Save Time With Fractional Factorial DOEs - Example

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	Discussion Forum "A Placket-Burnam design is certainly the way to go for a screening DOE, but it depends on the number of main effects that you want to study. If you have 6 or 7 factors that you want to study, then a 8-run PB is the way to go. Remember a 8-run PB is a resolution III design with a complex alias structure. To increase your design resolution (IV), it is a good idea to fold your design as a follow up experiment (another 8-run PB). This foldover will clear up the picture as to how main effects are affected by your design (i.e separating main effects from 2 factor interactions). If you have only have 3-5 factors to study, you may get a lot more value out of doing a full factorial or fractional factorial depending on how much it costs to setup and conduct your experiment." Optimizing A Microlectronic Device Response Using DOE

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By Shree Padnis

Page 1 > Save Time With Fractional Factorial DOEs

An experiment was conducted in a winding section for the winding of yarn with neps (small tangled fiber knot often caused by processing) formed on the yarn being the response. Below is the design matrix that was used. The experiment was conducted in two blocks to remove the effect of humidity on the formation of neps. It is a Resolution IV Experiment that means that two factor interactions are confounded with two factor interactions. Seven factors at two levels with two replications were experimented with.

The data was analyzed using Minitab, but many other statistical software programs can help perform this analysis. At first all the factors and possible two factors are selected to identify the significant effects. Below is shown the pareto chart and normal plot for the effects at an alpha value of 0.1.

Both the graphs indicate that the main factors of significance are:

• Factor A - speed
• Factor F - initial yarn quality, and
• Interaction B*D (package and bobbin setting)

However let us recollect that this is a Resolution IV design so confounding exists. We need to study the alias structure carefully before making any conclusions. The alias structure is given below:

By utilizing the hierarchical ordering principle we can conclude that speed and initial yarn quality are significant factors as they are confounded by three factor interactions.

On viewing the interaction between package and bobbin setting we find that it is aliased as below:

package*bobinset + tension*initial + cradlepr*tensiond

By utilizing the effect heredity principle we see that although [package*bobbin setting] was shown as significant in the pareto diagram, neither package or bobbin setting are themselves significant ruling out the possibility of this interaction being significant. The only interaction in the alias which has at least one factor significant is [tension*initial] wherein initial yarn is significant. Thus the significant interaction is [tension*initial] and not [package*bobbin setting].

The information gathered above is further validated by Interaction graphs and engineering logic.

As seen above it was relatively easy to analyze the experiment by use of the three principles. Based on this information we can now fit a reduced model. We can validate our various assumptions by verifying the residuals as below:

In residual analysis there are primarily three points to look at

1. In residual versus time order check for stability over time. The residuals should indicate a random scatter. There should not be any trends in the residuals.
2. In the Residuals versus fitted values we check for constancy of variance. This graph should also show a random scatter. If any trends observed, some transformation of data technique will probably need to be applied.
3. The residuals should be normally distributed as the third requirement.

It is essential to validate these assumptions before attempting to build the mathematical model.

In our example the residual analysis show no alarms, so we can now look at the mathematical model fitted which is given below:

Fractional Factorial Fit: resp_1_1 versus speed, tension, initial yarn

Based on the mathematical model fitted above we can make predictions about the response:

We can now run our confirmation trial and verify whether the predictions made by the model and the actual conditions tally.

Thus if we utilize the three fundamental principles of factorial effects:

• Hierarchal Ordering Principle
• Effect Sparsity Principle
• Effect Heredity Principle

It is possible to analyze most Resolution III and IV Design of Experiments without having to run the equal number of earlier run trials but with the signs reversed (also known as a fold over).

References
Wu, C. F. Jeff Wu and Michael Hamada, 2000. Experiments: Planning, Analysis, and Parameter Design Optimization. 1 edition: Wiley-Interscience.

About The Author
Shree Phadnis is a Master Black Belt at KPMG India. Mr. Phadnis is an ASQ certified Quality Manager and ASQ Certified Quality Engineer. Mr. Phadnis can be reached at shreephadnis@usa.net.

Page 1 > Save Time With Fractional Factorial DOEs

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