QPRC 2016

Restricted‐Randomization Optimal Design of Experiments Combining Mixture and Non‐Mixture Factors

Martin Bezener Patrick Whitcomb Wayne Adams Henry Anderson
Stat‐Ease, Inc. Stat‐Ease, Inc. Stat‐Ease, Inc. Stat‐Ease, Inc.

Mixture designs are common in a number of industries, particularly in the process industries‐‐chemical, food, pharmaceutical, etc.. These designs are employed in settings where at least two of the experimental factors under consideration must sum to a fixed total. Non‐mixture factors, or process factors, may also be included in the experimental design. Classical randomized designs require that each run use an independent preparation of the mixture, and that the process factors be completely re‐set between runs. This requirement, however, is usually impractical. It is often only possible to prepare batches of the mixture to be tested under varying settings of the process factors. In other situations, the mixture may be easy to prepare, but it is expensive or time‐consuming to reset the process factors. This restriction on randomization induces a split‐plot structure in the experimental design which, unfortunately, is often ignored.  Despite the prevalence of this situation in practice, designs of this type have not received much attention in the literature. In this talk, we briefly review the classical mixture design and common scenarios where it is used. We then discuss a split‐plot version of this design and briefly mention a mixed model approach for statistical analysis. We continue with a discussion of a novel algorithm for the optimal construction of these designs, and numerical issues that may be encountered. Attention is also given to practical issues such as sample size and the optimality criterion throughout the talk. We conclude by illustrating a real‐world mixture‐process experiment involving the optimization of coffee made from various beans and brewed up in our company’s cafeteria.