Type 1 optimal 2^m 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 2^m 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).