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Central Composite Response Surface Method
|
This example validates the calculation of the central composite response surface method in DOE++.
Reference Case
The data is from Example 11-1 on page 431 in the book Design and Analysis of Experiments by Douglas C. Montgomery, John Wiley & Sons, 2001.
Data
| Natural Variables
|
Coded Variables
|
Responses
|
| A (time)
|
B (temperature)
|
A
|
B
|
Y (yield)
|
| 8076.580 |
170 |
-1 |
-1 |
76.5
|
| 80 |
180 |
-1 |
1 |
77
|
| 90 |
170 |
1 |
-1 |
78
|
| 90 |
180 |
1 |
1 |
79.5
|
| 85 |
175 |
0 |
0 |
79.9
|
| 85 |
175 |
0 |
0 |
80.3
|
| 85 |
175 |
0 |
0 |
80.0
|
| 85 |
175 |
0 |
0 |
79.7
|
| 85 |
175 |
0 |
0 |
79.8
|
| 92.07 |
175 |
1.414 |
0 |
78.4
|
| 77.93 |
175 |
-1.414 |
0 |
75.6
|
| 85 |
182.07 |
0 |
1.414 |
78.5
|
| 85 |
167.93 |
0 |
-1.414 |
77
|
Result
From the book, the ANOVA table is:
| Source
|
Sum of Squares (Partial SS)
|
DF
|
Mean Square
|
F value
|
Prob > F
|
| Model |
28.25 |
5 |
5.65 |
79.85 |
<0.0001
|
| A |
7.92 |
1 |
7.92 |
111.93 |
<0.0001
|
| B |
2.12 |
1 |
2.12 |
30.01 |
0.0009
|
| A∙A |
13.18 |
1 |
13.18 |
186.22 |
<0.0001
|
| B∙B |
6.97 |
1 |
6.97 |
98.56 |
<0.0001
|
| A∙B |
0.25 |
1 |
0.25 |
3.53 |
0.1022
|
| Residual |
0.5 |
7 |
0.071 |
|
|
| Lack of Fit |
0.28 |
3 |
0.094 |
1.78 |
0.2897
|
| Pure Error |
0.21 |
4 |
0.053 |
|
|
| Total |
28.74 |
12 |
|
|
|
The final equation in terms of the actual values of these two factors is:
- [math]\displaystyle{ \begin{align}
Yield= -1430.52285+7.80749 * time + 13.27053 * temp-0.05505 * time^2 - 0.04005 * temp^2 + 0.01 * time * temp
\end{align} }[/math]
The maximum yield is achieved at 80.21 with time = 87 minutes and temperature = 176.5 F.
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 yield is achieved at 80.21, as shown in the optimization plot. The values at the red dash line are the optimal values for factor A and factor B. The blue line is corresponding to the maximum Y value.
