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Central Composite Designs

Central Composite Designs: Example

The data set used in this example is available in the example database installed with the software (called "DOE10_examples.rsgz10"). To access this database file, choose File > Help, click Open Examples Folder, then browse for the file in the DOE sub-folder.

The name of the example project is "Response Surface Method - Central Composite Design."

A chemical engineer is interested in determining the operating conditions that maximize the yield of a process.* Two controllable variables influence process yield: reaction time and reaction temperature.

A central composite design with five center points and alpha = 1.414 is used to conduct the experiment. A full quadratic model is fitted to the data.

Designing the Experiment

The engineer uses DOE++ to design a central composite design then performs the experiment according to the design and enters the response values into DOE++ for further analysis. The design matrix and the response data are given in the "Central Composite Design" folio. The following steps describe how to create this folio on your own.

Analysis and Results

The data set for this example is given in the "Central Composite Design" folio of the example project. After you enter the data from the folio, you can specify the settings for the analysis by doing the following:

Note: To minimize the effect of unknown nuisance factors, the run order is randomly generated when you create the design in DOE++. Therefore, if you followed these steps to create your own folio, the order of runs on the Data tab may be different from that of the folio in the example file. This can lead to different results. To ensure that you get the very same results described next, show the Standard Order column in your folio, then click a cell in that column and choose Sheet > Sheet Actions > Sort > Sort Ascending. This will make the order of runs in your folio the same as that of the example file. Then copy the response data from the example file and paste it into the Data tab of your folio.

In the window that appears, select the All Terms check box then click OK.

Then choose Pareto Charts - Regression from the Plot Type drop-down list. The following plot appears.

The vertical blue line in the plot marks the critical value determined by the risk level specified on the Analysis Settings page of the control panel. If the bar goes past the blue line, then the effect is considered significant.

From these results, effects A, B and AA and BB will be included in the reduced model. In fact, you could also include term AB in the model. From the Pareto chart, you can see that AB is only slightly below the critical value. The inclusion or exclusion of AB is a personal decision that should be made based on the knowledge of the experiment and the statistical results. For this example, only A, B, AA and BB will be included in the model.

Optimization

The results for the reduced model (which only includes the terms that were found to be significant) are given in the "Reduced Model" folio. The following steps describe how to create this folio on your own.

The optimum settings for factors A and B are shown next.

You also can use the surface plot to visually identify the optimal settings for factors A and B.

The 3D Plot folio will open and display the surface plot, as shown next.

Conclusions

From the surface plot, you can see that the maximum yield occurs at Time = 86.8 and Temperature = 176.3° F, which is the same as the result from the optimization. The predicted maximum yield is 80.1861. Keep in mind that it is necessary to conduct an experiment using these settings to confirm this conclusion.

* Montgomery, D. C. Design and Analysis of Experiments, 5th edition, John Wiley & Sons, New York, 2001.

 

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