Linearizing min function Problem How can I linearize $\min(x_1,x_2,x_3)$ in a maximization linear programming problem? Please help me. I've tried many things but I didn't solve.. My LP equations are as follows:

Objective function is:  maximize $z=\min(x_1,x_2,x_3)$

Constraints:

$0.6 x_1 + 0.8 x_2 \leq 4500 \times 20$

$0.2 x_2 \leq 4500 \times 3$

$0.3 x_2 \leq 4500 \times 6$

$0.4 x_1 + 0.6 x_2 \leq 4500 \times 10$

$0.3 x_1 \leq 4500 \times 5$
How you implement min($x_1, x_2, x_3$) in an LP solver depends on what you are trying to do with it.  Since you are maximizing it you can do the following 

$$
{\rm maximize } \ z
$$
subject to
$$
z \le x_1 \\
z \le x_2 \\
z \le x_3
$$
$z$ will take the value of min($x_1, x_2, x_3$) in any optimal solution.
assume $y=\min\{x_1,x_2,x_3\}$

Now we need to maximize $y$

since value of $y$ can either be $x1$, $x2$ or $x3$, we can safely add the following constraints:

$$x_1\geq y \qquad x_2 \geq y \qquad x_3\geq y$$

or

$$x_1-y\geq 0 \qquad x_2-y \geq 0 \qquad x_3-y\geq 0$$

now solve the problem.