A posterior predictive approach to multiple response surface optimization

In this paper we present an approach to multiple response surface optimization that not only provides optimal operating conditions, but also measures the reliability of an acceptable quality result for any set of operating conditions. The most utilized multiple response optimization approaches of "overlapping mean responses'' or the desirability function do not take into account the variance-covariance structure of the data nor the model parameter uncertainty. Some of the quadratic loss function approaches take into account the variance-covariance structure of the predicted means, but they do not take into account the model parameter uncertainty associated with the variance-covariance matrix of the error terms. For the optimal conditions obtained by these approaches, the probability that they provide a good multivariate response, as measured by that optimization criterion, can be unacceptably low. Furthermore, it is shown that ignoring the model parameter uncertainty can lead to reliability estimates that are too large. The proposed approach can be used with most of the current multiresponse optimization procedures to assess the reliability of a good future response. This approach takes into account the correlation structure of the data, the variability of the process distribution, and the model parameter uncertainty. The utility of this method is illustrated with two examples.