Selecting mating pairs with linear programming techniques.

Mate selection can increase progency merit if overall merit is nonlinear for one or more component traits. An index of expected progeny merit could be calculated for all possible mating pairs, and the set of pairs with the highest progeny mean could be selected. There are serious computational problems for more than a few males and females. To select and mate f, females, and m, males, from n of each, with k0 females per male, would require (nf)(nm)f!/(k0!)m evaluations. Linear programming algorithms can determine the optimal strategy efficiently by considering only a subset of these possibilities. Let pi ij be the index of progency merit of the ith sire mated to the jth dam and Xij be the decision variable for that mating (restricted to 0 or 1). Then the problem of selecting mating pairs can be stated as: maximize sigma i sigma j pi ij Xij, subject to sigma i Xij less than or equal to 1, sigma j Xij less than or equal to k0, sigma i sigma j Xij = f, and Xij = 0 or 1. By including an artificial sire and an artificial dam and choosing appropriate merit values for the artificial matings, this problem can be solved by efficient "transportation" algorithms. These algorithms could be used to develop rational mating packages for dairy artificial insemination studs provided that an objective evaluation of progeny merit could be formulated, provided that merit is not simply additively inherited.