# Brief background to feed formulating

The following simple example is used to illustrate the basic feed formulation problem.

Suppose a laying hen needs exactly 18g of protein and 1.3MJ of energy each day; that only three ingredients (fishmeal, maize and soya) are available; and that 1kg of these ingredients contain the following:

PROTEIN (g) | ENERGY (MJ) | |
---|---|---|

Fishmeal | 650 | 13.70 |

Maize | 80 | 14.34 |

Soya | 420 | 10.29 |

The problem can be expressed as follows:

Let x_{1}, x_{2} and x_{3} be the amount of fishmeal, maize and soya respectively. Then:

650x_{1} + 80x_{2} + 420x_{3} = 18

13.70x_{1} + 14.34x_{2} + 10.29x_{3} = 1.3

(x_{1}, x_{2} & x_{3}≥ 0, for the answer to be meaningful)

Solving the above two linear equations is relatively straightforward, and since the number of ingredients is larger than the number of nutrients, there are an infinite number of solutions of how the ingredients could be combined to satisfy the hen’s protein and energy requirements simultaneously.

(e.g., x_{1} = 7.48E-03, x_{2} = 7.08E-02, x_{3} = 1.78E-02; x_{1} = 6.20E-03, x_{2} = 7.05E-02, x_{3} = 1.98E-02 are both solutions)

In the above example, the weights of nutrients required each day are used to determine a daily feed allocation. However, farm animals are not usually fed in this way. Instead of weighing out the required quantities of the three ingredients each day it is more practical to formulate a feed to meet required nutrient concentrations, and then allow the animals *ad libitum* access to the food, assuming a given daily food intake. In this case the nutrients in each ingredient as well as the nutrients required by the animal are expressed as concentrations (in the above example g/kg or percent for protein, and MJ/kg for energy). To calculate these it is necessary to know both the daily amount of each nutrient required by the animal (e.g. mg lys/bird d) and the food intake (g/d) of the animal.

The standard method of describing the nutrient requirements of an animal is to specify the quantity of each nutrient required by the animal per kg of feed. Whether the animal eats the expected amount of this food each day is not part of the linear programming problem. However, a successful nutritionist would be one who is capable of accurately predicting the food intake of the animals so that nutrients are not oversupplied in the feed resulting in wastage and reduced profits. An undersupply of nutrients in the feed will invariably result in a higher than expected food intake, resulting also in reduced profits.

Using this more acceptable method of describing an animal’s nutrient requirements, the protein and energy requirements of the laying hen, above, assuming an intake of 120g food per day, would be specified as 150g of protein (18*1000/120) and 10.8MJ of energy (1.3*1000/120) per kg of feed. The linear equations would change to:

650x_{1} + 80x_{2} + 420x_{3} = 150

13.70x_{1} + 14.34x_{2} + 10.29x_{3} = 10.8

1x_{1} + 1x_{2} + 1x_{3} = 1

The last equation is added to ensure that the sum of the three ingredients is 1kg.

## Constraints and objective function

In the above example the nutrient requirements of the laying hen are exact, but normally an animal’s nutrient requirements need not be exact and can be relaxed to fall within an acceptable range (e.g. having the protein between 140 and 160g/kg and the energy between 10.0 and 11.5MJ/kg might be acceptable for a flock of laying hens having different daily food intakes). The linear equations can now be expressed as the following three constraints:

140 ≥ 650x_{1} + 80x_{2} + 420x_{3}≥ 160

10.0 ≥ 13.70x_{1} + 14.34x_{2} + 10.29x_{3}≥ 11.5

1x_{1} + 1x_{2} + 1x_{3} = 1

This relaxation of the constraints increases the number of potential solutions even further. As there are many potential solutions, it makes sense to find the best of these solutions, the most common being the least-cost solution. In this case a feed containing the nutrients in the required concentrations is formulated to contain the least expensive combination of ingredients. This can be accomplished by adding another linear expression, called an objective function, which can be minimised or maximised using a complex mathematical algorithm that was developed in the 1940’s. This objective function must be a linear expression of the ingredients, expressing their relative costs in terms of what is to be minimised or maximised. For example if cost is to be minimised, the objective function would be the sum of all the ingredient prices, whereas to maximise the energy, the energy values for each ingredient would be summed.

In our example, if the cost of the feed is to be minimised and the three ingredients cost R2300, R600 and R950 per ton respectively, our objective function would be:

2300x_{1} + 600x_{2} + 950x_{3}

It is important to realise that it is not only the cost that can be minimised, but that any nutrient can be minimised or maximised.

## Nutrient Independence

Ideally, the desired concentration of one nutrient should be independent of the concentration of another nutrient in the feed. Unfortunately this is not always the case. Often the amount of one nutrient will influence the necessary concentration of another nutrient in a feed. For example, bone growth is maximised when the ratio between calcium and phosphorus is 2:1; as the energy content of a feed is increased so the amino acid content should also increase, hence some nutritionists express the amino acid content as a proportion of the energy content; and there is evidence in broiler nutrition that to maximise performance, amino acid concentrations should be expressed as a proportion of the protein content of the feed.

WinFeed provides two methods by which you can express nutrient dependencies:

*Nutrient ratios*

A simple example is the calcium to phosphorus ratio, which should be maintained at approximately 2:1.*Linear expressions of nutrients*

One nutrient can be expressed in terms of another nutrient. For example, the amino acid requirement is sometimes expressed in units per MJ metabolisable energy (ME) i.e. the lysine content of a broiler starter feed might be expressed as 1g per MJ ME, and methionine, 0.45g per MJ ME.