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Transactions on Engineering and Computing Sciences - Vol. 12, No. 2
Publication Date: April 25, 2024
DOI:10.14738/tecs.122.16835.
Oladejo, N. K., & Kyere, G. (2024). Linear Programming Technique and its’ Application in Minimizing Piggery Feed Meals formulation
in a Piggery Farm in Ghana. Transactions on Engineering and Computing Sciences, 12(2). 217-225.
Services for Science and Education – United Kingdom
Linear Programming Technique and its’ Application in
Minimizing Piggery Feed Meals formulation in a Piggery Farm in
Ghana
N. K Oladejo
Department of Industrial Mathematics,
C. K. Tedam University of Technology and Applied Sciences Ghana
G. Kyere
Department of Industrial Mathematics,
C. K. Tedam University of Technology and Applied Sciences Ghana
ABSTRACT
In this paper, we develop an optimal feed mix formulation model for the piggery
farm industry utilizing locally and accessible feed ingredients in other to achieve an
optimal productivity in piggery farming in Ghana using information from a typical
standard farm to form the Linear Programming model for the feed composition
created and parameterized by considering both the decision and constraints
variables, to determine the optimal solution. Results from the values measured and
post optimality shows that the optimal reduction in the pig’s ration of 8.89%
decrease in the pigs feed formulation related to the farm’s existed procedure.
Keywords: Linear, Programming, Application, Model, Decision, Constraints, Variable
INTRODUCTION
Piggery business has a considerable impact on the national economy; it is a well-liked sector
for smallholders and makes a significant contribution to GDP and job growth. Over seventy 70
percent of the cost of raising pigs goes toward their feed, so an effective feed formulation
process is hereby necessary for a viable piggery sector. However, many piggery farmers use
ineffective techniques to address feed formulation issues such as firsthand knowledge, and
intuition thereby making their effort unfruitful.
According to [1] approximately the majority of rural people in developing world depend on
animals for a living, either partially or entirely. Raising livestock has many advantages such as
earnings, meals, animal waste, draft power, transporting services, investments, insurance, and
status in society as opined by [2]
According to [3] the rise in pork intake is one of the livestock industry’s fastest-growing
divisions is the intensive livestock sector, which supports family income as [4] opined that pork
output rises from 11,173 metric tons in 1999 to 17,512 metric tons in 2009, with production
indexes one rising from 98 to 154.
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Many farmers rely on piggery for meat, work opportunities, nutrition security, and investments
for storage or trade, and it is an important part of their socioeconomic and cultural identities.
It also provides food, work opportunities, and nutrition security, as well as store or trade assets
as opined by [5]
According to [6] Ghana piggery farmers have the ability to produce 66.8% of the country’s GDP
from livestock and crops as stated by minis try of food and agriculture as. Growing pigs and
other indigenous livestock species is essential to Ghanaian farming techniques. Systems which
may be categorized as free-range, intensive, or semi-intensive farming [7] Opined that lack of
better breeders, a compromise between accessibility of land and water constraints brought on
by urbanization processes, pollution, and an absence of managerial expertise have all impeded
the growth of Ghana’s domestic piggery farm business.
For these reasons [8] Suggested that government should create a number of research and
development (RD) programs, grants, research institutions, and other initiatives in other to
assist the piggery farmers as raising pigs is a crucial part agricultural sector and a significant
source of income for many business people seeking alternate routes to successful operations as
opined by [9]
LINEAR PROGRAMMING
Here we formulate and adopt the linear programming (LP) method which is one of the most
widely used optimization techniques and perhaps the most effective method. The term linear
programming was coined by [1] to refer to problems in which both the objective function and
constraints are provided.
We consider the following standard form of linear programming
Maximize F= ∑ cjui
n
j=0
(1)
Subject to
∑ a(i,j)uj = bi
, i = 1,2 ... ... . n
n
i=0
(2)
lj ≤ Uj ≤ tj, j = 1,2, ...., n (3)
Where Cj are the n objective performance coefficient a (i, j) and bj are parameters in the m linear
inequality constraints and lj and tj are lower and upper bounds with lj≤ tj. Both lj and tj may be
positive or negative.
Linear Programming Model Formulation Models for Pig Feed Mix
Here we apply the above Mathematical models for pigs’ meal mix ration using limited and
available ingredients in other to minimize the cost of producing a particular diet and meals after
satisfying a set of constraints. These restrictions mostly came from [12] restrictions on
ingredients and the pigs' nutrient needs. According to [13] the parameters in the models were
the cost and nutrient value of each component, whereas the variables in the models were those
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219
Oladejo, N. K., & Kyere, G. (2024). Linear Programming Technique and its’ Application in Minimizing Piggery Feed Meals formulation in a Piggery
Farm in Ghana. Transactions on Engineering and Computing Sciences, 12(2). 217-225.
