I am working on a project where the customer requested to change the current objective function $f$ to another function $g$ (both linear). It is easy to prove that $f \le g$ and as both are linear funcions my guess is that the hyperplane each define are parallel so there is no need to replace the current objective function to $g$ as the solution to the problem should be the same. Is this right?
If $g=f+c$ for some constant $c \ge 0$, then the optimal solutions will be the same, and this does not depend on linearity of either function. But if $f \le g$ just in the feasible region and maybe not elsewhere, optimizing $g$ is not necessarily equivalent to optimizing $f$.