Linear constraints in mixture design

Principles77

I am facing difficulties in implementing the constraints for my mixture design. The design consists of water as a filler and four additives, namely A1, A2, A3, and A4, which have specified value ranges. I need to impose a combination constraint that allows a maximum of two out of the three factors, A2, A3, and A4, to be used simultaneously. A1 and water are not important in constraining, but it is only possible to add either A2 with A3, A2 with A4, or A3 with A4.

Can you provide guidance on how to effectively implement these constraints in the design?

 

 

drdrf

My first question here is whether this experiment really needs to be designed as a mixture DOE.  Is water expected to only act as an inert filler?  Are the very small differences in the level of water as A1-A5 levels change expected to impact the properties of the resulting formulation?  If the answer to the first question is yes and the answer to the second question is no,  I would be inclined to treat A1 to A5 as independent continuous variables rather than mixture variables and do a standard factorial or custom design.  The consequence of this will be that the level of water will not appear the the predictive formulation for whatever properties are of interest, but if the above assumptions are correct that should not matter.

 

If you are happy to proceed in this way, the disallowed combinations filter could then be used to give a design where at least one of A2, A3 or A4 are always set to zero.

 

The only thing left to do would be to manually reset the very low levels in the resulting data table to zero (e.g. 6.028e-6 for A4 in run 4 should be reset to zero.

It might also be a good idea to add an additional disallowed combination filter to specify that A2, A3 and A4 can't all be very low at the same time.  This will avoid runs that contain zero or very low total additive levels.

 

 

There are probably more elegant solutions but this should work.  As always it would be worth testing the design with some simulated data before running the experiment to be sure the experiment is capable of giving useful analysis.