URL: http://dx.doi.org/10.14738/tecs.122.16835
two factors. These restrictions mostly came from [12] restrictions on ingredients and the pigs'
nutrient needs. According to [13] the parameters in the models were the cost and nutrient value
of each component, whereas the variables in the models were those two factors. Here we apply
the linear Programming models to optimize pigs feed ration using limited and available
ingredients in other to minimize cost of producing a particular diet and meals satisfying a set
of constraints.
Minimize P=∑ cjui
n
j=0
(4)
Subject to
(5)
Where P = Cost of the ration, Cj = Ingredient price, j=1,2,3.....m ui = Ingredient quantity,
i=1,2,3.....n ai = Technical coefficients of nutrient components in feedstuffs, bi = Constraints of
the ration.
Assumptions
The following were the assumption made in formulating the linear programming model:
1. All the projects and constraints are independent on each other.
2. Equal investment opportunities are assumed for the projects in each period.
3. The financial cash flows. Resources and constraints are known with certainty
DATA COLLECTION AND ANALYSIS
Recommended nutrient requirements schedule for both sow and winners’ pigs were collected
from Ghana Veterinary service limited which includes feedstuffs and nutrients used in ration
formulation in piggery farms and mathematically designated as shown in table1 and 2 below:
maize ( , Rice Bran ( , Wheat Flour ( , Soya bean Meal ( , Meat Meal ( , Cotton ( ,
Fish Meal ( , Cassava/starch ( , Methionine ( , Molasses ( .
Table1: Standard feedstuffs and Nutrient levels ingredients in Pig feed meal
Ingredient Cost/kg Protein Fat Lysine Methionine Phosphorus Calcium Fiber ME
u1 2.60 8.8 4.0 2.0 0.1 0.34 0.4 0.18 3432
u2 2.40 48 3.5 6.5 0.2 0.37 3.2 0.59 2557
u3 2.00 13 0 5.1 0.05 1.20 0.5 0.42 3153
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u4 2.80 60 4.5 1.0 6.5 3.5 4.5 1.8 2950
u5 7.00 95 0 0 0 0 100 0 0
u6 2.60 12 0.25 4.75 1.50 1.50 0.2 0.15 1260
u7 3.00 0 0 0 0 0 0 0 0
u8 1.40 39.6 0 0 0.15 0.48 62.8 71.9 2350
u9 0.24 0 0 0 38 0 0 0 0
u10 7.00 60 0 0 0 0 0 10 0
Table 2: Least-cost formulation restrictions on nutrients and feedstuffs for wiener and
sow ration
Nutrients Wiener Sow
Max Min Constraints Max Min Constraints
Crude protein (%) - 23 23 - 18 18
ME(Kcal/kg) 3200 2800 230 3400 3200 3200
Calcium (%) 15 10 15 25 10 10
Phosphorus (%) - 45 40 - 35 35
Fat (%) 50 - 50 60 - 60
Crude Fiber% 30 - 30 50 - 50
Lysine% - 10 11 - 11 11
Methionine% - 5 5 - 5 5
Formulation and Implementation of Linear Programming Model Implementation
We develop two Linear Programming model for the pig feed meals as follows:
1. Linear Programming model for least cost of wiener ration
2. Linear Programming model for least cost sow ration
We then insert the value of the ingredients obtained from the farmer and that of the Veterinary
service in tables 1 and 2 into the model in equation (4) as shown in table 3a, b and 4 below:
Min(P) = 2.6u1 + 2.4u2 + 2u3 + 2.8u4 + 7u5 + 2.6u6 + 3u7 +1.4u8 +0.24u9 + 7u10
Subject to:
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221
Oladejo, N. K., & Kyere, G. (2024). Linear Programming Technique and its’ Application in Minimizing Piggery Feed Meals formulation in a Piggery
Farm in Ghana. Transactions on Engineering and Computing Sciences, 12(2). 217-225.
