The Dial-a-Ride Problem with Synchronized Visits

The limited capacity of drones and future one- or two-seat modular vehicles requires multiple units to serve a single large customer (i.e., a customer whose demand exceeds a single vehicle's capacity) simultaneously, whereas small customers (i.e., those whose demand can be served by a single vehicle) can be consolidated in one trip. This motivates the Dial-a-Ride Problem with Synchronized Visits, where a fleet of drones must be routed and scheduled to transport orders at minimum cost. We propose four formulations: arc-based, event-based, time-space event-based (TSEF), and time-space fragment-based (TSFrag). An event is defined as a tuple of a location and a set of onboard customers, while a fragment represents a partial path. For TSEF and TSFrag, we also employ the dynamic discretization discovery (DDD) algorithm, which iteratively refines an initial low-resolution time-space network to obtain a continuous-time optimal solution. Computational results show that the event-based formulation performs best under low request intensity (few customers per unit time), whereas TSFrag with DDD excels with high request intensity; both substantially outperform the arc-based formulation. When implemented with DDD, TSFrag also requires less time and fewer iterations than TSEF. We also apply our methods to the classical dial-a-ride problem, where we find that that TSFrag with DDD can replace callbacks in case of high request intensity, and that using DDD is more beneficial to this problem than to the pickup-and-delivery problem with time windows.