Research Article
Linear Programming Application to Utilization of Duckweed (Lemna paucicostata) in Least-cost Ration Formulation for Broiler Finisher
Computer Centre, Adekunle Ajasin Umversity P.M.B. 001, Akungba-Akoko, Ondo State, Nigeria
Malnutrition has been one of the major problems besetting developing countries all over the world. The bulk of food consumed by the populace of these countries are made up of carbohydrate. Morna et al. (1993) reported that despite the fact that a hard-working adult - a farmer, for example - needs approximately 3,500 calories and 50 grams of protein per day; a one-year-old child needs about 1,000 calories and 15 grammies of protein per day; these quantities of essentials nutrients are missing in the diets of many rural Africans, which are based on staples of grains, such as maize, or tubers, such as cassava.
The protein usually needed for growth and replacement of various cells of the body is often lacking in the diets of most Africans. Protein intake can be increased through adequate consumption of crops such as soya beans, ground nut, pigeon peas, beans; and also animal protein such as poultry products, beef, pork, mutton, goat meat and other bush meats. While proteinous crops are still affordable in most of these countries, this can not be said of animal protein whose consumption can be said to be per acutely low.
In a World Bank (1991) report on the average percentage of population food insecure each year between 1980 and 1990, 17% of Nigerias populace were considered food insecure, that is-those who can not obtain the calories they need, either because they do not have the means or because adequate foods is not available in their area. The situation is further worsened by the deterioration in the production of livestock and crops in recent years, thereby leading to increases in price of meat and crop products. Obasanjo (1990) stated that before there can be any food security, there must be adequate access by all people at household levels to adequate food at all times. Food must not only be available, it must be cheap and affordable - particularly for low-income group.
There are many problems facing agricultural production in Nigeria. Obasanjo (1990) also stated that the basic problems of agrarian practice were compounded by insufficient and inadequate infrastructure, land tenure, poor research and extension services and suffocating marketing policy. Ilemobade (1985) reported that the problem facing livestock industry does not only lie with insufficient number of livestock, on insufficient number of specialists, but with inefficient production, cultural practices of livestock farmers, shortage of feed and water and wide-spread occurrences of animal diseases, aggravated by large human population whose percentage increase must be a matter of great concern.
A fast means of producing animal protein is through the use of monogastric animals such as pig and poultry birds because of their higher fecundity rather than ruminants. Whereas the production of pork and bacon is only limited to countries or areas where there are no religious norm against their production, but there is no known significant religious norm against poultry production hence the production of poultry products and by-products is unrestricted.
The fastest means of producing animal protein remains broiler production. The recent outbreak of bird flu (Avian influenza) in Nigeria (Anonymous, 2006a, b and c) has led to a very serious drop in the patronage of poultry and its allied by-products. This zoonotic disease has led to the destruction of millions of birds in the country thereby leading to loss of income to poultry farmers and also leading to severe shortage in the supply of poultry meat and its by-product that are already inadequate before now.
Although the major problem with poultry production can be streamlined to feeds in all its ramifications (David-West, 1989) however this does not rule out the outbreak of contagious diseases. The nutritional problem through inadequate consumption of qualitative feeds at affordable price can not be overemphasized. Also Ensminger (1980) stated that feed accounts for 70% of the recurrent cost of producing broiler chickens. Fetuga (1989) also reported that in Nigeria, feed costs represent between 70-80% of the total cost of producing various livestock products.
Before any feed can be formulated for poultry birds, the nutritionist takes a lot of factors into consideration such as the available ingredients, nutrient composition of available ingredients, inclusion rate restriction of available ingredients, and most importantly the nutrient requirements of each species of bird, doing this manually can be very cumbersome and tedious. Also at the end of the day the nutritionist may end up formulating lowest-priced feed instead of least-cost.
Patrick and Schaible (1980) defined least-cost feeds as the lowest-cost formular that contains all the nutritional elements needed for maximum performance while lowest-priced feeds may not be the formulation that will produce maximum performance.
The problem of least-cost ration formulation can be effectively managed with the application of computer using linear programming technique (Olorunfemi et al., 2001, 2006). Simplex method has been chosen to solve the mathematical models.
Most conventionally used protein and energy feedstuffs in poultry feeds have limited supplies and their costs are usually high because of the demands from both humans and livestocks. The use of cheaper and non-conventional feedstuffs such as duckweed-Lemna paucicostata, that grows into profusion in most tropical water bodies are aimed at minimizing the production cost of broiler finishers feeds.
