SOLVING LINEAR PROGRAMMING PROBLEMS USING THE SIMPLEX METHOD: THEORETICAL BASIS AND PRACTICAL APPLICATION

Аннотация

The article covers the theoretical foundations and practical applications of the Simplex method used in solving linear programming problems. The Simplex method is defined as the process of finding an optimal solution that provides the maximum or minimum value of the objective function through successive approximations. The article describes in detail the two main stages of the method - finding an initial base solution and determining the optimal solution. The structure of the simplex table, algorithms for selecting the decisive element, column and row, as well as the table replacement process are explained. A step-by- step solution of the linear programming problem given as an example using the Simplex method is presented and the practical application of the method is demonstrated. The article serves as a useful resource for students and specialists working in the field of mathematical optimization. Keywords:Simplex method, linear programming, optimal solution, basicplan, Simplex table, decisiveelement, objective function, canonical form, free terms, iteration. Literature review The literature on linear programming and the Simplex method has been extensively studied in the fields of mathematical optimization, operations research, and decision making. The following are the main literature sources on this topic and their analysis: Dantzig, GB (1963). Linear Programming and Extensions. Analysis : Written by George Dantzig, the founder of the Simplex method, this book provides a detailed explanation of the theoretical foundations of linear programming and an early form of the Simplex method. The book explains the algorithmic structure of the method, the processes of finding a basic solution, and the determination of an optimal solution with examples. Dantzig's work is still considered a fundamental source on linear programming today. The book's shortcomings can be said to be its lack of adaptation to modern computing technologies, but it is theoretically important. Chvátal, V. (1983). Linear Programming. Analysis : This book is written in a simple and understandable language for students of linear programming. Practical applications of the simplex method, including the construction of the simplex tableau and iterative processes, are extensively covered through examples. The book is accessible to students and beginners and helps them understand the algorithms. However, it does not provide in-depth information on complex problems and modern optimization methods. page1232 INTERNATIONAL JOURNAL OF ARTIFICIAL INTELLIGENCE ISSN:2692-5206,ImpactFactor:12,23 AmericanAcademicpublishers,volume05,issue04,2025 Journal: https://www.academicpublishers.org/journals/index.php/ijai Luenberger, DG, & Ye, Y. (2008). Linear and Nonlinear Programming. Analysis : This book aims to study linear and nonlinear programming together. The theoretical foundations of the simplex method and its practical application in a modern context are analyzed. The book explains the canonical form of the simplex method, the selection of the decisive element and the iterative processes with concrete examples. The advantage of the book is its adaptation to modern computational techniques and its relevance to nonlinear programming. As a disadvantage, some parts are intended for readers with a high level of mathematical knowledge. Taha, HA (2017). Operations Research: An Introduction. Analysis : Hamdy Taho's book on operations research aims to teach the Simplex method from a practical perspective. The book explains the structure of the Simplex table, the algorithms for selecting the optimal row and column, and real-life problems through examples. The book is useful for students and practitioners, and pays special attention to the application of the Simplex method using software (e.g. LINDO, Excel Solver). As a drawback, it can be said that the theoretical aspects are sometimes presented in a simplified form. Vanderbei, RJ (2020). Linear Programming: Foundations and Extensions.Analysis : This book includes modern forms of the Simplex method, including a comparison with interior point methods. The book clearly and systematically explains the two stages of the Simplex method (finding an initial base solution and finding an optimal solution). In addition, the canonicalization of the problem and the structure of the Simplex table are explained through examples. The advantage of the book is that it covers modern algorithms used to solve complex problems. As adrawback, someparts of thebookmay becomplicated forbeginners. Research methodology This study aims to apply linear programming methods to formulate an optimal production plan for a confectionery factory. The research methodology includes empirical and theoretical analysis. During the study, existing scientific sources on the formulation of an optimal production plan are studied through literature analysis. Various mathematician modeling methods , including simplex​ method , graphic method and dual method usin