Relaxation methods for fixed route demand responsive transit

This article examines a partially responsive ride-sharing transit service with predetermined cyclic routes and deterministic travel times. The main operational challenge is setting the time schedule, which varies from day to day in response to upfront passenger requests. The need of passengers to adjust their departure times from their desired value causes inconvenience, which we wish to minimize. A previous study of the fixed route dial a ride problem (FRDARP) considered strict fleet constraints. We  solve two relaxed formulations, related to the one-dimensional p-median location problem, by efficient dynamic programming embedding the SMAWK algorithm. Numerical results show the potential benefit of the FRDARP compared to fixed schedule (traditional) service and illustrate the impact of demand level and fleet constraints. In addition, based on these results, we characterize the wide range of scenarios where an easier to solve relaxed formulation can be nearly as useful as the constrained formulation.