A non-linear constraint on mixture DOE

[#1]

I want to add a non-linear constraint on the DOE mixture-process problem.

For examle, lets say that we have a mixture of A, B and C with limits on amount of each given [amin, amax], [bmin, bmax], [cmin, cmax].

Now there is the fourth component in the mixture (water) which is used to control the density of the formulation. We put more water to lower the density and decrease the amount of water to increase the density. There is a bound on minimum amount of water which the formulatin needs to have otherwise the process will not be able to mix the formulation. So i am handling the amount of water as a density process variable which has to follow the following constraint:

Density>(1+A*wa+B*wb+C*wc)/(A/da+B/db+C/dc+(A*wa+B*wb+C*wc)/dw),

where, wa, wb and wc are constants. A, B and C refer to mass fraction of each component.

da, db ad bc and dw are also constants.

How can i make sure that the DOE which is designed can take into account such a constraint.

Any help will be appreciated.

PS. i am mostly a jmp newbie so not very adept at scripting. But i have good programming background so i should be able to handle scripting if needed to achieve this.

[#2]

I would suggest you try the custom designer (first choice in the DOE menu).  

The Custom Design Help is here.

And an example of a mixture design with a process variable (not as complex as your process variable) is here.

[#3]

Thanks so much for the response.

I am using the Custom designer from the DOE menu. I just dont know how to add the non-linear constraint. Any help on that will be appreciated.

[#4 · Accepted solution]

To my knowledge, JMP's DOE only allows linear constraints. As you use JMP's Custom Designer, there is a location to add a LINEAR constraint. You may wish to think about this problem in some other way to remove that nonlinearity. Typically, you try to linearize the constraint. This link discusses some approaches.

http://www.academia.edu/7919285/Using_JMP_to_Create_Experiment_Designs_with_Non-Linear_Constraints_T...

Perhaps you could take an iterative approach to the design: create linear constraints that are as close to your nonlinear constraint as possible. Create the design. If a trial appears that is not possible, remove it through disallowed combinations. You may need to iterate a few times to keep removing points that don't fit your true nonlinear constraint, but perhaps this approach would work.

Maybe someone else could think of some other approach.

[#5]

Thank you, Dan.

This helps. I will try to see how i can linearize my constraint.

I wonder why the 'Disallowed Combination script' for non-linear constraints only works for non-Mixture type problems? (Did i understand the paper correctly?)

[#6]

This is not a direct answer to your question. I just want to mention for your awareness that there is a new book (2016) by Ron Snee and Roger Hoerl about mixture designs. It is called "Strategies for Formulations Development: A Step-by-Step Guide Using JMP." You can find it here: Strategies for Formulations Development. It inclides a section with three chapters devoted to the topic of constraints in mixture designs.