Optimizing resources with Linear Programming

Graphical Method

Graphical Interpretation of the Simplex Method

The resolution of linear problems which contains two or three decision variables can be illustrated graphically, pointing like a visual aid to understand a lot of concepts and terms that are employed and formalized with more sophisticated methods, for example The Simplex Method, needed for resolving problems with several variables.

In reality, although rarely problems have only two variables of decision, but this solution and interpretation methodology is very useful, because typical solutions will be shown, like the existence of only one optimal solution, alternative optimal solutions, no solution, and unbounded solution. We describe the phases of the Graphic Method procedure. These are:

  1. Draw a coordinate system Cartesian, in which each decision variable be represented by an axis, with the measure scale, that fits properly to his associated variable.
  2. Draw on the coordinate system the restrictions of the problem (including no negativeness). For it, we observe that if a restriction is an inequality, circumscribe a region that will be the semi-diagram limited by the straight line that must be considered when the restriction regarded as an equality. If restriction is an equation, the definitional region sketches like a straight line. The intersection of all these regions determines the feasible region or space of solutions (that is a convex set). If this region is not empty, go for the following phase. Alternatively, solution does not exist that can satisfy ( simultaneously ) all the restrictions and the problem does not have solution, calling it infeasible.
  3. Determine the extreme points (points that are not located in in-line segments that other ones join the convex set's two points) of the feasible region (that, as we will try at the following section, are candidates to optimal solution). Evaluate the objective function in these points and that one or those that they maximize (or minimize) the objective, they correspond to optimal solutions of the problem.

In order to understand easily the application of this method, will show an example of Graphical Method.

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Copyright ©2006-2015. All rights reserved.

Developed by:
Daniel Izquierdo Granja
Juan José Ruiz Ruiz

English translation by:
Luciano Miguel Tobaria

French translation by:
Ester Rute Ruiz