Transcript Reduced Factorial Design (RFD)

Process exploration by Fractional
Factorial Design (FFD)
Number of Experiments
Factorial design (FD) with variables at 2
levels.
The number of experiments = 2m,
m=number of variables.
m=3 : 8 experiments
m=4 : 16 experiments
m=7 : 128 experiments
The relationship between the number of
variables and the required number of
experiments
140
Number of experiments
120
100
80
60
40
20
0
1
2
3
4
5
Number of variables
6
7
Full design vs. Fractional design
First order model with 7 input variables:
y  b0  b1x1  b2 x2  b3 x3  b4 x4  b5 x5  b6 x6  b7 x7  
8 parameters have to be decided.
27 Factorial Design
 128 experiments
27-4 Fractional Factorial Design
 8 experiments
(example of saturated design)
Screening designs
• 27-4 Reduced Factorial Design
7 factors in 8 experiments
• 211 Plackett-Burman
11 factors in 12 experiments
7-4
2
Fractional Factorial Design
[x4]=[x1] [x2]
[x5]=[x1] [x3]
[x6]=[x2] [x3]
[x7]=[x1] [x2] [x3]
Generators:
[xi]2 = [I]
[xi] [xj] = [xj] [xi]
[I]=[x1] [x2] [x4]
[I]=[x1] [x3] [x5]
[I]=[x2] [x3] [x6]
[I]=[x1] [x2] [x3] [x7]
Defining relation
Def.: The generators + all possible combination products
27-4:
4
generators
6
products of 2 generators
4
products of 3 generators
1
products of 4 generators
15
Defining relation for
7-4
2
FFD
[I] = [x1] [x2] [x4] = [x1] [x3] [x5] = [x2] [x3] [x6]
= [x1] [x2] [x3] [x7] = [x2] [x4] [x3] [x5] = [x1] [x4] [x3] [x6]
= [x4] [x3] [x7] = [x1] [x2] [x5] [x6] = [x2] [x5] [x7]
= [x1] [x6] [x7] = [x4] [x5] [x6] = [x2] [x4] [x6] [x7]
= [x1] [x4] [x5] [x7] = [x3] [x5] [x6] [x7]
= [x1] [x2] [x3] [x4] [x5] [x6] [x7]
The confounding pattern appears by multiplying the defining
relation with each of the variables.
Process capacity
Variable
-1
+1
X1
Water Source
City reservoir
Private well
X2
Raw material
Local
Other
X3
Temperature
Low
High
X4
Recycling
Yes
No
X5
Caustic Soda
Fast
Slow
X6
Filter
New
Old
X7
Waiting time
Short
Long
Box and Hunter, 1961, Technometrics 3, p. 311
The Design Matrix
Response Variation
Estimates of the effects
1
2
3
4
5
6
7
Effect
Estimate
X1+ X2X4 + X3 X5+ X6 X7
X2 + X1X4 + X3 X6+ X5 X7
X3 + X1X5 + X2 X6+ X4 X7
X4 + X1X2 + X3 X7+ X5 X6
X5 + X1X3 + X2 X7+ X4 X6
X6 + X1X7 + X2 X3+ X4 X5
X7 + X1X6 + X2 X5+ X3 X4
-5.4
-1.4
-8.3
1.6
-11.4
-1.7
0.26
Estimates of the effects
1
2
3
4
5
6
7
Effect
Estimate
X1+ X2X4 + X3 X5+ X6 X7
X2 + X1X4 + X3 X6+ X5 X7
X3 + X1X5 + X2 X6+ X4 X7
X4 + X1X2 + X3 X7+ X5 X6
X5 + X1X3 + X2 X7+ X4 X6
X6 + X1X7 + X2 X3+ X4 X5
X7 + X1X6 + X2 X5+ X3 X4
-5.4
-1.4
-8.3
1.6
-11.4
-1.7
0.26
Plausible interpretations
There are four likely combinations of
significant effects:
1. Variable X1, X3 and X5
2. Variable X1, X3 and the interaction X1X3
3. Variable X1, X5 and the interaction X1X5
4. Variable X3, X5 and the interaction X3X5
New experimental series
It is desirable to separate the 1- and 2- factor
effects.
A new 27-4-design
with a different set
of generators is
generated:
[x4]= - [x1] [x2]
[x5]= - [x1] [x3]
[x6]= - [x2] [x3]
[x7]= - [x1] [x2] [x3]
The Design Matrix
-new experimental series
Estimates of the effects
1
2
3
4
5
6
7
Effect
Estimate
X1- X2X4 - X3 X5- X6 X7
X2 - X1X4 - X3 X6- X5 X7
X3 - X1X5 - X2 X6- X4 X7
X4 - X1X2 - X3 X7- X5 X6
X5 - X1X3 - X2 X7- X4 X6
X6 - X1X7 - X2 X3- X4 X5
X7 - X1X6 - X2 X5- X3 X4
-1.3
-2.5
7.9
1.1
-7.8
1.7
-4.6
Estimates of the effects
1
2
3
4
5
6
7
Effect
Estimate
X1- X2X4 - X3 X5- X6 X7
X2 - X1X4 - X3 X6- X5 X7
X3 - X1X5 - X2 X6- X4 X7
X4 - X1X2 - X3 X7- X5 X6
X5 - X1X3 - X2 X7- X4 X6
X6 - X1X7 - X2 X3- X4 X5
X7 - X1X6 - X2 X5- X3 X4
-1.3
-2.5
7.9
1.1
-7.8
1.7
-4.6
Estimate of the effects
(by combining the two series)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Effect
Estimate
X1
X2
X3
X4
X5
X6
X7
X2X4 + X3 X5+ X6 X 7
X1X4 + X3 X6+ X5 X 7
X1X5 + X2 X6+ X4 X 7
X1 X2 + X3 X7 + X5 X 6
X1X3 + X2 X7+ X4 X 6
X1X7 + X2 X3+ X4 X 5
X1X6 + X2 X5+ X3 X 4
-3.3
-1.9
-0.2
1.4
-9.6
-0.03
-2.2
-2.1
0.6
-8.1
0.2
-1.8
-1.7
2.4
Estimate of the effects
(by combining the two series)
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Effect
Estimate
X1
X2
X3
X4
X5
X6
X7
X2X4 + X3 X5+ X6 X 7
X1X4 + X3 X6+ X5 X 7
X1X5 + X2 X6+ X4 X 7
X1 X2 + X3 X7 + X5 X 6
X1X3 + X2 X7+ X4 X 6
X1X7 + X2 X3+ X4 X 5
X1X6 + X2 X5+ X3 X 4
-3.3
-1.9
-0.2
1.4
-9.6
-0.03
-2.2
-2.1
0.6
-8.1
0.2
-1.8
-1.7
2.4
Contour plot
Filter time, y (min)
Interpretation
Slow
65.4
42.6
Caustic
Soda
Fast
68.5
City
Water
Reservoir source
78.0
Private
Well
i) Slow addition of NaOH
improves the response (shorten
the filtration time)
ii) The composition of the
water in the private wells (pH,
minerals etc.) is better than the
water from the city reservoir
with respect to the response
(tends to shorten the filtration
time)
Bottom line...
Univariate optimisation of the speed used for
adding NaOH!
This result would not have been obtained by a
univariate approach!