INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
Journal:
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page 1183
OPTIMAL SOLUTION PLAN FOR MANUFACTURING ENTERPRISES USING
THE SIMPLEX METHOD
Ruzaliyev Sherzodjon Avazjonovich
Head of the Department of Information Technologies, Fergana State University ,
Doctor of Philosophy (PhD) in Pedagogical Sciences
E-mail: sherzodjonruzaliyev@gmail.com
ORCID ID 0000-0002-0019-8446
Ismoiljonova Odina Isroiljon kizi
Fergana State University, 3rd year student, Applied Mathematics Department,
student of group 22-08
E-mail: ismailjonovaodina88@gmail.com
Abstract:
This article discusses the process of solving linear programming problems using
the simplex method. The main steps of the simplex algorithm, their mathematical foundations
and practical applications are considered. In addition, information is provided about the
advantages and limitations of the method.
Keywords:
Linear programming, simplex method, optimization, mathematical model,
constraints, basic solution.
Abstract:
This article discusses the process of solving linear programming problems using
the simplex method. The main steps of the simplex algorithm, its mathematical foundations,
and practical applications are examined. Additionally, the advantages and limitations of the
method are also presented.
Key words:
Linear programming, simplex method, optimization, mathematical model,
constraints, basic solution.
Annotation:
V dannoy state rassmatrivaetsya process solution of linear programming
problem with simplex method. Description of basic stages of simplex-algorithm, its
mathematical basics and practical application. Takje privedeny svedeniya o
preimushchestvax i ogranicheniyax metoda.
Key words:
Linear programming, simplex method, optimization, mathematical model,
limitation, basic solution.
Login .
Modern economic under the circumstances working release of enterprises
main purpose from resources effective use through maximum benefit to take or expenses
from minimizing Especially food
industry one part
was
confectionery factories for working release plans right designation , product types
between balance storage and there is resources ( raw materials , labor) power , time ,
work release capacities ) optimal in a way distribution big importance profession will reach .
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
Journal:
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Linear programming is one of the powerful mathematical methods used to solve
such problems. In particular, the Simplex method is a widely used and practically effective
algorithm in this area. In this article, the problem of optimizing the production process of a
confectionery factory is formulated on the basis of a linear programming model and solved
using the Simplex method. The goal is to allocate production volumes within the existing
constraints in such a way that the total profit is maximized.
Literature analysis
The theoretical foundations of linear programming and the Simplex method were
created in the middle of the 20th century by the American scientist George Dantzig, and
this method is still used to solve many economic, engineering and logistics problems. Many
scientific works and textbooks (for example, Taha XA “Operations Research”, R. Hilliyer
and G. Lieberman “Introduction to Operations Research”) deeply analyze the theoretical
foundations and practical applications of the Simplex method.
A number of studies on linear programming have also been carried out by Uzbek
scientists. In particular, the effectiveness of the Simplex method is emphasized in articles
and textbooks on modeling, planning, and optimal resource allocation in economics. In
recent years, the implementation of these methods in software based on computer
technologies (such as MS Excel Solver, MATLAB, Python-PuLP) has further expanded the
possibilities of research.
Despite the lack of specialized literature on modeling production processes in the
confectionery industry, there is an opportunity to apply general linear optimization
approaches to real practice by adapting them.
Research methodology
In this study, the simplex method of linear programming was used to optimize the
activities of a confectionery factory. The methodological basis of the study is mathematical
modeling of the production process, that is, expressing a real problem using equations and
inequalities. To do this, first of all, the main types of products produced in the factory, the
types of raw materials required for them (for example, sugar, flour, oil and other
components), the available resource reserves and the net profit from each unit of product
were determined. After that, variables were introduced for each product and the objective
function — maximizing total profit — was formulated. Resource constraints were
expressed in the form of inequalities. The resulting linear programming model was made
ready to be solved using the simplex method. The Simplex algorithm was implemented step
by step: first, an initial basic solution was found, and then the pivot elements were used to
move towards the optimal point. With each iteration, the value of the objective function
was observed to improve. In order to verify and ensure the reliability of the practical results
of the study, the model was programmed in a computer program - in particular, using the
PuLP library in the Python programming language. Through this automated approach, the
optimal production plan was determined, the total profit value was calculated, and the level
of utilization of each resource was revealed. Analysis of the results showed that optimizing
the production process using the Simplex method not only increases efficiency, but also
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
Journal:
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page 1185
serves to increase the economic efficiency of the enterprise. At the same time, by adding
new conditions to the model, it can be adapted to various real situations.
