AN APPROACH TO THE CONSTRUCTION OF ASYMMETRICAL ORTHOGONAL ARRAYS

Use of asymmetrical orthogonal arrays (OA's) for planning industrial experiments with mixed levels has become increasingly popular. In addition to applications to experimental investigations in many disciplines, they can also be used in the balanced repeated-replications method of inference from general stratified survey samples. In this article, we propose a general approach to the construction of asymmetrical OA's with economic run size and flexibility in the choice of factor levels. As applications of this approach, we construct several general classes of arrays that include numerous existing classes of arrays as special cases. A large number of known asymmetrical OA's can be reproduced by our approach in a unified and simple manner. Many new arrays are obtained. A list of asymmetrical OA's with moderate run size is given. Our approach consists of three steps. We first construct an OA as the generalized Kronecker sum of an OA and some difference matrices. We then add, to the constructed OA, columns that are based on another OA. Since the choice of the second OA is quite flexible, the resulting array can take several forms. As the third step we use the replacement method to substitute p-level columns by p(r)-level columns, r > 1. Use of the methods in steps 2 and 3 greatly increases the variety of asymmetrical OA's. As an example, for 48 runs, we found thirteen asymmetrical OA's with factor levels equal to 2, 3, 4, 6, 8, or 12. Out of these OA's seven are apparently new.