# Central Composite Response Surface Method

Central Composite Response Surface Method |

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

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]\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 corresponds to the maximum Y value.