Example 3 - Two Level Fractional Factorial Design

Download Example File for Version 9 (*.rsr9)

Background

Consider a manufacturing process for an integrated circuit.* The objective is to improve the process yield. The five factors that may affect the process are:

Factor Name Unit Low High
A Aperture Setting - small large
B Exposure Time min 20  30
C Develop Time s 30 45
D Mask Dimension - small large
E Etch Time min 14.5 15.5

It is too expensive to run a full factorial design, which would require 25 = 32 runs. Therefore, the engineers decide to run a half fractional factorial design using generator E = ABCD.

Experiment Design

To find the important factors, the experimenters use the standard design folio in DOE++ to create a two level factorial design. On the Design tab of the folio, the following design settings are specified. The design matrix and response data are given on the Data tab of the "Fractional Factorial Design" folio.

Summary of design settings

Analysis Part I - Screening

Step 1: When the design is built, the experiment runs are shown in a randomized order on the Data tab. After performing the runs in the displayed order and recording the results, the experimenters enter the data set into the folio, as shown next.

Figure 1: Data set entered on the Data tab of the folio.

Step 2: Next, the analysis settings are specified. The data set will be analyzed using the default risk (significance) level of 0.1. In addition, the test statistic for the effects will be calculated using the partial sum of squares, and the ANOVA table will show the results for each individual term.

Analysis settings

Figure 2: Analysis settings.

Step 3: An effect probability plot is created, as shown next. The plot shows that the main effects A, B and C, and the interaction effect AB, are significant.

Effect probability plot

Figure 3: Effect probability plot indicating the significant effects.

Analysis Part II - Optimization

The results for the reduced model are given in the "Reduced Model" folio.

Step 1: The design folio is duplicated and the copy is renamed to "Reduced Model."

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

Select Terms window

Figure 4: Select Terms window showing the terms included in reduced model.

Step 3: The reduced model is calculated. The Regression Information table in the detailed summary of results shows the following coefficients for the included terms.

Term Coefficient
A 5.5625
B 16.9375
C 5.4375
AB 3.4375

Conclusions

Each of the effects in the reduced model has a positive effect on the yield (as indicated by the positive coefficients). Therefore, in order to increase the yield, the high levels of factors A, B, and C should be applied. For the remaining factors, the low or high levels could be used.

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