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| [[Image:behnken_anova.png|center]] | | [[Image:behnken_anova.png|center]] |
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| The final equation in terms of actual factors is: | | The final equation in terms of the actual factors is: |
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| [[Image:behnken_final.png|center]] | | [[Image:behnken_final.png|center]] |
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This example validates the calculation of the Box-Behnken RSM method in DOE++.
Reference Case
The data is 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]\displaystyle{ A^2 }[/math] |
200.83 |
1 |
200.83 |
19.59 |
0.0069
|
| [math]\displaystyle{ B^2 }[/math] |
12.98 |
1 |
12.98 |
1.27 |
0.3115
|
| [math]\displaystyle{ 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]\displaystyle{ \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:
The final equation in terms of the actual factors is:
The maximum viscosity is shown in the following optimal solution plot.
