Least-Cost Ration Formulations for Holstein Dairy Heifers By Using Linear and Stochastic Programming

Abstract

Four mathematical programming models were developed to formulate rations for large breed replacement dairy heifers in each of 11 different weight classes from 50 to 550 kg and daily growth rates of 600, 700, and 800 g, with the objective of achieving a final calving weight of 600 kg. First, a base linear programming model was developed; then, to account for variability in the crude protein content of ration ingredients three other methods were used: right-hand side adjustment, incorporation of a safety margin, and stochastic programming. The average daily cost to calving, given a daily gain of 600, 700, and 800 g, was $0.62, $0.64, and $0.68, respectively. The total feed cost to 600 kg was $89.87 more for a growth rate of 600 over 800 g/d. The stochastic programming model performed better, on the bases of cost and protein-feeding, than did the right-hand side adjustment or the safety margin methods. The stochastic programming model over-adjusted crude protein by 5% and cost an average of 3.5% more than the linear programming solution for a dairy heifer growing at 800 g/d with a desired probability of 80% of crude protein intake achieving the NRC minimum. The other two methods over-adjusted crude protein by 10 and 13% and cost an extra 5.5 and 7.6%, respectively, for the right-hand side adjustment and the safety margin methods.