URL: http://dx.doi.org/10.14738/tecs.122.16835
Table 3a: Feasible and initial Tableau for the least cost wiener
Implementation of Linear Programming Model for the Least Cost Sow Ration
Here, we insert the value of the ingredients obtained from the farmer and that of the Veterinary
service as shown in tables 1 and 2 into the model as follows:
Min(Z) = 2.6u1 +2.4u2 + 2u3 + 2.8u4 + 7u5 + 2.6u6 + 3u7 +1.4u8 + 0.24u9 + 7u10
Subject to:
Table 3b: Feasible and initial Tableau for the least cost of Sow pigs
Z -2.6 -2.4 -2 -2.8 -7 -3 -3 -1.4 -0.2 -7 0 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 0 0 1 0 0 0 0 0 0 0 0 1000
8.8 48 13 60 95 12 0 40 0 0 0 1 0 0 0 0 0 0 0 21
4 3.5 4.5 0 0 0.3 0 0 0 0 0 0 1 0 0 0 0 0 0 60
2 6.5 5.1 1 0 5 0 0 0 0 0 0 0 1 0 0 0 0 0 50
0.1 0.2 0.1 7 1.5 0 0 0.2 38 0 0 0 0 0 1 0 0 0 0 15
0.4 0.4 1.2 4 0 2 0 0 0 .5 0 0 0 0 0 1 0 0 0 45
0.4 3.2 0.5 6 100 .2 0 63 0 0 0 0 0 0 0 0 1 0 0 11
0.2 0.6 0.4 1.8 0 0.2 0 72 100 0 0 0 0 0 0 0 0 1 0 5
3432 2557 3153 2950 0 2350 0 2350 0 0 0 0 0 0 0 0 0 1 0 3200
Z -2.6 -2.4 -2 -3 -7 -3 -3 -1.4 -0.2 -7 1 0 0 0 0 0 0 0 0 0
1 1 1 1 1 1 1 1 0 0 0 1 0 0 0 0 0 0 0 1000
8.8 48 13 60 95 12 0 40 0 0 0 0 1 0 0 0 0 0 0 23
4 3.5 4.5 0 0 0. 0 0 0 0 0 0 0 1 0 0 0 0 0 50
2 6.5 5.1 1 0 4.8 0 0 0 0 0 0 0 0 1 0 0 0 0 50
0.1 0.2 0.1 6.5 2 0 0 0.2 38 0 0 0 0 0 0 1 0 0 0 15
0.3 0.37 1.2 3.5 0 2 0 0 0 .5 0 0 0 0 0 0 1 0 0 45
0.4 3.2 0.5 4.5 100 .2 0 63 0 0 0 0 0 0 0 0 0 1 0 11
0.2 0.59 0.4 1.8 0 .2 0 72 100 0 0 0 0 0 0 0 0 0 1 5
3432 2557 3153 2950 0 2350 0 2350 0 0 0 0 0 0 0 0 0 0 0 2800
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MATLAB PACKAGE SOLUTION OF LEAST COST FOR PIGS FEED MEAL RATION
Table 4: LP Model for Least Cost for pigs rationing solution using MATLAB Package
Decision Unit cost Solution variable Total cost Reduced cost
Winners Sow Winner Sow Winners Sow
Maize(u1) 2.6 510 656 1326 1706.64 -26 -68.64
Rice Bran (u2) 2.4 200 80.9 480 194.16 00 165.94
Wheat (u3) 2.0 00 00 00 00 146 60
Soya bean(ul4) 2.8 00 00 00 00 144 70
Meat meal(u5) 7.0 1.2 00 8.4 00 -1.4 17.5
Cotton(u6) 2.6 100 119 260 308.62 -78 --9.62
Fish meal(u7) 3.0 1 1 3 3 00 00
Cassava(u8) 1.4 00 00 00 00 35 5.6
Methionine(u9) 0.24 86 40.7 20.4 9.77 -8.64 -.17
Molasses(u10) 7.0 1.8 3.4 12.6 23.8 -9.06 -1.4
Total Optimal cost 205.90 239.11
Discussion
From the MATLAB software result, the amounts of wheat bran, soya bean, beef meal, and
cassava are reduced zero (0 kg) this is due to the presence of nutrients value other ingredients.
To compensate for the lack of nutrients provided by the wheat bran, soya bean, and cassava,
more maize, cotton, methionine, and molasses should be added to the diet. The amount of
nutrients in methionine was decreased, while the nutrient in the fishmeal was increased. In this
case the model minimized the cost of the feeding to GHc205.90.per feed formulation.
Likewise, the quantity of wheat bran, soya bean, beef meal, and cassava were reduced to zero
kilograms (kg) because the other components (a constraint) already contain nearly all of the
nutritional value they provide. To make up for the lack of nutritional value that would have
been provided by the wheat bran, soya bean, beef meal, and cassava, more maize, cotton,
methionine, and molasses were added. To balance the feed's nutritional content, the amount of
rice bran was decreased, and fishmeal was increased. The feed cost is decreased by this model
by nearly GH 239.11
Determination of the Cost-Effectiveness in the Pigs Feed Meal Ration
Here we determine the most cost-effective in the current feed meal procedures for the pig’s
ration using mathematical models as given in the Tables 5 below:
Table 5: shows cost effectiveness of the Pigs feed meal ration
Ingredients Cost Wiener Sow
Current Proposed Current Proposed
Value Cost Value Cost Value Cost Value Cost
(u1) 2.6 500 1300 510 1326 630 1638 510 1581
(u2) 2.4 200 480 200 480 150 360 200 580
(u3) 2 73 146 0 0 30 60 0 0
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Oladejo, N. K., & Kyere, G. (2024). Linear Programming Technique and its’ Application in Minimizing Piggery Feed Meals formulation in a Piggery
Farm in Ghana. Transactions on Engineering and Computing Sciences, 12(2). 217-225.
