Central Composite Response Surface Method: 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 central composite response surface method.
This example validates the calculation of the central composite response surface method in Weibull++.


{{Reference_Example_Heading1}}
{{Reference_Example_Heading1}}
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.       
The data are from Example 11-1 on page 431 in the book ''Design and Analysis of Experiments'' by Douglas C. Montgomery, John Wiley & Sons, 2001.       


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!colspan="2"|Responses
!colspan="2"|Responses
|-
|-
| A (time)||B  (temperature)||A||B||Y( yield)
!style="background:#f0f0f0;" |A (time)
!style="background:#f0f0f0;"|B  (temperature)
!style="background:#f0f0f0;"|A
!style="background:#f0f0f0;"|B
!style="background:#f0f0f0;"|Y (yield)
|-
|-
| 8076.580||170||-1||-1||76.5
| 8076.580||170||-1||-1||76.5
Line 39: Line 43:
| 85||167.93||0||-1.414||77
| 85||167.93||0||-1.414||77
|-
|-
|
|}
|}
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{{Reference_Example_Heading3}}


From the book, the ANOVA table is:
 
{| {{Table}}
 
!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>\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.
 


{{Reference_Example_Heading4|DOE++}}
{{Reference_Example_Heading4|DOE++}}
The software results match the book results. The ANOVA table is:
[[Image:composite_anova.png|center]]
The final equation in terms of the actual factors is:
[[Image:composite_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  corresponds to the maximum Y value. 
[[Image:composite_plot.png|center|750px]]

Latest revision as of 21:08, 18 September 2023

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Central Composite Response Surface Method

This example validates the calculation of the central composite response surface method in Weibull++.

Reference Case

The data are 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:

Composite anova.png

The final equation in terms of the actual factors is:

Composite final.png

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 corresponds to the maximum Y value.

Composite plot.png
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