Minimization of layer feed cost using linear programming

1. Introduction

The cost of poultry feed is a significant expense in the production of poultry birds in Ghana, accounting for approximately 70% of the total cost [1]. Sudden increases in the price of poultry feed have caused many commercial farms to fail, and many others are struggling to survive. To maintain a good profit margin, despite rising raw material costs and a shortage of feed ingredients, a linear programming (LP) model must be developed to reduce production costs while maintaining high levels of bird performance. Otherwise, the prices of poultry products (eggs and meat) will rise, as an increase in raw materials would almost certainly cause the price of finished feeds to grow. However, a perfect ration that provides the necessary quantity of nutrients at the lowest possible price is also necessary. This can be achieved with a computerized least-cost ration formulation, which is popular in producing large quantities of compound feed due to its advantages in reducing human calculation and timing errors.

Least expensive ration formulation techniques have been used to create diets that meet specific requirements from readily available ingredients [2]. Commercial enterprises and small animal holders require profit optimization with limited investment in feeding costs, as it is the significant cost of livestock production. Feed mixing has progressed from simple to complex operations with the help of computer programs in modern times. In the past, farmers used to mix a few feed ingredients on the floor to give supplements to pastured livestock and poultry. However, with the increased commercialization of animal farming, demands have increased, and ingredients are now mixed on a large scale at factories. A fully computerized mixing plant for livestock and poultry can use thousands of tons of feed ingredients per week [3].

The poultry industry in Ghana faces serious challenges, which have resulted in the collapse of some poultry farms, and the few remaining farms find it difficult to generate high profits. Among these challenges, feed cost is predominant, affecting poultry farms in Ghana. The production cost of feed is prohibitive due to high prices, high demand, and the unavailability of certain ingredients. Controlling poultry feed costs is important since it represents 60-70% of the total production cost [1]. For this reason, farmers are switching to low-cost feed substitutes and blending feed ingredients to produce their feed. Because they cannot rely on the quality of feed available in the local market, more than 70% of poultry farmers make their feed. Also, the feed available in the local market is costly. Most of these farmers have little experience formulating feed, making it laborious to produce quality and nutritious feed for the birds. Using low-quality diets can negatively affect the health and growth of the birds, which would then affect the entire poultry industry. This research seeks to minimize feed costs in poultry farms by formulating a feeding formula yielding good production. Specifically, the paper aims to do a quantitative analysis of blending poultry feed mix and formulate a linear programming model for minimizing the cost of producing or making one ton of feed.

Note that ration formulation determines how feed ingredients are blended in a proportion essential to provide animals with the right amount of nutrients needed at a specific production stage [4]. Effective ration formulation requires thorough knowledge of the feed materials, the nutrients they contain, and the type of animal that will be fed with the ration to ensure maximum production at a fair price. The animal should be able to consume the feed sufficiently, and efforts should be made to ensure that the feed does not harm the animal’s digestion or have any toxic effects [5]. Animals of different species, strains, or classes require different amounts of energy (carbohydrates and fat), proteins, minerals, and vitamins to maintain their various physiological processes, such as body upkeep, reproduction, egg production, milk production, and meat production, among others. The process of formulating poultry feed is complex, as it entails choosing a blend of feed components that sufficiently meet the stated nutritional needs as well as the needs of other birds. To provide the least expensive ration, the formulation of poultry rations places a strong emphasis on the use of linear programming and a computer. However, because the results of the calculations may be impractical and unsuitable for feeding the birds, ration formulation is more than just performing mathematical calculations to satisfy the needs of the birds. Therefore, before the feed formulation is administered to the birds, it must be evaluated [2].

Feed costs have a significant impact on chicken farm profitability. The high feed cost is connected to the diet’s calorie and protein levels. The feed would be more expensive in an unbalanced diet with an excess of protein, raising production expenses. Low protein diets would cause chickens to grow more slowly and make them more susceptible to disease. Chickens have different nutrient (feed) requirements depending on their type, age, and sex. Rations formulated to meet nutrient requirements produce faster-growing and healthier chickens and thus better products and more profits [6]. Excess dietary nutrients are often excreted in the faeces. The excess nitrogen and phosphorus in faeces could threaten the environment. For this reason, managing feed formulas for accuracy is an important step in poultry farm management to safeguard the environment and reduce operating costs [7].

The remaining sections of this paper are organized as follows: The mathematics of linear programming is presented in Section 2. Section 3 discusses the analyzed results and highlights major findings. A comprehensive summary, serving as the conclusion, is provided in Section 4.