Duckweed is a group of small floating aquatic plants found worldwide. It is a potentially important new crop due to its high productivity and nutritional value. The plant has a rapid growth rate and high yield with high protein content of 20-40%, low fibre content, high mineral content, non toxicity and only few known pest (Lemna Corporation, 1992). Duckweed meal had been fed to layers at up to 40% of total feed with satisfactory results (Skillicorn et al., 1993).
This study is therefore focussed on the use of a computer-based system which will analyze and indicate how best the available local ingredients can be combined together with duckweed to formulate least-cost feeds for broiler finishers. The result of this study is aimed at helping poultry nutritionists, feed-millers and also poultry producers in knowing the least-cost combination of feedstuffs to meet specific nutrient requirements, acceptance or rejection of ingredients based on their cost and nutritive value. This will enhance greater supply of poultry products and by-products at lower prices to the teeming populace thereby alleviating the malnutrition problem facing the developing countries all over the world.
Data collection for this study were based on raw material (feedstuffs) specification, constraints imposed on the selected raw materials and the dietary nutrient requirements for broiler finisher. Two models were created with one containing duckweed and the other diet without duckweed meal serving as the control.
Costs of raw materials used in the diet formulation were obtained from the prevailing market prices of feedstuffs in Nigeria through survey. Approximate constituents, limiting amino acids, calcium and phosphorus contents, minimum and maximum dietary inclusion levels of various feedstuffs used in diet formulation were obtained from standard tables and sources (Aletor, 1986; Mbagwu et al., 1990; Lemna Corp, 1992; Aduku, 1993a; Skillicorn et al., 1993; Tacon, 1993; NRC, 1994). Estimation of metabolizable energy of feedstuffs for the diets were calculated by converting the gross energy using the following equation as described by Miller and Payne (1959).
ME | = | Metabolizable Energy |
GE | = | Gross Energy |
N% | = | Dietary Nitrogen Percent |
Table 1 and 2 summarized the raw material specifications and restrictions imposed on selected raw materials used in the computerized linear programming for formulating a least-cost diet for broiler finishers.
Table 1: | Cost implications of raw materials and nutrient levels of feed ingredients |
1US$ = N130.00 |
Table 2: | Constraints imposed on the selection of feedstuffs by computerized linear programming for Broiler finisher's mash |
Table 3: | Least-cost formulation restrictions on nutrients for Broiler Finisher's mash |
Restriction data on the nutrient requirements of broiler finishers were obtained from literature (Fetuga, 1989; Oluyemi, 1989; Aduku, 1993b; NRC, 1994). These values are presented in Table 3.
Data analysis: The method of data analysis employed in this study is Linear Programming (LP) model. The model was designed to reflect various feedstuff combinations used in the diet formulation, current market prices, nutrient composition and range of inclusion to obtain a least-cost ration.
Linear programming is a computational method of selecting, allocating and evaluating limited resources with linear, algebraic constraints to obtain an optimal solution for a linear, algebraic objective function. Patrick and Schaible (1980) stated that linear programming is technically a mathematical procedure for obtaining a value - weighting solution to a set of simultaneous equations.
A linear programming model consists of three major different parts namely; an objective function, a series of equations which are alternative methods of attaining the objectives and the resources which are non negative variables. Kuester and Mize (1973) stated that the classical linear programming problem has the following form:
Subject to:
Where Aij, Bi and Cj are given constraints and Xj are the decision variables. From the above format, we are seeking the values of the Xj which will optimize (maximize or minimize) the objective function, F.
In a more compact form, the problem can be rewritten as:
Subject to:
where j = 1, 2, , n and i = 1, 2, , m.
The numbers of unknown is usually greater than the number of equations (n>m). Taha (1987) stated that by setting (n - m) variables to 0, the unique solutions are called basic solutions and that if a basic solution satisfies the non negativity condition, it is called a feasible basic solution.
Saaty (1955) stated that the upper bounds on the maximum number of iterations required to solve a general problem containing n variables and m inequality constraints are determined solely by the total number of such points, or vertices. Taha (1987) stated that the maximum number of iterations cannot exceed:
Model construction: Mathematical models were constructed for the two types of ration using a limited ingredients with one model containing duckweed and the control feed without duckweed. The objective of the model was to minimize cost of producing a particular diet after satisfying a set of constraints. These constraints were mainly those from nutrient requirements of the bird and also ingredients constraints. The variables in this model were the ingredients while the cost of each ingredients and the nutrient value of each ingredient was the parameter. The specified LP model for the attainment of this objective is given by Eq. 1 i.e., the objective function, through Eq. 9.
