Food and Agriculture Organization of the United Nations works towards ending hunger and poverty while using precious natural resources sustainably. For fisheries and aquaculture maximizing the sector’s contribution to food and nutrition security, ensuring that all people have access to good-quality, nutritious food, while simultaneously supporting the livelihood of hundreds of millions of people around the world. Maximizing the benefits of the fisheries and aquaculture sectors can only be achieved by carefully balancing environmental, social and economic sustainability principles in the management of our natural aquatic resources ^(1-2), the world fish consumption risen from about 45 million metric tons in 1973 to over 91 million metric tons in 1997 and this consumption of fish trends is increasing in the developing and decrease in developed world due to negative population growth rates in developed countries in contrast to the developing world ⁽³⁾. According to ⁽³⁾ report, Africa produces only 1% of world aquaculture fish with Nigeria being the major producer, followed by Egypt, Uganda and Kenya ⁽³⁾. Aquaculture is a feed-based industry, with over 50% of operational cost coming from feed source. The country often suffers risks of food insecurity largely due to lack of modern and integrated farming system., most of Ethiopian agriculture is dominated by small scale farmers who are farming using traditional system leading to low productivity, and the nation’s inland capture fishery has been out of control and failed to sustain fish resources. Because of most lakes experiencing on overexploitation, biological diversity degradation and reduction in fish population and becomes crucial to realize a sustainable aquaculture as a counter measure for ensuring food security and poverty alleviation. However, the actual natural potential of Ethiopia is good to develop aquaculture, thereby building effective strategy and feasible approach for effective aquaculture development in Ethiopia ⁽⁴⁾.

Feed is one of the factors which play an important role in determining a successful development of an aquaculture industry. It is always critical to produce the best aquaculture diet at a minimum cost in order to trim down the operational cost and gain more profit. However, the feed mix problem becomes increasingly difficult since many issues need to be considered simultaneously ⁽⁵⁾. Modified extensive and semi-intensive culture systems depend primarily on steady supply of supplementary artificial feeds. The use of artificial feed is balanced in protein, lipid, carbohydrate, fiber, amino-acids, digestible energy, vitamins and mineral is an obvious approach to realize genetic potential of the animal for survival, immunity, growth and reproduction. Aquaculture nutrition is a vital area for maintaining the sustainability of aquaculture industry. There is an emerging need to produce fish in quality and quantity with increasing demand in the global market. Fish feed formulation is the process which has to take into account the objective of achieving optimal fish production, consumer preference and net benefits that can be earned an aqua culturist to determine the types and quantities of ingredients to be mixed to produce a complete feed at possible low cost. The feed preparation thus becomes a major consideration for the process ofcombining different feed ingredients in a suitable way so as to achieve the specific goals of culture fish production. 

Feed is the most expensive component in aquaculture, particularly intensive culture, where it accounts for over 50% of operating costs ⁽⁶⁾. In pond culture, fish can obtain their energy and nutrients from natural food in ponds, from feed supplied by the farmer or from a combination of both sources ⁽⁷⁾. If a farmer manages to reduce the cost of feed then a significant amount of profit can be realized. It is important to develop methods that can effectively cut down the costs of feeds. In addition, feed ration formulation is a particular challenge especially if new nutritive mixtures are involved. The practical goals of feed formulation is for rapid growth rate, and successful reproduction and also the experimental goals like induction of a vitamin deficiency or establishment of a minimum dietary nutrient requirement. Several considerations need to be taken into account as the feed nutritionally viable, physically acceptable, practically applicable and economically feasible. To achieve optimal production most feed formulations fall between two extremes. These are formulation primarily based on cost and chemical composition, producing a feed that is less expensive than other feeds and the other extreme is formulation primarily based on nutritional value producing more expensive feed that is more productive, thus requiring less feed per unit of fish production. But the most favored one is to determine exact optimum proportions of different feed ingredients, which meets the necessary nutritional requirements for protein, lipid, carbohydrate, fiber, essential amino acids, calcium and phosphorus ratio, digestible energy, etc. for a particular fish species so that the total unit cost of the feed formulated will be the least possible. 

