Type 1 optimal 2m fractional factorial plans with n≡ℓ (mod 8) runs, ℓ=1,2

This paper considers the issue of optimality of fractional factorial experiments involving m factors each at two levels. The optimality criteria used here is the type 1 criteria, due to Cheng (1978), which include the D- and A-criteria. It is shown that if there exists an orthogonal array OA(n−ℓ,m,2,3), ℓ=1,2, then there exists an n-run type 1 optimal fractional factorial plan for a 2m experiment under a model that includes the mean, all main effects and all two-factor interactions with a factor in common. These plans are obtained by adding any one run to an OA(n−1,m,2,3) for n≡1 (mod 8) and two specific runs to an OA(n−2,m,2,3) for n≡2 (mod 8).