Mixture design or non-mixture design

[#1]

I have a two step process that I am trying to conduct a DOE. The first step is a polymerization process and it has three components adding up to 100%. After the polymerization is completed, the resulted solution will be extruded out to the second step for fiber formation. Because the polymerization step is a batch process and I am not able to vary the composition ratio of the three components until I complete the second step (fiber formation process), I define the factors for polymerization step as very hard to change. There is one hard to change factor and four easy to change factors in the second step (fiber formation). The two steps cannot be separated from each other meaning the solution polymerized from the first step will be extruded out for fiber formation directly from the polymerization reactor. 

Since there are three components in the first step for polymerization and they add up to 100%, the first thing coming into my mind is to use a mixture design with X11, X12, X13 as the three mixture factors. However, I am not able to get the main effects for the factors in the second step fiber formation process (X21, X22, X23, X24, and X25). This is the design on the left in the picture. The design on the right of the picture is to define two continuous factors X11 and X12 as the percentage of two out of three components and bound their levels within the interest of study and just make up to 100% with the third component based on X11 and X12. This design can allow for main effect as well as two way interaction for all the factors from the fiber formation process and has higher power than the first design. To me, the non-mixture design is the better one but I am not sure whether I am missing anything or there is a better design than both of the two illustrated here. BTW, for both designs, I am only interested in the main effect for X25, which is a hard to change factor from the second step, to control the reasonable size of experiment runs. Thanks!

[#2 · Accepted solution]

Hi,

You have an example of mixture-process experiment here, with both mixture factors and process factors. Mixture factors are a bit strange, in that you can't control them completely independently.

One of the results of this strangeness is that it is not possible to estimate both the mixture-process 2-factor interactions and also the process factor main effects. The reason is a little complicated and is explained in a case study of a mixture-process experiment in Optimal Designs: A Case Study Approach by Goos and Jones.

You should not let this stop you from using the mixture-process design. If the proportions of the polymerization ingredients are what matters, it would be a mistake to pretend that they are continuous factors instead. Treating ingredient amounts as continuous factors only really works when there is one ingredient that is the inactive bulk of the mixture.

I hope that helps.

Phil 

[#3]

Thank you Phil for the feedback!

[#4 · Accepted solution]

I only want to add that the two steps involved in this experiment necessitates a 'split-plot design.' The first step occurs before the process (extrusion) factors are involved, so the runs cannot be completely randomized. You indicate this by designating the mixture components as hard to change.