Liver transplantation has been a critical issue in the U.S. healthcare system for decades, and the region redesign aims to ameliorate this issue. This paper revisits two mixed integer programming (MIP) formulations of the liver region redesign problem proposed by Akshat et al. We study their first formulation considering two different modeling approaches: one compact formulation and one with exponentially many constraints. We also propose a set of variable fixing procedures and conduct a polyhedral study on their second formulation. Our computational results show that multiple unsolved instances are solved to optimality.