1
1
-1
-1
bc
1
-1
1
1
-1
-1
1
-1
abc
1
1
1
1
1
1
1
1
With this array:
- balance for A, B, C
- all but A B C are orthogonal
- defining relation I=ABC
23-1III
Manufacturing
e.g. aliases of A:
A*ABC=A*I
A*A = I
BC aliased with A
Main effects aliased
with interactions only
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Aliases:
A BC
B AC
C AB
I ABC
16
Smaller Fraction 2k-p
• p=1
• p=2
• p
Manufacturing
1/2 fraction
1/4 fraction
1/2p
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A Different Fraction
Consider I = AC
Test
I
A
B
AB
C
AC
BC
ABC
(1)
1
-1
-1
1
-1
1
1
-1
a
1
1
-1
-1
-1
-1
1
1
b
1
-1
1
-1
-1
1
-1
1
ab
1
1
1
1
-1
-1
-1
-1
c
1
-1
-1
1
1
-1
-1
1
ac
1
1
-1
-1
1
1
-1
-1
bc
1
-1
1
-1
1
-1
1
-1
abc
1
1
1
1
1
1
1
1
Manufacturing
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A Different Fraction
Consider I = AC
Test
I
A
B
AB
C
AC
BC
ABC
(1)
1
-1
-1
1
-1
1
1
-1
b
1
-1
1
-1
-1
1
-1
1
ac
1
1
-1
-1
1
1
-1
-1
abc
1
1
1
1
1
1
1
1
I=AC Aliases
Balance?
A with C
Orthogonality?
B with ABC
AB with BC
Manufacturing
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How Decide What Aliasing To Choose?
• Prior knowledge of process
• Rules of thumb
– Sparsity of effects
– Hierarchy of effects
– Inheritance of effects
Manufacturing
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Sparsity of Effects
• An experimenter may list a
large number of effects for
consideration
• A small number of effects
usually explain the majority of
the variance
A
B
C
D
Courtesy of Prof. Dan Frey
Manufacturing
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Hierarchy
• Main effects are usually
more important than twofactor interactions
• Two-way interactions are
usually more important than
three-factor interactions
A
AB
B
C
AC AD
D
BC BD CD
• And so on
ABC
ABD
ACD
BCD
ABCD
Courtesy of Prof. Dan Frey
Manufacturing
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Inheritance
• Two-factor interactions are
most likely when both
participating factors
(parents?) are strong
• Two-way interactions are
least likely when neither
parent is strong
• And so on
A
AB AC
ABC
B
AD
ABD
C
D
BC
BD
ACD
BCD
CD
ABCD
Courtesy of Prof. Dan Frey
Manufacturing
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Design Resolution
• Resolution III
2
3−1
III
I = ABC
– No main aliases with other main effects
– Main - interaction aliases
• Resolution IV
2
4 −1
IV
I = ABCD
– No alias between main effects and 2 factor effects,
but others exist
• Resolution V
2
5 −1
V
I = ABCDE
– No main and no 2 factor aliases ….
Manufacturing
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24-2
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
Manufacturing
A
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
C
-1
-1
-1
-1
1
1
1
-1
-1
-1
-1
-1
1
1
1
1
D
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
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Four Main Effects
Four tests?
Suppose we want to
alias A with BCD and
ABC
What are the defining
relations?
25
24-2
Suppose we want to alias
A with BCD and ABC
-1
a
b
ab
c
ac
bc
abc
d
ad
bd
cd
abd
acd
bcd
abcd
I
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
A
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
-1
1
B
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
AB
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
1
-1
-1
1
C
-1
-1
-1
-1
1
1
1
-1
-1
-1
-1
-1
1
1
1
1
AC
1
-1
1
-1
-1
1
-1
-1
1
-1
1
-1
-1
1
-1
1
BC
1
1
-1
-1
-1
-1
1
-1
1
1
-1
-1
-1
-1
1
1
ABC
-1
1
1
-1
1
-1
-1
-1
-1
1
1
-1
1
-1
-1
1
D
-1
-1
-1
-1
-1
-1
-1
-1
1
1
1
1
1
1
1
1
AD
1
-1
1
-1
1
-1
1
-1
-1
1
-1
1
-1
1
-1
1
BD
1
1
-1
-1
1
1
-1
-1
-1
-1
1
1
-1
-1
1
1
CD
1
1
1
1
-1
-1
-1
1
-1
-1
-1
-1
1
1
1
1
ABD
-1
1
1
-1
-1
1
1
-1
1
-1
-1
1
1
-1
-1
1
ACD
-1
1
-1
1
1
-1
1
1
1
-1
1
-1
-1
1
-1
1
BCD ABCD
-1
1
-1
-1
1
-1
1
1
1
-1
1
1
-1
1
1
1
1
-1
1
1
-1
1
-1
-1
-1
1
-1
-1
1
-1
1
1
Run only (1), bc, ad and abcd
A BCD = I
A ABC = BC = I
Manufacturing
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24-2
Suppose we want to alias
A with BCD and ABC
(1)
bc
ad
abcd
I
1
1
1
1
A
-1
-1
1
1
B
-1
1
-1
1
AB
1
-1
-1
1
C
-1
1
-1
1
AC
1
-1
-1
1
BC
1
1
1
1
ABC
-1
-1
1
1
D
-1
-1
1
1
AD
1
1
1
1
BD
1
-1
-1
1
CD
1
-1
-1
1
ABD
-1
1
-1
1
ACD
-1
1
-1
1
BCD ABCD
-1
1
-1
1
1
1
1
1
Defining Relations
A BCD = I
I=BC
A ABC = BC = I ( NB AD = I also)
I=AD
Aliases?
