(r5) LinearProgramming < OpsRes

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---+ 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

%BEGINLATEX% \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*} 
%ENDLATEX%

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

The process of solving a linear programme is called _linear programming_. Linear programming is done via the [[https://en.wikipedia.org/wiki/Simplex_algorithm#Revised_simplex_algorithm][Revised Simplex Method]] (also known as the Primal Simplex Method), the [[https://www.egwald.com/operationsresearch/lpdualsimplex.php][Dual Simplex Method]] or an [[https://en.wikipedia.org/wiki/Interior_point_method][interior point method]]. Some software packages, such as [[https://www.cplex.com][CPLEX]], allow you to specify which method you use, but we won't go into further detail here.

---++ Linear Programming Topics
   * [[BlendingModels][Blending or Mixing Models]]

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

%SEARCH{search="LinearProgramming +\"Case Study:\"" type="keyword" casesensitive="on" nosearch="on" nosummary="on" excludetopic="%TOPIC%"}%

-- Main.MichaelOSullivan - 16 Feb 2008
Topic revision: r5 - 2008-02-21 - MichaelOSullivan
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