process exploration by fractional factorial design (ffd)
TRANSCRIPT
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
1 2 3 4 5 6 70
20
40
60
80
100
120
140
Number of variables
Num
be
r o
f exp
eri
me
nts
Full design vs. Fractional design
First order model with 7 input variables:
776655443322110 xbxbxbxbxbxbxbby
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 Design7 factors in 8 experiments
• 211 Plackett-Burman11 factors in 12 experiments
27-4 Fractional Factorial Design
[x4]=[x1] [x2]
[x5]=[x1] [x3]
[x6]=[x2] [x3]
[x7]=[x1] [x2] [x3]
[I]=[x1] [x2] [x4]
[I]=[x1] [x3] [x5]
[I]=[x2] [x3] [x6]
[I]=[x1] [x2] [x3] [x7]
[xi]2 = [I]
[xi] [xj] = [xj] [xi]
Generators:
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 27-4 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
Box and Hunter, 1961, Technometrics 3, p. 311
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
Estimates of the effects
Effect Estimate
1 X1+ X2X4 + X3 X5+ X6 X7 -5.4 2 X2 + X1X4 + X3 X6+ X5 X7 -1.4 3 X3 + X1X5 + X2 X6+ X4 X7 -8.3 4 X4 + X1X2 + X3 X7+ X5 X6 1.6 5 X5 + X1X3 + X2 X7+ X4 X6 -11.4 6 X6 + X1X7 + X2 X3+ X4 X5 -1.7 7 X7 + X1X6 + X2 X5+ X3 X4 0.26
Estimates of the effects
Effect Estimate
1 X1+ X2X4 + X3 X5+ X6 X7 -5.4 2 X2 + X1X4 + X3 X6+ X5 X7 -1.4 3 X3 + X1X5 + X2 X6+ X4 X7 -8.3 4 X4 + X1X2 + X3 X7+ X5 X6 1.6 5 X5 + X1X3 + X2 X7+ X4 X6 -11.4 6 X6 + X1X7 + X2 X3+ X4 X5 -1.7 7 X7 + X1X6 + X2 X5+ X3 X4 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.
[x4]= - [x1] [x2]
[x5]= - [x1] [x3]
[x6]= - [x2] [x3]
[x7]= - [x1] [x2] [x3]
A new 27-4-design with a different set of generators is generated:
Estimates of the effects
Effect Estimate
1 X1- X2X4 - X3 X5- X6 X7 -1.32 X2 - X1X4 - X3 X6- X5 X7 -2.5 3 X3 - X1X5 - X2 X6- X4 X7 7.9 4 X4 - X1X2 - X3 X7- X5 X6 1.1 5 X5 - X1X3 - X2 X7- X4 X6 -7.8 6 X6 - X1X7 - X2 X3- X4 X5 1.77 X7 - X1X6 - X2 X5- X3 X4 -4.6
Estimates of the effects
Effect Estimate
1 X1- X2X4 - X3 X5- X6 X7 -1.32 X2 - X1X4 - X3 X6- X5 X7 -2.5 3 X3 - X1X5 - X2 X6- X4 X7 7.9 4 X4 - X1X2 - X3 X7- X5 X6 1.1 5 X5 - X1X3 - X2 X7- X4 X6 -7.8 6 X6 - X1X7 - X2 X3- X4 X5 1.77 X7 - X1X6 - X2 X5- X3 X4 -4.6
Estimate of the effects (by combining the two series)
Effect Estimate1 X1 -3.32 X2 -1.9 3 X3 -0.24 X4 1.4 5 X5 -9.6 6 X6 -0.037 X7 -2.2 8 X2X4 + X3 X5+ X6 X7 -2.19 X1X4 + X3 X6+ X5 X7 0.610 X1X5 + X2 X6+ X4 X7 -8.111 X1X2 + X3 X7+ X5 X6 0.212 X1X3 + X2 X7+ X4 X6 -1.813 X1X7 + X2 X3+ X4 X5 -1.714 X1X6 + X2 X5+ X3 X4 2.4
Estimate of the effects (by combining the two series)
Effect Estimate1 X1 -3.32 X2 -1.9 3 X3 -0.24 X4 1.4 5 X5 -9.6 6 X6 -0.037 X7 -2.2 8 X2X4 + X3 X5+ X6 X7 -2.19 X1X4 + X3 X6+ X5 X7 0.610 X1X5 + X2 X6+ X4 X7 -8.111 X1X2 + X3 X7+ X5 X6 0.212 X1X3 + X2 X7+ X4 X6 -1.813 X1X7 + X2 X3+ X4 X5 -1.714 X1X6 + X2 X5+ X3 X4 2.4
Interpretation
Water source
65.4 42.6
68.5 78.0
Caustic Soda
Slow
FastCity
ReservoirPrivate
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)