This paper considers the global optimization of max-plus linear systems with affine equality constraints. For both the cases that the variables are real and non-negative, the necessary and sufficient conditions for the existence and uniqueness of globally optimal solutions are given, respectively. The proposed approaches are constructive and yield two polynomial algorithms for checking the solvabilities of the global optimization problems and finding all globally optimal solutions, in which the analytic expressions of general solutions are presented. The global optimization is then applied in the load scheduling of distributed systems with different processor capacities. The optimal allocation scheme is designed to minimize the completion time of the overall task. Some illustrative examples are presented to demonstrate the results.