CC BY
0
УДК 53
Allanazarov O.,
Student.
Soltyyeva M.,
Teacher
Oguzhan Egineering and Technology University of Turkmenistan.
Ashgabat, Turkmenistan.
TRANSPORT PROBLEM AND THE SIMPLEX METHOD Abstract
The transport problem is an important topic in operations research, focusing on optimizing the distribution of goods from several sources to multiple destinations while minimizing cost. The simplex method, a mathematical approach, provides an efficient way to solve such problems. This article introduces the transport problem, explains the basics of the simplex method, and demonstrates its application in solving transportation issues.
Keywords:
transport problem, simplex method, optimization, operations research, cost minimization.
The transport problem is a classical optimization problem that arises in various industries such as logistics, supply chain management, and production. It deals with determining the most cost-effective way to transport goods while considering constraints such as supply and demand. The simplex method is a widely used tool in linear programming, offering a systematic way to find optimal solutions to such problems.
In this article, we will discuss the transport problem and how the simplex method is applied to solve it. We aim to provide a clear explanation of the concepts and steps involved.
The Transport Problem
The transport problem involves the allocation of goods from several suppliers to multiple consumers. The objective is to minimize transportation costs while satisfying supply and demand constraints.
For example: There are _m_ supply points, each with a specific supply capacity. There are demand points, each requiring a certain quantity of goods.
Transportation costs are given for each route between supply and demand points.
The problem can be represented in a tabular form, where rows correspond to supply points and columns represent demand points. The values in the table indicate the transportation costs.
The Simplex Method: The simplex method is an iterative process used to solve linear programming problems. When applied to the transport problem, it involves the following steps:
1. Formulate the Problem: Represent the transport problem as a linear programming model with an objective function and constraints.
- Objective function: Minimize total transportation cost.
- Constraints: Ensure supply equals demand.
2. Initial Feasible Solution: Use methods like the Northwest Corner Rule or the Least Cost Method to find an initial feasible solution.
3. Optimality Check: Apply the simplex method to determine whether the current solution is optimal. This involves:
- Calculating opportunity costs for unallocated routes.
- Identifying potential improvements in the solution.
4. Iterative Improvement: Adjust allocations iteratively to reduce overall cost while maintaining feasibility.
5. Final Solution: Conclude the process when no further improvements can be made. Example: Consider a transport problem with two suppliers and three consumers. The supply capacities, demand requirements, and transportation costs are as follows:
| Consumer 1 | Consumer 2 | Consumer 3 | Supply |
| Supplier 1 | 5 | 3 | 6 | 20 | | Supplier 2 | 4 | 7 | 2 | 30 | | Demand | 10 | 25 | 15 | |
By applying the simplex method, we can find the optimal allocation that minimizes cost. The transport problem is a key area in optimization and has wide applications in real-world scenarios. The simplex method provides a robust approach to solving such problems, ensuring cost efficiency and effective resource allocation. Understanding these methods is essential for tackling complex logistics and supply chain challenges. References
1. Dantzig, G. B. (1963). _Linear Programming and Extensions_. Princeton University Press.
2. Hillier, F. S., & Lieberman, G. J. (2001). _Introduction to Operations Research_. McGraw-Hill.
3. Taha, H. A. (2017). _Operations Research: An Introduction_. Pearson.
© Allanazarov O., Soltyyeva M., 2024
УДК 53
Bayramov A.,
student.
Oguzhan Egineering and Technology University of Turkmenistan.
Ashgabat, Turkmenistan.
USING MATHEMATICS ON A MAP TO MEASURE THE AREA OF LAND
Abstract
Mathematics has been an invaluable tool in cartography, enabling the accurate measurement of land areas on maps. By employing scaling techniques, geometric methods, and digital tools, land area calculations have become more accessible and reliable. This article discusses how mathematics is used to measure land areas, highlighting practical applications in urban planning, agriculture, and environmental management. In recent decades, the advent of digital tools has revolutionized the field of cartography. Technologies like Geographic Information Systems (GIS) and computer-aided mapping have incorporated advanced mathematical algorithms to analyze spatial data with unprecedented precision. These tools utilize techniques such as triangulation and interpolation to measure areas, even in rugged or inaccessible terrains. Modern mapping software also integrates satellite imagery and remote sensing data, further enhancing the accuracy and scope of land area calculations.
Keywords:
map measurement, land area, cartography, scaling, GIS, urban planning, environmental conservation.
Maps are more than just visual representations of geography; they are tools for analysis and decisionmaking. Measuring the area of land on maps is crucial in various fields such as agriculture, real estate, and
