Implementation of a genetic algorithm on MD-optimal designs for multivariate response surface models

This study presents a genetic algorithm (GA) for identifying the exact D-optimal design for multivariate response surface models (called MD-optimal design). The MD-optimal design minimizes the volume of joint confidence regions of model parameters. The covariance between any two responses is assumed to be identical in the two examples of four responses and biresponse problems considered in this study. We have also provided an example of covariance estimation. In order to obtain the initial candidate set, we first obtain a D-optimal design for each response model by using a conventional approach; then, the set of solutions obtained from the individual model is treated as the initial set in the GA. This shows that the MD-optimal designs converge toward the same D-optimal design in a single response linear model; however, the different variance–covariance matrices attain dissimilar objective values. The GA exhibits stable representation in multiple response design problems and performs better than the US algorithm, which is generated only near the MD-optimal design. It is possible for an experimenter to set a high crossover rate except full crossover, and estimate the variance–covariance matrix in the preprocess or set it as an identity matrix in the process of the GA.