Rebalancing Modular Transit Systems: A Hierarchical Graph-Based Optimization Framework for Fleet Sizing and Routing

This study addresses the rebalancing of empty modular transit pods between scheduled service trips in fixed-route bus systems. A two-stage hierarchical optimization framework is proposed. The first stage determines the minimum fleet size and initial vehicle assignments by solving a maximum matching problem on a bipartite graph, using a GPU-accelerated push-relabel algorithm. The second stage formulates detailed routing as a series of minimum-cost flow problems on time-space networks. To manage memory usage in large instances, a capped-interval heuristic limits the size of each network by dividing long scheduling intervals into subintervals. Computational experiments on the Manhattan bus network show that the proposed method achieves performance comparable to the full-scale time-space network formulation in terms of objective value, while enabling the solution of instances that are otherwise intractable due to memory limitations.