Improving the space-filling behavior of multiple triple designs

A space-filling (SF) design gives a good representation of experimental region with even fewer points by selecting its points everywhere in the region with as few gaps as possible. Elsawah (J Comput Appl Math 384:113164, 2021) presented the multiple tripling (MT) technique for constructing a new class of three-level designs, called multiple triple designs (MTDs). The MT technique showed its superiority over the widely used techniques by constructing new large optimal MTDs in an efficient manner using small initial designs (InDs). This paper gives a closer look at the SF behavior of MTDs after all of its factor projections and level permutations (FPs-LPs) that alter their statistical inference abilities. The selection of optimal designs by FPs-LPs of such large MTDs needs millions of trials to test all the possible cases. This paper tries to solve this hard computational problem by building theoretical bridges between the SF behavior of the MTD after all of its FPs-LPs and the behavior of the corresponding InD, which is investigated based on the similarity among its runs, confounding among its factors, and uniformity of its points. This study provides benchmarks to guide the experimenters before using FPs-LPs for improving the SF behavior of MTDs in the full dimension and any low dimension. Moreover, the construction of non-isomorphic MTDs is discussed and a lower bound of the number of non-isomorphic MTDs is given. Finally, numerical studies to support the interesting theoretical findings are provided.