Uniformity and projection uniformity of combined designs

The purpose of this paper is to study the issue of employing the uniformity criterion measured by the mixture discrepancy to assess the optimal foldover plans for q-level factorials. The average mixture discrepancy and the average projection mixture discrepancy based on the level permutation method are respectively defined for combined designs, and the optimal foldover plan in terms of the overall uniformity and the uniformity of each dimension are also explored. The tight lower bounds of the average mixture discrepancy and the uniformity pattern under the general foldover plan are obtained respectively, which can be used as a benchmark for searching optimal foldover plans. Some illustrative examples are provided to show the theoretical results.