Standard response surface methodology experimental designs for estimating location models involve the assumption of homogeneous variance throughout the design region. However, with heterogeneity of variance these standard designs are not optimal. Using the D and Q-optimality criteria, this dissertation proposes a two-stage experimental design procedure that gives more efficient designs than the standard designs when heterogeneous variance exists. Several multiple variable location models, with and without interactions, are considered. For each the first stage estimates the heterogeneous variance structure, while the second stage then augments the first stage to produce a D or Q-optimal design for fitting the location model under the estimated variance structure. However, there is a potential instability of the variance estimates in the first stage that can lower the efficiency of the two-stage procedure. This problem can be addressed and the efficiency of the procedure enhanced if certain mild assumptions concerning the variance structure are made and formulated as a prior distribution to produce a Bayes estimator. With homogeneous variance, designs are analyzed using ordinary least squares. However, with heterogeneous variance the correct analysis is to use weighted least squares. This dissertation also examines the effects that analysis by weighted least squares can have and compares this procedure to the proposed two-stage procedure. / Ph. D.