We have the following theorem:
For a BFS $x^{0}$, if $z_{j}-c_{j} \leq 0, \forall j \in J_{N}$ (N is the set of all non-basic variables), Then $x^{0}$ is an optimized solution for this problem.
Now using this fact, I want to show using an example that the reverse is not always true. But I can not find a suitable example.
My attempt: I need to find a problem that has the following properties:
$\exists j \in J_{N}$ s.t. $z_{j}-c_{j} > 0$. So for that $j$, $z_{j}>c_{j}$.
$c_{B}B^{-1}a_{j}>c_{j}$ $(c_{B}y_{j}>c_{j})$. So for that purpose, all I need is to find a problem having these properties, But I can't go further. Any help would be appreciated.