Several mathematical tools have been used to solve feed ration formulation problems such as Pearson’s Square method ⁽⁸⁾, goal programming (GP) ⁽⁹⁾, multi-objective goal programming (MOP) ⁽¹⁰⁾, nonlinear programming (NLP) ⁽¹¹⁾, chance constrained programming (CCP) ⁽¹²⁾, quadratic programming, and linear programming (LP) ^(13-15). However, most of these methods have been found to have difficulties in finding optimal solutions because of the following variables. Ingredient variability due to varying, fluctuating, and unstable nutrient components has made it difficult to find optimal solution of the feed mixing problems ⁽¹⁶⁾. Variability of price of the ingredient due to changes in price of the ingredients makes the feed formulation problem complex. Optimisation algorithms have been also used, to a limited extent, for animal diet formulation such as genetic algorithm (GA) ⁽¹⁷⁾ and evolutionary algorithm ⁽¹⁸⁾. Therefore, the purpose of this paper is to review stochastic and linear mathematical programming models on aquafeed formulation in solving variability of essential nutrients and least-cost fish feed formulation.

In aquaculture, it is important to determine the production costs of supplementary formulated feeds, nutrients found in the feed ingredients to know the nutritional composition in the formulated feed, and formulating methods of fish feed. The aquaculture industry is primarily based on fishes, cultivated in intensive (commercial) and semi-intensive (artisanal) production systems. Both systems involve the input of supplementary formulated feeds which account for up to 40 and 60% of production costs, respectively ⁽¹⁹⁾. Feed formulation is therefore, a central operation in fish production, ensuring that feed ingredients are economically used for optimum growth ⁽²⁰⁾. The purpose of commercial feed formulation is to balance nutrients in diets to meet the nutritional requirements of animals at least cost ⁽²¹⁾. However, the high profit margins enjoyed by the aquafeed manufacturers have diminished during recent years due to the impact of high raw material cost and lower profitability of the farming sector itself. As a result managing feed formula cost is high on the agenda of aquafeed business managers, nutritionists and formulators ⁽²²⁾.

2.1. Essential nutrients for aquafeed formulation: There are necessary nutrients that required for the formulation of fish feed for the best growth performance, healthy and reproduction of fish, such as water, proteins (amino acids), lipids, carbohydrates, vitamins, minerals, and pigments for the diet of aquarium’ fishes. Therefore, before formulating fish feed, it is very important to have/analyse the nutritional composition of feed ingredient to know the amount of those essential nutrients in fish feed formulation.

2.1.1. Nutritional composition of aquafeed ingredients: The official methods of chemical composition analysis of feed ingredients used in this study was determined using validated analytical methods published by the Association of Analytical Communities International ⁽²³⁾. The first step in the chemical evaluation of a feed ingredient is usually the Weende or proximate analysis used to determine the content of moisture, crude protein, lipid, crude fiber, ash and digestible carbohydrate. However, in this study the mean value of nutrient contents result of ingredients were used to formulate aquafeed using linear programming model. To use a least-cost computer program to formulate feeds, manufacturers must know the cost of feed ingredients and nutrient concentrations in feedstuffs, nutrient requirements and nutrient availability from feedstuffs and nutritional and non nutritional restrictions ⁽²⁴⁾. Nutrient composition of feed ingredient obtained from proximate analyses of the ingredients done following the methods of ⁽²³⁾. The analyzed nutrients include Dry Matter (DM), Crude Protein (CP), Crude Fibre (CF), Fat, Carbohydrate (C), Calcium (Ca), and Phosphorus (P). Calcium and the phosphorus mineral element content of the ingredients can be determined using a Perkin-Elmer Model 5000 Atomic Absorption Spectrophotometer. The amino acid profile of the ingredients can be determined using methods described by ⁽⁶⁾. The samples dried to a constant weight, defatted, hydrolyzed, evaporated in a rotary evaporator and loaded into the Technicon Sequential Multi-sample Amino Acid Analyzer (TSM) using ion-exchange chromatography ⁽²⁵⁾. Data collected from secondary sources; Mean and Standard deviations thus calculated from data from both sources. The method of data analysis employed is linear programming (LP) and stochastic programming (SP) models. All the data generated were computed in computer software with Simplex algorithm for linear and stochastic feed formulation ⁽²⁶⁾.