URL: http://dx.doi.org/10.14738/tecs.122.16835
(u4) 2.8 80 144 0 0 25 70 0 0
(u5) 7 1 7 1.2 8.4 2.5 17.5 1.2 9
(u6) 2.6 70 182 100 260 115 299 100 310
(u7) 3 1 3 1 3 1 3 1 3.5
(u8) 1.4 25 35 0 0 4 5.6 0 0
(u9) 0.24 50 12 86 20.64 40 9.6 86 63.64
(u10) 7 1 7 1.8 12.06 3.2 22.4 1.8 13.5
1001 2316 900 2110.1 1000.7 2485.1 900 2560.64
Based on the formulated Linear Programming model, the cost of new formulated Wiener feeds
is GH¢ 2110.10 against the existing cost of feed formulation which was GH¢2316.00 with a
difference of GH¢205.90 per feed meal formulation. This result yields an impressive saving of
8.89% per feeds formulation. While the Sow feed formulation costs GH¢2245.99 against
GH¢2485.10 with a difference of GH¢239.11 which represents a considerable decrease of
approximately 9.62%. Per feed formulation.
Sensitivity Analysis on the Pigs Ration
Here we examine the effects of changes or variations in selected key parameters on the model.
Here the cost of the various ingredients/kg was increased and decreased by 5% The sensitivity
analysis demonstrate that the proposed pigs ration is robust to changes in the unit costs of the
ingredients, and strictly adapt to cost fluctuations while maintaining the same total cost and
nutritional quality for the Pigs rations. The effects of variations of the cost of the ingredients
use in the feed mix was analyzed as presented in Table 6.
Table 6: shows the effect of variation in the Pig feed ration
Increased Cost by 5% Decreased Cost by 5%
Current Proposed Ingredient. Cost Current Proposed
Ingre dient Cost Value Cost Value Cost Value cost Value Cost
(u1) 3.10 500 1550 510 1581 2.10 630 1323.00 656.4 1378
(u2) 2.90 200 580 200 580 1.90 150 285.00 80.9 154
(u3) 2.50 73 182 0 0.00 1.50 30 45.00 0 0.00
(u4) 3.30 80 264 0 0.00 2.30 25 57.50 0 0.00
(u5) 7.50 1 7.50 1.2 9.00 6.50 2.5 16.25 0 0.00
(u6) 3.10 70 217 100 310 2.10 115 241.50 119 249
(u7) 3.50 1 3.50 1 3.50 2.50 1 2.50 1 2.50
(u8) 1.90 25 47.50 0 0.00 0.90 4 3.60 0 0.00
(u9) 0.74 50 37 86 63.64 -0.26 40 -10.40 40.7 -10.58
(u10) 7.50 1 7.50 1.8 13.50 6.50 3.2 20.80 3.4 22.10
Opt. value 1001 2897 900 2561 2.10 1000.7 1984.75 900.1 1795.44
CONCLUSION
This paper deal with optimizing feed mix formulation for the piggery farm industry utilizing
locally and accessible feed ingredients to achieve an optimal productivity in piggery farming in
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Ghana using information from a typical standard farm to form the Linear Programming model
for the feed composition created and parameterized by considering both the decision and
constraints variables, to determine the optimal solution. Results from the values measured and
post optimality shows that the optimal reduction in price of the pigs’ ration of 8.89% and 9.62%
decrease in wiener and sow’s ratio respectfully Aside, the new price per ton of the ration cost
GH¢2110.10 against the existing cost of feed formulation which was GH¢2316.00 with a
difference of GH¢205.90 per feed meal formulation. This result yields an impressive saving of
8.89% per feeds formulation. While the Sow feed formulation costs GH¢2245.99 against
GH¢2485.10 with a difference of GH¢239.11 which represents a considerable decrease of
approximately 9.62%. Per feed formulation.
It was therefore recommended that since the viability of every piggery farm depends on cost
management application of linear programming model approach in feed formulation and
implementation will increase production and minimize cost of production in any industry.
Conflicts of Interest
The authors hereby declare that there is no conflict of interests in the preparation and
submission of this manuscript
Funding Statement
The authors declare that there is no funding from either an organization or an individual.
Preparation and submission of this manuscript is so responsibility of the authors
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Oladejo, N. K., & Kyere, G. (2024). Linear Programming Technique and its’ Application in Minimizing Piggery Feed Meals formulation in a Piggery
Farm in Ghana. Transactions on Engineering and Computing Sciences, 12(2). 217-225.
URL: http://dx.doi.org/10.14738/tecs.122.16835
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