We develop D-optimal designs for linear models with first-order interactions on a subset of the $$2^K$$2K full factorial design region, when both the number of factors set to the higher level and the number of factors set to the lower level are simultaneously bounded by the same threshold. It turns out that in the case of narrow margins the optimal design is concentrated only on those design points, for which either the threshold is attained or the numbers of high and low levels are as equal as possible. In the case of wider margins the settings are more spread and the resulting optimal designs are as efficient as a full factorial design. These findings also apply to other optimality criteria.

中文翻译：

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对称受限设计区域上具有一阶交互作用的 K 因子两水平模型的优化设计

我们为在 $$2^K$$2K 全因子设计区域的子集上具有一阶交互作用的线性模型开发 D 最优设计，当因子数设置为较高水平时，因子数设置为较低级别同时受到相同阈值的限制。事实证明，在窄边的情况下，优化设计仅集中在那些达到阈值或高低电平数量尽可能相等的设计点上。在边距较宽的情况下，设置会更加分散，最终的优化设计与全因子设计一样有效。这些发现也适用于其他最优性标准。