Solving linear programming problems using the simplex method
Solving linear programming problems using the simplex method yields the following
We will get to know each other in detail as we solve the problem.
A confectionery factory produces 3 different products - energy drink (A), fruit
juice (B), mineral water (C) . Production is limited by water, sugar, and labor time (hours).
The goal of the enterprise is to maximize profit .
Information provided:
Product
Water (liters)
Sugar (kg)
Working time
(hours)
Profit
(thousand
soums)
Energy
drink(A)
2
4
3
5
Fruit juice (B)
3
1
4
4
Mineral water
(C)
1
2
2
3
You have 3 main resources for production:
1. Water (liters) – no more than 8 liters
2. Sugar (kg) – not more than 10 kg
3. Working time (hours) – should not exceed 12 hours
Linear programming model:
Variables:
1
P
– Energy drink
2
P
- Fruit juice
3
P
– Mineral water
We are given the following constraints and objective function
let it be:
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ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
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Limitations:
1
2
3
1
2
3
1
2
3
1
2
3
2
3
8
4
2
10
3
4
2
12
, ,
0
x
x
x
x x
x
x
x
x
x x x
+
+
+
+
+
+
Objective function:
1
2
3
5
4
3
max
Z
x
x
x
=
+
+
®
For each inequality in the given system, one basic variable is used.
We can write these inequalities in equation form by introducing and thereby obtain a linear
we get the canonical programming problem:
1
2
3
4
1
2
3
5
1
2
3
6
1
2
3
2
3
8
4
2
10
3
4
2
12
, ,
0
x
x
x x
x x
x x
x
x
x x
x x x
+
+ +
=
+
+
+
=
+
+
+
=
1
2
3
5
4
3
max
Z
x
x
x
=
+
+
®
We write the resulting system of equations in vector form:
1 1
2 2
3 3
4 4
5 5
6 6
0
x P x P x P x P x P x P P
+
+
+
+
+
=
Here
1
2
4
3
P
=
,
2
3
1
4
P
=
,
3
1
2
2
P
=
,
4
1
0
0
P
=
,
5
0
1
0
P
=
,
6
0
0
1
P
=
,
0
8
10
12
P
=
Basis vectors:
4
P
,
5
P
,
6
P
Base plan:
* (0,0,0,8,10,12)
X
=
Based on this data, we construct a simplex table,
0
F
and
i
i
i
z c
D = -
We calculate the
values of the and initial
*
X
We check the base plan for optimality.
Step 1. Enter initial data into the table
Iteration 1
Table 1
Basis
1
P
2
P
3
P
4
P
5
P
6
P
Right
side
4
P
2
3
1
1
0
0
8
5
P
4
1
2
0
1
0
10
6
P
3
4
2
0
0
1
12
Z
-5
-4
-3
0
0
0
0
Step 2. Initial
*
X
Check the base plan for optimality
Iteration 2
Table 2
Basis
1
P
2
P
3
P
4
P
5
P
6
P
Right
side
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ISSN: 2692-5206, Impact Factor: 12,23
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page 1187
4
P
2
3
1
1
0
0
8
5
P
4
1
2
0
1
0
10
6
P
3
4
2
0
0
1
12
Z
-5
-4
-3
0
0
0
0
The base plan is inevitably not optimal, because row 5 contains negative elements,
which by convention must all be nonnegative. So, we look for a new base plan. To do this,
we first find the leading column and leading row in this table.
To find the reference column, look in row 5 of table 1. we take the modulus of the
values, choose the value with the largest modulus, and
We mark the cell (cell) where this number is located, the column where the cell is located
is the reference column. In our example, it will be and the reference column is located
Step 1: Choose the most negative value
Our goal is to maximize Z, so we need to choose the most negative value in the Z′ column. In
the table:
1
P
value in column: -5 (most negative)
2
P
Value in column : -4
3
P
Value in column : -3
Therefore,
the pivot column
1
P
will be the top.
Step 2: Select the pivot row
To select the pivot row, we divide the value in each constraint
1
P
column by the values
in the RHS column:
(
)
1
8 4
2
P qator
=
-
2
10 2.5(
)
4
P qator
=
-
(
)
3
12 4
3
P qator
=
-
The smallest value is 2.5, so
the pivot row is
5
P
will be.
