Stochastic Feed Formulation

The purpose of commercial feed formulation is to balance nutrients in diets to meet the nutritional requirements of animals at least cost. Almost all commercial feed formulation software use Linear Programming for feed formulation. In real life nutrient composition is highly variable. This variation is associated with variety of factors which include variation of nutrient content of ingredients coming from different batches and sources and variation attributed to the laboratory procedure and human error. For example if a same sample of soybean is analyzed multiple times for protein content, it is very likely that every time a slightly different value will be obtained. Same sample analyses in different laboratories or by different persons usually results in highly variable values. In Linear Programming method a mean value of these analytical values is used for formulation. Statistically, these mean values are associated with only 50% confidence of meeting the requirements in prepared formula.

Most feed manufacturers want to minimize the risk of not meeting the nutrient requirements of the animal. The following two methods have been proposed to minimize this risk.

• Application of safety margin in linear formulation
• Use of Stochastic Programming

In the first solution diets are formulated at 5-10% higher than requirement. This is an unsatisfactory solution from quality control and economic points of view because it does not account for variation level. Nutrient variation could be higher enough to exceed safety margin level. In case where variation level is low, the formulated diet will be unnecessarily expensive. This will result in economical loss of the Feed manufacturer or livestock farmer.

The second solution Stochastic Programming has been widely recommended for feed formulation. The term Stochastic comes from the Greek word meaning skilful at aiming. In modern terms, stochastic has become a statistical word referring to variables that are random or uncertain.

The standard form of constraints in linear programming is as follows.

subject to

If a user wishes to increase the success rate of meeting the ith nutrient in the diet up to or to fall below the level bi, to a probability of P > Øi, then both constraints will be modified as follows

Because the above constraints are nonlinear so they could not be solved using software that formulates least cost diets by linear programming. This form of constraint is solved by stochastic programming. Thus Stochastic Programming provides assurance of meeting the requirement of animals to a grater probability.

WinFeed provides an excellent utility to solve Stochastic constraints. It uses standard deviation of variability to meet required degree of assurance, yet it is very simple to use.