DOE Mixtures with disallowed combinations of components

[#1]

Hi,

I have read one of the documents in the site (7 Component Mixture Design with Additional Constraints)

I have a similar problem; I have one mixture component (A) for which i have three different types (A1, A2 and A3). I have similar constraints to what include in your paper (min/max % of A as a whole). However i also want to evaluate mixtures with 

* only A1 or A2 or A3 within total limits min/max for A

* a combination of A1+A2 or A1+A3 or A2+A3 within total min/max for A

Is there a way to put these constraints in JMP? I have found that the disallowed combination filter does not work for continous mixture factors so i dont really know how to do this.

I suppose a way around is to do 3 different designs (A1+A2) and (A1+A3) and  (A2+A3) but i was wondering if there a way to do only 1 design with those constraints.

Any suggestions?

[#2]

You should be able to apply a linear constraint to mixture designs (e.g. to make it so that the 3 "A" components cannot exceed 70% of the total mixture, set A1 + A2 + A3 < 0.7).  You would just need to specify all 3 components.  It gets a bit trickier if you want to make it so only 2 of the 3 are used in any mixture.  One idea would be to only specify 2 generic "A" components and then include a categorical factor that specifies *which* substances are A1 and A2.  Here's what I mean:

[#3]

Hi Cameron,

Thank you for your reply!

I saw a similar reply to yours in a different discussion. So I had a go at it in JMP. I put a few screens shots below to

clarify (factors, constraints, model and design).

The dummy variables (1&2) are only there as you said for the combination of component A (I want only two types of A at the same time in the mixture).

Then, i have also defined a "allowed combination of A" categorical variable where i have the 3 possible combinations of component A (A1+A2, A1+A3, A2+A3).

I have selected only main factors just for the sake of the exercise. 

I get a design and yes, in this case, with the categorical value I can see which 2 A components I will put in the mixture. However, this design is not really telling me anything about how I have to combine the different quantities for A (1,2 or 3) for each experiment. For example, for run 1. I will only use A1 and A2 to a total of 33% but i dont get any information about how much of A1 or A2 I should put in that experiment.

Do you have any suggestion?

Thanks

[#4]

The intention in my suggestion was that Dummy 1 and Dummy 2 would correspond respectively to the first and second "A" component in the categorical factor Combination.  So, row 1 indicates 0% A1 and 33% A2, row 4 is 36% A1 and 0% A2.  Since you only have main effects, you're not going to get blends of Dummy 1 and Dummy 2 in your example.  Add an interaction between Dummy1 and Dummy2 and you should see some blends.

[#5]

Hi Cameron,

Thank you again for your reply!

So Dummy1 is component A1 and Dummy2 is component A2

Then from that design table i get 

                 A1        A2         A3       Combination

Run1       33%      0%       0%        A1+A2

Run4       36%      0%       0%        A1+A2

Run7       0%      35.64%   0%       A2+A3

Run14     35.64%  0%       0%       A1+A3

etc..

But when&how do I get to put some component A3 then? 

in Run7 and A1 (0%) and

in Run14 as A2 (0%)?

Is this what you mean? or do I get this wrong?

[#6]

No, dummy 1 and dummy 2 are whatever components are in the row for Combination.
In row 2, Combination is A2+A3. That means Dummy1 will indicate how much A2 for that row, and Dummy2 will indicate how much A3. In that row, A1 is 0% because Combination indicates it is not part of the mixture, A2 = 0% (Dummy1), and A3 = 33% (Dummy2).

For row 7, it’s the same thing except now A3 = 35.6% and A1 and A2 are 0%.

In short, if Combination = A1+A2 then Dummy1 is A1 and Dummy2 is A2, if Combination = A1+A3 then Dummy1 is A1 and Dummy2 is A3, and if Combination is A2+A3 then Dummy1 is A2 and Dummy2 is A3.

[#7]

Asier,

In reading your requirements on constraints for A1, A2, and A3

"* only A1 or A2 or A3 within total limits min/max for A" and

"* a combination of A1+A2 or A1+A3 or A2+A3 within total min/max for A"

I found that the original ranges and constraint in my example 

A1: 0 to 0.36

A2: 0 to 0.36

A3: 0 to 0.36

with constraint 0.2 <= A1 + A2 + A3 <= 0.36

yielded a design (supporting 2nd order model, with 56 trials) that contains trials with only A1, A2, or A3, as well as trials with the three pairwise sums of A1 + A2, A2+ A3, and A1 + A3, i.e. the case that A1, A2, or A3 = 0.

Separate constraints on the pairwise sums were not required.  The key I believe is allowing zero as the low limit for each components range.  That may not be a guarantee that all three pairwise sums show up, but it worked.  I hope this helps and that you can get your design to work.

Thanks for following my example and for your question. 

Good luck,

Tom

[#8]

Hello Tom,

Thank you for your reply!

Indeed, I can use your example as guidance (and I did). I put below, some of the screenshots to help the discussion.

In this case, i have 1 ingredient A with 3 different subtypes (A1, A2 and A3). If possible I would like to evaluate combinations of maximum 2 A ingredients (A1+A2, A1+A3, A2+A3) and obviously mixtures with only one type of A.

The total sum of ingredients in this case is only 44.34% (the other ingredients are not changed).Something similar to another of your examples. 

I only put main factors in the model. I just wanted to know what I would get. In this case, for some reason, I only get mixtures with only 1 A ingredient (out of the 3 A ingredients that i have) . Do you know why we get this type of design? 

Another question that I have (and this is more fundamental i believe) is about the correlations. From the correlations map, I would understand that this DOE is not great. Is that right? What can we do in general in this type of situations?

Similary, with the Power, I have the feeling that this is not great either? (Although I have to admit that I dont fully grasp the concept of power :-(

Again, what can we do in this type of situations?

Thank you again for any help!

[#9]

The correlation of the estimates and the power are related. The correlation inflates the variance of the estimates. The inflated variance decreases the power.

This behavior (high correlation of estimates, low power) is characteristic of mixture designs because of the ever-present constraint that the components must sum to 1. This behavior is exacerbated by restricting the range of the component (not full 0-1 range) and by adding further constraints among the components.

[#10]

Thanks Mark for responding to the power and correlation questions.
As for the design not having two As, I believe part of the problem is the ranges of all the factors particularly the non-A factors may be overly constraining the design space. The last four factors min and max ranges sum to 0.06 and 0.1604 respectively. When added to the 0.5566 proportion being held constant, the total range available for As is 0.283 to 0.3834, AND the max of the As is 0.36 making the actual sum of the mins of the last four factors 0.0834 AND when the minimum A of 0.33 is used the max sum of the last four is 0.1134. What we have is a design in the sum of the As or in the sum of the other 4 factors is only 0.03 wide. In my example highly constrained design I had at 2 "slack" variables ranging from 0 to 1 even though I knew we will never hit those levels. But these wide ranges give the As some room to have a broader range than the current situation we have here. I will try to work out a design and respond later today.
Tom