Below you will find the earlier parts of this article:

PART 1
PART 2

Mathematical model describing the influence of tested factor settings on humidity

In our example, we consider the effects of factors A, B, and C, as well as the A*B interaction, to be important. From these effects, we will construct an equation that will show the humidity level achieved for different settings of these three factors.

In experiment 23=8, each factor is tested four times at its -1 level and four times at its +1 level. Therefore, each factor is measured at the same mean. This mean is designated β0 in the prediction equation. β0 is the starting point for how humidity will change depending on how we set factors A, B, and C. Depending on the magnitude of these factors’ effects, their impact on humidity change relative to β0 will vary. This impact is measured by βA, βB, and βC, respectively. For example, if we set each factor exactly midway between the tested levels (level designated 0), humidity would equal β0. The beta coefficients calculated for all effects are included in the table below:

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The equation showing what the moisture of the powder at the outlet of the dryer will be, after taking into account significant factors, looks as follows:

To predict humidity, replace A, B, and C with the levels of these factors you want to predict. How does this work? Let’s calculate the humidity when:

A: Temperature 1 = 45°C
B: Temperature 2 = 37.5°C
C: Water Dose = 55%

We don’t enter actual values ​​into the equation, but encode them in a range from -1 to 1, using the testing levels used in the DOE.

Factor A: Temperature 1, in the DOE, was tested at -1 = 30°C and +1 = 60°C. 45°C is halfway between -1 and +1, meaning it’s 0 on this scale.

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Factor B: temperature 2, was tested at -1 = 30° and +1 = 60o. 37.5° is 1/4 of the distance between -1 and +1, so it is -0.5 on this scale.

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Factor C: water dose, was tested at -1 = 40% and +1 = 60%, 55% is in 3/4 of the distance between -1 and +1, i.e. it is 0.5 on this scale.

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Therefore:

For:

  • A = 0
  • B= -0,5
  • C= 0,5
  • A*B = 0 * – 0,5 = 0

The ending of this story is in part four which you will find below:

PART 4
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Author: Katarzyna Kornicka i Wojciech Florek, OpEx Six Sigma Master Black Belts