# Elements of the Mathematical Theory of Linear Programming

• Sven Danø

## Abstract

The general problem of linear programming can be formulated as follows: Find a set of numbers x 1, x 2,.., x n which satisfy a system of linear equations (side conditions)
$$\begin{array}{*{20}{c}} {{a_{11}}{x_1} + {a_{12}}{x_2} + .... + {a_{1n}}{x_n} = {b_1}} \\ {{a_{21}}{x_1} + {a_{22}}{x_2} + .... + {a_{2n}}{x_n} = {b_2}} \\ {....} \\ {{a_{m1}}{x_1} + {a_{m2}}{x_2} + .... + {a_{mn}}{x_n} = {b_m}} \end{array}$$
(1a)
and a set of sign restrictions (non-negativity requirements)
$${x_1} \geqslant 0,{\kern 1pt} {\kern 1pt} {x_2} \geqslant 0,{\kern 1pt} ....,{\kern 1pt} {x_n} \geqslant 0$$
(1b)
and for which the linear function
$$f = {c_1}{x_1} + {c_2}{x_2} + .... + {c_n}{x_n}$$
(1c)
has a maximum.

## Keywords

Feasible Solution Basic Solution Preference Function Fundamental Theorem Simplex Method
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.