Traditional methods for solving interval linear programming problems often involve transforming the problem into two sub-problems to determine the interval objective value. However, these methods generally apply only when the best and the worst sub-problems are bounded. This work presents a solution to the problem when the best sub-problem is unbounded and the worst sub-problem is bounded. In a situation when interval parameters in the constraints could be changed, we apply a perturbation method to convert the unbounded best sub-problems into a bounded one, based on vector projection and with the minimal changes of the objective coefficient. This method also covers solutions when the new coefficients fall outside of the original range. Numerical examples are provided to demonstrate the method.