Achievement of balanced ration of animals is very important from livestock as well as commercial point of view. Lack of balanced ration causes malnutrition in animals and they are not able to perform well in terms of high yields and reproductively. Therefore various mathematical models depending on different techniques have been widely used for animal ration formulation. Objective of animal ration formulation is to get a balanced ration at least cost which fulfill energy and nutrient requirements of animal. Standard linear programming technique has been extensively used for least cost ration formulation in 19th century (Deseit, 2009). Several mathematical models were formulated using linear programming method to get ration which supply required nutrients at minimum cost. A linear programming model was developed to get the least cost ration for drought maintenance of dry adult sheep. Results shows that sheep can be fully maintained at an approximate cost of one cent per head per day (Vere, 1972). A model, named the Grange Beef Model, was presented to find optimal beef production systems in Ireland. A series of scenarios concerning variation in beef and concentrate prices were used to explain model applications (Crosson, Kiely, Mara & Wallace, 2006). Linear programming technique was used for finding the least cost ration for broilers of age 6 to 10 weeks for the utilization of locally available and non- conventional feed stuff-Duckweed (Lemna paucicostata). The result shows the reduction of cost of feed by 20.82% with utilization of feed mix having 29.50% of duckweed and therefore improve profitability in broiler production (Olorunfemi & Temitope, 2006). A linear model was developed for the Nigerian poultry industry and data for this has been taken from a typical commercial farm. It has been observed that cost was reduced by 9% compared to the existing practice(Oladokun & Johnson, 2012).
An LP model was developed for least cost poultry ration in which elements of the tableau are stochastic. Results shows that less time and complexity is involved compared to alternative methods. The limitations imposed by lack of data and of biological minima was the largest obstacle in the examination of related problems(Rahman & Bender, 1971). An LP model with 8 ingredients and 18 constraints was developed for getting optimized shrimp feed mix. Growth tests were carried out with Penaeus japonicus over periods of more than 30 days. To determine the physiological state of shrimps after the experiment, some analyses were conducted on haemolymph. Results showed reduction in cost of nearly 30%, without any significant loss in growth performance (Barbieri & Cuzon, 1980). LP technique with the aid of an electronic computer was used to develop the models for the optimized fattening rations of weaner calves. Nutrients used in the model are digestible energy or estimated net energy, crude protein, crude fiber, calcium and phosphorus. Chemical analysis demonstrate a quite good agreement between specifications and analyses for crude protein, crude fiber and phosphorus. Animal performance data showed that estimated net energy was superior to digestible energy as a basis of ration formulation (Church, Brown & Ralston, 1963). An optimization method was used for a beef cattle fattening system to determine the utilization of food by-products under various situations. Later on reduction of feed costs and nitrogen and phosphorus excretions were also discussed in the method (Jayasuriya).
A model was developed using linear approximation of chance-constrained programming (CCP). In this model, linear codes are used to approximate the non-linear form of chance constraints (Olson & Swenseth, 1987).