Optimal foldover plans for two-level fractional factorial designs

A commonly used follow-up experiment strategy involves the use of a foldover design by reversing the signs of one or more columns of the initial design. Defining a foldover plan as the collection of columns whose signs are to be reversed in the foldover design, this article answers the following question: Given a 2 ^k-p design with k factors and p generators, what is its optimal foldover plan? We obtain optimal foldover plans for 16 and 32 runs and tabulate the results for practical use. Most of these plans differ from traditional foldover plans that involve reversing the signs of one or all columns. There are several equivalent ways to generate a particular foldover design. We demonstrate that any foldover plan of a 2 ^k-p fractional factorial design is equivalent to a core foldover plan consisting only of the p out of k factors. Furthermore, we prove that there are exactly 2 ^k-p foldover plans that are equivalent to any core foldover plan of a 2 ^k-p design and demonstrate how these foldover plans can be constructed. A new class of designs called combined-optimal designs is introduced. An n-run combined-optimal 2 ^k-p design is the one such that the combined 2 ^k-p+1 design consisting of the initial design and its optimal foldover has the minimum aberration among all 2 ^k-p designs.