Process exploration by Fractional Factorial Design (FFD)

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Presentation transcript:

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

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

Full design vs. Fractional design First order model with 7 input variables: 8 parameters have to be decided. 2 7 Factorial Design  128 experiments Fractional Factorial Design  8 experiments (example of saturated design)

Screening designs Reduced Factorial Design 7 factors in 8 experiments 2 11 Plackett-Burman 11 factors in 12 experiments

2 7-4 Fractional Factorial Design [x 4 ]=[x 1 ] [x 2 ] [x 5 ]=[x 1 ] [x 3 ] [x 6 ]=[x 2 ] [x 3 ] [x 7 ]=[x 1 ] [x 2 ] [x 3 ] [I]=[x 1 ] [x 2 ] [x 4 ] [I]=[x 1 ] [x 3 ] [x 5 ] [I]=[x 2 ] [x 3 ] [x 6 ] [I]=[x 1 ] [x 2 ] [x 3 ] [x 7 ] [x i ] 2 = [I] [x i ] [x j ] = [x j ] [x i ] Generators:

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

Defining relation for FFD [I] = [x 1 ] [x 2 ] [x 4 ] = [x 1 ] [x 3 ] [x 5 ] = [x 2 ] [x 3 ] [x 6 ] = [x 1 ] [x 2 ] [x 3 ] [x 7 ] = [x 2 ] [x 4 ] [x 3 ] [x 5 ] = [x 1 ] [x 4 ] [x 3 ] [x 6 ] = [x 4 ] [x 3 ] [x 7 ] = [x 1 ] [x 2 ] [x 5 ] [x 6 ] = [x 2 ] [x 5 ] [x 7 ] = [x 1 ] [x 6 ] [x 7 ] = [x 4 ] [x 5 ] [x 6 ] = [x 2 ] [x 4 ] [x 6 ] [x 7 ] = [x 1 ] [x 4 ] [x 5 ] [x 7 ] = [x 3 ] [x 5 ] [x 6 ] [x 7 ] = [x 1 ] [x 2 ] [x 3 ] [x 4 ] [x 5 ] [x 6 ] [x 7 ] The confounding pattern appears by multiplying the defining relation with each of the variables.

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

The Design Matrix

Response Variation

Estimates of the effects EffectEstimate 1X 1 + X 2 X 4 + X 3 X 5 + X 6 X X 2 + X 1 X 4 + X 3 X 6 + X 5 X X 3 + X 1 X 5 + X 2 X 6 + X 4 X X 4 + X 1 X 2 + X 3 X 7 + X 5 X X 5 + X 1 X 3 + X 2 X 7 + X 4 X X 6 + X 1 X 7 + X 2 X 3 + X 4 X X 7 + X 1 X 6 + X 2 X 5 + X 3 X

Estimates of the effects EffectEstimate 1X 1 + X 2 X 4 + X 3 X 5 + X 6 X X 2 + X 1 X 4 + X 3 X 6 + X 5 X X 3 + X 1 X 5 + X 2 X 6 + X 4 X X 4 + X 1 X 2 + X 3 X 7 + X 5 X X 5 + X 1 X 3 + X 2 X 7 + X 4 X X 6 + X 1 X 7 + X 2 X 3 + X 4 X X 7 + X 1 X 6 + X 2 X 5 + X 3 X

Plausible interpretations There are four likely combinations of significant effects: 1. Variable X 1, X 3 and X 5 2. Variable X 1, X 3 and the interaction X 1 X 3 3. Variable X 1, X 5 and the interaction X 1 X 5 4. Variable X 3, X 5 and the interaction X 3 X 5

New experimental series It is desirable to separate the 1- and 2- factor effects. [x 4 ]= - [x 1 ] [x 2 ] [x 5 ]= - [x 1 ] [x 3 ] [x 6 ]= - [x 2 ] [x 3 ] [x 7 ]= - [x 1 ] [x 2 ] [x 3 ] A new design with a different set of generators is generated:

The Design Matrix -new experimental series

Estimates of the effects EffectEstimate 1X 1 - X 2 X 4 - X 3 X 5 - X 6 X X 2 - X 1 X 4 - X 3 X 6 - X 5 X X 3 - X 1 X 5 - X 2 X 6 - X 4 X X 4 - X 1 X 2 - X 3 X 7 - X 5 X X 5 - X 1 X 3 - X 2 X 7 - X 4 X X 6 - X 1 X 7 - X 2 X 3 - X 4 X X 7 - X 1 X 6 - X 2 X 5 - X 3 X

Estimates of the effects EffectEstimate 1X 1 - X 2 X 4 - X 3 X 5 - X 6 X X 2 - X 1 X 4 - X 3 X 6 - X 5 X X 3 - X 1 X 5 - X 2 X 6 - X 4 X X 4 - X 1 X 2 - X 3 X 7 - X 5 X X 5 - X 1 X 3 - X 2 X 7 - X 4 X X 6 - X 1 X 7 - X 2 X 3 - X 4 X X 7 - X 1 X 6 - X 2 X 5 - X 3 X

Estimate of the effects (by combining the two series) EffectEstimate 1X X X X X X X X 2 X 4 + X 3 X 5 + X 6 X X 1 X 4 + X 3 X 6 + X 5 X X 1 X 5 + X 2 X 6 + X 4 X X 1 X 2 + X 3 X 7 + X 5 X X 1 X 3 + X 2 X 7 + X 4 X X 1 X 7 + X 2 X 3 + X 4 X X 1 X 6 + X 2 X 5 + X 3 X 4 2.4

Estimate of the effects (by combining the two series) EffectEstimate 1X X X X X X X X 2 X 4 + X 3 X 5 + X 6 X X 1 X 4 + X 3 X 6 + X 5 X X 1 X 5 + X 2 X 6 + X 4 X X 1 X 2 + X 3 X 7 + X 5 X X 1 X 3 + X 2 X 7 + X 4 X X 1 X 7 + X 2 X 3 + X 4 X X 1 X 6 + X 2 X 5 + X 3 X 4 2.4

Contour plot Filter time, y (min)

Interpretation Water source Caustic Soda Slow Fast City Reservoir 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!