# 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 x1, x2,.., xn which satisfy a system of linear equations (side conditions)
$$\begin{array}{*{20}{c}} {{a_{11}}{x_1} + {a_{12}}{x_2} + \ldots . + {a_{1n}}{x_n} = {b_1}} \\ {{a_{21}}{x_1} + {a_{22}}{x_2} + \ldots . + {a_{2n}}{x_n} = {b_2}} \\ {\begin{array}{*{20}{c}} { \ldots .} \\ {{a_{m1}}{x_1} + {a_{m2}}{x_2} + \ldots . + {a_{mn}}{x_n} = {b_m}} \end{array}} \end{array}$$
(1a)
and a set of sign restrictions (non-negativity requirements)
$${x_1}0,{x_2}0, \ldots .,{x_n}0$$
(1b)
and for which the linear function
$$f = {c_1}{x_1} + {c_2}{x_2} + \ldots . + {c_n}{x_n}$$
(1c)
has a maximum.

## Keywords

Feasible Solution Basic Solution Linear Programming Problem Preference Function Fundamental Theorem
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