Central Composite Response Surface Method: Difference between revisions

From ReliaWiki
Jump to navigation Jump to search
No edit summary
No edit summary
 
(15 intermediate revisions by 2 users not shown)
Line 1: Line 1:
{{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.       


{{Reference_Example_Heading2}}
{{Reference_Example_Heading2}}
Line 15: Line 15:
!style="background:#f0f0f0;"|A
!style="background:#f0f0f0;"|A
!style="background:#f0f0f0;"|B
!style="background:#f0f0f0;"|B
!style="background:#f0f0f0;"|Y( yield)
!style="background:#f0f0f0;"|Y (yield)
|-
|-
| 8076.580||170||-1||-1||76.5
| 8076.580||170||-1||-1||76.5
Line 46: Line 46:
{{Reference_Example_Heading3}}
{{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

DOE Reference Examples Banner.png


New format available! This reference is now available in a new format that offers faster page load, improved display for calculations and images and more targeted search.

As of January 2024, this Reliawiki page will not continue to be updated. Please update all links and bookmarks to the latest references at DOE examples and DOE reference examples.



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
Retrieved from "https://www.reliawiki.com/index.php?title=Central_Composite_Response_Surface_Method&oldid=66100"