Maximum related to Nonnegative Matrix

Maximum related to Nonnegative Matrix

Suppose $A$ is a fixed nonnegative $n\times n$ matrix (i.e. $A_{ij}\geq0$
for all $i,j$). Then for any arbitrary $n$ positive numbers
$x_1,\ldots,x_n$, we denote:
$$F(x_1,\ldots,x_n)=\min_i\frac{1}{x_i}\sum_{j=1}^n x_jA_{ij}$$ Frobenius
Theorem tells us that there's always an inequality
$$F(x_1,\ldots,x_n)\leq\rho(A)$$ where $\rho(A)$ is the spectral radius of
$A$.
Here my question is, what is the maximum of $F(x_1,\ldots,x_n)$ ? Is it
just $\rho(A)$ ? Or there's no maximum but a supremum? Thanks a lot for
your help.