I=ABCD
A - ABC
B-C
C - ABD
D - ABC
A-D
B - ABD
C - ACD
D - BCD
A - BCD
B - ACD
Manufacturing
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Outline
• Fractional Factorial Designs
• Aliasing Patterns
• Implications for Model Construction
• Process Optimization using DOE
Manufacturing
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28
Consider Higher Order Model
y = β0 + β1 x1 + β21x1
2
Quadratic Model
Now we need all 3 tests
12.00
y = -0.5258x
11.00
2
+ 4.8551x +
10.00
SSResiduals
=SSwithin
y
9.00
8.00
7.00
6.00
5.00
4.00
0
Manufacturing
1
2
3
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4
5
6
29
General Quadratic Equation
k
k
k
i =1
i =1
k
ηm = β 0 + ∑ βi xi m + ∑ β 2i x 2im + ∑ ∑ β ij x im x j m + h.o.t. + ε m
j =1 i =1
j <i
32 Problem
ŷ = β0 + β1 x1 + β2 x2 + β12 x1 x2 + β x + β22 x2
2
11 1
2
+ β21 x1 x2 + β12 x1 x2 + β x x
2
2
2
2
222 1 2
• How many levels for each input?
Manufacturing
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30
Quadratic Solution
• Same as before with
matrix equation: η=Xβ+ε
η1 1 x11
η2 1 x12
η2 = 1 x13
M
ηN
M M
1 x11
x 21
x 22
x 23
M
x11
2
11
2
12
2
13
x
x
x
M
x112
2
11
2
22
2
23
x
x
x
M
x112
x11x 21
x12 x 22
x11x 21
M
x11x 21
β0
ε1
L β1
ε2
L β2
L β11 + ε 2
M
O β 22
L β12 ε N
M
Manufacturing
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31
Experimental Design for Quadratic:
• Full factorial 3k
– Three levels per test
• Central Composite Design
– adding to 2x2 design
• Partial Factorials and Aliases
Manufacturing
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32
Consider a Quadratic Model w/Interaction
• Includes linear terms, quadratic terms and all
first and second-order interactions
• =3k
N
Manufacturing
k
No Interactions
Full Model
1
3
3
2
5
9
3
7
27
4
9
81
5
11
243
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33
32 Full Factorial – Quadratic Model
ŷ = β0 + β1x1 + β2 x2 + β12 x1x2 + β11x12 + β22 x22 + β21x12 x2 + β12 x1x22 + β222 x12 x22
(1)
A
B
AB
A2
B2
A2B
B2A
A2B2
y1
1
-1
-1
1
1
1
-1
-1
1
y2
1
0
-1
0
0
1
0
0
0
y3
1
1
-1
-1
1
1
-1
1
1
y4
1
-1
0
0
1
0
0
0
0
y5
1
0
0
0
0
0
0
0
0
y6
1
1
0
0
1
0
0
0
0
y7
1
-1
1
-1
1
1
1
-1
1
y8
1
0
1
0
0
1
0
0
0
y9
1
1
1
1
1
1
1
1
1
Manufacturing
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34
Which Partial Fraction?
ŷ = β0 + β1x1 + β2 x2 + β12 x1x2 + β x + β22 x2
2
11 1
2
(1)
A
B
AB
A2
B2
A2B
B2A
A2B2
y1
1
-1
-1
1
1
1
-1
-1
1
y2
1
0
-1
0
0
1
0
0
0
y3
1
1
-1
-1
1
1
-1
1
1
y4
1
-1
0
0
1
0
0
0
0
y5
1
0
0
0
0
0
0
0
0
y6
1
1
0
0
1
0
0
0
0
y7
1
-1
1
-1
1
1
1
-1
1
y8
1
0
1
0
0
1
0
0
0
y9
1
1
1
1
1
1
1
1
1
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35
Which Partial Fraction?
