Optimal use of dominance information requires a mating system and predictions of specific combining abilities for each set of prospective parents. Current evaluation procedures provide such predictions only for a limited number of parents. A procedure is described that predicts the specific combining ability for any parents. In this procedure, for each set of parents and their ancestors, the additive relationship matrix is created as a dense matrix. This matrix is then used to create a parental dominance matrix in a sparse matrix form, in which the rows of the matrix correspond to all parental combinations for which predictions are already available. Each new prediction requires a solution of the system of equations with the parental dominance matrix as the left-hand side. The efficiency of the mating system that accounts for dominance was evaluated in a simulation study. The simulated data files varied with respect to proportion of males and females selected, proportion of cattle born through embryo transfer, and additive and dominance variance. Sires and dams were preselected based on the additive merits only, but specific matings were arranged based on the combined additive plus dominance merit. The response to selection with consideration of dominance increased from 3.8 to 16.6% of the response from one generation of additive selection. The response was greater when the additive variance was smaller, the dominance variance was larger, the intensity of additive selection was lower, and the proportion of full sibs was greater. Use of dominance in the mating system is feasible and results in an additional genetic response to selection.