Incorporation of Uncertainty in Composition of Feeds into Least-Cost Ration Models. 1. Single-Chance Constrained Programming

Abstract

Ration formulation by linear programming techniques assumes a perfect knowledge of all coefficients involved. Because of variation in composition of feedstuffs, this assumption cannot hold in practical situations. However, the problem can be reformulated as a chance-constrained programming model for which a deterministic equivalent can be derived. In the case where only one nutrient is considered random, the methodology is labeled as single-chance constrained programming. The exact deterministic equivalent requires nonlinear programming methods for which algorithms are not available to the feed industry. Three linear approximation algorithms were derived and compared with the exact solution given by nonlinear programming in the formulation of a premix feed. Prices from 1970 to 1979 for 17 feed ingredients were used. An algorithm based on a linear function of the standard deviation of the random nutrient within each feedstuff, for which a penalty parameter is iterated in a search for a desired probability, yielded results very similar to the exact solutions found by nonlinear programming methods. This methodology could be implemented easily by the feed industry.