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We study a variant of the multi-robot goal assignment problem where a unique goal to each robot needs to be assigned while minimizing the largest cost of movement among the robots, called makespan. A significant step in solving this problem is to find the cost associated with the robot-goal pairs, which requires solving a complex path planning problem. We present OM, a scalable optimal algorithm that solves the multi-robot goal assignment problem by computing the paths for a significantly less number of robot-goal pairs compared to the state-of-the-art algorithms, leading to a computationally superior mechanism to solve the problem. We extensively evaluate our algorithm for hundreds of robots on randomly generated and standard workspaces. Our experimental results demonstrate that the proposed algorithm achieves a noticeable speedup over two state-of-the-art baseline algorithms.