Theoretical determination of the optimum number of parents for synthetics.

From the consideration of the expressions of the mean and of the variances amongk-parent synthetics, it is possible, in the absence of epistasis, to give an algebraic determination of the optimum number of parents to include in a synthetic. The knowledge of four components of variance of inbred populations is necessary. Such components can be replaced by four simple statistics for the plant breeder: variances of general and specific combining abilities, variance among S1 families from the parents, and covariance between S1 value and general combining ability (GCA). A numerical study shows that this optimum is rather broad for a number of parents greater than four. As expected, the optimum tends to be higher for greater inbreeding depression. With inbreeding depression greater than 0.30, the maximum gain, in comparison to the random mating population with realistic selection intensity, would be less than 5%. In such a situation it will be better to use as synthetic the population improved by recurrent selection.