Herd Sire Portfolio Selection: A Comparison of Rounded Linear and Integer Programming

Abstract

Use of linear programming to select portfolios of herd sires can result in solutions containing noninteger units per bull. Because semen cannot be purchased in partial units, two alternatives were investigated: integer programming and rounded linear programming. Expected net revenue based on PTA dollars for milk, fat, and protein; udder depth; teat placement; foot angle; and semen price were calculated for 383 Holstein bulls. Maximization of expected net revenue was subject to constraints: required lots of 5 units, maximum average price per lot, minimum and maximum lots per bull, minimum average reliability, minimum number of lots from bulls that transmit calving ease, and minimum number of bulls. Multiple formulations were used for expected net revenue and constraint combinations. Eighteen of 36 linear programming portfolios contained noninteger lots per bull. Five rounded linear programming portfolios were identical to their respective integer programming portfolios. Differences in lots per bull between the remaining 13 pairs of portfolios were small. Linear programming portfolios that were unconstrained by semen price tended to be integer. Six of 18 rounded linear programming portfolios did not stay within bounds set by constraints. For the model used, integer programming did not offer a significant advantage over rounded linear programming.