Linear Programming

The simplest type of mathematical programme is a linear programme.

For your mathematical programme to be a linear programme you need the following conditions to be true:

  1. The decision variables must be real variables;
  2. The objective constraint must be a linear expression;
  3. The constraints must be linear expressions.

Linear expressions are any expression of the form

 \begin{equation*} a_1 x_1 + a_2 x_2 + ... + a_n x_n \left\{ \begin{matrix} \leq \\ = \\ \geq \end{matrix} \right\} b \end{equation*}

where $a_1, a_2, \ldots, a_n$ and $b$ are known quantities and $x_1, x_2, \ldots, x_n$ are variables.

The general form of a linear programme is

 \[ \begin{array}{rr@{}l} \min & c^\top & x \\ \text{subject to} & A & x \left\{ \begin{matrix} \leq \\ = \\ \geq \end{matrix} \right\} b \end{array} \]

The process of solving a linear programme is called linear programming. Linear programming is done via the Revised Simplex Method (also known as the Primal Simplex Method), the Dual Simplex Method or an interior point method. Some software packages, such as CPLEX, allow you to specify which method you use, but we won't go into further detail here.

Linear Programming Topics

To see linear programming in action, check out some of the linear programming case studies:

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Number of topics: 9

-- MichaelOSullivan - 16 Feb 2008

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Topic revision: r6 - 2008-02-21 - MichaelOSullivan
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