Accepted Solutions
My next set of thoughts/questions. My first thought is that you'll want to plan several iterations rather than one big experiment...
1. First do you understand the noise (ingredient quality, contaminates, measurements, etc.)?
2. Of course you'll need to anticipate interactions. I suggest you rank order your predicted model effects to at least 2nd order.
3. Is there any other advantages to 1 B over another or to the mixture of B's (practically: cost, availability, ease of use, shelf life, etc.)? Are there interim response variables that can be measured that correlate to the final product measurements particularly related to B?
4. How do you know Note 2?
5. The type of packaging...is it dependent on the final formulation (e.g, is the formulation acidic or does it need to be sealed or kept cool)? Perhaps you should first figure out formulation and then experiment with packaging?
The go to book on Mixtures is Cornell "Experiments with Mixtures". There are lots of ways to proceed, but it is challenging when we don't know what type of reactions might occur in the formulation and what are the response variables.
There are 3 strategies I have used regarding mixture type designs over and above simplex or extreme vertices.
1. Component Proportion: The quantity of the "batch" is held constant and the active ingredients total is fixed. For example, if the batch size is 19 liters (including A) and there is 1 liter of active ingredients (I), so total (T) is 20 and I: A+B1+B2 =1.
This is a three-component mixture design and the 1st order design matrix is:
B1 | B2 | B3 | Total | Ratio |
1 | 0 | 0 | 20 | 19:1 |
0 | 1 | 0 | 20 | 19:1 |
0 | 0 | 1 | 20 | 19:1 |
.33 | .33 | .33 | 20 | 19:1 |
The changes in the response variable are a function of the 4 combinations (proportions) and is not influenced by the amounts of each B.
2. Mixture Amount Factorial: T and I are varied at 2 levels (T= 20 & 25) and (I = 1 & 2).
This is a factorial design and the resulting design matrix is:
W | A | Total | Ratio (T/I) |
-1 | -1 | 21 | 120:1 |
1 | -1 | 26 | 25:1 |
-1 | 1 | 22 | 20:2 |
1 | 1 | 27 | 25:2 |
In this strategy the changes in the response are a function of the changing amounts of ingredients with varying proportions
3. Mixture + Factorial: A design where processing factors (X1 and X2 for example packaging) are manipulated factorially and the simplex design (Z is A or B's) is replicated at each factorial combination
What about this design?
Here is the script to set up the Custom Design platform as I did:
DOE(
Custom Design,
{Add Response( Maximize, "Y", ., ., . ), Add Factor( Mixture, 0.6, 1, "A", 0 ),
Add Factor( Mixture, 0, 0.4, "B1", 0 ), Add Factor( Mixture, 0, 0.4, "B2", 0 ),
Add Factor( Mixture, 0, 0.4, "B3", 0 ), Add Factor( Mixture, 0, 0.4, "B4", 0 ),
Add Factor( Mixture, 0, 0.4, "B5", 0 ), Set Random Seed( 343091112 ),
Number of Starts( 305 ), Add Term( {1, 1} ), Add Term( {2, 1} ),
Add Term( {3, 1} ), Add Term( {4, 1} ), Add Term( {5, 1} ), Add Term( {6, 1} ),
Add Term( {1, 1}, {2, 1} ), Add Term( {1, 1}, {3, 1} ),
Add Term( {1, 1}, {4, 1} ), Add Term( {1, 1}, {5, 1} ),
Add Term( {1, 1}, {6, 1} ), Add Term( {2, 1}, {3, 1} ),
Add Term( {2, 1}, {4, 1} ), Add Term( {2, 1}, {5, 1} ),
Add Term( {2, 1}, {6, 1} ), Add Term( {3, 1}, {4, 1} ),
Add Term( {3, 1}, {5, 1} ), Add Term( {3, 1}, {6, 1} ),
Add Term( {4, 1}, {5, 1} ), Add Term( {4, 1}, {6, 1} ),
Add Term( {5, 1}, {6, 1} ), Set Sample Size( 42 ), Simulate Responses( 0 ),
Save X Matrix( 0 ), Make Design}
);
The minimum number of mixtures is 21. I used the default number of 42. The choice is yours, of course.