Model process parameter interactions with process time

[#1]

Hello everyone, 

I have a 3 step process with 7 factors A, B, C, D, E, F, G and a single response Y.

Step 1: A and B are some continuous machine settings and C is the time for which those settings are applied

20 <= A <= 100

300 <= B <= 700

30 <= C <= 180

Step 2: D and E are continuous factors for a treatment that is applied to the samples after step 1. 

Step 3: F and G are discrete numeric factors for a post conditioning that is applied done to the samples after step 2. 

I am trying to estimate main effects and 2 way interactions within the above limits as well as 3 way interactions for step 1. I am also interested in what happens if I omit step 1 altogether, which is the giving me the headache here. My naive approach was extending the limits of time C in step 1 to 0, i.e. 0 <= C < 180, but that of course creates nonphysical runs where C = 0 but A, B are at some factor level. It feels wrong to me to have such runs, but then again, that would probably result in A and B being insignificant factors , but the interactions with C might then be significant, which does not sound to far off from the physical reality of the problem.

I feel like I am missing something obvious here in how to construct an appropriate design for such a problem and I'd be grateful for any help I can get!

Many thanks! 

[#2]

Welcome to the community.  I don't have enough information about your situation to suggest an appropriate strategy, but I do have some thoughts/questions:

1. If you really want to omit step 1, can you measure the current output of step 1?  Can you treat the output of step 1 as a covariate?  If you want to omit it, why are you doing it?  What is being machined?  I'm trying to guess what you are manipulating and why there might be a time effect? Is it a lapping/honing operation (surface finish)?  Or a curing operation? 

2. Can anything be measured after each step in the sequence?

3. Have you tried directed sampling vs. DOE?  You may be able to use a nested sampling plan to provide clues.

4. Is the measurements system for the response adequate?  Has it been studied?

5. You say your are interested in the factorial effects from step 1, but no mention of what effects you want from steps 2 or 3?

6. Sequential step experiments are ideal for split-plot designs.

[#3]

I completely concur with @statman. Please see the JMP Help about split-plot designs. You will treat the upstream factors as hard-to-change and very hard-to-change factors because they are randomized less often that the factors in the last step. The randomization aspect of how you conduct your experiment is important in the modeling of the random effects of changing factor levels.

[#4]

Thanks for your thoughts. I have tried to reduce the overall process as much as possible already above, it is as follows: 

Step 1: Plasma treatment of surfaces

Step 2: UV fix of adhesive used to bond two surfaces

Step 3: Final curing of the adhesive

Response: Die shear strength in shear test

I am considering omitting step 1 in case I can get a response in the desired range without it as it is costly. I can measure some output of step 1, e.g. surface energy, but only whether or not it is above a threshold, not the exact value (ink test). I understand you are suggesting to treat surface energy as a covariate factor instead of having the plasma process parameters in the model itself, which is a good hint, thanks. 

There are more steps involved in the overall process, I have reduced them to 3 to keep it short here. I suspect strong interactions between some of the steps, e.g. UV and final curing, adhesive layer thickness and UV curing. Unfortunately, a meaningful measurement of the response can only be conducted after the final curing. Of course, other measurements could be conducted after the individual steps, e.g. polymer chain length, degree of polymerization, but I lack the equipment to do so, which is why we kind of try to condense everything into one response measurement at the end of the process, which is of course less than ideal.

I am not entirely sure what direct sampling refers to in this context.

[#5]

Maybe I'd like to take a step back and rephrase my question as follows: 

In an effort to optimize my step 1 (plasma treatment) with the factors A (plasma power), B (gas flow), C (process time), how do I include the case of no treatment at all, that is, how do I include a control runs in the model where the samples are untreated during step 1 and hence factors A, B, C are irrelevant? I found a similar question here  . The poster retreated to make this a categorical question by first taking one specific set of factor levels and test against no treatment to see if the effect is beneficial for the goal. 

Currently, my best idea is to generate 2 separate designs: 

Design 1: Steps 1 - 3 including all 7 factors, where the plasma step is always done with factor levels within the physical limitations.

Design 2: Steps 2 - 3 including only factors D, E, F, G, where those are at the same levels as for Design 1. That way, I am generating a negative control group for the plasma treatment

I feel this is rather clumsy and I'd be more than happy to hear a better suggestion. 

[#6]

There are, of course, multiple options.  Each may differ in their inference space, what can be separated, what is confounded.  And the potential information that can be gained will be weighed against the resources necessary to accomplish.  My advice is always to design multiple plans, predict all possible outcomes and what each plan will "get you" in terms of knowledge.  The pick one. It is virtually impossible to know what the "best" plan is á priori.  In addition, you should always consider iteration, so how will the plan you select help you design your next plan?

I'm not sure what your plasma operation is intended to do, but if it is a "cleaning" operation, why not first study the post first step factors while blocking on surface cleanliness.  If you get a significant block effect or block-by-factor interactions, then perhaps spend some time optimizing the first step.  If not, your post first step process may be robust to the first step.

[#7]

Let me address directed sampling (component of variation study).  Currently in your process there is variation in the surface condition as a result of the plasma etching, there is also likely variation in thickness of adhesive, and extent of UV curing and then "final" curing.  Lastly you have variation in the measurement system.  Sampling will help understand which sets of variables (perhaps which steps in the process) contribute most to the overall variation in the response as well as determine whether those source are consistent or not.  I don't know what you are trying to apply the adhesive to nor what is being joined, but I would have questions whether the variation in the response is greater sample to sample in time (e.g., over changing ambient conditions or lot-to-lot of the raw materials), sample-to-sample (selected to expose the processing variables, deposition of the adhesive and amount of cure), within sample (lack of homogeneity of the application of adhesive or exposure to UV) or measurement system (particularly challenging as your dest is destructive).