Augmenting Mixture-Process DOE

[#1]

3 component mixture,

0 < A < 0.4

0 < B < 0.1

0 < C < 1

with 1 linear constraint,

0 < (A+B) < 0.4

and 1 categorical process factor with 4 levels,

P1 = {1,5,10,15}.

I setup the experiment using JMP 12 and this tutorial (example 1),

Mixture Experiments

Now, JMP has calculated that the default number of runs is 18, using points at vertex, edge and centroid.

How can I augment this by adding additional axial points (similar to a "space filling" scheme)? I want to have more data on the interior region as this is the expected optimization zone.

Can I change the process factor to continuous while using multiple levels? Why does JMP only allow hi/lo?

Thanks a million!

[#2]

In response to my second question, "Can I change the process factor to continuous while using multiple levels? Why does JMP only allow hi/lo?", see below.

35417 - How do I create a design with a continuous factor that has more than 2 levels in JMP®?

[#3 · Accepted solution]

You can get a design with more interior region points by changing the optimality criterion from D-optimal to I-optimal under the red triangle. Additionally choosing a Scheffe cubic can sometimes drive more internal points in your design space.

The discete numeric factors can be used wrt your 4 level categorical factor.

[#4]

Hello Lou,

Thank you for the response. I understand that I-optimal will put more points inside the design space and less at the extremes, I will give that a try. As for the Scheffe cubic, how will the third-degree polynomial term affect my DOE?

My objective is to test for factor-response, and not factor-factor, interactions. Do you think RSM would be appropriate for this type of experiment?

All the best.

[#5]

Hello acid1c,

The higher order Scheffe cubic model will generally give more runs inside the design space because it is a more complex model. I am not suggesting to use that model to analyze the design but use it to give you a candidate design that you can compare to the other potential designs.

Due to the interdependent nature of mixture designs in essence a model with main effects and non-linear blends (2-way interactions) is like an RSM.

The D-optimal and I optimal fraction of design space plots for 18 runs are shown below.

[#6]

The settings you suggested, I-Optimal and Sheffe Cubic, added a handful of extra points in the middle of the design space. But sadly the maximum desirability increase two-fold and the average predictive variance increased by about 0.1 (when compared against I-Optimal with secondary interactions and quadratic model). Maybe I'll try changing the number of runs.

[#7]

The behavior you saw should be expected. I-optimal or D-optimal are OPTIMAL for the number of runs you picked. By building a more complex model to force interior points with the same number of runs will have to increase the prediction variance.

There are several points here to keep in mind. I will discuss first by ignoring the process factor as I think there may be some confusion about the mixture model.

Mixture models are very different from the standard regression models. As Lou indicated, a 2-way interaction in a Scheffe mixture model (which is what JMP is fitting) is equivalent to a squared term for a continuous term). However, the two-way interactions in a Scheffe model are only looking at a curvature effect along the axis between the two components that are in that interaction term. For your three component case, that would be curvature along the edge of the design space.

So given that, the I-optimal or D-optimal designs will choose the best design to estimate your model. If you are only trying to estimate curvature along the edges of the design space, there is no need to put points in the interior of the design. Anything on the interior will do little to help you understand what is happening at the edges of the design space. Think of just a single continuous X. If you only specify a straight-line model, you only need experiments at the ends of the range. You need to specify a quadratic to force a third level.

The model that is needed to start putting points in the interior of a mixture design space is to add 3-way interactions. These are called cubic terms (actually, special cubic terms) in the Scheffe mixture models. They estimate curvature between 3 components at a time. Since you have a three-component mixture that will force at least one point into the interior of your design space. The I-optimal design will tend to put more points in the interior of the design space in order to better estimate that three-way term.

Adding the process factor does not change things too much to what I typed. However, that model gets more complex and there may be some things you want to do to ensure you are getting at least three levels for that process variable.

[#8]

acid1c,

One thing to check is if the design evaluation was comparing apples to apples? After generating the design with the Scheffe cubic to drive more interior design points you would have to prune back the model to evaluate the same model that was applied to analyze the D-optimal and I-optimal scenarios, namely main effects and two-way interactions (non-linear blends).

[#9]

I've created another thread on this topic, perhaps you can lend some expertise to the topic:

Save all settings from custom doe design and design report

[#10]

Previously I was using Minitab, which is restricted to simple mixture designs. I've had to take a couple steps back and re-learn some concepts to use the full power behind JMP. Great application and user community, thanks!