On the optimality of orthogonal and balanced arrays with $$N\equiv 0$$ $$(\text {mod}$$ 9) runs

Abstract

We investigate the role of orthogonal arrays (OAs) and balanced arrays (BAs) in both full and fractional factorial designs with N runs and m three-level quantitative factors. Firstly, due to the non-existence of the OA(18, 8, 3, 2), we find and construct a BA(18, 8, 3, 2) that represents the E-, A-, D-optimal design with $$N=18$$ runs and $$m=8$$ three-level factors under the main-effect model. Also, we are interested in comparing the OA(N, m, 3, 2)s with the BA(N, m, 3, 2)s, when they represent designs with $$N\equiv 0$$$$(\text {mod}$$ 9) runs and m three-level factors with respect to the E-, A-, D-criteria under the second-order model. We provide a generalized definition of balanced arrays. Moreover, we find and construct the OA(N, m, 3, 2)s and the BA(N, m, 3, 2)s that represent the E-, A-, D-optimal designs with $$N=9$$, 18, 27, 36 runs and $$m=2$$ three-level factors under the second-order model. Furthermore, it is shown that the BA(18, m, 3, 2)s, $$m=3$$, 4 and a BA(27, 3, 3, 2) perform better than the OA(18, m, 3, 2)s, $$m=3$$, 4 and the OA(27, 3, 3, 3), respectively, when they represent the corresponding designs with respect to the E-, A-, D-criteria under the second-order model.