Robust and Flexible Microtransit Design: Chance-Constrained Dial-a-Ride Problem with Soft Time Windows

Microtransit offers a promising blend of rideshare flexibility and public transit efficiency. In practice, it faces unanticipated but spatially aligned requests, passengers seeking to join ongoing schedules, leading to underutilized capacity and degraded service if not properly managed. At the same time, it must accommodate diverse passenger needs, from routine errands to time-sensitive trips such as medical appointments. To meet these expectations, incorporating time flexibility is essential. However, existing models seldom consider both spontaneous and heterogeneous demand, limiting their real-world applicability. We propose a robust and flexible microtransit framework that integrates time flexibility and demand uncertainty via a Chance-Constrained Dial-A-Ride Problem with Soft Time Windows (CCDARP-STW). Demand uncertainty is captured through nonlinear chance constraints with controllable violation probabilities, while time flexibility is modeled with soft time windows and penalized cost. We develop a bounded-support relaxation using limited statistical information to linearize the chance constraints and solve the model using a tailored Branch-and-Cut-and-Price (BCP) algorithm with a probabilistic dominance rule. This rule improves computational efficiency, reducing explored labels by 17.40% and CPU time by 22.27% in robust cases. A case study based on real-world Chicago data shows our framework yields 11.55 minutes and 11.13 miles of savings versus conventional microtransit, and achieves the highest service reliability (96.46%) among robust models.