Screening Design and Response Surface Design with Multiple Responses

[#1]

Hi, 

I use JMP Pro 17 for response surface design. I have 13 design variables. First I perform a Screening Design to decide on which 8 parameters are the most effective on the response, then I vary those 8 parameters to conduct a Response Surface Design. (8 is the maximum number of variables JMP accepts for Response Surface Design.) I conduct 33 experiments (with 1 center point) for Screening Design and 81 experiments (with 1 center point) for Response Surface Design.

The optimization has three goals with equal importance.

1. Minimize A

2. Minimize B

3. Match a target value of 0 for (C-D) (I want C=D, therefore (C-D)^2 should be zero.)

I (will) have the values of A, B, C and D for all 33 and 81 experiments.

For both Screening Design and Response Surface Design, I have thought of two different options to work on the response of the problem.

- Option 1: Defining a single response (Option1_singleresponse.JPG) as:

Y = A/A_centerpoint + B/B_centerpoint + (C-D)^2/(C_centerpoint-D_centerpoint)^2

In this option, I non-dimensionalized three terms with respect to their center point values. However, I am still not sure if I reflect the mathematics of the problem correctly. I want three goals to have equal importance on Y, but the terms A/A_centerpoint, B/B_centerpoint and C-D)^2/(C_centerpoint-D_centerpoint)^2 have different order of magnitudes. (For example; 0.1<A<1.3 and 2000<B<50000.) Also minimizing (C-D)^2/(C_centerpoint-D_centerpoint)^2 would not be enough since I want C to be equal to D.

- Option 2: Defining multiple responses (Option2_multiresponse_3minimize.JPG or Option2_multiresponse_2minimize_1target.JPG) for three optimization goals and following what is described in this video: Optimizing Multiple Responses . 

Which option would work better for me?

If I continue with Option 2, do you think Option2_multiresponse_3minimize.JPG and Option2_multiresponse_2minimize_1target.JPG would make a difference?

(I would appreciate your time and ideas, @MRB3855 and @statman . I tagged you in order to notify you.)

Thank you so much in advance to anyone who want to contribute!

[#2]

I would love to help, but there just isn't enough context.  You have a multivariate responses.  How do they correlate? Should they correlate?  If not, it may be difficult to optimize all. Make sure you code the level setting to prevent bias. I'm not sure how you know you will need an 8 factor response surface design?  That is a huge design space with 8 dimensions.  Unlikely that would be useful.  I suggest you modify the current plan and plan on a more sequential approach.  It is unlikely you will know what the next design will be as it will be a function of the analysis of the first design.

[#3]

Hi @statman , thank you for your reply.

I am performing an aircraft wing geometry optimization study.

A: Wing weight 

B: Drag force

C: Aircraft weight (C=Some constant value + A)

Lift force

So I am trying to achieve an aircraft wing geometry with minimized drag force and weight, where lift force should be equal to aircraft weight.

All A, B, C, D are functions of the wing geometry, but it is difficult to relate them with mathematical expressions. So I guess no, they should not correlate.  

"Make sure you code the level setting to prevent bias.": I am performing a 2-level optimization. I only take the minimum and maximum limits of the design variables in order to form the design space.

You may be right, I do not need to use all the 8 variables that JMP lets me use for Response Surface Design. After performing the Screening Design, I checked the effect summary for the effects with Individual p-Value less than 0.10. (https://www.jmp.com/support/help/en/17.2/index.shtml#page/jmp/contrasts.shtml) There are more than 8 variables with p-Value<0.1. However, the maximum number of variables to perform a Response Surface Design in JMP is 8. Do you have a suggestion on how I can justify the selection of the number of variables I chose for response surface design? (If I continue with 8 variables, I can say that this number is chosen since it is the upper limit for the number of variables. But I do not know how to justify if I select a number of 5 variables, for example.)

Thank you again, in advance.

[#4]

Sorry, but I still don't understand the entire situation.  Is this being done with simulation?  Are you actually building multiple wings?  My guess is you are using some simulation software to model the parameters you input.  There are already known theoretical relationships between weight and drag (see Boldmethod:  https://www.boldmethod.com/learn-to-fly/performance/why-aircraft-a-weight-increase-affects-climb-per...

And lift (see Nasa:  https://www1.grc.nasa.gov/beginners-guide-to-aeronautics/lift-to-drag-ratio/

and of course, Bernoulli's Principle

"All A, B, C, D are functions of the wing geometry, but it is difficult to relate them with mathematical expressions."

The simulation software already has the algorithm loaded into the software.  You may be able to "uncover" the algorithm.  Since it apparently is a simulation, there is no need to fractionate, nor to minimize the number of levels unless you are running into computing power issues.

I'm not sure how you use p-values when there is no "real" estimate of the MSE.  How is "experimental error" estimated with the simulation software? (e.g., some sort of Monte Carlo with a selected probability distribution of errors?)

[#5]

Hi @ceydakavak : To follow on to what @statman  said; if it is true that there is some deterministic function/model (e.g., finite element method) that takes your inputs and spits out  A, B, and D (C=Some constant value + A, so C is known if you know A) where there is no error (i.e., replicates will return the exact same values of A, B, and D each time), then the only "error" in your model is Lack-Of-Fit error. That doesn't necessarily mean the model isn't fit for purpose...it may well be. But p-values etc. are specious.