2.1.2. Feeding specializations, feed bulk and pigments in fish feed formulation: In their natural environment, fish have developed a wide variety of feeding specializations (behavioral, morphological and physiological) to acquire essential nutrients and utilize varied food sources. Based on their primary diet, fish are classified as: carnivorous (consuming largely animal material), herbivorous (consuming primarily plant and algae), or omnivorous (having a diet based on both plant and animal materials). However, regardless of their feeding classification, in captivity fish can be taught to readily accept various prepared foods which contain the necessary nutrients ^(27-7).

Feed bulk, many fish farmers buy the bulk of their feed made commercially. However, small quantities of specialized feeds are often needed for experimental purposes, feeding difficult-to maintain aquarium fishes, larval or small juvenile fishes, brood fish conditioning, or administering medication to sick fish ⁽²⁸⁾. Small fish farmers, and laboratory technicians are left with the option of buying large quantities of expensive feed, which often goes to waste. Small quantities of fish feeds can be made quite easily in the laboratory, classroom, or at home, with common ingredients and simple kitchen or laboratory equipment. Nutrients essential to fishare the same as those required by most other animals. The essential nutrients to fish are the same as those required by most other animals, which include water, proteins (amino acids), lipids (fats, oils, fatty acids), carbohydrates (sugars, starch), vitamins and minerals. In addition, pigments (carotenoids) are commonly added to the diet of aquarium’ fishes to enhance their flesh ^(27-7). 

Ingredients, such as fish meal, soybean meal, fish hydrosylate, skim milk powder, legumes, and wheat gluten are excellent sources of protein. Free amino acids (building blocks of proteins) such as lysine and methionine are commercially available to supplement the diet. Thiaminase, an enzyme that destroys thiamine (vitamin B1, one of the essential water-soluble vitamins), is mostly found in freshwater fish. It is destroyed by the heat (i.e., cooking) and diets include the spread of infectious diseases such as Mycobacterium and botulism. Lipids: Oils from marine fish, such as menhaden, and vegetable oils from canola, sunflower and linseed, are common sources of lipids in fish feeds. Cooked carbohydrates from flours of corn, wheat or other ‘breakfast’ cereals, are relatively inexpensive sources of energy that may spare protein (which is more expensive) from being used as an energy source. The variety and number of vitamins and minerals are so complex that they are usually prepared commercially as a balanced and premeasured mixture of a vitamin or mineral premix to be added to the diet in generous amounts to ensure adequate levels of vitamins and minerals are supplied to meet dietary requirements ^(27-7). 

Pigments: a variety of natural and synthetic pigments or carotenoids are available to enhance coloration in the flesh of fish and the skin of fresh water and marine ornamental fish. The pigments most frequently used supply the colours red and yellow. The synthetically produced pigment, as taxanthin, is the most commonly used additive (100-400mg/kg). Cyanobacteria (blue green algae such as Spirulina), dried shrimp meal, shrimp and palm oils, and extracts from marigold, red peppers and Phaffia yeast are excellent natural sources of pigments. Another important ingredient in fish diets is a binding agent to provide stability to the pellet and reduce leaching of nutrients into the water. Beef heart has traditionally been used both as a source of protein, and as an effective binder in farm made feeds. Carbohydrates (starch, cellulose, pectin), and other polysaccharides, such as extracts or derivatives from animals (gelatin), plants (gum arabic, locust bean) and sea weeds (agar, carageenin and other alginates) are also popular binding agents ^(27-7).

Preservatives: Preservatives, such as antimicrobials and antioxidants are often added to extend the shelf life of fish diets and reduce the rancidity of the fats. Vitamin E is an effective, but expensive, antioxidant that can be used in laboratory prepared formulations. Commonly available commercial antioxidants are butylated hydroxyanisole (BHA), or butylated hydroxytoluene (BHT) and ethoxyquin. BHA and BHT are added at 0.005% of dry weight of the diet or no more than 0.02% of the fat content in the diet, while ethoxyquin is added at 150 mg/kg of the diet. Sodium and potassium salts of propionic, benzoic or sorbic acids are commonly available antimicrobials added at less than 0.1% in the manufacture of fish feeds. Other common additives incorporated into fish feeds are chemo-attractants and flavorings, such as fish hydrosylates and condensed fish soluble (typically added at 5% of the diet). The amino acids glycine and alanine, and the chemical betaine are known to stimulate strong feeding behavior in fish. The attractants enhance feed palatability and its intake ^(27-7).

