Simplex method and its application

Any graphic solution of the problems posed in linear programming determines that the most correct (optimal) solution of any of the problems is fully associated with the extreme point of the set (or the corner point of space). This idea is based on an algebraic general simplex method of solving problems that allows solving absolutely any programming problem.

To move from the geometric method of solving problems to a solution using the simplex method of linear programming, it is necessary to describe all extreme points of space using algebraic methods. To perform this transformation, you need to bring any programming task into a standard form (also called canonical).

To do this, you need to take the following steps:

• Transform all inequalities of constraints into equalities (realized by introducing additional new variables);
• The maximization problem must be transformed into a minimization problem;
• It is necessary to obtain non-negative variables, transforming all free variables into them.

The form of a standard form problem obtained as a result of all transformations will allow us to determine the basic solution. Which, in turn, clearly defines all the corner points of space. Subsequently, the simplex method will allow us to find the most optimal solution from all the basic ones obtained.

The main thing that accomplishes this method of solving algebraic tasks in practice is a consistent and constant improvement in the implementation of the plan, the result of which is the implementation of the tasks with the maximum efficiency. The main thing that you need to do to get the desired result is to correctly implement it in mathematical and program form.

The result of all 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 of the plane and finding the optimal one.

It has long been proven that the entire search for an optimal solution (in the case, if any) is completed for a whole and a finite number of steps. The only exception that the simplex method can not handle is the "degenerate problem". In this case, there is a so-called "looping", which leads to a 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 US George Danzig. In view of 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 society. It can be used in both scientific and production spheres. Its wide application will help to make mathematically justified correct solutions to complex problems.