0
$\begingroup$

What happens when a variable is negative?

An example would be:

$$\begin{array}{ll} \underset{x_1, x_2 \in {\Bbb R}}{\text{maximize}} & 3 x_1 + 4 x_2 \\ \text{subject to} & 2 x_1 + 3 x_2 \leq 10 \\ & 2 x_1 - 4 x_2 \leq 20 \\ & x_2 \leq 10 \\ & x_1 \geq 0 \end{array} $$

To set up a Linear Programming problem in Standard Form, I learned that it must be of maximization type. The constraints must be $\leq$ (which is good as 1) and 2) agree with that). However, $x_1$ is $\geq 0$, but $x_2$ is not. What would one do in this case? I tried introducing slack variables, namely $x_2'$, but I don't know where to go from here, any hints/help would be appreciated. I am just confused on what to do when one of the variables does not satisfy the positivity constraint, here $x_2 \leq 10$.

I think I can rewrite $x_2 \leq 10$ as $x_2 - 10 \leq 0$. Then introduce $x_2' = x_2 - 10$. Then, can I replace $x_2$ with $x_2'+10$?

$\endgroup$

2 Answers 2

Reset to default
1
$\begingroup$

Replace $x_2$ with $x_2^+ - x_2^-$, with $x_2^+ \geq 0$ and $x_2^- \geq 0$.

$\endgroup$
0
$\begingroup$

Note that $10-x_2 \ge 0$, so let $y_2=10-x_2$ and replace each $x_2$ with $10-y_2$.

$\endgroup$

You must log in to answer this question.

Start asking to get answers

Find the answer to your question by asking.

Ask question

Explore related questions

See similar questions with these tags.