Many computer algorithms have been developed to construct experimental designs that are D-optimum for the fixed parameters of a statistical model. However, the case when the interest is in the variance components has not received much attention. This problem has similarities with that of designing experiments aiming at D-optimality for the fixed parameters of nonlinear models as its solution depends on the values of the unknown parameters that need to be estimated. An algorithm that can be used to construct locally and pseudo-Bayesian A- and D-optimum designs for the variance components in a linear mixed effects model, or for variance ratios, when there is a three-stage crossed or nested variability structure is proposed. Suitable visualizations of the results in order to help the assessment of the robustness of the designs against possible inaccuracies of the assumptions about the true values of the variance components used in the selection of the designs are recommended. © 2013 Elsevier Inc. All rights reserved.