Bayesian modeling and optimization for split-plot experiments with multiple responses

Abstract

In many industrial processes, cost or time constraints make some input variables harder to change or control than others. An appropriate experimental design method is restricted randomization, which results in split-plot experiments. Empirical models that connect multiple quality characteristics with input variables play a crucial role in robust parameter design for split-plot experiments. At present, many modeling methods typically adopt the single response model for analyzing industrial processes in the split-plot experiments without considering correlation among multiple responses, correlation among whole plots, and uncertainty of model parameters. However, ignoring these issues can lead to poor product or process design. To solve these issues, this paper suggests a novel Bayesian modeling and optimization approach. We first construct a Bayesian multi-response linear mixed-effects model and obtain the posterior distribution for model parameters by employing Bayesian theorem. Then, the Gibbs sampling procedure is employed for the estimation of model parameters. Finally, the overall weighted desirability optimization function meeting the specification is developed to avoid acquiring ideal input settings with outliers. A simulation and an engineering case study demonstrate the validity of the proposed method. In comparison to existing methods, the optimization results given the proposed method are more robust and reliable.