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Box Behnken RSM
This example validates the calculation of the Box-Behnken RSM method in DOE++.
Reference Case
Data is from Example 7.5 on page 347 in the book Response Surface Methodology, process and product optimization using design experiments, second 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
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
The viscosity of the resin was recorded as an indirect measure of molecular weight.
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
〖 A〗^2
200.83
1
200.83
19.59
0.0069
B^2
12.98
1
12.98
1.27
0.3115
C^2
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
Results in DOE++
The software results match the book results. The ANOVA table is:
The final equation in terms of 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.
