Page 88
CENTRAL ASIAN JOURNAL OF ACADEMIC
RESEARCH
IF = 5.441
Volume 3, Issue 05, Part 2 May 2025
www.in-academy.uz
OPTIMIZATION OF RESOURCES IN PRODUCTION
SYSTEMS USING LINEAR PROGRAMMING
Ro‘zaliyev 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
Xaydarova Pokizaxon Avazjon qizi
Third-year student, Applied Mathematics major, Group 22.11, Fergana
State University
E-mail: pokizaxonxaydarova@gmail.com
https://doi.org/10.5281/zenodo.15424463
ARTICLE INFO
ABSTRACT
Qabul qilindi: 05-May 2025 yil
Ma’qullandi: 10- May 2025 yil
Nashr qilindi: 15-May 2025 yil
This article discusses the problem of optimal distribution
of production products using the method of linear
programming (LP). Through linear programming, the
documents and objective functions involved in the
production process are represented by corresponding
linkage indicators. The article analyzes the general
linear programming model and its practical applications,
including solving optimization problems in production
and resource allocation using methods such as the
graphical method and the Simplex method. The role of
linear programming methodology in economics is
demonstrated through practical examples, particularly
concerning production and resource allocation. The
paper explores project-based planning and the
application of economic principles in optimizing
production plans..
KEYWORDS
Linear
programming,
optimization,
resource
allocation, Simplex method,
production planning, practical,
economic optimization
Introduction
In modern production processes, the efficient allocation of production resources holds
critical importance. Linear programming (LP) is widely used to solve this issue. This article
addresses the problem of modeling production processes based on linear programming and
finding optimal solutions.
Fundamental Concepts of Linear Programming
Linear programming (LP) is a method of mathematical optimization in which the
objective function and related constraints are expressed in linear form. This methodology
aims to solve mathematical, economic, and applied problems by finding optimal solutions. In
linear programming, the objective function is typically to maximize or minimize quantities
such as profit, cost, or other economic indicators.
A linear programming model consists of the following components:
- Objective Function: A linear function representing profits, costs, or other economic
indicators in relation to decision variables.
Page 89
CENTRAL ASIAN JOURNAL OF ACADEMIC
RESEARCH
IF = 5.441
Volume 3, Issue 05, Part 2 May 2025
www.in-academy.uz
- Constraints: Constraints are related to the limited availability of resources. They are
usually expressed in the form of inequalities or equations, representing the maximum
availability of resources or conditions that must be met.
- Decision Variables: These are the elements whose values must be optimized, such as
production volumes, production methods, or management decisions.
The general form of a linear programming model is as follows:
Maximize: Z = c1x1 + c2x2 + ... + cnxn
Subject to constraints:
a11x1 + a12x2 + ... + a1nxn ≤ b1...
am1x1 + am2x2 + ... + amnxn ≤ bm
Main Methods of Solving Linear Programming Problems
There are several methods for solving linear programming problems. The most widely
used methods include:
- Graphical Method: Suitable for problems with two variables. By plotting the objective
function and the constraints on a graph, the optimal solution can be visually identified.
Although intuitive and simple, it is limited to two-variable problems.
- Simplex Method: The Simplex method is one of the most efficient techniques for solving
LP problems, especially those involving multiple variables. It iteratively moves toward the
optimal solution by adjusting the objective function at each step.
- Dual Method: The dual method works based on the primal model. It provides solutions
from a different perspective, helping to better understand resource allocation and optimized
strategies.
- Production and Distribution Models: Various models are used for resource allocation
and production optimization, such as production scheduling and market-oriented production
planning.
The Role of Linear Programming in Economic Optimization
Linear programming serves as an essential tool in economic optimization. It enables
efficient management of production and economic processes.
In economic optimization, linear programming is widely applied in the following areas:
- Resource Allocation: Linear programming tools allow for the most efficient distribution
of resources (such as labor time, raw materials, and workforce).
- Production Planning: Enterprises and organizations use linear programming methods
to optimally structure their production plans.
- Market Pricing and Demand Management: Linear programming can be applied to
control production in accordance with market demand and pricing.
Constraints in Linear Programming and Their Management
Constraints are one of the key aspects defining the production processes in linear
programming. Constraints can be of the following types:
- Resource Limitations: Each production resource (e.g., raw materials, labor time,
workforce) has its maximum capacity.
- Restrictions on Decision Variables: Decision variables must stay within specific limits
(e.g., production volume, quantity of goods).
- Market Supply and Demand: Production volume and resource allocation must be
optimized according to market supply and demand.
Page 90
CENTRAL ASIAN JOURNAL OF ACADEMIC
RESEARCH
IF = 5.441
Volume 3, Issue 05, Part 2 May 2025
www.in-academy.uz
Practical Example
Suppose a company produces five types of products: "Apple," "Banana," "Pomegranate,"
"Persimmon," and "Orange."
Available resources and profits are:
- Labor Time (hours): Apple (2), Banana (1), Pomegranate (3), Persimmon (4), Orange
(2) — Total 100 hours
- Raw Material (kg): Apple (1), Banana (2), Pomegranate (3), Persimmon (1), Orange (2)
— Total 80 kg
Profits per product:
- Apple: 40,000 UZS
- Banana: 30,000 UZS
- Pomegranate: 50,000 UZS
- Persimmon: 20,000 UZS
- Orange: 60,000 UZS
Model: Maximize Z = 40x1 + 30x2 + 50x3 + 20x4 + 60x5
Subject to:
2x1 + 1x2 + 3x3 + 4x4 + 2x5 ≤ 100
1x1 + 2x2 + 3x3 + 1x4 + 2x5 ≤ 80
Thus, the total profit is calculated as Z = 1,700,000 UZS.
Conclusion
Using linear programming methods to optimize resource allocation is a crucial tool in
efficiently managing production systems. This method ensures the most effective use of
available resources. Practical examples demonstrate that linear programming helps find
optimal solutions and enables efficient resource distribution.
References:
1.
Vasilev, V.P. (2015). Theory and Practice of Linear Programming. Moscow: MGU.
2.
- Zhuravlev, A.A. (2017). Mathematical Modeling: Linear Programming and Its Applications.
Saint Petersburg: Polytechnic Publishing.
3.
- Klimov, V.A., & Yuldashev, I.A. (2019). Methods of Linear Programming in Economic
Systems. Tashkent: Sharq Publishing.
4.
- Rakhmanov, D.I., & Komilov, S.I. (2020). Production Management and Optimal Resource
Allocation. Tashkent: University Press.
5.
- Mordukhovich, V.P. (2014). Optimal Control and Resource Allocation. Moscow: Nauka
Publishing.