2.2. Mathematical programming models of aquafeed formulation: There are various methods of feed formulation and least-cost fish feed manufacturing techniques, and some of them are linear and stochastic programming.Nutrient requirements used in linear programming feed formulation are usually fixed for maximum rate of growth. This may not be the best decision from economic point of view. Nutrient constraints may be relaxed to bring down feed cost while still achieving acceptable lower growth. Given certainty in estimates of cost and nutrient content for each ingredient, the problem can be formulated as a standard linear programming model. Solution of the model minimizes the cost for the final mixture so that the specified nutrient requirements are satisfied, but the nutrient content of ingredients is likely not known with certainty.A Major problem associated with the application of conventional linear programming in determining the least-cost feed mixture which satisfies certain specific nutritive and nonnutritive requirements is the risk associated with the nutritive content of each ingredient. However, stochastic programming is a better approach when compared to linear programming in solving nutrients variability, minimizing cost and variability of nutrients is in the formulation. Though, in both cases to use a least-cost computer program to formulate feeds, manufacturers must know the cost of feed ingredients and nutrient concentrations in feedstuffs, nutrient requirements and nutrient availability from feedstuffs and nutritional and non nutritional restrictions ⁽²⁴⁾. A number of methods have been defined for the formulation of animal diet; square method, two by two matrix methods, simultaneous equation method, trial and error method and linear programming method to formulate least cost diet. Therefore, this review paper explained about linear and stochastic programming technology, and also weaknesses of linear programming technology.

2.2.1. Linear programming technology in aquafeed formulation: Linear programming was first introduced to the animal compound feed industry in the mid fifties and then, its application in least cost formulation of feed for livestock and poultry has gained widespread acceptance in most countries with well-developed compound feed industries. There are various studies on Linear programming formulation, the linear programming solution may not provide right decisions from economic point of view. For example, nutrient requirements determined to achieve maximum growth rate using linear programming may not be the best from economic considerations. Relaxing nutrient constraints while still achieving acceptable lower growth may bring down feed cost. Such studies includes, used linear programming model to examine the economic viability of fish production strategies in the context of fed farming systems. explained computerized linear programming formulation and quadratic programming formulation techniques as different types of aqua feed formulations besides discussing about selection of ingredients. applied linear programming technique to identify the least cost strategy. described about the application of linear programming for optimization of income in integrated rice-fish duck-vegetable farming system and for the optimal combination of different farming sub-systems for adoption, revealing the results of increased farm income, additionally generated employment and reduced farm business risk. In addition, ⁽²⁹⁾ also described the application of linear programming for feed formulation. Furthermore, Thuleswar and Ashok (2014) also study about Linear programming technique in fish feed formulation on how its application approach to feed formulation lead to higher productivity.

⁽³⁰⁾on the linear programming technique of fish feed formulation,  linear programming technique has gained grounds solution for various problem around warehouse, caterer, personnel, advertising media selection, crude oil refining and gasoline blending, manpower planning, allocation of aircraft to different routes, water quality management, traffic light control, etc. In addition, linear programming technique also made a considerable impact on agricultural, livestock and animal husbandry research and implemented in determining fish feed compounds for fish farmers to improve the productivity of fishes. In addition, diet formulated by linear programming is based on assumption of linearity between animal yield and nutrient ingredients included in the diet. The feed formulation model seeks the optimum combination of available feed ingredients that will satisfy the nutritional requirements of the animal at the least cost possible. The model has to satisfy a set of constraints on nutritional levels, availability restrictions, special ingredients to be included, budget or fund constraints. The generic mathematical model which is applicable to each type of ration using the available ingredients is prevalent in the industry, nutrient contents and availability of feed ingredients and their prevalent market prices. The decision variables, objective function and problem constraints were defined and a mathematical model of the feed formulation problem was developed. The technique is thus highly maneuverable and the nutritionist should take full advantage to investigate the varied types of alternatives for deciding on the best scheme for implementation. Fishing contributes a lot to the socio-economic development in terms of employment, food security in a form of protein intake and also for poverty reduction in both rural and urban communities. But the industry faces a lot of challenges including high input cost, lack of infrastructure, destructive fishing methods and dominance of foreign markets.

Formulation rules of the linear programming system, with respect to a linear equation, require that: (a) only one variable may appear on the RHS. While row variables, if there are more than one, may appear on the LHS along with column variables, column variables may not appear on the RHS of any problem; (b) the same variable may not appear on the RHS of more than one equation; (c) the coefficient of the variable on the RHS of an equation must be 1.0. In formulating feeds by LP, the nutritionist first lays down a set of constraints. He then lists all available raw materials which he wishes to be considered for selection by the computer to achieve his objective (Al-Deseit, 2009).

