This paper focuses on developing the optimal solution or a lower bound for N-job, M-machine Permutation Flowshop Scheduling (PFS) problem in a manufacturing system with the objective of minimizing the makespan using Lagrangian Relaxation (LR) technique. Even though LR technique is considered, in general,as a good method to obtain a lower bound, research in this direction with respect to our problem under study appears scarce. We address this gap by developing two MILP based Lagrangian Relaxation models, namely, Lagrangian Relaxation Method 1 (called Proposed Lagrangian Lower Bound Program (PLLBP)) and Alternate Lagrangian Relaxation Method 1 (called ALR) to find the optimal solution or a lower bound on the makespan. Basically, we develop these LR methods to overcome the possible limitation of the general LR procedureinvolving the sub-gradient approach. Benchmark PFS problem instances are used to evaluate the performance of these methods. It is observed that the PLLBP outperforms the ALR, and it provides better lower bounds than thelower bounds (in most instances) reported in the literature. Even though the PLLBP is superior in terms of solution quality, it has a limitation in that it cannot execute problem instances beyond 500 jobs due to the associated computational effort.