Any graphical solution to the problems posed in linear programming determines that the most correct (optimal) solution to any of the problems is completely associated with the extreme point of the set (or the corner point of space). The algebraic general simplex method for solving problems is based on this idea, which allows us to solve absolutely any programming problem.
To move from the geometric way of solving problems to a solution using the simplex method of linear programming, it is necessary to describe all the extreme points of space using algebraic methods. To perform the indicated transformation, it is necessary to bring any programming task into a standard form (also called canonical).
To do this, take the following steps:
- convert all inequality constraints into equalities (implemented by introducing additional new variables);
- the maximization problem must be transformed into a minimization problem;
- it is necessary to obtain non-negative variables by converting all free variables into them.
The form of the standard form problem obtained as a result of all the transformations allows us to determine the basic solution. Which, in turn, clearly defines all the corner points of space. Subsequently, the simplex method will allow you to find the most optimal solution of all the obtained basic ones.
The main thing that performs a similar method of solving algebraic tasks in practice is a consistent and continuous improvement in the implementation of the plan, the result of which is the implementation of the tasks with the maximum share of efficiency. The main thing that needs to be done to obtain the desired result is to correctly implement it in mathematical and programmatic form.
The result of all the developments should be a simplex method, which is a special computational procedure based on the continuous improvement of each subsequent solution. This happens by pairwise comparing all points on the plane and finding the optimal one.
It has long been proven that the entire search for an optimal solution (if one exists) is completed in an integer and a finite number of steps. The only exception that the simplex method cannot handle is the “degenerate problem”. In this case, the so-called "looping" occurs, which leads to the constant repetition of the same tasks an infinite number of times.
The simplex method was developed back in 1947. His "parent" was a mathematician from the United States George Danzig. Due to the fact that the simplex method has such a long history, now it is one of the most studied and most effective for finding optimal solutions to any problems facing a person.
The method of step-by-step optimization greatly simplifies any activity of the company. It can be used in both scientific and industrial fields. Its widespread use will help to make mathematically sound correct solutions to complex problems.