Background
General full factorial design is used when there are several factors (<5) that have multiple levels. If there are many multiple level factors, the size of a general full factorial design will be prohibitively large. In such cases, Taguchi OA design should be used.

Consider that a soft drink bottler is interested in obtaining more uniform fill heights in the bottles.* The filling machine theoretically fills each bottle to the correct target height, but in practice, there is variation around this target, and the bottler would like to understand better the sources of this variability and eventually reduce it. There are three control factors.

There are two replicates at each factor setting, making a total of 24 runs. The response is the average deviation from the target fill height observed in a production run of bottles. Positive deviations are fill heights above the target.

Experiment Design
The experimenters use DOE++ to design a general full factorial design. The design-specific settings, the factor properties and the response properties used are shown next.

The design matrix and the response data are given in the "Soft Drink Bottling Experiment" Folio.

Analysis Part I
Step 1: After performing the experiment according to the design and recording the results, the experimenters enter the data set into the Standard Folio, as shown next.

[Click to Enlarge]

Step 2: The data set is analyzed with the default risk (significance) level of 0.1, using individual terms. The ANOVA table from the Analysis tab is shown next.

This table shows that effects A, B, C and AB are significant.

Analysis Part II
The results for the reduced model and the optimization are given in the "Reduced Model" Folio.

Step 1: The design Folio is duplicated and the copy is named "Reduced Model."

Step 2: In the Select Effects window, only the significant effects are selected to calculate the new model, as shown next.

Step 3: The reduced model is calculated.

Step 4: To identify which factor settings can provide the smallest height deviation, the Diagnostics window is used, as shown next.

Step 5: These results are copied to a Spreadsheet for further analysis. The data are given in the "Best Setting" Spreadsheet, as shown next.

Conclusions
The Spreadsheet shows that the runs at run orders 12 and 23 (i.e. standard orders 5 and 17) are at the best combination of settings, which is A = 12, B = 25, C = 200. At these settings, the expected deviation is lowest.

In general full factorial design, all factors are assumed to be qualitative factors, which means that the factors can only take the discrete values defined in the design, which in this case are:

If the factors can be treated as quantitative factors, meaning they can take any value within a range, further analysis using response surface methodology should be conducted to find the optimal settings for the manufacturing process.

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