GFMT Article: Synthesis of animal feed formulation techniques

Reading about Evonik and Biomin's respective expansions in feed additives today, this story from the GFMT archives caught my attention.

Written by Dr Pratiksha, Assistant Professor, School of Applied Sciences, Gautam Buddha University, Greater Noida, India, the features explains the differences between linear and non-linear models of feed formulation.  

Read the article as it appears in the magazine here or scroll down for just the text.

Synthesis of animal feed formulation techniques: Linear and Non-Linear model

by Dr Pratiksha, Assistant Professor, School of Applied Sciences, Gautam Buddha University, Greater Noida, India

It is well known fact that nutrition is the most important factor for animal growth, reproduction and proper maintenance.

Dietary information is essential for conducting research and performing experiments. The diet should supply all the essential nutrients and adequate amount of energy to satisfy requirement for body growth, well health conditions and animal yields.

Another important aspect of diet is to maintain environmental and flesh quality. A number of feeding standards have been defined and practiced for a long time.

Conventional and non-conventional feed resources are used to formulate the diet for ruminant livestock in developing countries. While formulating the diet, tendency is to reject the poor quality feeds that are available in vast quantities.

The objective is to use locally available feed resources effectively by applying basic nutrition principles to optimise animal yield, health and weight. The alternative approach is to use feeding standards that would ensure that the production system matches the available resources.

The diets should be formulated so as it contain all essential nutrients in adequate quantities. The diet should be supplied in a form, which is easily accepted by the cultivated animal and should have little adverse environmental impact. The strategy of choosing the feed ingredients is based on locally available feed resources and it requires the understanding of the relative roles and nutrient needs of the two-compartment system represented by the microorganisms in the rumen and the host animal.

Levels of nutrients

A lot of work has been done in the area of finding different levels of nutrient ingredient in diet and formulation. On the basis of percentage of nutrient level, components are included in animal diet.

No one feed ingredient can supply all of the nutrients and energy need for the best growth. Diet should contain a mixture of feedstuffs, vitamin and mineral premixes that provide the right essential nutrients as well as the energy necessary to use the nutrients.

The amount of each feed ingredient depends on several factors, including nutrient requirements, ingredient cost, availability of each ingredient, and processing characteristics.

For formulating diets for experimental purposes, it is necessary that all ingredients are controlled for all essential nutrients.

But, under practical conditions, such a control is difficult to set forth and mostly restricted to rapid proximate composition analyses. Specific attention should however be given to obtain guarantees for absence of anti-nutritional factors.

Diet given to laboratory animal may be of two types: Natural Ingredients or Purified Ingredients. Natural ingredient diets support reproduction, growth, and maintenance of laboratory animals. Purified diets are made of refined ingredients to minimise nutrient variation, certain environmental contaminants and the presence of active compounds naturally occurring in plants.

Generally, purified diets are used for diet formulation as they can be manipulated to contain very high or low levels of specific macro-nutrients (that is, 60 percent of kcal from fat or six percent protein) and micro nutrients (that is, two percent calcium or vitamin A deficient).

Two types of formulations are used in diets; fixed formulation and variable formulation. Fixed formulations are used for the diets where the ingredient composition is known and is not altered. It decreases the amount of variation in dietary constituents that could jeopardise experimental results or have a negative influence on the well being of animals.

To assure nutrient specifications, ingredient standards for nutrient concentrations are established prior to the procurement of the ingredients. Dietary contaminants are controlled by procuring ingredients according to strict contaminant standards and by testing ingredients for contaminants of concern.

Variable formulations may allow for changes in ingredient composition or concentration. These changes in formulations could lead to the incorporation of lower quality, less costly ingredients, where undesirable, non-assayed components, such phytoestrogens, may be introduced.

Variable formulated diet makes necessary adjustments according to raw material macronutrient variability, which could cause significant variation in the finished product. The largest volume of laboratory animal diet produced is comprised of agricultural commodities like corn, wheat, plant by-products, soybean meal, oats, alfalfa meal, and animal derived ingredients such as fish meal and meat and bone meal.

There are different methods to formulate animal diet.

Diet formulation includes balance mixture of ingredients which are economically sustainable and provides nutrient and energy requirements of a given species for a given response.

