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Process exploration by Fractional Factorial Design (FFD)

Process exploration by Fractional Factorial Design (FFD). Number of Experiments. Factorial design (FD) with variables at 2 levels. The number of experiments = 2 m , m =number of variables. m =3 : 8 experiments m =4 : 16 experiments m =7 : 128 experiments.

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Process exploration by Fractional Factorial Design (FFD)

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  1. Process exploration by Fractional Factorial Design (FFD)

  2. 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

  3. The relationship between the number of variables and the required number of experiments

  4. Full design vs. Fractional design First order model with 7 input variables: 8 parameters have to be decided. 27 Factorial Design  128 experiments 27-4 Fractional Factorial Design  8 experiments (example of saturated design)

  5. Screening designs • 27-4 Reduced Factorial Design7 factors in 8 experiments • 211 Plackett-Burman11 factors in 12 experiments

  6. [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] 27-4 Fractional Factorial Design [xi]2 = [I] [xi] [xj] = [xj] [xi] Generators:

  7. Defining relation Def.: The generators + all possible combination products

  8. Defining relation for 27-4FFD [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.

  9. Process capacity Box and Hunter, 1961, Technometrics 3, p. 311

  10. The Design Matrix

  11. Response Variation

  12. 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

  13. 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

  14. 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

  15. 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]

  16. The Design Matrix-new experimental series

  17. Estimates of the effects Effect Estimate 1 X1- X2X4 - X3 X5- X6 X7 -1.3 2 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 1.7 7 X7 - X1X6 - X2 X5- X3 X4 -4.6

  18. Estimates of the effects Effect Estimate 1 X1- X2X4 - X3 X5- X6 X7 -1.3 2 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 1.7 7 X7 - X1X6 - X2 X5- X3 X4 -4.6

  19. Estimate of the effects (by combining the two series) Effect Estimate 1 X1 -3.3 2 X2 -1.9 3 X3 -0.2 4 X4 1.4 5 X5 -9.6 6 X6 -0.03 7 X7 -2.2 8 X2X4 + X3 X5+ X6 X7 -2.1 9 X1X4 + X3 X6+ X5 X7 0.6 10 X1X5 + X2 X6+ X4 X7 -8.1 11 X1X2 + X3 X7+ X5 X6 0.2 12 X1X3 + X2 X7+ X4 X6 -1.8 13 X1X7 + X2 X3+ X4 X5 -1.7 14 X1X6 + X2 X5+ X3 X4 2.4

  20. Estimate of the effects (by combining the two series) Effect Estimate 1 X1 -3.3 2 X2 -1.9 3 X3 -0.2 4 X4 1.4 5 X5 -9.6 6 X6 -0.03 7 X7 -2.2 8 X2X4 + X3 X5+ X6 X7 -2.1 9 X1X4 + X3 X6+ X5 X7 0.6 10 X1X5 + X2 X6+ X4 X7 -8.1 11 X1X2 + X3 X7+ X5 X6 0.2 12 X1X3 + X2 X7+ X4 X6 -1.8 13 X1X7 + X2 X3+ X4 X5 -1.7 14 X1X6 + X2 X5+ X3 X4 2.4

  21. Contour plot Filter time, y (min)

  22. Slow 65.4 42.6 Caustic Soda Fast 68.5 78.0 City Reservoir Water source Private Well Interpretation 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)

  23. Bottom line... Univariate optimisation of the speed used for adding NaOH! This result would nothave been obtained by a univariate approach!

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