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## Formulation DOE

I have to define DOE for  2 continuous factors and 1 qualitative factor with 2 levels (A and B)  tested at different concentrations. My constraints are the following:

- if the A concentration is >0 then B=0

- if the B concentration is >0 then A=0

The objective is to optimize the formulation.

I try to do a custom design but I didn't find the way to define my constraint.

The optimum should select the best level for the qualitative factor (A or B) and the concentration level between the range -1 to 1.

I will appreciate your help for this topic.

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## Re: Formulation DOE

One additional approach that could be used if the ranges are very different is to use a coded range from -1 to 1. When you encounter a treatment at (A,-1), it will mean you run A at 10. If it is (B,-1), you run B at 800. You would need to keep in mind the translation of the results back to the appropriate scales.

Dan Obermiller
7 REPLIES 7
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## Re: Formulation DOE

Do you need a constraint? It seems that you could have two factors: a categorical factor for ingredient (A or B) and a continuous factor for concentration. This way, you would never have A and B in the same formulation.

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## Re: Formulation DOE

I forgot to add that the concentration range is different between both ingredient.

In fact my constraint are the following:

if CA>0 then CB=0

or if CB>0 then CA=0

where CA (resp. CB)  is the concentration of the A (resp. B) ingredient.

When I try to implement this cosntraint, I have the following message:

"The selected terms in the Model outline are linearly dependent on the previous terms. Please press the Remove Term button and try again."

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## Re: Formulation DOE

Please see this discussion. Define concentration to have a wide range that spans both A and B. Set one constraint so that when factor set to A, then exclude one portion of the concentration range. Do the same for factor set to B.

I think that it should answer your question, but if you still need help, let us know.

Learn it once, use it forever!
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## Re: Formulation DOE

Thanks a lot for your response, it is an interesting approach to solve my issue (I was a little bit desperate). Just a comment and maybe a piece of advice from you, how do you manage when A concentration range (10-15) is really different than the B concentration range (800-1000). The number of runs between A and B are really unbalanced (3 runs with A and 17 with B). It seems to me strange. Here below an example of code.
What do you think?
Thanks again for your prompt response.

DOE(
Custom Design,
{Add Response( Maximize, "Y", ., ., . ),
Add Factor( Continuous, -1, 1, "X1", 0 ),
Add Factor( Continuous, -1, 1, "X2", 0 ),
Add Factor( Categorical, {"A", "B"}, "X3", 0 ),
Add Factor( Continuous, 10, 1000, "Conc", 0 ), Set Random Seed( 2111591509 ),
Number of Starts( 1453 ), Add Term( {1, 0} ), Add Term( {1, 1} ),
Add Term( {1, 1}, {2, 1} ), Add Term( {2, 2} ), Add Term( {1, 1}, {3, 1} ),
Add Term( {2, 1}, {3, 1} ), Add Term( {1, 1}, {4, 1} ),
Add Term( {2, 1}, {4, 1} ), Add Term( {3, 1}, {4, 1} ), Add Term( {4, 2} ),
Add Alias Term( {1, 1}, {2, 1} ), Add Alias Term( {1, 1}, {3, 1} ),
Add Alias Term( {1, 1}, {4, 1} ), Add Alias Term( {2, 1}, {3, 1} ),
Add Alias Term( {2, 1}, {4, 1} ), Add Alias Term( {3, 1}, {4, 1} ),
Set Sample Size( 20 ), Disallowed Combinations(
X3 == 1 & Conc > 15 | X3 == 2 & Conc < 800
), Optimality Criterion( 2 ), Make Design, Set Run Order( Randomize ),
Make Table}
)
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## Re: Formulation DOE

Custom Design did not converge for me, so I cannot inspect the resulting design. I believe your observation about the imbalance of the A and B levels.

I removed the constraint and made the design, enabled Simulate Responses, made the data table, manually changed 505 to 12.5 and 1000 to 15 for A and changed 10 to 800 and 505 to 900 for B, and then simulated the response.

I suspected that this data would not actually support estimating the model parameters due to a singularity. It turns out that the situation isn't that bad, it is just really bad. The desired concentration levels for A and B produce very strong correlations among the estimates for X3, Conc, and any interaction with X3. This correlation in turn leads to unstable estimates. I included strong non-linear effect of Conc and a large X3*Conc interaction in the simulated response but they were not detected at all.

I attached this data table for you to inspect.

Learn it once, use it forever!
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## Re: Formulation DOE

Thanks for this additional information. So, if I well understood, it is not really recommanded to use this doe tip when range are really different.
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## Re: Formulation DOE

One additional approach that could be used if the ranges are very different is to use a coded range from -1 to 1. When you encounter a treatment at (A,-1), it will mean you run A at 10. If it is (B,-1), you run B at 800. You would need to keep in mind the translation of the results back to the appropriate scales.

Dan Obermiller
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