• Formulation of linear programming (LP) Model for fishes:

Example: Consider formulation of a feed mixture with ingredients shown in the following table and subject to minimum nutrient contents. The quantity of the mixture to be prepared is 100 kg.

• Let (X1) , (X2) , (X3), (X4), (X5), (X6), (X7) and (X8) be the respective quantities in kg of groundnut cake, soyabean cake, rice-bran, wheat bran, fish meal, brewer waste, till cake required for the mixture and market price (MP) (Thuleswar and Ashok, 2014). In addition, the nutrient contents such as protein (Pr), lipid (L), carbohydrate (C), calcium (Ca), and phosphorous (P), represented by their initial letters used to replace the required nutrients for calculation of constraints. 

• The minimum-cost linear programming model can be written as.

• Cost function min=MP*X1+MP*X2+MP*X3+MP*X4+MP*X5+MP*X6+MP*X7;

• Protein requirement=Pr*X1 + Pr*X2 + Pr*X3 + Pr*X4 + Pr*X5 + Pr*X6 + Pr*X7; 

• Lipid requirement=L*X1 + L*X2 + L*X3 + L*X4 + L*X5 + L*X6 + L*X7; 

• Carbohydrate requirement=C*X1 + C*X2 + C*X3 + C*X4 + C*X5 + C*X6 + C*X7; 

• Calcium requirement=Ca*X1 + Ca*X2 + Ca*X3 + Ca*X4 + Ca*X5 + Ca*X6 + Ca*X7; 

• Phosphorous requirement=P*X1 + P*X2 + P*X3 + P*X4 + P*X5 + P*X6 + P*X7;

• Quantity requirement=X1 + X2 + X3 + X4 + X5 + X6 + X7 =100; (Thuleswar and Ashok, 2014). Therefore, any researcher can simply apply the above formulas of linear programming technology by considering cost and nutrient content of ingredients.

Therefore, Linear programming is the simplest way of a mathematical modeling technique used to maximized or minimized when subjected to various constraints such as guiding quantitative decisions in business planning, in industrial engineering, and to a lesser extent in the social and physical sciences. In addition, it has made a significant impact on agricultural, livestock and animal husbandry research and also it can be implemented in determining fish feed compounds for fish farmers to improve the productivity of fishes. 

2.2.2. Stochastic programming models on aquafeed formulation: Stochastic programming is a framework for modeling optimization problems that involve uncertainty and it deals with a class of optimization models and algorithms in which some of the data may be subject to significant uncertainty. An optimization model is a translation of the key characteristics of the business problem you are trying to solve and it consists of three elements: the objective function, decision variables and business constraints. Algorithmic model (programming) is a method of estimating software cost using mathematical algorithms based on the parameters (such as brittle, sensitive to step size choice and other ⁽³¹⁾ which are considered to be the major cost drivers ⁽³²⁾. Based on economic applications, stochastic dynamic programming is a useful tool in understanding decision making under uncertainty. The accumulation of capital stock under uncertainty is one example often it is used by resource economists to analyze bioeconomic problems where the uncertainty enters in such as weather, etc ⁽³³⁾. In addition,the goal of stochastic programming is to find a decision which both optimizes some criteria chosen by the decision maker, and appropriately accounts for the uncertainty of the problem parameters and its applications found in a broad range of areas ranging from finance to transportation to energy optimization ⁽³⁴⁾.

In addition, almost all-commercial feed formulation software use Linear Programming (LP) 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 ⁽³⁵⁾. 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 ⁽²⁸⁾. Most feed manufacturers want to minimize the risk of not meeting the nutrient requirements of the animal. One of the methods to achieve this is by the use of Stochastic Programming ⁽³⁶⁾. Stochastic formulation, a non-linear optimization program to manage risk in ingredient variability ⁽¹¹⁾, is not used widely but has the potential to provide more cost effective formulas than those who use conservative approaches in linear optimization programs to reduce the risk of under formulation and over formulation. Therefore, stochastic programming is a better approach as compared to linear programming especially in solving nutrients variability, minimizing cost and variability of nutrients in the formulation.

