How to obtain the optimum formulation

[#1]

I have a new material made of different oxides. All of them have a fixed proportion, with only two oxides changing in a reverse manner, so their final proportion is fixed. Finally, I have six samples. I have used online software to study a specific response; hence, I finally have an energy column with 25 different values, with each one having its own unique response value for the specific composition. I'm trying to produce a predictive model so I can find the optimum formulation for the composition to give the best response for the whole range of energy. How to deal with such an issue?

[#2]

Hi @Sherif_96,

To make sure I understood your topic, you have 6 independent observations of different oxides mixtures, and only 2 oxides quantity can vary, but not independently as the sum is fixed ? I haven't understand the situation about the 25 energy values and their links to the composition, but I may still provide some comments at this stage.
If you only have 2 factors (oxides quantity) that can vary with a fixed sum (since the others oxides mentioned are fixed), that means you only have 1 independent factor that can be set as the 2 oxides ratio. This simplify the study you're trying to do, as you may have access to different platforms depending on your objectives, data, needs and complexity :

• You can start first with a visualization, and assessing how this oxides ratio affect the response(s), using Graph builder,
• You can also analyze if there is trend linking the response(s) to the oxides ratio by using the platform Fit Y by X (Y: your response(s), X: oxides ratio) and trying a large variety of relationships/equations,
• You may try the analysis in the same way using Fit Model platform, which may help you fitting a regression equation involving main effect, and possibly higher order terms (like quadratic and cubic effect).

If you need to optimize your formulation for several energy levels and you have JMP Pro, maybe the use of Functional Data Explorer could help, setting the energy range as X variable, the response as Y, and your formulation parameter as Z supplementary variables.

There are a lot of options, but without any example dataset and further explanations/informations, this is the best I can think of.
Hope this first discussion starter will help you,

[#3]

I apologize for the information that was not clear and appreciate your response.

As you said, the sum of the mixture as a whole must equal one, and the sum for the two different oxides must equal 0.45.

The energy comes from the test, which exposes the samples to particular levels of radiation. To evaluate the sample's ability to resist radiation, each radiation value is represented by a distinct energy value. By manually comparing the response values of the six manufactured samples, I have already identified which sample is the best to withstand radiation. However, what if I wanted to examine additional samples without actually manufacturing them in order to determine their response and, consequently, the best formulation to withstand the response?

The attached file contains responses and the varied oxide ratios.

[#4]

Hi @Sherif_96,

Some ideas to help you and get you started :

• It may be easier to work with a stacked datatable for your data : one column for Sample ID, one column for ratio, one column for Energy levels and one column for Response to the energy levels by sample :
  
• You can plot your data with Graph Builder, using Energy levels as X, the response as Y, and Sample ID as Overlay variable to get a better sense of the behaviour of your data/curves :
  
  The curves look very similar with an exponential shape, which leads me to the next point :
• One possible way to deal with the curves data is to use the Fit Curve platform, with column Response as "Y, Response", Energy as "X, Regressor", Sample as "Group", and V/B Ratio as "Z, Supplementary". You can then model the curves using a model Exponential 2P, which gives you for each sample the Growth rate and scale parameters, according to this equation :
  

  Using these parameters in regression model as Y and V/B ratio as X could help understand the link between the shape of the curves and the V/B ratio :

• Using a table with the extracted parameters scale and growth rate from the previous analysis, and with the Platform Fit Y by X, you can link these two curves parameters to your V/B ratio factor. It looks like there is a linear trend between the curves parameters and the V/B ratio (a polynomial degree 2 equation might also work): 
  
• You can then use the equations linking these curve parameters to the V/B ratio in order to predict and modelize any curves in the range of the V/B ratio analyzed to create a prediction table :
  

  This prediction table with "virtual samples" may help you "examine additional samples without actually manufacturing them in order to determine their response and, consequently, the best formulation to withstand the response". You can also use the Response formula and create a Profiler to better assess the changes in curves and values based on the scale and growth rate values : 

  
  

  Or expand the formula and integrate the equation of scale and growth rate directly in the formula to have the prediction profiler based only on V/B ratio :
  
  

  

These are only suggestions based only on the data and a quick analysis path (which needs better verification and check with domain expertise), there are a lot of other possible analysis. Use domain expertise in accordance with your needs and objectives to choose and define the right analysis path.

One recommandation : Maybe it could be interesting if possible to expand the range of the V/B ratio, to make sure that the trends linking curves energy response to the V/B ratio remains the same outside of the analyzed range.

Please find attached the datatable used with all the analysis scripts. 

Hope these few suggestions may help you,

[#5]

Thank you so much, Victor, for your excellent and comprehensive guidance. I sincerely appreciate your efforts.