ECONOMIC PROBLEMS INTO LINEAR PROGRAMMING PROBLEMS AND SOLVING THE SIMPLEX METHOD

Аннотация

This article deals with bringing economic problems to the problem of linear programming and solving them graphically. Also, at first, linear programming will be touched upon, its solution methods and mathematical interpretation of the given problem: a special emphasis will be placed on solving it through a linear function. As an example, the issue of optimal production planning for the “Olmos” furniture factory is given. Key words: Multi-argument linear function, argument, linear constraints, extremum, graphical method, mathematical modeling, objective function. If the number of variables in the mathematical model of a linear programming problem is more than two (with some exceptions), the problem cannot be solved graphically. The simplex method is used to solve such problems. The simplex method is a method of successively moving from one basic solution (one end of the solution polygon) to another until the objective function of a linear programming problem takes the optimal (maximum or minimum) value. This Выпуск журнала №-27 Часть–5_Июнь –2025 111 MODERN EDUCATION AND DEVELOPMENT method is a universal method that allows you to solve any linear programming problem, unlike the graphical method, which is designed to solve problems with only two variables. is considered . The simplex method was proposed in 1947 by the American mathematician R. Danzig, and since then it has been widely used in industrial production to solve linear programming problems involving thousands of variables and constraints. Before describing the simplex method, let us recall some concepts of a system of linear equations. To us Выпуск журнала №-27 Часть–5_Июнь –2025 112 n variable m Given a system of equations: a a 1 2 m x 1 1 x 1 1 x 1 1 a x 1 2 a x 2 2 . . . . . . . a m 2 2 2 . . . . x 2 . . .  a x 1 n . . .  a 2 n . . . . . . . . . . .  . . .  a m n x n n x n b 1 b 2 b m (1) In linear programming problems A  ( a ij ) ( i  1 , 2 , . . . , m ; j  1 , 2 , . . . , n ) matrix colorr m is,mn The situation is interesting. If (1) the systemm If the determinant of the matrix formed from the coefficients before the variables is different from zero, then the basis for such variables is are called variables . remaining n  m variables are either independent or non-independent. are called variables . If (1) the system ( x 1 , x 2 , . . . , x n ) solutions x j  0 (j 1, 2,..., n)If the condition is satisfied, such solutions are called feasible solutions, otherwise they are called impossible solutions . A solution to a system in which the non-basic variables are zero is called a basic solution . Analysis and results For convenience in calculations, it is advisable to present the simplex method in tabular form. Simplex table. The following algorithm is used to solve linear programming problems using the simplex method. MODERN EDUCATION AND DEVELOPMENT Step 1. Construct an initial simplex tableau; Step 2. Check the solution for optimality. End the process when an optimal solution is found; Step 3. Find the state that leads to optimality; Step 4. Switch to a new solution and return to step 2. The general form of a simplex table is given in Table 1 ( m– number of conditions, Выпуск журнала №-27 Часть–5_Июнь –2025 113 n – number of variables) Objective function f n n s o r o o g e s c t i u t i s i s n n t n l a h a i e f u s o e b B a s i s c C o e f f i c e c t i v e B a s i s d i n t h Coefficients of the terms of the problem b j d e s o u e u l e c l h n a t i v F j C j  F j line definition C F A string defining the optimality criterion j j Table 1. Simplex table view • The first row of the table records all (main and additional) variables; • TableB The first column, separated by the letter , lists the basic variables. • The second row of the table, starting from cell 3, lists the coefficients of the objective function. •C The coefficients of the variables included in the basis are placed in b the column (except for the last two rows). • The coefficients of the conditions are given in the rows dedicated to the basic variables. •P The column contains the values of the basic variables. 0 MODERN EDUCATION AND DEVELOPMENT • Last Выпуск журнала №-27 Часть–5_Июнь –2025 114 C j  F j The line is aimed at determining the optimality criterion. • F j using the latest information C j  F j is a row, the last cell of which contains the current value of the objective function. This F x x  1 2 x  1  0 1 2 x 1 2 , x x 3 2 2 x 2 x 2 1 1 0 4 0 m a x We determine the solution of the problem using the simplex method. To bring the problem into canonical form, we use the following addition s , 1 s 2 We introduce variables: x 2x  s 10 1 2 1 2x  x  s 14 1 2 2 x 0, x 0, s 0, s 0 (2) 1 2 1 2 F  2x 3x 0s 0s max 1 2 1 2 There are a total of 4 equations in the system of equations, i.e. 2 basic ones x 1 , x 2 and 2 additional ones s 1 , s 2 There are variables. The vector of coefficients of the objective function is C, and the matrix of coeffi