Incorporation of Uncertainty in Composition of Feeds into Least-Cost Ration Models. 2. Joint-Chance Constrained Programming

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Abstract
Least-cost formulation of livestock rations and mixes by linear programming techniques assumes a perfect knowledge of the composition of each available ingredient. This assumption cannot hold in practical situations. When more than one nutrient is considered imperfectly known (random), there exists a deterministic equivalent from the family of joint-chance constrained programming problems. The theory is reviewed first; from this theory four models are proposed for the formulation of a supplement feed. The first model represents what is considered an industry standard. The second model is a linear approximation to the original problem and is solved by linear programming techniques. The last two models make use of a Bonferroni inequality but must be solved by nonlinear programming techniques. Fifteen ingredients were considered; their prices from 1970 to 1979 were published. Over the 10-yr period, cost of ingredients per ton of mix averaged $135.27, $124.82, $127.05, and $126.24 for the four models, respectively. Therefore, a $9.03 (6.7%) reduction in ingredient costs can be expected for the feed industry with the adoption of these techniques. However, a nonlinear programming algorithm must be used which is not available and "friendly" as current least-cost computer software.