Mixture design with 11 ingredients, including 4 under study

[#1]

Hello everyone,

Not having found a similar case on the forum, I am seeking your help and opinions regarding my mixture design.

I aim to maximize the stability duration of a formula containing 11 ingredients, 4 of which (X1 to X4) I want to vary and study.

- X1 between 1 and 4%
- X2 between 3 and 8%
- X3 between 7 and 12%
- X4 between 0.4 and 0.8%
- X5 = 4%
- X6 = 2%
- X7 = 1%
- X8 = 2%
- X9 = 0.5%
- X10 = 0.8%
- X11 = water = sufficient quantity to complete the formula to 100%

The sum of the percentages of X1 to X4 ranges between 11.4% and 24.8%.

The sum of the percentages of X5 to X10 is 10.3%.

Water (X11) ranges between 64.9% and 78.3%.

The sum of the percentages of X1 to X4 and water has a fixed value of 89.7%.

Using JMP version 18, I attempted to create an optimal design.

I entered the factors X1 to X4 as well as the factor X11 (water) with their minimum and maximum values.

In the advanced options, I specified that the total mixture sum is 0.897.

Given that the objective is to study the influence of factors X1 to X4 and not water, does the approach I have followed so far seem appropriate?

Would another approach—where only X1 to X4 are defined as factors—have been possible?

The desired modeling is either a quadratic model or a Scheffé cubic model. The choice will depend in particular on the number of experiments to be conducted.

If water is defined as a factor, but its influence is not to be studied, is it possible to simply remove from the model's effect list all the effects that include water?

I hope the description of my case is complete enough.

As I have little experience with mixture designs, your help would be invaluable in designing the best possible plan.

[#2 · Accepted solution]

Hello @Kyle_Katarn.

For the situation you describe it sounds like there are two viable options here.

Option 1: Treat this as a true mixture design. With this approach you could enter X1-X4 and X11. You can adjust the mixture total as you described. If you take this approach, you would use the Scheffe special cubic model (removing the unusual ternary terms of the form X1*X2*(X1-X2) to save experiments). That model will have X11 in it and it should stay in it to keep the integrity of the mixture design and the way a Scheffe model works.

Option 2: Ignore the water. Create a design with just X1-X4 and declare them as continuous, not mixture factors. If you take this approach your model would just be the quadratic model. Since you have little experience with mixture experiments, this may be a better option for you to take.

[#3]

I agree with Dan.  His second approach would be fine. You don't actually have a mixture restriction since you can adjust the water to meet the 100%.

[#4]

Thank you both for your help and advice.

I hadn't thought of using continuous factors.

I agree, it seems to be the easiest way.