Risk Management in Herd Sire Portfolio Selection: A Comparison of Rounded Quadratic and Separable Convex Programming

Abstract

Quadratic programming, which does not guarantee integer solutions, may be used to consider risk when herd sires are selected. Because semen cannot be purchased in partial units, two alternatives were investigated: separable convex programming (a linear approximation of quadratic programming) and rounded quadratic programming. Expected net revenue and its variance were calculated for 383 Holstein bulls. Expected net revenue was based on semen price; PTA dollars for milk, fat, and protein; and three linear type traits: foot angle, udder depth, and teat placement. Maximization of utility (expected net revenue less its variance times a risk aversion factor) was subject to constraints: lots of 5 units, maximum average price per lot, minimum number of lots from bulls that transmit calving ease, and minimum number of different sires of selected bulls. Multiple formulations for expected net revenue, constraint combinations, and risk aversion factors were used. Twenty-two of 60 pairs of sire portfolios from rounded quadratic and separable convex programming were identical. Semen price per lot and lots from bulls that transmit calving ease were influential constraints. At the two lowest amounts of risk aversion examined, results supported common recommendations of selection of three to seven herd sires, regardless of herd size.