Fig. 5

Illustration of the set of permissible breeding goals. Illustrative example with a two-dimensional space of phenotypes. The level sets of the objective function indicate the position of the optimum breeding goal \({\dot{\varvec{\upmu} }}_{b}^{s}\) in the search area \({\mathcal{S}}_{b}\) for the breed. The search area includes the response area of the breed but excludes putative breeding goals that violate ethical constraints. The set \({\mathcal{U}}_{b}\) of permissible breeding goals is included in the response area. The optimum permissible breeding goal \({\dot{\varvec{\upmu}}}_{b}^{i}\) has the property that the level sets of the objective function are tangential to the boundary of \({\mathcal{U}}_{b}\). The vector from the current trait means \({\varvec{\upmu}}_{b}^{c}\) to the optimum permissible breeding goal \({\dot{\varvec{\upmu}}}_{b}^{i}\) defines the optimum selection index