Constructing Two-Level Designs by Concatenation of Strength-3 Orthogonal Arrays

Abstract

Two-level orthogonal arrays of N runs, k factors and a strength of 3 provide suitable fractional factorial designs in situations where many of the main effects are expected to be active, as well as some two-factor interactions. If they consist of N/2 mirror image pairs, these designs are fold-over designs. They are called even and provide at most N/2 − 1 degrees of freedom to estimate interactions. For k < N/3 factors, there exist strength-3 designs that are not fold-over designs. They are called evenodd designs and they provide many more degrees of freedom to estimate interactions. For N ≤ 48, attractive even-odd designs can be extracted from complete catalogs of strength-3 orthogonal arrays. However, for larger run sizes, no complete catalogs exist. In order to construct even-odd designs with N > 48, we develop an algorithm for an optimal concatenation of strength-3 designs involving N/2 runs. Our approach involves column permutations of one of the concatenated designs, as well as sign switches of the elements of one or more columns of that design. We illustrate the potential of the algorithm by generating two-level even-odd designs with 64 and 128 runs involving up to 33 factors, because this allows a comparison with benchmark designs from the literature. With a few exceptions, our even-odd designs outperform or are competitive with the benchmark designs in terms of the aliasing of two-factor interactions and in terms of the available degrees of freedom to estimate two-factor interactions.

A nonlinear multidimensional knapsack problem in the optimal design of mixture experiments

“…VNS has been successfully applied to a wide variety of optimization problems such as vehicle routing (Kytöjoki et al, 2007), project scheduling (Fleszar and Hindi, 2004), automatic discovery of theorems (Caporossi and Hansen, 2004), graph coloring (Avanthay et al, 2003), the synthesis of radar polyphase codes (Mladenović et al, 2003), and the molecular distance geometry problem (Liberti et al, 2009). In the field of experimental design, VNS has been used to determine optimal run orders for the design points of standard experimental designs in the presence of serially correlated responses (Garroi et al, 2009), to assign treatments to whole plots in orthogonal split-plot designs (Sartono et al, 2015) and to optimize the concatenation of two orthogonal experimental designs (Vazquez et al, 2019). In each of these three cases, solving the optimal experimental design problem at hand involved the identification of an optimal permutation.…”

Orthogonal Minimally Aliased Response Surface Designs for Three-Level Quantitative Factors and Two-Level Categorical Factors

“…Schoen et al (2010) presented an enumeration algorithm and a large catalog of OAs. Schoen et al (2017), Schoen and Mee (2012) and Vazquez et al (2019), among others, performed detailed studies concerning two-level OAs. Sartono et al (2012) investigated a large collection of three-level OAs assuming all factors are categorical.…”

“…This result suggests that one method for constructing large designs, with or without two-level categorical factors, might be to concatenate two definitive screening designs or, more generally, two OMARS designs. Such constructions have been proven to be successful for large two-level OAs(Li and Lin (2016);Vazquez et al (2019)). Design 1 allows the estimation of any full second-order model in any three quantitative factors and one categorical factor.…”

On Construction of Nonregular Two-Level Factorial Designs With Maximum Generalized Resolutions

“…Nonregular designs from quaternary codes were considered by, for example, Xu and Wong (2007) and Phoa and Xu (2009). A recent work on constructing strength-three orthogonal arrays is that of Vazquez, Goos and Schoen (2019). We refer to Xu, Phoa and Wong (2009) for a general review of topics on nonregular designs.…”