Order-of-addition orthogonal arrays to study the effect of treatment ordering

The effect of the order in which a set of m treatments is applied can be modeled by relative-position factors that indicate whether treatment i is carried out before or after treatment j, or by the absolute position for treatment i in the sequence. A design with the same normalized information matrix as the design with all m! sequences is D- and G-optimal for the main-effects model involving the relative-position factors. We prove that such designs are also I-optimal for this model and D-optimal as well as G- and I-optimal for the first-order model in the absolute-position factors. We propose a methodology for a complete or partial enumeration of nonequivalent designs that are optimal for both models.