Use of nonlinear programming to determine the economically optimal energy density in laying hens diet during phase 1

This study was designed to show the advantage of nonlinear programming in diet formulation. A nonlinear programming Excel workbook was developed that used the Excel solver to optimize energy density and bird performance. In this study 6 dietary treatments (include 2.515, 2.615, 2.715, 2.815, 2.915 and 3.015 Mcal of metabolizable energy per kilogram) were fed to Hy-line W-36 laying hens (n = 192) in phase 1 (from 24 to 32 weeks of age). Data were fitted to quadratic equations to express egg mass, feed consumption and objective function return over feed cost in terms of energy density. Nutrient: energy ratio constraints were transformed into equivalent linear constraints. To demonstrate the capabilities of the model, the prices for egg, corn and soybean meal were increased and decreased by 25% and the program solved for the maximum profit and optimized feed mix. Formulations were identical in all other respects. By increasing egg price, the model changed the optimal diet formulation and energy density in such a way as to improve performance and feed consumption and accepted a higher energy concentration. In order to make nutritional and economical decisions for a given feed formulation problem, the sensitivity analysis was performed. The sensitivity analysis for linear programming showed that if the protein level of the diet were to change from 16.046 to 15.046%, the cost of the diet would decrease by $0.0293 from $0.4089 to $0.3799 kg-1. The sensitivity analysis for nonlinear programming showed that if the protein level of the diet were to change from 14.96 to 13.96%, the cost of the diet would decrease by $0.0272 kg-1 from $0.3547 to $0.3275 kg-1. Results indicated that there are considerable savings to be made by egg producers from the use of the nonlinear programming model described here as opposed to a linear one with fixed minimums for energy and other nutrients. These savings result from the nonlinear programming modelsʹ ability to determine the most profitable energy density that should be fed as energy and protein prices change.