New foundations for designing U-optimal follow-up experiments with flexible levels

Abstract

Follow-up experiments are used extensively to provide precious information about the relationships between inputs and outputs to gain a better understanding of a process or system under study. This article gives a new look at designing optimal follow-up experiments that involve any number of factors with any number of different levels in light of the uniformity behaviour of the corresponding two stage sequential experiments, which are composed of initial experiments and follow-up experiments. Novel analytical expressions and lower bounds of the wrap-around $$L_2$$-discrepancy, as a uniformity measure, for sequential experimental designs are proposed for evaluating the optimality of the follow-up experimental designs. Finding equivalent follow-up experimental designs is investigated, which can be used to reduce the computational complexity. Our results show that two stage sequential experimental designs give greater precision than single stage experimental designs with the same size.