Bayesian Optimal Single Arrays for Robust Parameter Design

It is critical to estimate control-by-noise interactions in robust parameter design. This can be achieved by using a cross array, which is a cross product of a design for control factors and another design for noise factors. However, the total run size of such arrays can be prohibitively large. To reduce the run size, single arrays are proposed in the literature, where a modified effect hierarchy principle is used for the optimal selection of the arrays. In this article, we argue that effect hierarchy principle should not be altered for achieving the robustness objective of the experiment. We propose a Bayesian approach to develop single arrays which incorporate the importance of control-by-noise interactions without altering the effect hierarchy. The approach is very general and places no restrictions on the number of runs or levels or type of factors or type of designs. A modified exchange algorithm is proposed for finding the optimal single arrays. MATLAB code for implementing the algorithm is available as supplemental material in the online version of this article on the Technometrics web site. We also explain how to design experiments with internal noise factors, a topic that has received scant attention in the literature. The advantages of the proposed approach are illustrated using several examples.