How to set the range for mixture components and linear constraint in Extreme Vertices Design

[#1]

Hi, I am trying to do a screening of mixture with 12 components using extreme vertices mixture design. I have the specific range to test: 

X1: 0.541 to 0.784

X2: 0 to 0.147

X3: 0 to 0.08

For X4 to X 12, I am testing them using the same range between 0.216-0.324 to find which of them are suitable for my formulations. However, when I tried to key in the lower limit, only 2 components are possible. From X6 onwards, it did not allow me to set the lower limit at 0.216, but reset to 0 instead. I do not understand why is it so? I have no problem in setting the upper limit for all.

Hence, I tried to set the linear constraint as follow:

I am not certain if my linear constraint is correct 

I generated a subset table with 25 trials and realize that most runs will have 0 value in the multiple components. Also component X3 is only tested either 0 or 0.08. I felt that something is not quite right. Is there any advice for me to improve the design? 

Thanks in advance! 

Glenna

[#2 · Accepted solution]

Hi @LatinLinkages95,

Welcome in the Community !

Some answers regarding your questions :

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@LatinLinkages95 wrote:

For X4 to X 12, I am testing them using the same range between 0.216-0.324 to find which of them are suitable for my

formulations.

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I'm not entirely sure about what is your objective.

1. Would you like to have all mixtures from X4 to X12 in the formulation ?
2. Only one ?
3. Or between one and several (but not all) ?

If the answer is 1, a mixture design may be appropriate, but you have to check your ranges and design choice.

If the answer is 2, a Custom design may be more relevant, with a categorical factor "Component choice" (from X4 to X12) and another mixture factor with the possible amount in the mixture.

If the answer is 3, a mixture design is possible, but with an emphasis on screening and with a simple model, as the number of mixture factors is high. ABCD design, Extreme Vertices or the use of Optimal design may be interesting to consider. More infos on Mixture designs here : Overview of Mixture Designs

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@LatinLinkages95 wrote:

However, when I tried to key in the lower limit, only 2 components are possible. From X6 onwards, it did not allow me to set the lower limit at 0.216, but reset to 0 instead. I do not understand why is it so? 

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 It seems you have a problem in your constraints resulting in an impossible design if you want all factors from X4 to X12 with a lower limit at 0,216. Since all factors should sum up at 1, 9 factors with a lower limit at 0,216 results in 1,944 which is not possible (and there is also the lower limit of X1 that I didn't take into account). I would recommend to set the lower limit of all these components to 0 and use the linear constraints to specify that the sum should be more or equal to 0,216.

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@LatinLinkages95 wrote:

Hence, I tried to set the linear constraint as follow:

I am not certain if my linear constraint is correct 

---

Your linear constraint implies that the sum of the mixture factors from X3 to X12 should be between 0,216 and 0,324. I'm not sure why X3 is included in the constraint based on what you describe in your context.

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@LatinLinkages95 wrote:

I generated a subset table with 25 trials and realize that most runs will have 0 value in the multiple components. Also component X3 is only tested either 0 or 0.08. I felt that something is not quite right. Is there any advice for me to improve the design? 

---

Your design does what you specified, it explores the edges and vertices of your experimental space (so mostly at the lower and upper levels of your factors), and respecting your linear constraint. Since you have a lot of factors and this design has an emphasis on vertices, most of the factors will be at 0 for each runs, it will explore mostly the borders/edges/vertices of the design, not the inside with more sophisticated mixtures involving more components.

If you're not satisfied with this design, perhaps you could describe more what were your objectives, expectations and experimental budget constraints ?

I hope this answer will help you,

[#3]

Hi Victor,

Thank you for helping me to clear the doubts that I have. My objective is to screen the potential inclusion of X3 to X12 components based on the range values that I have indicated into a recipe formulation; some but not all. I hope to be able to analyze the experiment results with the recommended fit model and the gradients of best fit line to see which are the ones to be able to include (i.e., positive values) or exclude with negative values. I have 4 responses (Y). After which, I wish to run a RSM to optimize the formula. So far my only constraint is the number of runs I could go for.

I had an overlooked on the X3 component. I would like to include X3-X12 as a mixture range between 0.216 - 0.324. There are 2 things about the linear constrains that I am uncertain about:

(1) Based on the example of the mixture in mixtures design from the link info, there is a need to set a negative value for the linear constraint. I tried the following:

However when I try to apply this to my design, it prompted that is not possible. Will this affect my results outcome? Is there a need to include the negative values constraint?

(2) Based on the linear constraint from my 1st post, I could not generate the Ternary Plot. Hence, I tried to set 2 more linear constraint and able to view the Ternary Plot. Is this appropriate to set for my experiment design?

My experiment design requirements are:
Combine of X1 and X2 to be in the range of 0.676 - 0.784
Combine of X3 to X12 to be in the range of 0.216 - 0.324

(3) Based on my linear constraint, I would have expect there could be in between values for testing X3 as I could see that for X4 to X 12, there are suggested values (i,e., 0.136 and 0.244) between the range of 0.216 - 0.324. So I thought it could be the way I've set the constraint that doesn't allow X3 to be so.

