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# Elements of the Mathematical Theory of Linear Programming

Chapter
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## Abstract

The general problem of linear programming can be formulated as follows: Find a set of numbers x1, x2,.., x n which satisfy a system of linear equations (side conditions)
$${a_{11}}{x_1} + {a_{12}}{x_2} + .... + {a_{1n}}{x_n} = {b_1}\;{a_{21}}{x_1} + {a_{22}}{x_2} + .... + {a_{2n}}{x_n} = {b_2}\;....\;{a_{m1}}{x_1} + {a_{m2}}{x_2} + ....{a_{mn}}{x_n} = {b_m}$$
(1a)
and a set of sign restrictions (non-negativity requirements)
$${x_1} \geqslant 0,{x_2} \geqslant 0,...,{x_n} \geqslant 0$$
(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 Linear Programming Problem Preference Function Fundamental Theorem
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.

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## Copyright information

© Springer-Verlag Wien 1965

## Authors and Affiliations

1. 1.University of CopenhagenCopenhagenDenmark