Mixed-Level Column Augmented Uniform Designs

TL;DR

This paper introduces mixed-level column augmented uniform designs for follow-up experiments, optimizing uniformity and non-orthogonality, with analytical bounds and an effective construction algorithm demonstrated through examples.

Contribution

It proposes a novel design framework for mixed-level follow-up experiments using uniformity criteria and provides an efficient construction algorithm.

Findings

Lower bounds of WD are tight.

Column augmented uniform designs are optimal under $E(f_{NOD})$.

The construction algorithm is effective and practical.

Abstract

Follow-up experimental designs are popularly used in industry. In many follow-up designs, some additional factors with two or three levels may be added in the follow-up stage since they are quite important but may be neglected in the first stage. Such follow-up designs are called mixed-level column augmented designs. In this paper, based on the initial designs, mixed-level column augmented uniform designs are proposed by using the uniformity criterion, wrap-around $L_2$-discrepancy (WD). The multi-stage augmented procedure which adds the additional design points stage by stage is also investigated. We present the analytical expressions and the corresponding lower bounds of the WD of the column augmented designs. It is shown that the column augmented uniform designs are also the optimal designs under the non-orthogonality criterion, $E(f_{NOD})$. Furthermore, a construction algorithm for the column augmented uniform design is provided. Some examples show that the lower bounds are tight and the construction algorithm is effective.