The application of nonlinear programming on ration formulation for dairy cattle

J Li,E Kebreab,Fengqi You,J.G Fadel,T.L Hansen,C Vankerkhove,K.F Reed

doi:10.3168/jds.2021-20817

Abstract

The objective of this study was to compare the application of iterative linear programming (iteLP), sequential quadratic programming (SQP), and mixed-integer nonlinear programming-based deterministic global optimization (MINLP_DGO) on ration formulation for dairy cattle based on Nutrient Requirements of Dairy Cattle (NRC, 2001). Least-cost diets were formulated for lactating cows, dry cows, and heifers. Nutrient requirements including energy, protein, and minerals, along with other limitations on dry matter intake, neutral detergent fiber, and fat were considered as constraints. Five hundred simulations were conducted, with each simulation randomly selecting 3 roughages and 5 concentrates from the feed table in NRC (2001) as the feed resource for each of 3 animal groups. Among the 500 simulations for lactating cows, 57, 45, and 21 simulations did not yield a feasible solution when using iteLP, SQP, and MINLP_DGO, respectively. All the simulations for dry cows and heifers were feasible when using SQP and MINLP_DGO, but 49 and 11 infeasible simulations occurred when using iteLP for dry cows and heifers, respectively. The average ration costs per animal per day of the feasible solutions obtained by iteLP, SQP, and MINLP_DGO were $4.78 (±0.71), $4.45 (±0.65), and $4.44 (±0.65) for lactating cows; $2.39 (±0.52), $1.48 (±0.26), and $1.48 (±0.26) for dry cows; and $0.98 (±0.72), $0.97 (±0.15), and $0.91 (±0.14) for heifers, respectively. The average computation time of iteLP, SQP, and MINLP_DGO were 0.59 (±1.87) s, 1.15 (±0.62) s, and 58.69 (±68.45) s for lactating cows; 0.041 (±0.070) s, 0.76 (±0.37) s, and 14.84 (±39.09) s for dry cows; and 1.60 (±2.90) s, 0.51 (±0.19) s, and 16.45 (±45.56) s for heifers, respectively. In conclusion, iteLP had limited capability of formulating least-cost diets when nonlinearity existed in the constraints. Both SQP and MINLP_DGO handled the nonlinear constraints well, with SQP being faster, whereas MINLP_DGO was able to return a feasible solution under some situations where SQP could not.

Highlights

Feed costs account for around 50 to 70% of the expenses of operating a dairy farm (Bozic et al, 2012), which highlights the importance of minimizing this cost when feasible
The objective of this study was to compare the application of iterative linear programming, sequential quadratic programming (SQP), and mixedinteger nonlinear programming-based deterministic global optimization (MINLP_DGO) on ration formulation for dairy cattle based on Nutrient Requirements of Dairy Cattle (NRC, 2001)
Accepted October 26, 2021. *Corresponding author: kfr3@cornell.edu lating least-cost diets when nonlinearity existed in the constraints. Both SQP and MINLP_DGO handled the nonlinear constraints well, with SQP being faster, whereas MINLP_DGO was able to return a feasible solution under some situations where SQP could not

Summary

Introduction

Feed costs account for around 50 to 70% of the expenses of operating a dairy farm (Bozic et al, 2012), which highlights the importance of minimizing this cost when feasible. Diet formulation relies on the nutrient requirements of the animal, nutrient composition of available feeds, and nutrient interactions, all of which require systems for estimation, such as Nutrient Requirements of Dairy Cattle (NRC, 2001) and the Cornell Net Carbohydrates and Protein System (Fox et al, 2004). One method to formulate least-cost diets that fulfill an animal’s nutrient requirements is linear programming (LP), which optimizes a linear objective function subject to a set of linear constraints (Chandler and Walker, 1972; O’Connor et al, 1989). One limitation of LP is that it allows only linear objective functions and constraints, while some equations in the dairy nutrition model are nonlinear when adapted to an LP structure.

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Conclusion