2. Linear programming (LP) theory

High feed ingredient prices are a major issue faced by poultry producers [8]. The availability of high-quality feed at reasonable prices is critical to the success of poultry operations [9]. One of the most important techniques for allocating available feedstuffs in a least-cost layer ration formulation is linear programming [10].

Linear programming is a technique for optimizing a linear objective function that is constrained by linear equality and linear inequality [11]. Informally, linear programming determines the best way to achieve the best outcome, such as maximum profit or lowest cost, in a given mathematical model, given a set of requirements represented as linear equations [12]. Linear programming is a mathematical procedure for obtaining a value-weighted solution to a set of simultaneous equations. It was first used extensively during World War II to determine the most efficient way of deploying scarce resources such as troops, ammunition, and machinery [13]. Linear programming is used in agriculture in countless ways [14,15] thoroughly reviewed the use of linear programming in aquaculture least-cost ration formulation. They also used linear programming to incorporate duckweed into the least expensive feed formulation for broiler and layer starters.

An important category of optimization models is linear programming (LP) models, which are widely used for many types of operations design and planning problems that involve allocating limited resources among competing alternatives, as well as for many distribution and supply chain management designs and operations. The term programming is used because these models find the best “program” or course of action to follow.

An exemplary maximization LP problem is defined by the objective function: \[\label{key} \text{Max} \quad Z = C_1x_1 + C_2x_2 + …+ C_nx_n\,,\] subject to the following constraints: \[\begin{aligned} a_{11}x_1 + a_{12}x_2 + … + a_{1n}x_n \le b_1,\\ a_{21}x_1 + a_{22}x_2 + … + a_{2n}x_n \le b_2,\\ \vdots \qquad \qquad \qquad \vdots \qquad \qquad \\ a_{m1}x_1 + a_{m2}x_2 + …+ a_{mn}x_n \le b_m,\\ x_1,x_2,x_3,…x_n \ge 0\,.\\ \end{aligned}\] To create the simplex tableau (see Table 1) for a specific objective function and its constraints, we convert the function into a standard form which is defined as: \[\text{Max} \quad Z = c_1 x_1 + c_2 x_2 + . . . + c_n x_n + 0s_1 + 0s_2 + . . . + 0s_m\,,\] subject to linear constraints: \[\begin{aligned} & a_{11}x_1 + a_{12}x_2 + … + a_{1n}x_n + s_1 = b_1,\\ & a_{21}x_1 + a_{22}x_2 + … + a_{2n}x_n + s_2 = b_2,\\ & \qquad \vdots \qquad \qquad \qquad \quad \vdots \\ & a_{m1}x_1 + a_{m2}x_2 + …+ a_{mn}x_n + s_m = b_m,\\ & x_i,s_1,s_2,…,s_m \ge 0\,, \mbox{ for } i=1,2,…,n\,. \end{aligned}\]

• \(C_B\) represents the coefficients of the objective functions for each of the basic variables.

• \(Z_j\) denotes the reduction in the value of the objective function that would result if one unit of the variable corresponding to the \(j^{th}\) column of the matrix formed from the coefficients of the variables in the constraints is brought into the basis (i.e., if the variable is made a basic variable with a value of one).

• The net evaluation row, denoted by \(C_j – Z_j\), represents the net change in the value of the objective function if one unit of the variable corresponding to the \(j^{th}\) column of the matrix (formed from the coefficients of the variables in the constraints) is brought into the solution.

• The pivot column is the column in the \(C_j – Z_j\) row with the largest positive number. The pivot row is determined by dividing the value in the right-hand side (RHS) by the positive entry in each row’s pivot column (while disregarding all infinite or negative entries) and taking the ratio with the smallest value.

• The pivot is the number that appears at the intersection of the pivot column and pivot row.

• Aside from the pivot, row operations are used to reduce all other entries in the pivot column to zero. This is done by dividing the entries of that row in the matrix by the pivot.

• When all entries in the net evaluation row satisfy \(C_j – Z_j \le 0\), the optimal solution to the linear program problem is reached.

