Ways to solve a linear optimization problem

Question

What are the methods/techniques that one can use to solve linear(or integer) optimization problems ?

Googling around I stumbled upon the Simplex Method. But is that the only method used?

https://www.youtube.com/watch?v=2ACJ9ewUC6U <--- Is this an application of the Simplex Method ?

Let's take Excel Solver for instance. What method is this program using to tackle linear optimization problems?

Which are the techniques that we use today ?

Answer 1

Simplex is still a competitive method for many types of problems. That is the method that the Excel Solver uses.

I didn't watch the whole video but it doesn't look like they're using Simplex method - it looks like they are just graphing the constraints and identifying extreme points. There's a theorem that says if an optimal solution exists, then one of the extreme points is an optimal solution, so that's what they would be using there.

There are other methods for solving LPs. For very large problems, interior point methods are the favored method. These methods travel along the interior of the feasible region rather than along the extreme points. See for example this .pdf from MIT.

Answer 2

In the video the graph is almost like the graph below:

http://www.bilder-upload.eu/thumb/5ef2c5-1488338236.png

Additionally I have drawn the objective function (blue line). To do this you have to solve the objective function, at level $C=0$, for $y$.

$C=0.8x+0.5y$

$0=0.8x+0.5y$

$-0.8x=0.5y$

$-1.6x=y$

You need two points to draw it. One point is obviously $(0/0)$ and another point is for instance $(2/-3.2)$

Now you push the line right upwards parallel until the line touches the feasible region the first time.

The second picture shows the process.

http://img5.fotos-hochladen.net/uploads/grafikversch25hbd90nevu.png

The line touches the feasible region first at $(2.4/1.2)$. If you cannot identify the exact value from the graph you calculate the intersection of the two lines. Here they are the two constraints. But in many cases the intersection can be identified directly from the graph.

The more constraints a model has the more powerful is this method.

Is this an application of the Simplex Method ?

No, it is not an application of the Simplex Method.