Elements of the Mathematical Theory of Linear Programming

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.