Counting vectors for Orthogonal Fractional Factorial Design generation

Abstract

Generation of orthogonal fractional factorial designs (OFFDs) is an important and extensively studied subject in applied statistics.In this paper we present a methodology based on counting vectors and a methodology based on polynomial counting functions and strata,[1, 2, 3]. Both methodologies allow us to represent the OFFDs that satisfy a given set of constraints, expressed in terms of orthogonality between simple and interaction effects, as the positive integer solutions Y of a homogeneous system AY = 0 of linear equations.We show how to use this system AY = 0• to compute, for smaller cases, a minimal set of generators of all the OFFDs (Hilbert basis);• to obtain, for larger cases, a sample of OFFDs.Finally we describe a method to find minimum size OFFDs. We set up an optimisation problem where the cost function to be minimized was the size of the OFFD and the constraints were represented by the system AY = 0. Then we searched for a solution using standard integer programming techniques.It is worth noting that the methodology does not put any restriction on the number of levels of each factor and so it can be applied to a very wide range of designs, including mixed orthogonal arrays.