Equation 1: Minimize Z =∑CjXj
subject to:
Equation 2: CPi | = | ΣbijXj-Crude Protein |
Equation 3: EEi | = | ΣdijXj-Fat |
Equation 4: EFi | = | ΣeijXj-Crude Fibre |
Equation 5: CAi | = | ΣhijXj-Calcium |
Equation 6: PHi | = | ΣkijXj-Phosphorous |
Equation 7: MTi | = | ΣfijXj-Methionine |
Equation 8: LYi | = | ΣgijXj-Lysine |
Equation 9: MEi | = | ΣaijXj-Estimated metabolizable energy |
Where Z = Sum of the total cost of the various feedstuffs used in the diet formulation programme such as blood meal, soybean, maize, duckweed etc.
C = Per unit cost of the different feedstuffs aij,bij . kij = the coefficients (technical) of the component of the particular nutrient found in the given feedstuffs as obtained from proximate analysis.
The variables in this model are the ingredients while the cost of each ingredient and the nutrient value of each ingredient is the parameter.
We define the variables as follows:
x1 | = | Yellow Maize x2 = Groundnut cake x3 = Soyabeans meal |
x4 | = | Rice bran x5 = Brewers Grain x6 = Blood meal |
x7 | = | Fish Meal x8 = Bone meal x9 = Oyster shell |
x10 | = | Dl-methionine x11 = Lysine x12 = Vitamin/mineral premix |
x13 | = | Salt x14= Duckweed |
The constraints used in the construction of model for broiler finishers mash for both feeds are the same and is as explicitly stated below:
Crude Protein >= 200.00 kg
Fat <= 50.00 kg
Fibre <= 50.00 kg
Calcium >= 9.00 kg
Phosphorus >= 3.50 kg
Methionine>= 3.80 kg
Lysine >= 10.00 kg
Energy >= 3000ME Kcal kg-1
Vitamin/Mineral Premix = 2.50 kg
Salt = 2.50 kg
Blood Meal <= 50.00 kg
Total Weight = 1,000.00 kg
However the objective function for the two formulations differs.
Model construction for broiler finishers control mash: The linear programming model for broiler finishers control mash is as follows :
Min (-Z) =-45x1 - 39x2 - 66x3 - 6.5x4 - 12x5 - 27x6 - 145x7 - 23x8 - 10.5x9 - 700x10 - 650x11 - 450x12 - 20x13 |
Subject to:
1) | x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 = 1000 |
2) | 0.1x1 + 0.48x2 + 0.42x3 + 0.118x4 + 0.18x5 + 0.8x6 + 0.65x7 + 0.6x10 + 0.6x11 - s1 = 200 |
3) | 0.04x1 + 0.06x2 + 0.035x3 + 0.125x4 + 0.06x5 + 0.01x6 + 0.045x7 + s2 = 50 |
4) | 0.02x1 + 0.05x2 + 0.065x3 + 0.125x4 + 0.2x5 + 0.01x6 + 0.01x7 + s5 = 50 |
5) | 0.0001x1 + 0.002x2 + 0.002x3 + 0.0004x4 + 0.002x5 + 0.0028x6 + 0.0061x7 + 0.37x8 + 0.35x9 - s6 = 9 |
6) | 0.0009x1 + 0.002x2 + 0.002x3 + 0.0046x4 + 0.0016x5 + 0.0009x6 + 0.03x7 + 0.15x8 + 0.1x9 - s7 = 3.5 |
7) | 0.0018x1 + 0.0048x2 + 0.0059x3 + 0.0024x4 + 0.004x5 + 0.01x6 + 0.018x7 + x10 - s8 = 3.8 |
8) | 0.0025x1 + 0.016x2 + 0.028x3 + 0.005x4 + 0.009x5 + 0.069x6 + 0.045x7 + x11 - s9 = 10 |
9) | 3.432x1 + 2.64x2 + 2.42x3 + 2.86x4 + 1.98x5 + 3.08x6 + 2.86x7 - s10 = 3000 |
10) | x12 = 2.5 |
11) | x13 = 2.5 |
12) | x6 + s11 = 50 |
Where:
x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, s1, s2, s3, s4, s5,
s6, s7, s8, s9, s10, s11 >= 0
x1, x2,
, x12, x13 = Various ingredients
s1, s2,
, s10, s11 = Slack variables
Model construction for duckweed based broiler finisher's mash: Min (-Z) = -45x1-39x2-66x3-6.5x4-12x5-27x6-145x7-23x8-10.5x9-700x10-650x11- 450x12-20x13-6x14
Subject to :
1) | x1 + x2 + x3 + x4 + x5 + x6 + x7 + x8 + x9 + x10 + x11 + x12 + x13 + x14 = 1000 |
2) | 0.1x1 + 0.48x2 + 0.42x3 + 0.118x4 + 0.18x5 + 0.8x6 + 0.65x7 + 0.6x10 + 0.6x11 + 0.259x14 - s1 = 200 |