(1)
A
B
AB
A2
B2
A21
B21
AB22
y1
1
-1
-1
1
1
1
-1
-1
1
y2
1
0
-1
0
0
1
0
0
0
y3
1
1
-1
-1
1
1
-1
1
1
y4
1
-1
0
0
1
0
0
0
0
y5
1
0
0
0
0
0
0
0
0
y6
1
1
0
0
1
0
0
0
0
y7
1
-1
1
-1
1
1
1
-1
1
y8
1
0
1
0
0
1
0
0
0
y9
1
1
1
1
1
1
1
1
1
Manufacturing
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Which Partial Fraction?
(1)
A
B
AB
A2
B2
A21
B21
AB22
y1
1
-1
-1
1
1
1
-1
-1
1
y2
1
0
-1
0
0
1
0
0
0
y3
1
1
-1
-1
1
1
-1
1
1
y4
1
-1
0
0
1
0
0
0
0
y5
1
0
0
0
0
0
0
0
0
y6
1
1
0
0
1
0
0
0
0
y7
1
-1
1
-1
1
1
1
-1
1
y8
1
0
1
0
0
1
0
0
0
y9
1
1
1
1
1
1
1
1
1
+1
0
-1
(1)
A
B
AB
A2
B2
A21
B21
AB22
y1
1
-1
-1
1
1
1
-1
-1
1
y2
1
0
-1
0
0
1
0
0
0
y3
1
1
-1
-1
1
1
-1
1
1
y4
1
-1
0
0
1
0
0
0
0
y5
1
0
0
0
0
0
0
0
0
y6
1
1
0
0
1
0
0
0
0
y7
1
-1
1
-1
1
1
1
-1
1
y8
1
0
1
0
0
1
0
0
0
y9
1
1
1
1
1
1
1
1
1
Manufacturing
x2
-1
0
+1
x1
-1
0
+1
x1
x2
+1
0
-1
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37
Which Partial Fraction?
(1)
A
B
AB
A2
B2
A21
B21
AB22
y1
1
-1
-1
1
1
1
-1
-1
1
y2
1
0
-1
0
0
1
0
0
0
y3
1
1
-1
-1
1
1
-1
1
1
y4
1
-1
0
0
1
0
0
0
0
y5
1
0
0
0
0
0
0
0
0
y6
1
1
0
0
1
0
0
0
0
y7
1
-1
1
-1
1
1
1
-1
1
y8
1
0
1
0
0
1
0
0
0
y9
1
1
1
1
1
1
1
1
1
Manufacturing
(1)
A
B
AB
A2
B2
y1
1
-1
-1
1
1
1
y2
1
0
-1
0
0
1
y3
1
1
-1
-1
1
1
y4
1
-1
0
0
1
0
y5
1
0
0
0
0
0
y7
1
-1
1
-1
1
1
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x2
+1
0
-1
-1
0
+1
x1
38
Quadratic Solution
ŷ = β 0 + β1 x1 + β 2 x2 + β11 x12 + β 22 x2 2 + β12 x1 x2
y1
y2
1 x1
1
y3 1
=
y4
1
y5 1
y6 1
Manufacturing
x1
x1
x1
x1
x1
x2
x12
x2 2
x1 x2 β 0
x2
x2
x2
x2
x2
x12
2
x1
2
x1
2
x1
x12
x2 2
2
x2
2
x2
2
x2
x2 2
x1 x2
x1 x2
x1 x2
x1 x2
x1 x2
2.830J/6.780J/ESD.63J
β1
β2
β11
β 22
β12
y = Xβ
β = X −1 y
39
A Quadratic Surface
5
0
-5
-10
-15
-20
1
0.6
-25
0.2
-0.2
-0.6
-1
y = 1 + 5x1 + 5 x 2 + x1 x2 − 10 x1 − 5x 2
2
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40
A “Standard” 32 Full Factorial Design
Test
1
2
3
4
5
6
7
8
9
Manufacturing
x1
-1
0
1
-1
0
1
-1
0
1
X2
-1
-1
-1
0
0
0
1
1
1
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41
Central Composite Design
• Consider the case:
– First Experiment is 22 with 4 tests
– Model is shown to have poor fit
• High SS Quad for intermediate point
– Decide to go to Quadratic
– Not Sure of Shape of Surface
Manufacturing
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42
Central Composite Design
x2
+1
=1.414
-1
0
+1
• Add 5 additional points:
• One at center
• One equidistant
from center along
each axis
x1
-1
Manufacturing
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43
Central Composite
Test
1
2
3
4
5
6
7
8
9
Manufacturing
x1
-1
+1
-1
+1
0
0
1.414
0
-1.414
X2
-1
-1
+1
+1
0
1.414
0
-1.414
0
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original tests
additional tests
44
Outline
• Fractional Factorial Designs
• Aliasing Patterns
• Implications for Model Construction
• Process Optimization using DOE
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45
Process Optimization
• Create an Objective Function
“J”
Minimize or Maximize
max J
x
min J
x
J=J(factors) ; J(x); J(α)
Adjust J via factors
with constraints, such as….
