Abstract
We consider a parallel linear model with correlated dual responses on a symmetric compact design region and construct locally D-optimal designs for estimating the unknown parameters in the model, and locally optimal designs for estimating the location shift parameter. The D-optimal designs for the additive model are invariant under linear transformation of the design space but locally optimal designs for estimating the location shift do not share this property. The latter optimal designs depend on the correlation between the dual responses in an interesting and sensitive way.