The reliability of knowledge on the quality of ingredients and the constraints; both have an impact on the quality of diet formulation. Reliable and updated database on chemical composition, physical characteristics and bioavailability information on feed ingredients is essential for diet formulation.

Including all the information diet is formulated to achieve the objective of least cost with adequate nutrients. There are three important aspects while considering the diet formulation, cost, nutrient level and ingredients limit.

Different kinds of conventional methods to formulate the diets are:

Trial-and-error method

Two by two matrix method

Square method

Simultaneous equation method

Least cost formulation

Linear programming method

Two-by-two matrix method solves two nutrient requirements using two different feed ingredients.

A two-by-two matrix is set and a series of equations are established to find the solution of the problem. Square method is relatively easy and simple to work on. It is used with only two nutrient ingredients. To use this method, level of nutrient being computed should be intermediate between the nutrient concentrations of the two feed ingredients being used. 

This method is used to satisfy only one nutrient requirement. This method has limited use as it is based on certain limitations.

Simultaneous equation method is also has limited use, because it is used for two nutrient ingredient combination diet. It uses simple algebraic method to solve these equations.

Trial and error method is generally used to formulate rations for swine and poultry. This method tries different diets and manipulates it until the nutrient requirements of the animal are met. This method makes possible the formulation of a ration that meets all the nutrient requirements of the animal. But in practice, it is really not possible to use it always, as it is a time and money consuming method.

Linear programming method is widely used for animal diet formulation.

It is a method to determine the least cost combination of ingredients using a series of mathematical equations. This method provides a number of possible solutions to each series of equations, but when the factor of cost is applied, there can only be one least cost combination. 

This method is in practice for a long time to give solutions to the problem of diet formulation considering the cost factor associated with it. Before using this technique for ration formulation, certain information should be available about the important nutrient ingredient to be included in diet.

First, all available ingredients should be listed with associated cost factor. Tables representing the nutrient composition of feed ingredients should be analysed properly. Nutrient levels are estimated from a variety of sources including published commodity compendium data, wet chemistry testing of raw materials and finished product testing. Nutrient losses due to heat treatment and mechanical processes during manufacturing, or post-production effects of irradiation or autoclaving are not routinely taken into consideration in these estimates.

After this nutrient requirement for the particular species and ingredient limitation should be given proper consideration. After collecting all the necessary information, a mathematical model is derived with Linear Programming specifications. Now method of LPP is used to solve it and it provides solution for the feed mixture.

Let us consider an example to formulate a linear programming model for the diet formulation. Suppose 1 kg of feed mix must contain a minimum quantity of each of four nutrients as below:

Nutrient	A	B	C	D
Gram	70	30	20	4

The ingredients have the following nutrient values and cost -

	A	B	C	D	Cost/Kg.
Ingredient 1 (gram/kg)	140	90	40	----	50
Ingredient 2 (gram/kg)	200	120	20	30	60

Now objective is to find the amounts of active ingredients and filler in one kg of feed mix. Now it is considered as one kg of feed mix is made up of three parts - ingredient 1, ingredient 2 and filler so let:

x₁ = amount (kg) of ingredient 1 in one kg of feed mix
x₂ = amount (kg) of ingredient 2 in one kg of feed mix
x₃ = amount (kg) of filler in one kg of feed mix
where x₁ ≥ 0, x₂ ≥ 0 and x₃ ≥ 0

Now the nutrient constraints are set up according to given information as

And the objective function is to minimise:. It represents formulation of a complete linear diet model.

Well-balanced ration

A number of models have been derived for different objective of study and constrains.

The LP model can be solved for a complicated set of nutrient requirements to give a relatively well-balanced ration [VandeHaar and Black, M. J., 1991]_.

The principal objective in the application of LP to feed formulation is the production of least cost rations that will produce satisfactory results.

A nutrition program was developed for high producing dairy herds to attain efficient and profitable levels of milk production [Sklan, D. and Dariel, I., 1993].

A model was developed to represent the efficiency of nutrient use and its relationship to profitability on dairy farms [Tedeschi, L. O, 2004].

A cost analysis spreadsheet and validation of that spreadsheet on milking and custom heifer operations was developed [Guevara V.R., 2004].

