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

Design of experiments is an effective, generic methodology for problem solving as well as for improving or optimizing product design and manufacturing processes. The most commonly used experimental designs are two-level fractional factorial designs. In recent years, nonregular fractional factorial two-level experimental designs have gained much popularity compared to the traditional regular fractional factorial designs, because they offer more 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 of high-quality nonregular two-level designs 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 orthogonal designs in terms of the G-aberration criterion, and apply the best of these methods to 64- and 72-run orthogonal designs.