Exact optimality of augmented order-of-addition designs

Abstract

We prove that, for the pairwise-ordering model with at least three components, adding any run to an order-of-addition orthogonal array gives a $D$-optimal design among all exact designs of the enlarged size, allowing replication. We also establish the corresponding result for foldover designs: adding any reversal pair to a foldover order-of-addition orthogonal array gives a $D$-optimal design within the foldover class at the enlarged size. Both augmentations also minimise the average prediction variance and a covariance-scaled information discrepancy, and maximise the minimum relative efficiency. The key step represents the scaled discrepancy through signed comparisons of triples and derives a sharp integer lower bound; classical spectral inequalities then give the exact-size optima, and signed four-component relative-order counts characterise equality. Complete reversal pairs additionally protect the estimation of pairwise-ordering effects against arbitrary reversal-invariant nuisance means. At the smallest estimable five-component foldover size, exact enumeration identifies five determinant-optimal classes with different prediction and balance properties, and a seven-component example shows that deleting a reversal pair need not be optimal even when it minimises the discrepancy. Constructions through ten components illustrate the attained run sizes.