Box Behnken RSM

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Box Behnken RSM

This example validates the calculation of the Box-Behnken RSM method in DOE++.

Reference Case

The data are from Example 7.5 on page 347 in the book Response Surface Methodology: Process and Product Optimization Using Designed Experiments, 2nd Edition, by Raymond H. Myers and Douglas C. Montgomery, John Wiley & Sons, Inc. 2002.

Data

Level (coded value) Temperature (A) Agitation (B) Rate (C)
High (+1) 200 10 25
Center (0) 175 7.5 20
Low (-1) 150 5 15

The viscosity of the resin was recorded as an indirect measure of molecular weight.

Standard order A B C Y
1 -1 -1 0 53
2 +1 -1 0 58
3 -1 +1 0 59
4 +1 +1 0 56
5 -1 0 -1 64
6 +1 0 -1 45
7 -1 0 +1 35
8 +1 0 +1 60
9 0 -1 -1 59
10 0 +1 -1 64
11 0 -1 +1 53
12 0 +1 +1 65
13 0 0 0 65
14 0 0 0 59
15 0 0 0 62


Result

From the book, the ANOVA table (partial sum of squares) is:

Source Partial SS DF Mean Square F value Prob > F
Model 882.48 9 98.05 9.57 0.0114
A 8 1 8.00 0.78 0.4174
B 55.12 1 55.12 5.38 0.0681
C 45.13 1 45.13 4.40 0.09
[math]A^2[/math] 200.83 1 200.83 19.59 0.0069
[math]B^2[/math] 12.98 1 12.98 1.27 0.3115
[math]C^2[/math] 48.52 1 48.52 4.73 0.0816
AB 16.00 1 16.00 1.56 0.2668
AC 484.00 1 484.00 47.22 0.001
BC 12.25 1 12.25 1.22 0.3241
Residual 51.25 5 10.25
Lack of Fit 33.25 3 11.08 1.23 0.4774
Pure error 18 2 9.00
Cor total 933.73 14

The final equation in terms of actual factors:

[math]\begin{align} Viscosity=-58.875+2.65 * Temp-0.65 * Agitation-11.125*Rate-0.0118*Temp^2+0.3*Agitation^2 \\ -0.145*Rate^2-0.032*Temp*Agitation+0.088*Temp*Rate+0.14*Agitation*Rate \end{align}[/math]


Results in DOE++

The software results match the book results. The ANOVA table is:

Behnken anova.png

The final equation in terms of the actual factors is:

Behnken final.png

The maximum viscosity is shown in the following optimal solution plot.

Behnken plot.png