3) | 0.04x1 + 0.06x2 + 0.035x3 + 0.125x4 + 0.06x5 + 0.01x6 + 0.045x7 + 0.057x14 + s2 = 50 |
4) | 0.02x1 + 0.05x2 + 0.065x3 + 0.125x4 + 0.2x5 + 0.01x6 + 0.01x7 + 0.123x14 + s5 = 50 |
5) | 0.0001x1 + 0.002x2 + 0.002x3 + 0.0004x4 + 0.002x5 + 0.0028x6 + 0.0061x7 + 0.37x8 + 0.35x9 + 0.0011x14- s6 = 9 |
6) | 0.0009x1 + 0.002x2 + 0.002x3 + 0.0046x4 + 0.0016x5 + 0.0009x6 + 0.03x7 + 0.15x8 + 0.1x9 + 0.008x14 - s7 = 3.5 |
7) | 0.0018x1 + 0.0048x2 + 0.0059x3 + 0.0024x4 + 0.004x5 + 0.01x6 + 0.018x7 + x10 + 0.0044x14 - s8 = 3.8 |
8) | 0.0025x1 + 0.016x2 + 0.028x3 + 0.005x4 + 0.009x5 + 0.069x6 + 0.045x7 + x11 + 0.0178x14 - s9 = 10 |
9) | 3.432x1 + 2.64x2 + 2.42x3 + 2.86x4 + 1.98x5 + 3.08x6 + 2.86x7 + 3.45x14 - s10 = 3000 |
10) | x12 = 2.5 |
11) | x13 = 2.5 |
12) | x6 + s11 = 50 |
Where:
x1, x2, x3, x4, x5, x6, x7, x8, x9, x10, x11, x12, x13, x14, s1, s2, s3, s4, s5,
s6, s7, s8, s9, s10, s11 >= 0
x1, x2,
, x13, x14 = Various ingredients
s1, s2,
, s10, s11 = Slack variables
The optimum solution for the duckweed based ration was found at iteration 15 while the optimum solution for the control diet was found at iteration 19.
The results of optimal nutrient analysis produced by computerised linear programming for broiler finisher's mashes are presented in Table 4. The crude protein level of the least-cost diet for both diet was 20.00%. The energy content of both diet was the same (3000.00 kcal kg-1). The crude protein and the energy level of both diet satisfied the minimum constraint levels set for both. The lysine in the control diet (1.00%) was 90.09% of the duckweed based diet (1.11%). This means that the duckweed-based diet contains more of this essential amino-acid than the control diet. The methionine values of both diets was the same and fall within the recommended optimum values for broiler finishers mash (NRC, 1994). Fiber contents for the control was 4.99% while the duckweed-based diet was at the upper limit of 5.00%, the fiber is basically an energy diluent and it aids in digestion.
The results of least cost diet formulation produced by linear programming model are presented in Table 5. Diet containing duckweed had its cost at N30.19 kg-1 while the control diet gave the highest feed - cost of N38.13 kg-1. There was total exclusion of brewer's dried grain and rice bran (Table 5) in the duckweed based diet, these ingredients are basically energy diluents. While the calcium to phosphorus ratio for this work was 2.57, the duckweed based diet gave and higher ratio of 2.98 while the control diet gave a lower ratio of 2.37 but they were still within acceptable level of inclusion.
Table 4: | Optimal nutrient analysis of least-cost diets produced by computerized linear programming for Broiler Finisher |
1Duckweed based diet |
Table 5: | Ingredient composition of least-cost ration formulation produced by computerized linear programming for Broiler Finisher |
1Control diet, 2Duckweed based diet, 31US$ = N130.00 |
The result of this study shows that utilization of diet containing duckweed (Lemna paucicostata) at 29.50% is cost effective and may reduce cost of feeding by as much as 20.82% and this will invariably improve profitability in broiler finishers production.
The author expresses sincere appreciation to Dr. F.M. Aderibigbe and Prof. E.A. Fasakin for their guidance on this work.