Bayesian modeling and optimization for multi-response surfaces

Abstract This paper proposes a new Bayesian modeling and optimization method for multi-response surfaces (MRS). The proposed approach not only measures the conformance probability (i.e., the probability of all responses simultaneously falling within their corresponding specification limits) through the posterior predictive function but also takes into account the expected loss (i.e., bias, quality of predictions and robustness) with the expected loss function. Also, it is shown that the Bayesian SUR models can provide more flexible and accurate modeling than the standard multivariate regression (SMR) models with the same covariate structure across different responses. Besides, the proposed approach also takes into account the correlation structure of the response data, the variability of the process distribution, and the uncertainty of model parameters as well as the prediction performance of the response model. A Polymer experiment and a simulation experiment are used to demonstrate the effectiveness of the proposed approach. The comparison results show that the Bayesian SUR model has higher conformance probability and lower expected loss than the Bayesian SMR model.