mixture experiment and cost function

[#1]

Hello,

I have a mixture data with ingredients I_1, I_2, ... I_9. They add up to 1 as this is a mixture data. The response is y. Of the 9 ingredients 6 of them have cost associated with them, say I_1 - I_6. I creates a new column: C = p1*I_1 + p2*I_2 + ... + p6*I_6. This gives the total cost per unit weight. Now I run a mixture response surface model for y. But when I go to optimize it, I like to take into account the cost column C and find the optimal solution while also minimizing the cost C.

How would I do this? 

Thanks,

[#2 · Accepted solution]

Save the prediction formula for the fitted model. Select Graph > Profiler and select the model and cost formula columns. Now they both appear in the profiler for joint optimization.

[#3]

Thank you very much Mark!

[#4]

I have a question regarding the joint optimization as instructed in this post. 

1. what if we have 10 responses that need to be jointly optimized? Can JMP handle this?

2. The reason for question 1, is that when I run 6 optimize 6 responses jointly using Graph > Profiler, the values of some responses are grossly out of the range as specified in Column Property > Response Limit. For example when one response has limit (47, 52) the optimized value is 15298 while in other cases the value is negative which is not acceptable at all.

Thanks,

[#5]

1. Yes.

2. The Response Limits column property defines the desirability of the response. It does not constrain the optimization; it guides it. If the response with a desirable range of (47,52) aims to maximize the response, the solution can exceed 52. Sometimes, the unexpected range of the response is because the linear regression model assumes that the response can range from negative infinity to positive infinity without constraint. Is your example of (47,52) a case where you want to match a target?

[#6]

Yes that is correct. here is an example of one response:

I do the same for other responses with their own limits. But when I click on maximize desirability, I get negative values for some of the response for the choice of the ingredients settings. some other responses become extreme positive.

[#7 · Accepted solution]

First, I recommend establishing the Lower and Upper Desirability values using the Goal drop-down menu. This is a quirk of the Profiler. It would be best not to change them to 0 or 1. Doing so can defeat the internal algorithm that optimizes the factor settings. It is OK to change the Middle value if the target is not centered in the interval.

Your example of a response with a goal of matching a target is helpful. These values are a goal. They are not guaranteed. When you have multiple responses, the optimization may not be able to achieve the goal for every response.

You changed the Importance value from the default of 1. This value can be helpful to emphasize one or more responses. The joint desirability uses this value as a power in the product of the individual desirabilities, so a large number is unnecessary. A large number might allow a single response to hijack the optimization.

As to the unrealistic high predictions, that is a matter of the model that you fit and the fact that the response limits do not actually limit the range of the response here. A linear regression model assumes that the response can range from negative to positive infinity. Setting a goal of maximizing and a range of (lower, upper) means that the high limit is good enough if another response can be satisfied, but otherwise, it does not limit the response.

With your domain expertise, you might be able to manually adjust one or more factor settings to bring such a response down to a reasonable level. You can then lock this factor so that JMP may not change it during optimization. This approach forces JMP to only use the unlocked factors during optimization.

[#8]

Thank you so much!

[#9]

Hi Mark,

In regards to the discussion of mixture modeling, I am having hard time with JMP's consistency! 

The attached table is what I am using. The attached model is the mixture response surface after step-wise selection. You will notice the profiler looks unreasonable, and way out of bound. Each component has it's own limits. I am hoping I am doing something wrong and easily fixed.

In advance I appreciate your assistance.

[#10 · Accepted solution]

I have not seen this behavior before. I recommend sending your problem to JMP Technical Support (support@jmp.com) for resolution. Please report any solution they can give you here.