Exact Set Packing in Multimodal Transportation with Ridesharing System for First/Last Mile

We propose a centralized transportation system that integrates public transit with ridesharing to provide multimodal transportation. At each time interval, the system receives a set of personal drivers, designated drivers, and public transit riders. It then assigns all riders to drivers, ensuring that pick-ups and drop-offs occur at designated transit stations. This effectively replaces first-mile/last-mile (FM/LM) segments with a ridesharing alternative, reducing overall commuting time. We study two optimization problems: (1) minimizing the total travel distances of drivers and (2) minimizing the number of designated drivers required to serve all riders. We show the optimization problems are NP-hard and give hypergraph-based integer linear programming exact algorithm and approximation algorithms. To enhance computational efficiency, we introduce a clustering heuristic that utilizes both spatial and temporal aspects of the input data to accelerate rider-to-driver assignments. Finally, we conduct an extensive computational study using real-world datasets and surveys from Chicago to evaluate our model and algorithms at a city-wide scale.