LP: Simplex Method

Maximization

How lucky my sister was for waking me up at the middle at the night to teach her the simplex method, in the matrix way for her exam in a few hours. How luckier I am, it is one of the class topics we were about to discuss allowing me to recall on some things. 

Here is a simple maximization problem from her book. All we needed to do after the first table is to make the $2$ x $2$ matrix on the upper left an identity matrix and the last $1$ x $2$ matrix on the lower left, zeroes. Remember that these are equations so the "elimination via matrix" is applied.

Minimization

For minimization problems, a little tweaking is needed. Example, if you are given
  $$\text{min } z = 0.15x_1+0.12x_2$$
with constraints
$$\begin{split} 60x_1 +60x_2 & \geq 300 \\ 12x_1 + 6x_2 & \geq 36 \\ 10x_1 + 30x_2 & \geq 90 \end{split}$$
then we have the matrix
$$\begin{bmatrix} 60 & 60 & \vdots & 300 \\ 12 & 6 & \vdots & 36 \\ 10 & 30 & \vdots & 90 \\ \cdots &\cdots & \vdots & \cdots \\ 0.15 & 0.12 & \vdots & 0\\ \end{bmatrix}$$
The transpose of this matrix will give you a maximization problem and the rest of the process is the same as earlier.  

Application - Simplex method Application varies from business to business and are commonly used.

Links for more info is provided below: 
http://pages.intnet.mu/cueboy/education/notes/algebra/simplex.htm
http://college.cengage.com/mathematics/larson/elementary_linear/5e/students/ch08-10/chap_9_3.pdf
http://college.cengage.com/mathematics/larson/elementary_linear/5e/students/ch08-10/chap_9_4.pdf