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Second Order Design of Experiments

The most important objective of Design of Experiments theories is to decrease the time required to achieve six sigma levels of quality by providing people with tools to characterize and/or improve equipment, processes and products through the application of efficient experimentation and analysis techniques. When designing experimental runs to understand and characterize complex phenomena it is important to follow a logic experimental strategy.

First it is always important to perform screening runs on the edges of each factor (half factorial DOE) to identify the main X of a function, and then move in to the direction of increasing the goodness of Y, by planning the new DOE with its center point as the corner of the first design. Up to this point, the task of designing an experiment seems straight forward by lending themselves to sequential experimentation, with just the constraints inherent to optimize resources.

But, in opposition to the optimization purpose, we must think of what happens outside the region of interest and inside the region of operability. What can go wrong?  Suppose runs were made only at extremes, and linear behavior was assumed in the interior of the design space - may not be accurate. The response might be parabolic in the interior, but relatively flat near the edges.

The first approach to address this potential error is to add more exploration points to the experimental design. But let´s remember we always will have pressure to optimize resources when conducting experiments, so let´s explore some options to address this potential error while getting the most from the experiments by using Second Order Design of Experiments.

In order to explore the most part of the response surface and avoid missing important sections of the design space, the second order design of experiments can help. Central Composite Designs (CCD) are the most common and include the possibility of using data already obtained in a 1st order DOE. These are first order designs augmented with center points and star (or axial) points.

Corner Points
For the assessment of linear
and 2-way interaction terms.

Center Points
Used to detect curvature.
Replicated in experimental
DOE to estimate pure error

Star Points
For the assessment of
quadratic terms.

The star points are useful in this DOE designs because in multidimensional problems, both center an star points are necessary to define the response for surfaces with curvature. Consider the following example.

This design is build on a Full Factorial DOE, thus are classified in three different ways depending on the location of the star points. This yields to the definition of a variable to characterize the distance from the star points to the center point, known as alpha.

The three classes of CCD are:

Face centered (Alpha=1)
-      Does not have points outside the region of interest
-      Includes three levels for each factor.
-      Cannot be used for extrapolation.

Star points outside the region of interest (Alpha>1)
-      Used to explore beyond the area of interest

Star points inside the region of interest (Alpha < 1)
-      To improve accuracy near the edges of the region of interest.

In this moment, it is evident that the value of Alpha describes the geometry of a sphere (either inscribed when Alpha<1, or circumscribed to the CCD when Alpha>1) when the experiment is meant to be symmetric, thus, the variance of the prediction is a function of Alpha. When choosing the value of Alpha for each experiment the region of interest, operability and the portion of the design space to explore should be considered.

Whenever the corner points of the experimental region are not accessible, for physical limitations for instance, a second order experimental design without corners can be defined. Known as Box-Behnken it contains a similar number of runs as a CCD for the same number of factors.

The experts and Black Belts ate Terasigma can help you design and analyze your experiments plan in the most efficient way according to the specifics of your project. Should you have any question regarding this or other Six sigma topics  please don’t hesitate to contact us.

Best Regards

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