Optimization of family sizes in sets of crosses with a greedy allocation strategy based on automatic differentiation

Line breeding programs of autogamous crops like barley and wheat are characterized by multiple crosses with high variability in cross means and segregation variances. From theory of order statistics, we derived formulas for the expected selection differential, the response to selection, and the expected genetic value of the total selected fraction from all crosses. With these target functions, we developed a greedy optimization algorithm based on automatic differentiation to maximize the expected genetic value in the next breeding cycle by optimizing the allocation of a fixed total number of individuals to the various crosses. We show mathematically that the optimization algorithm yields the global optimum of progeny allocation for a fixed set of crosses under the constraint of fixed total population size and the given size of the selected fraction per cross. We tested the optimization algorithm with an experimental dataset from barley resistance breeding and with 60 simulated datasets characterized by different ratios of the variance of the segregation standard deviations and the variance of the cross means. For every scenario, we investigated population sizes of 400 and 1,200 individuals and heritabilities of 0.7 and 0.9. Optimized family sizes consistently outperformed constant family sizes, particularly in scenarios with high variability in segregation variances and a large ratio of the variance of the segregation standard deviations and the variance of the cross means. The relationship between the segregation variances and the optimal family sizes was approximately linear. The results indicate that the optimal allocation of individuals to crosses is dataset-specific and cannot be derived from a single heuristic criterion. The study highlights the importance of considering segregation variances in optimizing family sizes and offers user-friendly R code for improving the efficiency of line breeding programs.