Equation Form of a Standard Maximization Problem
#[Given a maximization problem,][Dado un problema de maximización,]#
#[Maximize][Maximizar]# $p = a_{1}x_{1} + a_{2}x_{2} + \cdots + a_{n}x_{n}$ #[subject to][sujeta a]#
$\quad b_{1}x_{1} + b_{2}x_{2} + \cdots + b_{n}x_{n} \color{blue}{{}\leq c} \quad$ #[or][o]# $\quad \color{red}{\geq c}$
$\quad b_{1}'x_{1} + b_{2}'x_{2} + \cdots + b_{n}'x_{n} \color{blue}{{}\leq c'} \quad$ #[or][o]# $\quad \color{red}{\geq c'}$
$\quad \vdots$
$\quad x_{1} \geq 0,$ $x_{2} \geq 0,$ $\ldots,$ $x_{n} \geq 0,$
its
equation form is the system of linear equations
| $b_{1}x_{1} + b_{2}x_{2} + \cdots + b_{n}x_{n}$ |
${} \pm s_1$ | | | ${}= c$ |
| $b_{1}'x_{1} + b_{2}'x_{2} + \cdots + b_{n}'x_{n}$ |
| ${} \pm s_2$ | | ${}= c'$ |
| $\vdots$ |
| $-a_1x_1 - a_2x_2 - \cdots - a_nx_n$ |
| | ${}+p$ | ${}=0$ |
#[where][donde]# $s_1,\ s_2, \ldots$ #[are
added when the corresponding inequality is $\leq$ , and
subtracted when the corresponding inequality is $\geq$.][se
suman cuando la desigualdad correspondiente es $\leq$, y
restan cuando la desigualdad correspondiente es $\geq$.]# #[When added, these additional variables are called
slack variables, and when subtracted, they are called
surplus variables.][Cuando se suman, estas variables adicionales se denominan
variables de holgura y cuando se restan, se denominan
variables de excedente.]#