Integer programming approaches to find row-column arrangements of two-level orthogonal experimental designs

Nonregular fractional factorial experimental designs offer flexibility in terms of run size as well as the possibility to estimate partially aliased effects. For this reason, there is much interest in finding good nonregular designs and in orthogonal blocking arrangements of these designs. In this contribution, we address the problem of finding orthogonal blocking arrangements in scenarios with two crossed blocking factors. We call these blocking arrangements orthogonal row-column arrangements. We propose two strategies to find row-column arrangements of given two-level orthogonal treatment designs such that the treatment factors’ main effects are orthogonal to both blocking factors. The first strategy involves a sequential approach which is especially useful when one blocking factor is more important than the other. The second strategy involves a simultaneous approach for situations where both blocking factors are equally important. For the latter approach, we propose three different optimization models, so that, in total, we consider four different methods to obtain row-column arrangements. We compare the performance of the four methods by looking for good row-column arrangements of the best two-level 24-run designs in terms of the G-aberration criterion. We compare the methods in terms of computing time and in terms of solution quality. We then apply the best approaches to 64- and 72-run orthogonal designs, and end the paper with a conclusion.