Determine equality constraints for related time/temperature variables in DOE

[#1]

Hi:

This seems like a dumb problem, but I can't seem to figure out how to build the constraints portion of a DOE model I'm trying to prepare.  It's a big recipe type optimization, but it runs through two distinct time/temperature windows.

The basis is that I have two (difficult to change) continuous variable time portions that each have an associated temperature range.  However, the second temp must be greater than the first.  In other words: Regime #1 can range from 15 to 60minutes, and the temperature can be (static) between 30 and 56°C.  Regime #2 can be from 15 minutes to 60minutes, at any temperature > Regime #1.  Regime#2 always follows Regime#1.  I can't just collate the experimental time variable and test different temperatures, because it is likely that recipe components will have optimal performance at distinct temperatures, and we need a model that can discriminate that.  It is very likely that running for x amount of time at a low temperature followed by y amount of time at a higher temperature will give the best performance.

I've gone through a few of Prof Goos lectures, but the temp regime dependence keeps throwing me.  FWIW, I'm using JMP 15.1, and I really don't understand scripting.

Thank you in advance for any advice you could offer!

Ted

[#2]

I did this in version 17 and it seems to work.  

Choose "RSM" design (since you said you are doing optimization)

Resulting Design

[#3]

(a small addition/clarification) The above will generate a design where Temp2 is >= Temp1. If you need Temp > Temp1, then you'll need to determine by how much Temp2 must be higher than Temp1, then put the negative of that number in the Linear Constraints section (example for 1° minimum difference below).

[#4]

OK, man was I overthinking this.  I had it in my head that I needed some way to actively associate the time stamp to the temp.  This works perfectly fine.  In my case, though, I think I need to set all four time/temp variables to 'hard?'  

This experiment is pretty involved - the time/temp are processing parameters, but we're trying to choose levels of 22 different additives at multiple pH levels.  These are the equivalent of cookie recipes - everything gets mixed and put into an oven for a period of time at a given temp, and then the oven temp gets turned up for the second period of time.  I have a lot of experiments, but very few ovens.  I'm worried about blocking effects, but at the same time, mapping all of these components introduces it's own variance. 

I'm thinking this means running as a d optimal design and planning for a follow up to augment the initial design, as running RSM pushes past my experimental budget.  Does this sound kosher?  Let me know if this is too much scope creep, and I'll repost as a new question!

In any case, thank you again, this was super helpful. 

Ted

[#5]

Hi @TedGHL ,

This sounds like a real challenge and very interesting for a DOE nerd like me!

A sequential approach is a good idea. (I'm very guilty of mostly presenting DOE as a one-step process - it is NOT).

You will almost certainly need to use optimal design, especially if you have hard-to-change oven time and temperature factors.

It is not totally clear from your description, but I assume that each "bake" in the oven will contain "cookies" with different levels of the additives. If that is the case, then the oven factors are your hard-to-change factors and it is a split plot situation where each bake is a "whole plot". The additive factors are easy to change and vary within each bake or whole plot.

I have covered an example of a split plot design in this presentation. It is a very different application (wind tunnel experiment) but hopefully you can see the same ideas might apply in your case.

It might be wise to get proper consultation from a DOE expert. There is only so much that we can do via JMP Community discussions. Your local JMP contact should be able to advise you on where you can get more in-depth help. 

I hope this helps,

Phil

[#6]

Thank you!  That video was a great follow-up and extension to Prof. Goos's introduction of the same response surface model!  I particularly like your 'all industrial experiments..." comment - it hits close to home.  My local jmp tech. support has been great, I'm just trying to diversify my use of resources - I'm generally trying to develop plausible models that we can compare/contrast for suitability.

[#7]

Here are my thoughts/questions:

1. I wonder how you "know" the the second temperature must be greater?

2. What are the response variables? Can the sample be measured between Regimes? Are you confident in the measurement system?

I agree with Phil in this seems like a perfect application of split-plot designs.  Though there are alternatives to how to setup and execute the design.  To describe it in lay terms, you have two sequential experiments.  I might suggest the first regime will create the first experiment (the whole plot).  This will be some type of factorial with potentially multiple samples for each treatment.  Subsequently, these samples will undergo a second experiment (subplot) another factorial which will result in multiple experimental units (the reason for multiple samples per treatment is to assess and separate within treatment variation (measurement, within sample and sample-to-sample)). This will provide a design with increased precision for the whole plot and subplot and provide excellent resolution across the two regimes.

I suggest you read:

Box, G.E.P., Stephen Jones (1992), “Split-plot designs for robust product experimentation”, Journal of Applied Statistics, Vol. 19, No. 1

[#8]

Good questions!  What I'm referring to as 'ingredients' actually include two classes of very different components.  One is a group of enzymes (each with known optimal temperature windows) and the other is a group of solution additives (salts, solvent additives, etc...).  Prev. OFAT experiments have verified the differing temp. optima.  As a practical matter, we can really only effectively run at multiple temperatures by adding thermal energy - our end device requirements won't allow for controlled cooling.  We considered just throwing the temp wide open in a single time window, but the concern  is that we would mask the contribution of enzymes that optimally function at low temperatures, but are deactivated at higher temperatures.  There is a solid hypothesis that the enzymatic effects are synergistic, so this is an attempt to expose such contributions.

Measurement between regimes is really not possible, as each experiment goes through a subsequent irreversible workup and evaluation.  We're evaluating 'recovery' of a cellular component through chemo-enzymatic treatment.  As for confidence in the measurement system... let's put that at a solid "kind of!"  Because the raw experimental material is fundamentally heterogeneous, the cellular component recovery is normalized against a standard treatment.  We can only run a half dozen or so experiments in parallel, so we're already deep into split plot country.  I've had problems in a very similar system in separating systematic variance between plots from the plot variable, and I suspect this variance is part of the recovery normalization.

Thank you very much for the advice and article recommendation, I'm trying to dive into a better understanding, but it's sometimes a bit of a slog for an old organic chemist trying to learn a few new tricks...

[#9]

No worries I'm an "O" too although not practicing.  I personally wouldn't trust OFAT results for anything, but that is my bias.  The split-plots I propose are not due to your typical restrictions on randomization.  They are purposeful restrictions (See the Box and Jones paper).  I usually create this manually in JMP as it has nothing to do with how hard the factors are to change.