Supersaturated designs with less β-aberration

Supersaturated design is an important class of fractional factorial designs in which the number of experimental runs is not enough to estimate all the main effects. These designs are widely used in screening experiments, where the primary goal is to find important active factors at a low cost. The minimum β-aberration criterion is an appropriate criterion for measuring designs with quantitative factors. In this article, we first establish the explicit expression of β2 for three-level designs based on the relationship between the wordlength enumerator and the β-wordlength pattern. It can reduce the computational complexity of the β-wordlength pattern, and help provide an effective way for finding designs under the minimum β-aberration criterion. Moreover, a sharper lower bound of β2 is obtained, which can be considered as a benchmark for constructing optimal supersaturated designs. We further provide a simulated annealing algorithm to construct three-level supersaturated uniform designs with less β2. Finally, numerical results verify that our lower bound is sharper than the existing lower bound.