Custom design factor constraints

[#1]

Good morning,

I create a DOE containing 2 answers (A and B) and 2 factors (C and D). The two factors are concentrations of, say, 5 to 9 for C and 10 to 25 for D. I would like to create a plan with constraints on the factors, the fact that the D/C ratio must be equal to or greater than 4. How should I realize this. Additionally, what types of factors should I encode? continuous or discrete? I am a little lost.

[#2 · Accepted solution]

Hello @SeanCor,

Welcome to the JMP Community!

To better understand how to manage constraints in the creation of DoE with JMP, I recommend reading the blog article by @Jed_Campbell : Demystifying Factor Constraints - JMP User Community

Regarding your questions, some answers, questions and comments:

• If the concentrations of factors C and D can vary continuously, then they are most likely continuous factors. “Discrete” type factors can only take a few ordered values ​​defined over the entire possible range. More info on factor types here: Factors (jmp.com)
• What is your goal with this plan? Study the effects of these two factors quantitatively? Have a predictive model? Do you want to analyze possible interaction or quadratic effects of these factors? These questions are only a starting point, but will allow you to better understand which type of plan is most suitable, and make good decisions on the construction and analysis of the plan.
• Concernant votre contrainte relationnel entre C et D, vous pouvez l'ajouter dans plusieurs plateformes de construction de plan. J'ai pris l'exemple ici avec la plateforme Custom Design :
  1. Enter your answers, factors and ranges:
     
  2. Dans la partie "Define Factor Constraints", cliquer sur "Specify Linear Constraints". Comme vous souhaitez que le ratio D/C soit supérieur ou égal à 4, une remise en forme de cette équation donne :
     

     4 * :C + -1 *  <= 0

     Vous pouvez donc entrer cette contrainte dans la partie dédiée :
     
  3. Define the type of effects you want to estimate using your plan. In the example here, I added the interaction between C and D, the main effects of C and D as well as the intercept are entered by default:
     
  4. You can then define a number of replicates, central points, and you also have a certain flexibility on the number of runs to carry out, then you can start generating the “Make Design” plan. In this example, I chose 3 central points to be able to evaluate whether a quadratic effect is possibly present and check the fit of the model with a Lack-Of-Fit test, for a recommended number of runs equal to 12:
     

Note that due to this relational constraint, your factors C and D are no longer independent. Estimating the individual effects of each may be more difficult.

It is important to clearly distinguish whether this relational constraint is physically necessary to carry out the experiments (in which case it must be taken into account in the plan to avoid generating experiments that are physically impossible to carry out or dangerous), or if it is a constraint "preferential", where you think you have good results. In the second case, it is better to remove this constraint to avoid any bias at the start and to have the most neutral and complete view of your experimental space possible. To understand where, how and why you have positive results in your experimental space, you need to also have negative results.

Finally, the question also arises as to whether the individual concentrations of your factors constitute the information of interest, or whether it is rather the ratio which seems to be of interest. In the second case, you could combine factors C and D into a factor (ratio), which you would make evolve over a certain range, and which would not require a design of experiments.

Here is the script used for generating the constrained experimental design mentioned here:

DOE(
	Custom Design,
	{Add Response( Maximize, "A", ., ., . ), Add Response( Maximize, "B", ., ., . ),
	Add Factor( Continuous, 5, 9, "C", 0 ), Add Factor( Continuous, 10, 25, "D", 0 ),
	Set Random Seed( 1047040825 ), Number of Starts( 55959 ),
	Add Constraint( [4 -1 0] ), Add Term( {1, 0} ), Add Term( {1, 1} ),
	Add Term( {2, 1} ), Add Term( {1, 1}, {2, 1} ), Center Points( 3 ),
	Set Sample Size( 12 ), Optimality Criterion( "Make A-Optimal Design" ),
	"A-Optimality Parameter Weights"n( [1 1 1 1] ), Simulate Responses( 0 ),
	Save X Matrix( 0 ), Make Design, Set Run Order( Randomize ), Make Table}
);

You will also find in PJ the plan generated to illustrate this response (with addition in the table of a calculated column D/C so that you can check that all the experiments generated by this method respect your constraint).

Hope this answer helps you,

[#3]

Good morning,

Thank you very much for this answer which helps me a lot. It is in fact the creation of a model with the aim of predicting output from input in order to offer packages, that is to say different possibilities depending on desires. Let us assume that A is a concentration yield and B is a percentage. I would like to predict the answers A and B based on the different concentrations of factors C and D. However, carrying out the model as you describe it to me results in very precise decimal point concentrations which are difficult to achieve in practice. In addition, the constraint requires the ratio to be equal to or greater than 4 and we only have ratios ranging from 4 to 4.91, no more, is this normal?

Thank you

[#4]

Hello @SeanCor,

Concerning the very precise factor values, this is a consequence of the algorithm used to create the optimal plan. See the answers on this topic:

Solved: Re: How are odd factor settings in D-optimal RSM generated - JMP User Community

Solved: Re: Random decimals incorporated in mixture screening design - JMP User Community

Concerning the ratio constraint, the latter is very demanding: if you take the extreme values ​​of D and C (max for D, min for C), you arrive at a maximum ratio equal to 25/5 = 5. So it is normal to find ratios between 4 and 5. Can this ratio constraint be modified, or the ranges of the factors enlarged (lower values ​​for C, larger values ​​for D)?

Concerning your prediction objective, a Space-Filling type plan is perhaps more appropriate, because it allows you to distribute the points in your experimental space and allows great flexibility in the modeling (in particular with Machine Learning algorithms) and the number of points required (no minimum, but you must choose a reasonable number of points).

You can build it in the Space-Filling platform available in "DoE", "Special Purpose", "Space-Filling":

You will find in PJ an example of a Space-Filling plan built with 20 experiments with your factors, responses and the ratio constraint.

A comparison of the distribution of points in the experimental space according to the type of plan chosen:

Hoping that this additional answer helps you,