Joint optimization of independent multiple responses

Most of the existing methods for the analysis and optimization of multiple responses require some kinds of weighting of these responses, for instance in terms of cost or desirability. Particularly at the design stage, such information is hardly available or will rather be subjective. An alternative strategy uses loss functions and a penalty matrix that can be decomposed into a standardizing (data-driven) and a weight matrix. The effect of different weight matrices is displayed in joint optimization plots in terms of predicted means and variances of the response variables. In this article, we propose how to choose weight matrices for two and more responses. Furthermore, we prove the Pareto optimality of every point that minimizes the conditional mean of the loss function.