Optimal restricted phenotypic selection.

Phenotypic selection is modified by introducing upper limits on the portion (P 1) of individuals selected from a family as well as on the portion (P 2) of family number that are allowed to contribute. At a preset selection proportion, P and P 1, the maximum genetic gain is obtained by finding an optimum restriction on family number (P 2 (*) ). A numerical procedure for solving the problem of optimization is developed for infinite populations. In small populations, maximum gain and P 2 (*) can be found by simply comparing all possible P2. Numerical examples are demonstrated for infinite breeding populations, assuming a normally-distributed family mean and within-family deviation. Selection and its simulation were applied to the fieldtest results of two tree species. Optimum restriction on family number is very close to P/P 1, especially when heritability is low. In the real world of tree breeding, P 2 (*) is given, or approximated, by P/P 1+1/ tm where m is the initial family number. The improvement of gain and the conservation of inbreeding effective population size are easy with high heritability and could be simultaneously obtained by using intense selection with a relatively low P 1.