2.1. Proposed LP model

Mathematical models for starter (chick mash), grower, and prelayer layer rations were created using limited ingredients. The models aimed to reduce the cost of producing a particular diet while satisfying a number of constraints related to the ingredients and the birds’ nutrient needs[16]. The cost and nutritional value of each ingredient were used as parameters, and the variables in the models were their feed components. The specified LP model for achieving the objective function is as follows: \[\text{Min} \quad Z = \sum_{j=1}^{n}C_jx_j \mbox{ where } j = 1, 2 , . . . , n\,,\] subject to the following constraints: \[\begin{aligned} x_1 + x_2 + x_3 + …+ x_7 &= b_1\,,\\ a_{11}x_1 + a_{12}x_2 + a_{13}x_3 + …+ a_{17}x_7 &\ge b_2\,,\\ a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + …+ a_{27}x_7 &\le b_3\,,\\ a_{31}x_1 + a_{32}x_2 + a_{33}x_3 + …+ a_{37}x_7 &\le b_4\,,\\ a_{41}x_1 + a_{42}x_2 + a_{43}x_3 + …+ a_{47}x_7 &= b_5\,,\\ a_{51}x_1 + a_{52}x_2 + a_{53}x_3 + …+ a_{57}x_7 &\ge b_6\,,\\ a_{61}x_1 + a_{62}x_2 + a_{63}x_3 + …+ a_{67}x_7 &\ge b_7\,,\\ a_{71}x_1 + a_{72}x_2 + a_{73}x_3 + …+ a_{77}x_7 &\ge b_8\,,\\ a_{81}x_1 + a_{82}x_2 + a_{83}x_3 + …+ a_{87}x_7 &\ge b_9\,,\\ a_{91}x_1 + a_{92}x_2 + a_{93}x_3 + …+ a_{97}x_7 &\ge b_{10}\,,\\ a_{101}x_1 + a_{12}x_2 + a_{103}x_3 + …+ a_{107}x_7 &= b_{11}\,.\\ x_1 , x_2 , … , x_n &\ge 0\,, \end{aligned}\] In the LP model, Z represents the total cost of the ration, where \(C_j\) denotes the cost per kg of each ingredient, \(x_j\) represents the quantity of ingredients, \(a_{ij}\) is the technical coefficient of nutrient components in ingredients, and \(b_i\) represents the constraints of the ration.

The paper affirms that the axioms of linearity, certainty, additivity, divisibility, non-negativity, finiteness, and proportionality for LP are valid.

3. Model Implementation and Results Analysis

The study collected data on feedstuff specifications, constraints on raw materials, and nutrient requirements for layer birds at each stage of their life cycle. The data was obtained from E-Konadu Farms in Sekyere South District, with feed information from Maridav Concentrate Limited (see Table 2) and the National Research Council’s nutrient requirements for poultry – Ninth Revised Edition (1994) (see Tables 3, 4, and 5). Feed prices were obtained by surveying the feedstuff market prices effective in Ghana in June 2022. The maximal and minimal quantities of different feed components used in the diet were determined by analyzing feed ingredients and using data from Tables 2,3,4, and 5 [17-19]. The feedstuffs used in the ration formulation for layer farms included maize (\(x_{1}\)), soybean (\(x_{2}\)), wheat bran (\(x_{3}\)), concentrate (\(x_{4}\)), premix (\(x_{5}\)), oyster shell (\(x_{6}\)), and methionine (\(x_{7}\)). This study utilized these feedstuffs. Tables 2 and 5 summarize the cost implications of feedstuffs and constraints imposed on selecting feedstuffs for layer rations, respectively. Table 3 provides details on layer nutrient requirements, and Table 4 shows nutrient levels in feed ingredients.

The LP model is presented in three different ways, that is the least cost chick mash, grower, and prelayer ration.

3.1 Implementation of the LP least cost model chick mash rationing

The various ingredients from Tables 2,4, and 5 are substituted into the model to obtain:

\[\text{Min} \quad Z= 3.6x_{1} + 5.7x_{2} + 2.1x_{3} + 6.9x_{4} + 10x_{5} + 0.8x_{6} + 20x_{7}\,,\] subject to \[ x_{1} + x_{2} + x_{3} + x_{4} + x_{6} = 1000 \tag{1}\]\[ 0.088x_{1} + 0.44x_{2} + 0.159x_{3} + 0.65x_{4} + 0.6x_{7} \ge 18 \tag{2}\]\[ 0.04x_{1} + 0.035x_{2} + 0.045x_{4} \le 40 \tag{3}\]\[ 0.02x_{1} + 0.065x_{2} + 0.105x_{3} + 0.01x_{4} \le 50 \tag{4}\]\[ 0.0001x_{1} + 0.0023x_{2} + 0.004x_{3} + 0.061x_{4} + 0.38x_{6} = 9.5 \tag{5}\]\[ 0.0009x_{1} + 0.002x_{2} + 0.0115x_{3} + 0.03x_{4} \ge 4.5 \tag{6}\]\[ 0.0038x_{1} + 0.0257x_{2} + 0.0063x_{3} + 0.045x_{4} + 0.05x_{5} \ge 4.5 \tag{7}\]\[ 0.0009x_{1} + 0.006x_{2} + 0.0024x_{3} + 0.018x_{4} + 0.007x_{5} + x_{7} \ge 3.5 \tag{8}\]\[ 0.022x_{1} + 0.004x_{2} + 0.017x_{3} \ge 9.5 \tag{9}\]\[ 3430x_{1} + 2730x_{2} + 1680x_{3} + 2860x_{4} \ge 2850 \tag{10}\]\[ x_{5} = 1 \tag{11}\]\[ x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} \ge 0 \tag{12}\] The equations listed above correspond to:

• (1): Minimum Demand Requirement,

• (2): Crude Protein Requirement,

• (3): Fat Requirement,

• (4): Crude Fibre Requirement,

• (5): Calcium Requirement,

• (6): Phosphorus Requirement,

• (7): Lysine Requirement,

• (8): Methionine Requirement,

• (9): Linoleic Acid Requirement,

• (10): Metabolizable Energy Requirement,

• (11): Premix Requirement,

• (12): Non-negativity Condition.

The amount of soyabean and concentrate in Table 6 was reduced to zero (0kg) because almost all the nutritional value they provide can also be found in the other ingredients (constraints). To supplement the nutritional value that the soyabean and concentrate would have provided, the amount of maize, wheat bran, oyster shell, and methionine was increased. However, the amount of premix used was kept constant in order to balance the nutritional value of the feed. This model saves approximately GH1,307 on feed costs.

3.2. Implementation of the LP least cost model grower rationing

The model equations are given by:

\[\text{Min} \quad Z = 3.6x_{1} + 5.7x_{2} + 2.1x_{3} + 6.9x_{4} + 10x_{5} + 0.8x_{6} + 20x_{7}\,,\] subject to \[\begin{aligned} x_{1} + x_{2} + x_{3} + x_{4} + x_{6} &= 1000\,,\\ 0.088x_{1} + 0.44x_{2} + 0.159x_{3} + 0.65x_{4} + 0.6x_{7} &\ge 16\,,\\ 0.04x_{1} + 0.035x_{2} + 0.045x_{4} &\le 35 \,,\\ 0.02x_{1} + 0.065x_{2} + 0.105x_{3} + 0.01x_{4} &\le 45\,, \\ 0.0001x_{1} + 0.0023x_{2} + 0.004x_{3} + 0.061x_{4} + 0.38x_{6} &= 8.5\,, \\ 0.0009x_{1} + 0.002x_{2} + 0.0115x_{3} + 0.03x_{4} &\ge 3.5 \,,\\ 0.0038x_{1} + 0.0257x_{2} + 0.0063x_{3} + 0.045x_{4} + 0.05x_{5} &\ge 6.5\,, \\ 0.0009x_{1} + 0.006x_{2} + 0.0024x_{3} + 0.018x_{4} + 0.007x_{5} + x_{7} &\ge 2.5\,, \\ 0.022x_{1} + 0.004x_{2} + 0.017x_{3} &\ge 9.5 \,,\\ 3430x_{1} + 2730x_{2} +1680x_{3} +2860x_{4} &\ge 2850\,,\\ x_{5} &= 1 \,,\\ x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} & \ge 0\,. \end{aligned}\]

The amount of soyabean in Table 7 was reduced to zero (0kg) because almost all the nutritional value they provide can also be found in the other ingredients (constraints). To supplement the nutritional value that the soyabean would have provided, the amount of maize and wheat bran was increased. To balance the nutritional value of the feed, the amount of methionine, concentrate, and oyster shell was reduced while the amount of premix was maintained. This model saves approximately GH750.

