Algebraic generation of minimum size orthogonal fractional factorial designs: an approach based on integer linear programming.

Generation of orthogonal fractional factorial designs (OFFDs) is an important and extensively studied subject in applied statistics. In this paper we show how searching for an OFFD that satisfies a set of constraints, expressed in terms of orthogonality between simple and interaction effects, is, in many applications, equivalent to solving an integer linear programming problem.We use a recent methodology, based on polynomial counting functions and strata, that represents OFFDs as the positive integer solutions of a system of linear equations. We use this system to set up an optimization problem where the cost function to be minimized is the size of the OFFD and the constraints are represented by the system itself. Finally we search for a solution using standard integer programming techniques. Some applications are also presented in the computational results section. It is worth noting that the methodology does not put any restriction either on the number of levels of each factor or on the orthogonality constraints and so it can be applied to a very wide range of designs, including mixed orthogonal arrays.