ILP:
$max$ $3x-2y+z$,
$st$ $x+5y-z <= 4$,
$2x-2y+4z <= 6$
$x,y,z >= 0$
I was trying the simplex method for a maximisation problem, and I got the solution $x=(19/6,1/6,0)$ and cost in tableau$=(-55/6)$.
Then I converted the objective function from max to min, and converted the sign as well. After doing the tableau, I found solution $x=(19/6,1/6,0)$ and cost in tableau$=(55/6)$.
Pluging the solution into the main objective function, will give me $55/6$.
My question if simplex method is only for min problem? Or I am missing something?
As you have noticed, $$\max c^Tx = - \min (-c^Tx)$$
It depends on the convention that you adopt. Look at the reduced cost $c_j-p'A_j$ in your algorithm. If it terminate when the reduced cost is is nonnegative, then it is written with minimization in mind.
What is inconsistent now is your main objective function and the cost of the tableu. Of which, I can't tell for sure unless you are willing to share your objective function.