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 beta-aberration criterion is an appropriate criterion for measuring designs with quantitative factors. In this article, we first establish the explicit expression of beta(2) for three-level designs based on the relationship between the wordlength enumerator and the beta-wordlength pattern. It can reduce the computational complexity of the beta-wordlength pattern, and help provide an effective way for finding designs under the minimum beta-aberration criterion. Moreover, a sharper lower bound of beta(2) 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 beta(2). Finally, numerical results verify that our lower bound is sharper than the existing lower bound.

Orthogonal array has been used in various fields, as it possesses attractive combinatorial properties. However, for a long time, combinatorially equivalent orthogonal arrays are thought to be indistinguishable, especially when an ANOVA model is established. Later on, some papers pointed out that permuting levels of orthogonal arrays will alter their statistical inference abilities. Criteria have been recommended for evaluating the different performance of orthogonal arrays. In this paper, a revised method is proposed to construct saturated orthogonal arrays when the level of the factors is an odd prime and properties of the related wrap-around L2-discrepancy will be investigated. Theoretical result shows that the wrap-around L2-discrepancies of the saturated orthogonal arrays constructed using the revised method are less than those of the original ones, and asymptotically attain the lower bounds. A series of numerical examples also confirms the effectiveness of the proposed method.

Covering arrays and locating arrays have been investigated in depth in various scenarios. Test suites based on locating arrays can not only test the presence of system defects, but also identify effects that cause failure according to outcomes. In this article, we first obtain a refined lower bound of test numbers for mixed-level locating arrays, which serves as a benchmark for constructing optimal locating arrays with specific parameters. We also provide an algorithm to construct mixed-level locating arrays aiming to reaching the lower bound. The goal to find a mixed-level locating array is transformed into solving an optimization problem. A simulated annealing algorithm is implemented to approximate a global solution. As a result, a series of mixed-level locating arrays are obtained, which shows that the proposed algorithm is quite effective.