Another setting that I could not find much information about is the "degree" to set for the extreme vertices design. Since I have 12 components, should I set the degree to 11? Or is there a limit to which the degree I could go for? So far I only seen 4 degree with 5 factors model.

Once again, thanks in advance!

Glenna

[#4 · Accepted solution]

Hi @LatinLinkages95,

Ok thanks for the clarification, I understand better your objectives and needs.

An Extreme Vertices design might be helpful for the first part (assessing the influence and relative importance/benefit of each mixture factors on the responses), for the second part you can use the Augment Design platform, select the important mixture factors detected in the first part and the responses, and either use a model-based approach (add runs to the design in order to fit a specific model, like Mixture Response Surface model) or a model-agnostic approach (add space filling points to a design in order to homogeneously improve the coverage of your experimental space) that will let you some flexibility in the modeling part (regression models, Machine Learning models, ... to fit a Response Surface).

About your questions :

1. You can add the constraints the way you intended to do it in your first post (with positive values for factors and inequality constraints >= and =<), it should work, no need to add the constraints with negative values. The negative values writing is used by JMP to handle any inferior linear contraints. In a simple example with 3 mixture factors, I specified that X1 + X3 should be between 0,5 and 0,8 and here is how this constraint is translated by JMP in the table :

   {1 * :X1 + 1 * :X2 - 0.8, -1 * :X1 + -1 * :X2 + 0.5}

   You can see that for the inferior linear constraint, JMP did rewrite it like - X1 - X2 =< -0,5 (or -X1 -X2 + 0,5 =< 0), it's just a way to have the same JMP scripting language writing convention as for the superior linear constraint with the same inequality sign, but you don't need to do it yourself. 
   
   Adding the same constraints with positive and negative values is helpful if you're in a situation of mixture of mixtures, where some components should sum up to a certain percentage (like A+B = 0,8), and adding the rest of the mixture components sum up to 1 (or 100%, like A+B+C+D).
2. Based on your requirements, I would set up the linear constraints like these :
   
3. X3 has a very narrow range (0 - 0,08) so that may explain why other levels are not proposed. Are you sure about specifying an individual (narrow) range for X3 in addition to the linear constraint for X3 to X12 ? Same question for X1 and X2 with their individual constraints and their linear constraint.
   Also you're right, the degree of the design is an indication about the maximum number of mixture factors that can be involved in the experiments (from 1 to the degree). But since your primary focus is evaluating the influence and importance of each mixture factors and the subset selection had a D-Optimality criterion (and you have a limited budget compared to the number of possible combinations), even with a higher degree the subset proposed with a 25-runs budget will put high emphasis on experiments with few components :

Here is the script to generate this design :

DOE(
	Mixture Design,
	{Add Response( Maximize, "Y", ., ., . ),
	Change Factor Settings( 1, 0.541, 0.784, "X1" ),
	Change Factor Settings( 2, 0, 0.147, "X2" ),
	Change Factor Settings( 3, 0, 0.08, "X3" ), Add Factor( Mixture, 0, 1, "X4", 0 ),
	Add Factor( Mixture, 0, 1, "X5", 0 ), Add Factor( Mixture, 0, 1, "X6", 0 ),
	Add Factor( Mixture, 0, 1, "X7", 0 ), Add Factor( Mixture, 0, 1, "X8", 0 ),
	Add Factor( Mixture, 0, 1, "X9", 0 ), Add Factor( Mixture, 0, 1, "X10", 0 ),
	Add Factor( Mixture, 0, 1, "X11", 0 ), Add Factor( Mixture, 0, 1, "X12", 0 ),
	Set Random Seed( 119094230 ), Add Constraint(
		[-1 -1 0 0 0 0 0 0 0 0 0 0 -0.676,
		1 1 0 0 0 0 0 0 0 0 0 0 0.784,
		0 0 -1 -1 -1 -1 -1 -1 -1 -1 -1 -1 -0.216,
		0 0 1 1 1 1 1 1 1 1 1 1 0.324]
	), Mixture Design Type( Extreme Vertices, 8 ), Find Subset( 25 ),
	Simulate Responses( 0 ), Set Run Order( Randomize ), Make Table}
)

And attached you'll find the datatable.

Hope these answers may help you better understand,

[#5]

Hi Victor,

Thanks a lot for your suggestions and explanation. It really makes all things much clearer for me. I would take into consideration on your suggestions for the part 2. 

Regarding the constraints for X3, I have some research info that it might not be appropriate to go beyond. I think I will stick to it if it does not affect my whole experiment. Similar to X1 and X2, I have to work within these reference ranges. 

Thanks very much for the suggested linear constraints setting, script and data table. Really appreciate for all the help and guidance provided!