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Level I

## Re: DOE continuous factor affects mixture property

Hi,

I'm trying to set up an experiment to understand how the concentration of a particular ingredient "A" and a particular ingredient "B" affect properties of a formulation.

Of interest is the:
Concentration of A in the formulation (eg. grams of A per liter)
Ratio of A content to B content in the formulation.

A and B are ingredients that come from purchased mixtures. Mixture 1 supplies A in some concentration. Mixture 2 supplies B in some concentration. So our formulation can be represented as:
x% Mixture 1 + y% mixture 2 + z% filler, where x+y+z = 100%.

Our objective is to understand how A and B concentrations frm these purchased mixtures affect the finished formulation. We thought to set up a study where we artificially inflate how much of A and B we use to hit the target A value and A/B ratio by treating A value and A/B ratio as continuous variables. For example if Mixture 1 contains 2 grams/ml of A, and we wanted to understand how more of A affects our formulation, we would just add more of Mixture 1 to the formula (if we doubled the amount of Mixture 1, we'd double the amount of A). To adjust the A/B ratio we'd need to inflate the amount of A used, then solve to find how much of Mixture 2 we need to hit the target ratio. The problem is that all of this is ultimately affecting the mixture; adding more of mixture 1 and 2 is reducing how much filler we end up using.

So I am not sure how the model can be set up so that I can separate the effect of manipulating A and B concentration from adjusting the filler concentration. Can the JMP community give nay guidance on this? Thank you!

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Staff

## Re: DOE continuous factor affects mixture property

Hi,

This sounds like a mixture design problem. I suggest that you search for "Mastering JMP Mixture Experiment" for a video that will introduce you to this topic. (These videos are periodically updated but currently there is a one here: https://www.jmp.com/en_in/events/ondemand/mastering-jmp/mixture-designs.html)

If I understand correctly, A, B and Filler are mixture factors: you want to explore the how changing the relative proportions of these ingredients affects the properties of the formulated material. And you are constrained that an increase in the amount of one ingredient requires a reduction in the amount of one more of the other ingredients.

I hope this helps.
Phil
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Level I

## Re: DOE continuous factor affects mixture property

Hi Phil,

Thanks for such a prompt reply!  Your summary of my requirement is mostly correct.

My formulation is basically:

x% Mixture 1 + y% Mixture 2 + z% filler

Mixtures 1 and 2 are purchased goods, and in those mixtures are crucial ingredients A and B, respectively.  What we're really interested in are the effects of concentration of A and B.  In a perfect world we'd be able to adjust the concentration of A and B for experimental purposes.  As it stands, we think we can "cheat it" by adjusting the concentration of the mixtures in the final formulation to simulate the effect.  In other words:

If we wish to add more of concentration of A --> Need to increase amount of Mixture 1

If we wish to add more concentration of B -->  Need to increase amount of Mixture 2.

I hope this is more clear!

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Staff

## Re: DOE continuous factor affects mixture property

Are you familiar with the Prediction Profiler in JMP? If so, imagine the profiler that you want to produce at the end of this work. It sounds like you would want it to have %A, %B and %filler as 3 factors on the x-axes. But you can't directly adjust %A and %B because they are supplied as mixtures. What else is in the A and B mixtures? Is it the same filler that you would be adding to make the final formulation?
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Staff

## Re: DOE continuous factor affects mixture property

I think Phil was on the right track, but let me take a stab at a different approach. First, let's see if I restated the problem correctly:

You have mixture 1. Let's suppose it is made up of 3 ingredients: A1, A2, and A3. You are primarily interested in A1. You cannot control the proportions within mixture 1.

You have mixture 2. Let's suppose it is made up of 3 ingredients: B1, B2, and B3. You are primarily interested in B1. You cannot control the proportions within mixture 2.

You are going to be blending mixture 1 and mixture 2 together, along with a filler. You wish to see how the amounts of A1 and B1 influence your final product.

Now an important question: will it be the AMOUNT of A1 that influences the product or the RELATIVE PROPORTION of A1 and B1 that influences the product? If it is the AMOUNTS, that will change everything. I will work from the proportions, since I believe that is what you are looking for.

The fact that mixture 1 and mixture 2 exist is of little use since you cannot adjust those. You have a three component mixture: Mixture1, Mixture2, and Filler. So in JMP you just specify three mixture components of Mixture1, Mixture2, and Filler. You know that when you increase Mixture1 you are increasing the amount of A1 (your ingredient of primary interest), but remember my assumption stated above: the proportions are all that matter, not the amount.

I hope that I have summarized everything correctly and this helps. If not, let me know!

Dan Obermiller
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Level I

## Re: DOE continuous factor affects mixture property

You asked: "Now an important question: will it be the AMOUNT of A1 that influences the product or the RELATIVE PROPORTION of A1 and B1 that influences the product? If it is the AMOUNTS, that will change everything. I will work from the proportions, since I believe that is what you are looking for."

Answer:  it the AMOUNT of A1 in the total formulation that matters and the ratio of the amount of A1/B1.  To be sure, it is a proportion only to the extent that we are interested in the amount of A1 and B1 per batch size.

It is correct that mixture's 1 and 2 are purchased, and for all intensive purposes, we cannot change them; we get what we get and suppliers do have variation.  When suppliers ship us mixtures 1 and 2, they report the content of A1 and B1 respectively.

We have a few extra lots of Mixture 1 with different levels of A1 in our lab.  At first I thought this could let me set up a mixture design with %x Mixture 1 + %y Mixture 2 + %z Filler as variables where x+y+z=1, then add categorical variables for A1 content.  Eg we have 3 lots of Mixture in the lab with A1 levels at 4, 6, 8.  This would be a discrete numeric with 3 levels at 4, 6, and 8.

This approach seems straight forward to me.  The problem is that these various lots come from more than one supplier, and I would really like to test to see if Mixture 1 performs differently for a given amount of A1 in the formulation depending on which supplier we use. That's why I thought I could "cheat" my way into adjusting absolute A1 and B1 contents using each supplier's material, then treat supplier of Mixture 1 as a categorical variable. That way we could see if the underlying model has any change when we use the other supplier's Mixture 1.  But then I get into that issue that started my post: adjusting the amount of A1 requires we add more mixture 1, which means we use less filler, so it seems like I would be confounding the effect of filler content vs A1 content.

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Staff

## Re: DOE continuous factor affects mixture property

Supplier is another factor. Is supplier confounded with the levels 4, 6, and 8 of A1? If so, you will not be able to separate those two without various A1 levels from each supplier.

I think the amount of A1 and the A1/B1 ratio will cause some issues, and the problem is the ratio. Suppose A1 is 4 and A1/B1 = 1. This implies that B1 is 4. You collect some data here and get a good response (let's just say Y=14). Now you increase A1 to 8 and A1/B1 =1 and you get a better response (say Y=18). Must be due to the increase in A1, right? No, maybe it is due to the increase in B1. Going further, suppose you run these four trials:

A1     A1/B1     Y

4          1          14

8          1          18

4          0.5       12

8          0.5       14

Notice that the effect of A1 looks exactly like the effect of A1/B1.

when in reality, Y = 10 + B1 is the model that generated the data.

Perhaps this could be simplified with another thought: could the filler be considered inert? If so, then just base your experiments around varying A1 and B1. Filler is just adjusted to maintain the proper batch size and is ignored because it is inert.

Dan Obermiller
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