Lead factors are used in computerised ration formulation programs developed at Virginia Tech to increase milk production above a herd or group average for which total mixed rations are formulated for group feeding [Stallings, C. C; Mcgilliard, M. L]. Chance constrained programming is used to formulate commercial feeds for animals [Britt, J. S; Thomas, R. C; Speer, N. C; Hall, M. B., 2003].

A stochastic-linear program Excel workbook was developed that consisted of two worksheets illustrating linear and stochastic program approaches.

Both approaches used the Excel Solver add-in. Excel spreadsheet was set up so that the calculated margin of safety (MOS) value, according to the requested probability, was the same for both the linear and stochastic programs.

A multiple-objective programming (MOP) model was applied to the feed formulation process with the objectives of minimising nutrient variance and minimising rations cost.

A study was conducted to introduce a dual model in an original linear program to obtain the shadow prices of resources that take part in optimisation, in feed formulation. The shadow prices of nutrients resourced showed degrees of influence of a diet's least cost when increasing or decreasing expected diet nutrient ‘b’ values of a diet.

The higher the shadow prices of a nutrient resource, the more obvious its influence on least cost. When the shadow price of a kind of resource equals to zero, it means that reaching this nutrient value does not have an influence on a special diet least cost within a particular ‘b’ value. This paper also discusses the development of direction of feed formulation-optimizing techniques in China [Xiong BenHai, Luo QingYao, Pang ZhiHong].

The importance of Non-linear Programming Applications is growing due to rapidly increasing sophistication of managers and operation researchers in implementing decision oriented mathematical models, as well as to the growing availability of computer routines capable of solving large-scale nonlinear problems.

While formulating a mathematical model related to real life problems, many different situations lead to non-linear formulation of constraints and objective function.

The application of non-linear programming to the field of animal nutrition is growing day-by-day. The main goal in making feedstuffs is to increase profits of animal production by increasing the nutritional value of the feedstuff or a mixture of feedstuffs.

Feedstuffs containing 20 percent crude protein or more are considered protein supplements. Protein supplements may be classified as animal or plant proteins. Animal proteins are generally considered to be higher quality than plant proteins.

The main plant protein sources used in catfish feeds are oilseed meals, such as soybean meal, cottonseed meal, and peanut meal. Some other oilseed meals could be used but are not generally available on a timely basis and at an economical cost per unit of protein. Table 1 represents levels of crude protein in different meals.

Table 1: Crude protein (CP)

CP level	Ingredients
<25%	Whole cereals, pulses, oil seeds
25-50%	Oil seed meals
>50%	Animal by-products (meat meal, blood meal), plant protein concentrates, isolates, extractives

Vitamins and minerals

Vitamin and mineral premixes are generally added to feeds. They provide more vitamins and minerals than what is needed for growth to make up for any losses that may occur during feed manufacture or storage.

They are made from high quality ingredients, using forms of vitamins and minerals which animal can readily digest.

The rate of growth and the efficiency with which the nutrients are utilized mainly depend on three factors, which may be used to maximize it⁸. Accounting all these facts, weight gain of an animal depends upon:

Digestible crude protein

Total digestible nutrient

Digestible dry matter

Metabolic weight is used as a base for whole of the calculations. Moir had earlier reported that a level between 200 and 300K. Cal DE per Kg^0.75 is generally encountered while studying the intake in growing animals.

A non-linear model is defined as:

To maximize

Subject to:

I=1,2,3,…………….m, j=1,2,3,……….l

Where f(x), g(x) and h(x) are functions defined on, X is a subset of  and x is a vector of n components.

Optimising feed for weight gain

Non-linear programming is used to maximise the body weight of sheep under the given experimental conditions and satisfying NRC feeding standards (Pratiksha, 2006).

To sum up, an effort has been made to give a new dimension to the already existing multi-dimensional non-linear models and its use to formulate a real-world problem of optimising the feed in terms of weight gain of the animal and to solve it as well.

This objective supports the all over effect of nutrient ingredients simultaneously on the animal yield and weight gain of an animal. It has already been accepted that non-linear programming has a great deal of future prospects as it has direct practical utility in the field of animal nutrition.