I know that we can solve preemptive goal programming using the preemptive simplex method.
But what if there exists a greater than or equal constraint that is not part of the goals? I know that instead of slacks, I shall subtract surplus variables. Should I still use the simplex method or the Big-M? If the Big-M then how? Since in the preemptive simplex method we are separating the objective functions into $z_i$s such that
$$ z_i = \text{ The objective function term involving goal } i. $$
Preemptive goal programming assumes that there are constraints on the decision variables (other than those created by higher priority goals than the current goal). Those constraints are treated exactly as they would be in an ordinary linear program. If solving by hand, you would add a slack variable ($\le$ constraint) or subtract a surplus variable ($\ge$ constraint). If using a solver, the solver would do that for you.