Manufacturing
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46
Methods for Optimization
• Analytical Solutions
– ∂y/ ∂x = 0
• Gradient Searches
– Hill climbing (steepest ascent/descent)
– Local min or max problem
– Excel solver given a convex function
Manufacturing
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47
Basic Optimization Problem
y0 = J0
y (or J)
x0
Manufacturing
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x
48
3D Problem
5
z0
0
-5
-10
-15
-20
1
0.6
-25
0.2
-0.2
-0.6
x20
Manufacturing
x10
-1
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49
Analytical
∂y(x)
=0
∂x
y
x
• Need Accurate y(x)
– Analytical Model
– Dense x increments in Experiment
• Difficult with Sparse Experiments
– Easy to missing optimum
Manufacturing
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50
Sparse Data Procedure
y
β1
x-
x+
x
• Linear models with small increments
• Move along desired gradient
• Near zero slope change to quadratic model
Manufacturing
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51
Extension to 3D
1
5
0.8
0.6
0
0.4
0.2
-5
0
-10
-0.2
-0.4
-15
-0.6
-20
1
-0.8
0.6
-1
-25
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Manufacturing
-1
1
0.8
0.6
-0.6
0.4
0.2
0
-0.2
-0.2
-0.4
-0.6
-0.8
-1
0.2
-1
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Linear Model Gradient Following
S21
S20
S19
3.5
S18
S17
S16
3.0
S15
S14
2.5
y
S13
y
2.0
S12
S11
S10
1.5
S9
S8
1.0
S7
S6
0.5
S5
S4
S19
0.0
S3
S16
S2
S13
S7
x2
S1
x1
S10
x1
S4
S1
x2
x1
yˆ = β 0 + β1 x1 + β 2 x 2 + β 12 x1 x2
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53
Steepest Descent
yˆ = β 0 + β1 x1 + β 2 x 2 + β 12 x1 x2
S20
S19
S18
S17
S16
S15
S14
y
S13
S12
S11
S10
S9
g2
Make changes in x1and x2 along G
Δx 2 =
Manufacturing
gx1
gx 2
G
S8
S7
S6
S5
S4
x2
S3
g1
x1
∂y
gx1 =
= β1 + β12 x 2
∂x1
∂y
gx 2 =
= β 2 + β 12 x1
∂x 2
S21
S2
S1
x1
Δx1
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54
Experimental Optimization
• WHY NOT JUST PICK BEST POINT?
• Why not optimize on-line?
– Skip the Modeling Step!
• Adaptive Methods
– Learn how best to model as you go.
• e.g. Adaptive OFACT
Manufacturing
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55
EVOP
• Evolutionary Operation
x2
y2
y3
y0
y4
±δx1
±δx2
y1
• Pick “best” yi
• Re-center process
• Do again.
x1
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56
Confirming Experiments
• Checking Intermediate points
•
•
•
•
15
10
5
Y
0
-5
0
+1
-10
-1
X1
+1
-1-0.2
Data only at corners
Test at interior point
Evaluate error
Consider Central
Composite?
X2
0.6
• Rechecking-1the “optimum”
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A Procedure for DOE/Optimization
• Study Physics of Process
– Define Important Inputs
– Intuition about model
– Limits on inputs
• Define Optimization Penalty Function
– J=f(x)
max J
x
min J
x
For us, x = u or α
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Procedure
• Identify model (linear, quadratic, terms to
include)
• Define inputs and ranges
• Identify “noise” parameters to vary if possible
(Δα’s)
• Perform Experiment
– Appropriate order
• randomization
• blocking against nuisance or confounding effects
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Procedure
• Solve for ß’s
• Apply ANOVA
– Data significant?
– Terms significant?
– Lack of Fit Significant?
• Drop Insignificant Terms
• Add Higher Order Terms as needed
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Procedure
• Search for Optimum
– Analytically
– Piecewise
– Continuously
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Procedure
• Find Optimum value x*
• Perform Confirming experiment
– Test Model at x*
– Evaluate error with respect to model
– Test hypothesis that
Manufacturing
y(x*) = yˆ(x*)
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Procedure
• If hypothesis fails
– Consider new ranges for inputs
– Consider higher order model as needed
– Boundary may be optimum!
Manufacturing
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Summary
•
•
•
•
Fractional Factorial Designs
Aliasing Patterns
Implications for Model Construction
Process Optimization using DOE
Manufacturing
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64