Step 3: Select the pivot element
To select the pivot element, we select the value in the row
1
P
in the column
5
P
. This value
is 4, which is
the pivot element
.
Step 4: Update the table
Now we need to set the pivot element to 1 and update all other values.
To set the pivot element to 1,
5
P
We split the array into a pivot element:
5
_
_
_
4
P qatoridagi har bir element
Now let's update the table:
INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE
ISSN: 2692-5206, Impact Factor: 12,23
American Academic publishers, volume 05, issue 04,2025
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page 1188
Basis
1
P
2
P
3
P
4
P
5
P
6
P
Right
side
4
P
0
2.5
0
1
-0.5
0
3
1
P
1
0.25
0.5
0
0.25
0
2.5
6
P
0
3.25
0.5
0
-0.75
1
4.5
Z
0
-2.75
-0.5
0
1.25
0
12.5
Pivot again
Now
Z
We find the most negative value in the column. This time the most negative value
is
2
P
in the column -2.75. So,
2
P
We enter .
The following:
4
3 1.2(
)
2.5
P qator
=
-
1
2.5 10(
)
0.25
P qator
=
-
6
4.5 1.38(
)
3.25
P qator
»
-
Pivot: 2.5 (1st row,
2
P
column
)
Step 3. (Update according to pivot)
Iteration Table 3
Basis
1
P
2
P
3
P
4
P
5
P
6
P
Right
side
2
P
0
1
0
0.4
-0.2
0
1.2
1
P
1
0
0.5
-0.1
0.3
0
2.2
6
P
0
0
0.5
-1.3
-0.1
1
0.6
Z
0
0
-0.5
1.1
0.7
0
15.8
Pivot selection
The only negative value: - 0.5(
3
P
(includes)
The following:
1
6
2.2 4.4(
)
0.5
0.6 1.2(
)
0.5
P qator
P qator
=
-
=
-
Pivot: 0.5 (3rd row,
3
P
column)
Step 4. (Final)
Iteration Table 4
Basis
1
P
2
P
3
P
4
P
5
P
6
P
Right
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ISSN: 2692-5206, Impact Factor: 12,23
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page 1189
side
2
P
0
1
0
0.4
-0.2
0
1.2
1
P
1
0
0
0.2
0.4
0
1.4
3
P
0
0
1
-2.6
-0.2
2
1.2
Z
0
0
0
-0.2
0.6
1
17.4
Step 5: Check
Z
row, which means that the optimal solution has been found:
1
P
= 7/5=1.4
2
P
= 6/5 = 1.2
3
P
= 6/5 = 1.2
Profit:
Z=5* (7/5)+4*(6/5)+3* (6/5)=7+ 4.8 +3.6= 17.4
(17.4 thousand soums)
Result:
The maximum profit is
17.4 thousand
soums, where:
1
P
= 1.4 (Energy drink)
2
P
= 1.2 (Fruit juice)
3
P
= 1.2 (Mineral water)
General Summary
In this study, the Simplex method was used to obtain maximum profit through the efficient
allocation of resources in a confectionery factory. As a result of the analysis, the main
constraints and the objective function affecting the production process were identified, and a
mathematical model was formed. Using the Simplex algorithm, this model was optimized
and the most optimal resource utilization plan was developed.
The results showed that by optimally determining the volume of products produced within
the specified constraints, it is possible to increase the economic efficiency of the factory. This
allows to reduce production costs, fully utilize available raw materials, and maximize profits.
In conclusion, the Simplex method is an effective tool for creating a production plan for a
confectionery factory, and its implementation in practice allows for optimization of the
production process and increased competitiveness.
Literature:
1. IG Bashmakov - "Optimal Production Systems" (2005). Theoretical and practical
approaches to optimizing modern production processes are presented.
2. GN Nemchinov - "Linear Economic Models" (1972). Dedicated to the application of
linear programming and analysis methods in economic systems.
3. Sharipov AA, “Optimization Methods” (Tashkent, 2019)
4. L. Kantorovich - "Mathematical Programming and Economic Analysis" (1959).
Describes methods for optimal planning and resource allocation in production processes.
5. G. Dantzig - "Linear Programming and Its Applications" (1963). The simplex method and
its application to real production conditions are discussed.