3.3. Implementation of the LP least cost model prelayer rationing

The model is given as: \[\text{Min} \quad Z = 3.6x_{1} + 5.7x_{2} + 2.1x_{3} + 6.9x_{4} + 10x_{5} + 0.8x_{6} + 20x_{7}\,,\] subject to: \[\begin{aligned} x_{1} + x_{2} + x_{3} + x_{4} + x_{6} &= 1000\,,\\ 0.088x_{1} + 0.44x_{2} + 0.159x_{3} + 0.65x_{4} + 0.6x_{7} &\ge 17\,,\\ 0.04x_{1} + 0.035x_{2} + 0.045x_{4} &\le 30\,,\\ 0.02x_{1} + 0.065x_{2} + 0.105x_{3} + 0.01x_{4} &\le 45 \,,\\ 0.0001x_{1} + 0.0023x_{2} + 0.004x_{3} + 0.061x_{4} + 0.38x_{6} &= 3.5\,,\\ 0.0009x_{1} + 0.002x_{2} + 0.0115x_{3} + 0.03x_{4} &\ge 3.5\,,\\ 0.0038x_{1} + 0.0257x_{2} + 0.0063x_{3} + 0.045x_{4} + 0.05x_{5} &\ge 5\,,\\ 0.0009x_{1} + 0.006x_{2} + 0.0024x_{3} + 0.018x_{4} + 0.007x_{5} + x_{7} &\ge 2.5\,,\\ 0.022x_{1} + 0.004x_{2} + 0.017x_{3} &\ge 8.5\,,\\ 3430x_{1} + 2730x_{2} + 1680x_{3} + 2860x_{4} &\ge 2850\,,\\ x_{5}& = 1\,,\\ x_{1}, x_{2}, x_{3}, x_{4}, x_{5}, x_{6}, x_{7} &\ge 0\,. \end{aligned}\]

The amount of soyabean in Table 8 was reduced to zero (0kg) because almost all of the nutritional value they provide can also be found in the other ingredients (constraints). To supplement the nutritional value that the soyabean would have provided, the amount of maize, wheat bran, and methionine was increased. To balance the nutrional value of the feed, the amount of concentrate and oyster shell was reduced while the amount of premix was maintained. This model saves approximately GH520.

3.4. Comparison of rations to current practice

To determine which method is more economical, we compared it to the current practices of layer chick mash, grower, and prelayer rations. The results are displayed in Table 9, 10 and 11.

In Table 9, the cost of creating a prelayer feed using the farm’s current technique is GH4320.00, compared to GH3045.005. If the feed formulation is based on our proposed mathematical model, this results in a 30% savings. Feed formulation is obviously more successful when based on a good linear programming model.

In Table 10, the cost of creating a prelayer feed using the farm’s current technique is GH4028, compared to GH3278.199. If the feed formulation is based on our proposed mathematical model, this results in a 19% savings. Feed formulation is obviously more successful when based on a good linear programming model.

In Table 11, the cost of creating a prelayer feed using the farm’s current technique is GH3719.5, compared to GH3199.065. If the feed formulation is based on our proposed mathematical model, this results in a 14% savings. Feed formulation is obviously more successful when based on a good linear programming model.

4. Conclusion

In this paper, we presented an optimal feed allocation for the Ghanaian poultry industry using linear programming techniques. The corresponding LP problem was solved using the Excel solver. Our results indicate that the cost of the least expensive starter ration was GH1307 per ton, representing a 30% reduction in layer starter feed formulation. The cost of the least expensive grower ration was GH750 per ton, representing a 19% reduction in layer grower feed formulation. Finally, the cost of the least expensive prelayer ration was GH520 per ton, representing a 14% reduction in layer prelayer feed formulation compared to the cost of some existing practices.

In terms of the birds’ diet formulation, the starter or “chick mash” ration was composed of 61.5% maize, 35.76% wheat bran, 0.1% premix, 2.44% oyster shell, and 0.21% methionine. This ration met all the nutritional requirements for the starter or chick mash layer with a cost of GH3045.05 per ton.

The grower ration consisted of 63.46% maize, 32.26% wheat bran, 4.72% concentrate, 0.1% premix, 1.43% oyster shell, and 0.03% methionine ingredients. All the nutritional requirements for the grower layer were satisfied by this ration with a cost of GH3278.199 per ton.

For the prelayer ration, the results showed 68.41% maize, 29.64% wheat bran, 1.05% concentrate, 0.1% premix, 0.7% oyster shell, and 0.097% methionine ingredients. This ration met all the nutritional requirements for the grower layer as well, with a cost of GH3199.065 per ton.

Keeping feed costs under control is essential for the sustainability of the Ghanaian poultry industry. Our study demonstrates that using a linear programming approach to determine feed formulation is more efficient than the trial-and-error method. These analyses can help increase production in the Ghanaian poultry industry.