Computational Statistics and Data Analysis 118 (2018) 84–97 Contents lists available at ScienceDirect Computational Statistics and Data Analysis journal homepage: www.elsevier.com/locate/csda Optimizing two-level orthogonal arrays for simultaneously estimating main effects and pre-specified two-factor interactions Ping-Yang Chen a , Ray-Bing Chen a, *, C. Devon Lin b a Department of Statistics, National Cheng Kung University, Tainan 70101, Taiwan b Department of Mathematics and Statistics, Queen’s University, Kingston, ON, Canada article info Article history: Received 16 November 2016 Received in revised form 7 July 2017 Accepted 9 August 2017 Available online 20 September 2017 Keywords: D-optimal design Fractional factorial design Hadamard matrix Swarm intelligence optimization abstract This paper considers the construction of D-optimal two-level orthogonal arrays that allow for the joint estimation of all main effects and a specified set of two-factor interactions. A sharper upper bound on the determinant of the related matrix is derived. To numerically obtain D-optimal and nearly D-optimal orthogonal arrays of large run sizes, an efficient search procedure is proposed based on a discrete optimization algorithm. Results on designs of 20, 24, 28, 36, 44 and 52 runs with three or fewer two-factor interactions are illustrated here to demonstrate the performance of the proposed approach. In addition, two cases with four two-factor interactions are also demonstrated here. © 2017 Elsevier B.V. All rights reserved. 1. Introduction Two-level orthogonal arrays are most commonly used in industrial experiments to identify active effects because they allow for the joint estimation of all main effects and interactions (Hamada and Wu, 1992). In some experiments, the experimenter wants to estimate a set of specified effects, which is known as a requirement set (Greenfield, 1976) that consists of all main effects and some specified two-factor interactions (2fis). For example, Wu and Chen (1992) provided an example that the quality of products is affected by the factors in two independent manufacturing processes. In addition to the main effects, the 2fis between factors from each of the two processes are necessary to take into account. For more details, please refer Wu and Chen (1992). There is ample research on obtaining two-level orthogonal arrays for estimating such a requirement set (see, for example, Franklin and Bailey, 1977; Wu and Chen, 1992; Dey and Suen, 2002; Ke and Tang, 2003; Cheng and Tang, 2005). Recently, Tang and Zhou (2009) investigated the existence of such orthogonal arrays and provided a construction strategy. Tang and Zhou (2013) further derived theoretical results for finding D-optimal orthogonal arrays for jointly estimating main effects and some specified 2fis, and showed the results of 12 and 20 runs for all the requirement sets with three or fewer 2fis. The strategy for obtaining D-optimal orthogonal arrays in Tang and Zhou (2009, 2013) is as follows. Let S denote the requirement set that includes m main effects and e specified 2fis, and C (S ) be a set that contains all 2fis in S and the main effects that are involved in at least one 2fi in S . Thus C (S ) is a subset of S which is called the core set of S . For example, for S ={F 1 , F 2 , F 3 , F 4 , F 5 , F 6 , F 1 F 2 , F 3 F 4 }, where F i F j is the 2fi of F i and F j , and then C (S ) ={F 1 , F 2 , F 3 , F 4 , F 1 F 2 , F 3 F 4 }. To find an orthogonal array of n runs with m columns that supports S , a saturated orthogonal array with more columns, say H, is considered, which can be expressed as (D 1 , D 2 , D 3 ). Here the matrix D 1 is the subarray with m 1 columns that supports C (S ), * Corresponding author. E-mail addresses: R28021017@mail.ncku.edu.tw (P.-Y. Chen), rbchen@mail.ncku.edu.tw (R.-B. Chen), cdlin@mast.queensu.ca (C.D. Lin). http://dx.doi.org/10.1016/j.csda.2017.08.012 0167-9473/© 2017 Elsevier B.V. All rights reserved.