Box Behnken RSM: Difference between revisions

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{{Reference Example|{{Banner DOE Reference Examples}}}}
{{Reference Example|{{Banner DOE Reference Examples}}}}
This example validates the calculation of the Box-Behnken RSM method in DOE++.
This example validates the calculation of the Box-Behnken RSM method in Weibull++.


{{Reference_Example_Heading1}}
{{Reference_Example_Heading1}}
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.       
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.       


{{Reference_Example_Heading2}}
{{Reference_Example_Heading2}}
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!Temperature (A)
!Temperature (A)
!Agitation (B)
!Agitation (B)
!Rate'''
!Rate (C)
|-
| (C)
|-
|-
| High (+1)||200||10||25
| High (+1)||200||10||25
Line 23: Line 21:
|}
|}


{| {{Table}}
The viscosity of the resin was recorded as an indirect measure of molecular weight.
 
{| {{Table|23%}}
!Standard order
!Standard order
!A
!A
Line 30: Line 30:
!Y
!Y
|-
|-
| 1||-1||-1||0||53
|style="width:100px;"|1||-1||-1||0||53
|-
|-
| 2||+1||-1||0||58
| 2||+1||-1||0||58
Line 59: Line 59:
|-
|-
| 15||0||0||0||62
| 15||0||0||0||62
|-
|}
|}


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


{{Reference_Example_Heading3}}
{{Reference_Example_Heading3}}
Line 68: Line 67:
From the book, the ANOVA table (partial sum of squares) is:
From the book, the ANOVA table (partial sum of squares) is:


{| {{Table}}
{| {{Table|50%}}
!Source
!Source
!Partial SS
!Partial SS
Line 74: Line 73:
!Mean Square
!Mean Square
!F  value
!F  value
!Prob>F
!Prob > F
|-
|-
| Model||882.48||9||98.05||9.57||0.0114
| Model||882.48||9||98.05||9.57||0.0114
Line 84: Line 83:
| C||45.13||1||45.13||4.40||0.09
| C||45.13||1||45.13||4.40||0.09
|-
|-
| 〖 A〗^2||200.83||1||200.83||19.59||0.0069
| <math>A^2</math>||200.83||1||200.83||19.59||0.0069
|-
|-
| B^2||12.98||1||12.98||1.27||0.3115
| <math>B^2</math>||12.98||1||12.98||1.27||0.3115
|-
|-
| C^2||48.52||1||48.52||4.73||0.0816
| <math>C^2</math>||48.52||1||48.52||4.73||0.0816
|-
|-
| AB||16.00||1||16.00||1.56||0.2668
| AB||16.00||1||16.00||1.56||0.2668
Line 102: Line 101:
| Pure error||18||2||9.00||||
| Pure error||18||2||9.00||||
|-
|-
| Cor total||933.73||14||||||
| 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>




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[[Image:behnken_anova.png|center]]
[[Image:behnken_anova.png|center]]


The final equation in terms of actual factors is:
The final equation in terms of the actual factors is:


[[Image:behnken_final.png|center]]
[[Image:behnken_final.png|center]]


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


[[Image:behnken_plot.png|center|750px]]
[[Image:behnken_plot.png|center|750px]]

Latest revision as of 21:08, 18 September 2023

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Box Behnken RSM

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

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]\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:

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
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