2.3. The weaknesses of linear programming technology in feed formulation: In the reputation of LP as a tool to find optimal solution in feed mix problem, some LP limitations have been identified ⁽³⁶⁾. There are three weaknesses observed in linear programming. The first weakness is that the LP models assume nutrients levels are fixed. Due to the linearity of the nutrient levels LP formulated feed only at 50% probability level which lead to under formulation. However, nutrients levels in feed ingredient are unstable and fluctuating. By comparing linear programming (LP) and stochastic programming (SP), the SP shows that the probability that nutritional requirements will be met increases from 60 up to 85%. The second weakness is that the LP method is regularly hard to determine a good balance of nutrients in the final solution. If only the minimum levels of nutrient requirements are placed, there is a probability for nutrients imbalance to arise in the final solution ⁽³⁷⁾. Balance nutrients and variability are related, and hence, when the variation is small, the quality of balance nutrients will improve (9). In comparison, SP tries to balance the nutrient as the probability level increases. The third weakness is the constraints rigidity in LP which means that no constraints violation is allowed. This weakness will normally lead to infeasible solution (9). Infeasibility problem was encountered in support of this assertion. For instance, when crude protein was raised above 32% there was infeasible solution in LP while SP would only adjust the probability level of the formulation in order to come out with a better formula. This makes formulation by LP cumbersome because the formulator would continue to adjust the constraints till the formulation is achieved.

Aquaculture nutrition is a vital area for maintaining the sustainability of aquaculture industry. Feed is one of the main factors which play an important role in determining a successful development of an aquaculture industry. Fish feeds, feed ingredients, some common conventional feedstuffs, animal and plant sources of unconventional feeds for culture fish, fish feed formulation, and feeding methods are important for the effective management of fish farming. Essential nutrients to fish are the same as those required by most other animals. Modified culture systems primarily depend on the supply of well balanced artificial feed include water, proteins (amino acids), lipids (fats, oils, fatty acids), carbohydrates (sugars, starch), vitamins, minerals, pigments (carotenoids), etc are very essential in fish farming to realize genetic potential of the animal for survival, immunity, growth and reproduction. In addition, it is imperative for fish feed farmers to search for alternative feedstuffs keeping in mind the high prices of fish feed ingredients and non availability without affecting the quality of feed and fish productivity. 

Fish feed formulation is important for achieving optimal fish production, consumer preference and net benefits to determine the types and quantities of ingredients to be mixed to produce a complete feed at possible low cost. Availability of quality feed at a reasonable cost is a key to successful fish farming operation and linear programming technique is one of the most important techniques to allocate the available feedstuffs in a least cost ration formulation. The optimal solution of the linear programming model gives reduction in feed formulation costs compared to the existing method on the farm. Fish feed cost represents over 70% of the total cost of production; consequently efficient feed formulation practice is required for a sustainable fish industry. Many Assamese fish farmers, however, employ inefficient methods like rule of thumb, experiences, and intuition to handle feed formulation problem. Similarly, a stochastic relationship can be incorporated in a quadratic programming model, solved iteratively for a least cost feed mix, subject to a probabilistic protein constraint and other linear inequality restrictions. 

Several mathematical modeling tools used to solve feed ration formulation problems such as Pearson’s Square method, two by two matrix methods, simultaneous equation method, trial and error method, goal programming, multi-objective goal programming, nonlinear programming, chance constrained programming, quadratic programming, linear programming, Optimization algorithms, and evolutionary algorithm to formulate least cost diet. However, most of these methods found difficulties on feed mixing problems in finding optimal solutions b/c of ingredient variability of price due to varying, fluctuating, and unstable nutrient components. 

In this article, the comparison of linear and stochastic programming technology using essential nutrients and least-cost fish feed formulation, using stochastic programming technology is better due to various weaknesses of linear programming technology. In addition, stochastic programming’s advantages are minimizing cost and the variability of nutrients is accounted for in the formulation. A major problem of application of linear programming in determining the least-cost feed mixture which satisfies certain specific nutritive and nonnutritive requirements is the risk associated with nutritive content of each ingredient. However, stochastic programming is a better approach as compared to linear programming, especially in solving nutrients variability which is the linear programming disadvantage, advantage of minimizing cost and the variability of nutrients is accounted for in the formulation.

Conflict of Interest: No

Funding: No funding sources

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