Orthogonal blocking of regular and nonregular strength-3 designs

There is currently no general approach to orthogonally block two-level and multi-level orthogonal arrays and mixed-level orthogonal arrays. In this article, we present a mixed integer linear programming approach that seeks an optimal blocking arrangement for any type of regular and nonregular orthogonal array of strength 3. The strengths of the approach are that it is an exact optimization technique which guarantees an optimal solution, and that it can be applied to many problems where combinatorial methods for blocking orthogonal arrays cannot be used. By means of 54- and 64-run examples, we demonstrate that the mixed integer linear programming approach outperforms two benchmark techniques in terms of the number of estimable two-factor interaction contrasts. We demonstrate the generality of our approach by applying it to the most challenging instances in the catalog of all orthogonal arrays of strength 3 with up to 81 runs. Finally, we show that, for two-level fold-over designs involving many factors, the only way to arrange the runs in orthogonal blocks of size four is by grouping two pairs of fold-over pairs in each of the blocks.