Mixture Design: Optimization of Component Ratios

That Day, When the Cpk Report Came Out, the Entire Room Was Silent for Three Seconds

I still remember a process integration meeting where we were discussing the yield problem of a new type of photoresist. At that time, the yield was consistently stuck at 92%, and no matter how much we adjusted, it wouldn't go up. The boss's face was grim, the QA manager's brows were deeply furrowed, and the atmosphere was so tense you could hear a pin drop. After Engineer A reported the latest Cpk value of only 1.08 and a DPMO as high as 6210, the entire conference room fell silent for three seconds. Then, the boss's command, "Figure out that photoresist formulation!" immediately ignited the discussion. To be honest, at that point, we had tried so many different ratios for the photoresist's additives that we didn't know what else to try.

What Was the Problem?

You must have encountered this situation before, right? A product's performance (yield, reliability, cost, etc.) results from a combination of several "components." These components could be the concentrations of chemical solutions, alloy ratios, or even mixtures of materials from different suppliers. The common mistake we make is changing only one component at a time, or making haphazard changes. But the problem is, these components often have interactions! If A increases, B might decrease, and C might even have a synergistic effect with A.

So, here's the key point: Mixture Design is specifically designed to solve this "component ratio optimization" problem. It doesn't simply look at the individual effects of A, B, and C, but rather how their proportions "mixed together" affect the final result. Simply put, it helps you find the perfect golden ratio to achieve your target product performance.

How Is It Actually Done?

We used Mixture Design at that time to find the optimal formulation for that photoresist. Specifically, you would do this:

1. Define Components and Total Amount: Assume your photoresist has three main components: A, B, and C. They must always add up to 100%. You cannot add only A without B and C.
2. Set the Experimental Range: You need to determine the "valid" proportion range for each component. For example, A must be between 20% and 60%, B between 10% and 40%, and C between 30% and 70%. Remember, these ranges do not necessarily add up to 100%; the key is their individual upper and lower limits.
3. Choose the Experimental Design Model: This part will involve statistical software (such as JMP or Minitab). Based on your number of components and objectives, they will automatically generate a set of experimental points. These points are not random; they effectively cover the entire experimental range, allowing you to obtain the most information with the fewest experiments.

For example, if we have three components A, B, and C, the software might suggest testing the following sets of proportions:

• A=50%, B=30%, C=20%
• A=20%, B=60%, C=20%
• A=40%, B=20%, C=40%
• ...and so on.

You follow these proportions to prepare the mixtures and conduct experiments, then input the results (e.g., new Cpk values) back into the software. The software can then plot a response surface map, showing you which combination of proportions can maximize the Cpk. In our case, we discovered that when the proportion of component B was slightly increased and the proportion of C slightly decreased, the yield surprisingly soared to 98.5%!

Common Pitfalls

To be honest, I made some mistakes the first time I did Mixture Design too.

1. Setting Arbitrary Limits: The biggest mistake is setting the limits for component proportions too rigidly. For example, if you think A is the most important, you might set A's upper proportion limit to 90%, thereby squeezing out the exploration space for other components. Later, you might discover that sometimes, that seemingly insignificant 5% of B is actually the key!
2. Ignoring Interactions: Some people treat Mixture Design as a general DOE, only looking at the impact of individual components. However, the essence of this design is to observe "interactions." If you only look at how A affects something and how B affects it, you will miss the unexpected chemical reactions that occur when A and B are mixed.
3. Insufficient Sample Size: The biggest fear in DOE is drawing conclusions after running only a few points. Especially when there are many components, you must at least run the "corner points" and "center points" suggested by the software to ensure the model is reliable enough. Otherwise, you'll only see the tip of the iceberg and think you've seen it all.

One Thing You Can Do Today

Think about whether there's a product in your hands whose "golden ratio" you haven't found yet. Today, open your DOE software and try planning an experiment using Mixture Design!