This tip is from Heath Rushing, coauthor of Design and Analysis of Experiments by Douglas Montgomery: A Supplement for Using JMP®.
After a recent design of experiments (DOE) course, a student asked about experiments with dependent factors. Throughout the two days of training, we spent considerable time designing experiments to determine the effect of one or more independent factors on multiple responses. However, in her particular experiment, she was mixing three different components in a formulation. The level of each of the three components were proportions of a mixture. This is known as a mixture design.
Her ultimate goal was to determine the proportion for each of the components to optimize multiple responses. Because she could not discuss the details of the responses or components, I referred to her formulation as Grandpa McCafferty’s cough syrup. The understanding of three key concepts was needed for her experiment:
Recognize a mixture design. In most designed experiments, the factor settings are assumed to be independent of the other factor settings; one factor can be varied independently of the level of the other factors. If the factors in the experiment are components in a mixture where the factor levels are proportions (of that particular component), the mixture factor cannot be varied independently of the other factors. Because the levels of mixture factors sum to 100%, by increasing or decreasing the proportion of one, the level of at least one of the others must decrease or increase.
Add constraints on mixtures. In many mixture designs, the proportion of components are constrained. For example, the formulation must have at least 20%, but no more than 80%, of component 1. Mixture designs allow for addition of constraints on levels of components.
Optimize multiple responses with specifications. Mixture designs are special cases of response surface designs where the primary objective is to optimize the proportions of multiple components for one or more responses. Because these responses are critical to the quality of the product, they often have specifications. Use the prediction profiler to find the optimal proportions of the components or the ternary plot and/or mixture profiler to determine optimal areas of operation.
Lucky for this student, all three of these concepts are covered in my book, Design and Analysis of Experiments by Douglas Montgomery: A Supplement for Using JMP®. The book provides step-by-step instructions on how to set up and analyze not only this type of mixture design, but also the popular factorial, fractional factorial, response surface, and custom designs.
We hope you found this tip useful!