A catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs

ABSTRACT

The foldover plan is a transformation map for adding a foldover design to the initial design, thus resulting in a combined design which can be used for breaking the links between aliased effects. This paper discusses the optimality of foldover plans for four-level designs via the most common criteria: the generalized word-length pattern (GWLP), Lee discrepancy (LD), wrap-around discrepancy (WD), centered discrepancy (CD) and mixture discrepancy (MD). We prove that the LD, WD and GWLP are equivalent for: any initial design and the corresponding foldover designs; any foldover design and its complementary foldover design; and any combined design and its complementary combined design. However, these interesting properties do not necessarily take place in the cases of the CD and MD. New analytical expressions and lower bounds of these discrepancies are given for initial and combined designs, which can be used as benchmarks for constructing uniform designs. For illustration of the usage of our theoretical results, a catalog of optimal foldover plans for constructing U-uniform minimum aberration four-level combined designs that involve $2\le m\le 10$ factors with $8\le n\le 52$ run is tabulated, which can be used for investigating either qualitative or quantitative factors.

KEYWORDS:

• Foldover plan
• Foldover design
• combined design
• optimal foldover plan
• discrepancy
• generalized word-length pattern

MSC:

• 62K05
• 62K15

Acknowledgments

The authors greatly appreciate valuable comments and suggestions of the referees and the Associate Editor that significantly improved the paper. We also greatly appreciate the useful comment of Dr. Y. D. Zhou about Lemma 4 in [9].

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

A. M. Elsawah  http://orcid.org/0000-0001-6116-4779

Additional information

Funding

This work was partially supported by the UIC Grant (Nos. R201409, R201712 and R201810) and the Zhuhai Premier Discipline Grant.