Introducción a la programación lineal y entera: métodos del Sı́mplex y de B&B

Abstract

[en] Since Dantzig published the simplex method in 1949 for the solution of linear programming problems (area of optimization) a great interest was sparked that allowed a vast theoretical growth, opening the door to the development of new methods and algorithms that are widely used today. The main goal of this work is to address, at an introductory level, the simplex method and the Branch and Bound method (or $\mathrm{B\&B}$ ). If $x=\left(x_{B}, x_{N}\right)$ is a basic feasible solution of a linear programming problem $\min \{f(x) \mid A x=b, x \geq 0\}$ with no degenerate solutions, throughout the iterative algorithm of the simplex it is determined through a finite quantity of steps whether the problem is already within an optimal solution, whether a finite optimal solution does not exist, or whether it determines the optimal solution. The Branch and Bound method is used for solving linear programming problems in which the solution (or part of it) must be an integer. The algorithm follows a binary tree structure and under ideal conditions finishes in a finite quantity of steps.