Interaction Graphs for Three-Level Fractional Factorial Designs
Process exploration by Fractional Factorial Design (FFD)
description
Transcript of Process exploration by Fractional Factorial Design (FFD)
![Page 1: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/1.jpg)
Process exploration by Fractional Factorial Design (FFD)
![Page 2: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/2.jpg)
Number of Experiments
Factorial design (FD) with variables at 2 levels.
The number of experiments = 2m, m=number of variables.
m=3 : 8 experimentsm=4 : 16 experimentsm=7 : 128 experiments
![Page 3: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/3.jpg)
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
ber o
f exp
erim
ents
![Page 4: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/4.jpg)
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)
![Page 5: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/5.jpg)
Screening designs
• 27-4 Reduced Factorial Design7 factors in 8 experiments
• 211 Plackett-Burman11 factors in 12 experiments
![Page 6: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/6.jpg)
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:
![Page 7: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/7.jpg)
Defining relation
Def.: The generators + all possible combination products
27-4: 4 generators6 products of 2 generators4 products of 3 generators1 products of 4 generators15
![Page 8: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/8.jpg)
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.
![Page 9: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/9.jpg)
Process capacity
Box and Hunter, 1961, Technometrics 3, p. 311
Variable -1 +1X1 Water Source City reservoir Private wellX2 Raw material Local OtherX3 Temperature Low HighX4 Recycling Yes NoX5 Caustic Soda Fast SlowX6 Filter New OldX7 Waiting time Short Long
![Page 10: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/10.jpg)
The Design Matrix
![Page 11: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/11.jpg)
Response Variation
![Page 12: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/12.jpg)
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
![Page 13: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/13.jpg)
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
![Page 14: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/14.jpg)
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
![Page 15: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/15.jpg)
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:
![Page 16: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/16.jpg)
The Design Matrix-new experimental series
![Page 17: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/17.jpg)
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
![Page 18: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/18.jpg)
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
![Page 19: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/19.jpg)
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
![Page 20: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/20.jpg)
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
![Page 21: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/21.jpg)
Contour plot
Filter time, y (min)
![Page 22: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/22.jpg)
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)
![Page 23: Process exploration by Fractional Factorial Design (FFD)](https://reader035.fdocuments.us/reader035/viewer/2022062222/568160f6550346895dd031f3/html5/thumbnails/23.jpg)
Bottom line...
Univariate optimisation of the speed used for adding NaOH!
This result would not have been obtained by a univariate approach!