A new look on optimal foldover plans in terms of uniformity criteria

In this paper, we develop a new mechanism for finding the optimal foldover plans (OFPs) which is based on the uniformity criteria measured by Lee discrepancy, wrap-around L₂-discrepancy, and centered L₂-discrepancy. For three-level fractional factorials as the original designs, general foldover plans and combined designs are defined, and lower bounds of these three discrepancies of combined designs under general foldover plans are also obtained, which can be used as benchmarks for searching OFPs. Illustrative examples with a comparison study between the foldover plans under these discrepancies are provided. Our results provide a theoretical justification for OFPs of three-level designs in terms of uniformity criteria.     

Lower bounds of various criteria in experimental designs

Criterion is essential for measuring the goodness of an experimental design. In this paper, lower bounds of various criteria in experimental designs will be reviewed according to methodology of their construction. The criteria include most well-known ones which are frequently used as benchmarks for orthogonal array, uniform design, supersaturated design and other types of designs. To derive the lower bounds of these criteria, five different approaches are explored. Some new results are given. Throughout the paper, some relationships among different types of lower bounds are also discussed.     

A lower bound for the centred L 2-discrepancy on combined designs under the asymmetric factorials

The foldover is a useful technique in the construction of two-level factorial designs for follow-up experiments. To search an optimal foldover plans is an important issue. In this paper, for a set of asymmetric fractional factorials such as the original designs, a lower bound for centred L ₂-discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. All of our results are the extended ones of Ou et al. [Lower bounds of various discrepancies on combined designs, Metrika 74 (2011), pp. 109–119] for symmetric designs to asymmetric designs. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.     

A Lower Bound for the Wrap-around L2-discrepancy on Combined Designs of Mixed Two- and Three-level Factorials

The objective of this article is to study the issue of employing the uniformity criterion measured by wrap-around L₂-discrepancy to assess the optimal foldover plans. For mixed two- and three-level fractional factorials as the original designs, general foldover plan and combined design under a foldover plan are defined, and the equivalence between any foldover plan and its complementary foldover plan is investigated. A lower bound of wrap-around L₂-discrepancy of combined designs under a general foldover plan is obtained, which can be used as a benchmark for searching optimal foldover plans. Moreover, it also provides a theoretical justification for optimal foldover plans in terms of uniformity criterion.