# PHPSimplex

## Optimizing resources with Linear Programming

### Modeling problems theory

To be able to solve a problem intervening an algorithm, first must extract all the information that contributes itself, and arranging the problem for the aforementioned algorithm.

The steps to model a problem are the following:

• Step 1: Determining decision variables and expressing them algebraically.

• X1,..., Xn

• Step 2: Determining the restrictions and expressing them as equations or inequalities in function of the decision variables:

• A11·X1 + A12·X2 + ... + A1n·Xn ≥, ≤, ó = b1
• A21·X1 + A22·X2 + ... + A 2n·Xn ≥, ≤, ó = b2
• ...
• Am1·X1 + Am2·X2 + ... + Amn·Xn ≥, ≤, ó = bm

• Step 3: Expressing all implicit conditions established by the origin of variables: negativeness, integer, only a few allowed values, ...

• X1,..., Xn ≥ 0
• X1,..., Xn are integers, or boolean,...

• Step 4: Determining Objective Function.

• Maximize or minimize Z = C1·X1 + C2·X2 + ... + Cn·Xn

As example, let see how to model some typical problems:

X

PHPSimplex
Version 0.81