GRAPHICAL METHOD FOR SOLVING LINEAR PROGRAMMING PROBLEMS

Аннотация

The graphical solution of linear programming problems is based on the geometric analysis of inequalities and the objective function in systems of two variables (n=2). Each inequality represents a half-plane bounded by a straight line, x 0 and the conditions j represent areas bounded by the coordinate axes. The solution domain of the system, which can have a common solution, is formed as a convex polygon, a single point, or an empty set. The graphical method includes the following steps: constructing graphs of inequalities, determining the solution domain, drawing the objective function vector, finding the extreme point by parallel displacement in the direction of the vector, and calculating the optimal solution.In theexample, thesystem of inequalities is analyzed graphically, and the maximum (3,0) and minimum (0,3) values of the Z function are determined. This method allows you to solve the problem visually and intelligibly, and practically demonstrates the main principles oflinear programming. Keywords:Linear programming, graphical method, system of inequalities, domain of possible solutions, convex polygon, objective function, extremum point, vector direction, optimalsolution,half-plane, coordinate axes,maximum value, minimumvalue. Literature review The following sources and their content are considered for the analysis of the literature on solving linear programming problems graphically. Below is a general analysis andmain directions, sincea specific list ofliterature is not given.If you provide aspecific list of sources, I can deepen the analysis based on them. Classical textbooks (for example, VA Yemelichev, MM Kovalev, LA Petrosyan, etc.) : In classical textbooks on linear programming, the graphical method is presented as the main approach for problems with two variables. In these sources, the system of inequalities, the concept of a convex polygon, and the process of determining the extreme points of the objective function are explained geometrically. As an advantage of the graphical method, its visual clarity is emphasized, but as a limitation, the impossibility of applying it to problems with many variables is noted. The practical significance of the method is shown through examples (for example, resource allocation or production optimization). Modern literature on mathematical optimization (e.g., D. Bertsimas, JN Tsitsiklis) : In modern sources, the graphical method is discussed not as the page1264 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue04,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai main theory of linear programming, but as a simplified approach used for educational purposes. In this literature, concepts such as the geometric properties of a convex polygon, the direction of the objective function vector, and the location of the optimal solution at the edge points are thoroughly covered. At the same time, cases of the graphical method with bounded and unbounded solution domains are analyzed. Examples are often focused on logistics, economic modeling, and other areas. Uzbek-language sources (e.g., textbooks of Uzbek universities) : Uzbek-language textbooks (e.g., textbooks on mathematical programming of TDTU or NUU) present the graphical method in a way that is understandable for students. They focus on more practical examples and explain step-by-step the process of constructing a system of inequalities graphically, determining the solution domain, and finding the optimal point. These sources often cite economic and technical issuesas examples,but theoretical depth may be limited. Research methodology Analysis of the effectiveness of solving linear programming problems graphically, its geometric andanalytical properties, as well as thestudy ofthe possibilities ofeducational and practical application of the method. The research is focused on assessing the advantages, limitations of the graphical method and its integration with modern technologies. Theoretical analysis : Literature analysis on the basic concepts of linear programming (system of inequalities, convex polygon, objective function). Study of the mathematical foundations of the graphical method, including the theory of convex sets and the properties of extreme points. Comparison of approaches to the graphical method from various sources (classical and modern textbooks, scientific articles). Mathematical modeling : Formulation of linear programming problems in two variables and their solution using the graphical method. Modeling of cases of various solution domains (bounded polygon, unbounded domain, singular point, empty set). Mathematical verification of the process of determining the minimumand maximumvalues oftheobjective function. Introduction. Process research and optimal control are fields of science focused on decision making and optimization of systems. Process research uses methods such as mathematical models, linear pr