Correct use of a mixture design for formulation optimisation

DeepDormouse199 — Sun, 09 Feb 2025 15:47:03 GMT

Hey, I'm a chemist trying to optimise three components in a formulation but have some limitations on them.

All 3 = max 20% of formulation

A = 0.5 - 10%

B = 0 - 5%

C = 0 - 5%

B + C = min 2.5 %

However, they don't all need to add to 20% e.g. can have an experiment with 0.5% A, 0% B, 0% C and then can balance with X.

I've used a mixed design for this.

A, mixture, 0.025, 0.5

B, mixture, 0, 0.25

C, mixture, 0, 0.25

X, mixture, 0, 0.975

Linear constraint, B: 1, C: 1, > 0.125

Is this the best way to do this for a DoE? How could I compare designs looking at 2nd order interactions?

Thanks!

Re: Correct use of a mixture design for formulation optimisation

Victor_G — Sun, 09 Feb 2025 21:26:30 GMT

Hi @DeepDormouse199,

The conditions you have at the beginning of the post and your mixture design conditions settings are different (particularly the min, max ranges of the components + the linear constraint for B+C). Also the example you provided for the total amount is not consistent with your constraint definition (as having only A at 0,5% will not respect your constraint of B+C > 2,5%).
I will do an example with the first settings provided, hoping that these conditions are the ones you're investigating.

It seems your formulation does involve a solvant/QS that enables you to have a total content at 100%.

As the relative quantities of your components A, B and C are low compared to the solvant X of your formulation, you may not be interested in the role/impact of the solvant, but more likely to the effect of dilution/quantity of the components A, B and C. Moreover, as the sum of the quantities for A, B and C is not limited to a specific value, you may have two solutions :

Either design it as you intended to do, with a Custom mixture design and adding solvant X as a mixture factor so that the total quantity is 100%. The drawback of this approach is that you use one factor in which you're not particularly interested in (ratio/quantity of X, that you could deduce from other components quantities, and that is not supposed to "interact" with other components). This mixture approach, combined with a model with main effects and 2-factors interactions, may require a recommended number of runs of 20 (if you add all 2-factors interactions between any component and X) :

Resulting 20-runs mixture design :
Another option could be to design an I-optimal design with the platform Custom Designs for A, B and C, setting the two linear constraints (B+C > 2,5% and A+B+C<20%), and specifying a Response Surface Model with all main effects, 2-factors interactions and quadratic effects :

The default recommended number of runs for this design is 16 runs :

You can create a formula in the datatable to calculate the quantity of X you need to add to have formulations at 100%.

As these designs are created with different Factors type, you won't be able to compare the two options directly using the Compare Designs platform. But you can still generate various Mixture vs. Optimal factorial designs and compare the pros and cons in each design category.

I attached the two designs options mentioned.

Hope this answer will help you,

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Correct use of a mixture design for formulation optimisation

Re: Correct use of a mixture design for formulation optimisation