Pareto-efficient designs for multi- and mixed-level supersaturated designs

Abstract

Supersaturated designs are used in science and engineering to efficiently explore a large number of factors with a limited number of runs. It is not uncommon in engineering to consider a few, if not all, factors at more than two levels. Multi- and mixed-level supersaturated designs may, therefore, be handy. While the two-level supersaturated designs are widely studied, the literature on multi- and mixed-level designs is still scarce. A recent paper establishes that the group LASSO should be preferred as an analysis method because it can retain the natural group structure of multi- and mixed-level designs. A few optimality criteria for such designs also exist in the literature. These criteria typically aim to find designs that maximize average pairwise orthogonality. However, the literature lacks guidance on the better or ‘right’ optimality criteria from a screening perspective. In addition, the existing optimal designs are often balanced and are rarely available. We propose two new optimality criteria based on the large-sample properties of group LASSO. Our criteria fill the gap in the literature by providing design selection criteria that are directly related to the preferred analysis method. We then construct Pareto-efficient designs on the two new criteria and demonstrate that (a) our optimality criteria can be used to order existing optimal designs on their screening performance, (b) the Pareto-efficient designs are often better than or as good as the existing optimal designs, and (c) the Pareto-efficient designs can be constructed using a coordinate exchange algorithm and are, therefore, available for any choice of the number of runs, factors, and levels. A repository of three- and four-level designs with the number of runs between 8 and 16 is also provided.

Appendix A The EPCEA algorithm

The modified EPCEA algorithm is provided in Algorithm 2. But first, we explain some notations and describe an algorithm that will be used in Algorithm 2. Also, note that every time the coordinate exchange operator is invoked, it makes all possible coordinate exchanges. In particular it adds +1 (mod $v_j$), +2 (mod $v_j$),..., + $(v_j-1)$ (mod $v_j$) to the original level in the jth factor. Recall that the values of our criteria in Sect. 3 are dependent on the chosen sets from all possible ${m \atopwithdelims ()k}$ sets. Let u be the maximum number of sets that are computationally feasible. We define $S_k = \min (u, {m \atopwithdelims ()k})$ to denote the sets to be used for computing criteria values for k. Define $S^u = (S_2, \ldots ,S_{\lceil n/3 \rceil })$ for the collection of sets used.

Algorithm 1

Steps of EPCEA

Algorithm 2

Modified EPCEA

For the results in this paper, we set $u=1000$, $ndes = 2$, $ntry = 500$, and the two starting designs are randomly created.