Multi-Optimal Designs for Second-Order Response Surface Models

A conventional single design optimality criterion has been used to select an efficient experimental design. But, since an experimental design is constructed with respect to an optimality criterion pre specified by investigators, an experimental design obtained from one optimality criterion which is superior to other designs may perform poorly when the design is evaluated by another optimality criterion. In other words, none of these is entirely satisfactory and even there is no guarantee that a design which is constructed from using a certain design optimality criterion is also optimal to the other design optimality criteria. Thus, it is necessary to develop certain special types of experimental designs that satisfy multiple design optimality criteria simultaneously because these multi-optimal designs (MODs) reflect the needs of the experimenters more adequately. In this article, we present a heuristic approach to construct second-order response surface designs which are more flexible and potentially very useful than the designs generated from a single design optimality criterion in many real experimental situations when several competing design optimality criteria are of interest. In this paper, over cuboidal design region for $3\;{\leq}\;k\;{\leq}\;5$ variables, we construct multi-optimal designs (MODs) that might moderately satisfy two famous alphabetic design optimality criteria, G- and IV-optimality criteria using a GA which considers a certain amount of randomness. The minimum, average and maximum scaled prediction variances for the generated response surface designs are provided. Based on the average and maximum scaled prediction variances for k = 3, 4 and 5 design variables, the MODs from a genetic algorithm (GA) have better statistical property than does the theoretically optimal designs and the MODs are more flexible and useful than single-criterion optimal designs.