Simplex method and its application

Any graphic solution of the problems posed inlinear 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 the algebraic general simplex method of solving problems, which allows solving absolutely any programming problem.

simplex method

To go from the geometric solution methodtasks 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 resulting form of all transformationsproblems of the standard form, 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.

simplex method of solving problems

The main thing is that this method of solvingalgebraic 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 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 simplexa method that 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 all search for the optimaldecisions (in the case, if any) are completed for the whole and the final 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.

simplex method of linear programming

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 step-by-step optimization method greatly simplifiesany activity of society. It can be used in both scientific and production spheres. Its wide application will help to make mathematically substantiated correct solutions to complex problems.