Rebalancing the Rebalancers: Optimally Routing Vehicles and Drivers in Mobility-on-Demand Systems

Abstract: In this paper we study rebalancing strategies for a mobility-on-demand urban transportation system blending customer-driven vehicles with a taxi service. In our system, a customer arrives at one of many designated stations and is transported to any other designated station, either by driving themselves, or by being driven by an employed driver. The system allows for one-way trips, so that customers do not have to return to their origin. When some origins and destinations are more popular than others, vehicles will become unbalanced, accumulating at some stations and becoming depleted at others. This problem is addressed by employing rebalancing drivers to drive vehicles from the popular destinations to the unpopular destinations. However, with this approach the rebalancing drivers themselves become unbalanced, and we need to "rebalance the rebalancers" by letting them travel back to the popular destinations with a customer. Accordingly, in this paper we study how to optimally route the rebalancing vehicles and drivers so that stability (in terms of boundedness of the number of waiting customers) is ensured while minimizing the number of rebalancing vehicles traveling in the network and the number of rebalancing drivers needed; surprisingly, these two objectives are aligned, and one can find the optimal rebalancing strategy by solving two decoupled linear programs. Leveraging our analysis, we determine the minimum number of drivers and minimum number of vehicles needed to ensure stability in the system. Interestingly, our simulations suggest that, in Euclidean network topologies, one would need between 1/3 and 1/4 as many drivers as vehicles.

• The paper introduces a bi-level rebalancing framework that couples vehicle repositioning with driver recirculation via customer-carrying taxi trips.
• It models MOD systems using continuous-time fluid dynamics and decoupled minimum cost flow linear programs to manage imbalances.
• Empirical results indicate a stable driver-to-vehicle ratio of 25–33%, guiding efficient fleet and staffing strategies under optimal rebalancing.

Optimally Routing Vehicles and Drivers in Mobility-on-Demand Systems

Introduction and Context

This paper addresses the fundamental operational problem in one-way Mobility-on-Demand (MOD) systems, such as car-sharing services, where asymmetric demand causes persistent imbalances of vehicles and drivers across service stations. The innovation of the proposed framework is its explicit coupling of two forms of rebalancing: (i) repositioning idle vehicles using a pool of employed drivers, and (ii) recirculating those drivers back to demand-heavy locations by assigning them to customer-carrying trips (i.e., as taxi drivers). This bi-level rebalancing problem—termed "rebalancing the rebalancers"—is modelled and solved in a way that ensures network stability and optimizes for both the minimal number of wasted (empty) trips and required drivers.

Fluid Model Formulation

The authors treat the MOD system as a continuous-time fluid model, assuming a network of $n$ stations, with customers, vehicles, and drivers represented as real-valued densities. Each customer arriving at a station can travel to any destination station, either by self-driving or by taxi mode (with an employed driver). The critical imbalance arises because the popularity of stations is heterogeneous: some accumulate excess vehicles while others face shortages.

Vehicles are rebalanced by assigning drivers to move them without customers to stations in need. The drivers themselves must also be rebalanced, achieved by assigning them to move back to their original stations via customer trips, effectively operating as taxi drivers for one-way rides. Notably, a fraction $f_{ij}$ of the OD trips are assumed amenable to taxi mode, subject to customer preferences.

Analytical Results: Well-Posedness, Equilibria, and Stability

The paper rigorously demonstrates that the continuous-time system, described by coupled nonlinear, time-delay differential equations, is well-posed in the Filippov sense. Conservation properties for the total numbers of vehicles and drivers are established, regardless of system evolution. The authors derive the necessary and sufficient conditions for the existence of system equilibria under any assignment of vehicle and driver rebalancing rates $(\alpha, \beta)$.

Key results include:

• Equilibrium conditions: Each station must satisfy a specific imbalance (or balancing flow) constraint; the total vehicles and drivers must exceed certain thresholds, corresponding to their system-wide circulation in equilibrium.
• Feasibility of assignments: There always exists a vehicle rebalancing policy to achieve station balance. Existence of a driver rebalancing policy depends on user tolerance for taxi rides ($f_{ij}$ values); the existence condition is equivalent to a classic minimum cost flow feasibility condition.
• Local stability: Under the derived optimal policies, the equilibrium configuration is locally asymptotically stable, ensuring bounded customer queues and sustainable operation in practical settings.

Optimal Rebalancing: Decoupled Linear Program Formulation

A central theoretical insight is that minimizing the number of in-transit rebalancing vehicles and the requisite number of drivers are not conflicting objectives; rather, they can be optimized independently. The optimal rebalancing strategies for vehicles and drivers are shown to be solutions to two decoupled minimum cost flow linear programs:

• Vehicle rebalancing ($\alpha$): Minimizes aggregate time spent by empty (rebalancing) vehicles, subject to flow conservation balancing net vehicle flows due to demand.
• Driver rebalancing ($\beta$): Minimizes aggregate driver circulation, subject to feasible customer willingness for taxi rides and ensuring sufficient flows to return drivers.

This structure enables efficient computation and optimal deployment policies for large MOD systems.

Empirical Insights and Numerical Results

Simulation results on Euclidean random graphs (networks of urban-like stations) yield several operationally significant findings:

• Ratio of drivers to vehicles: For system stability, the number of required full-time drivers is typically 25–33% of the fleet size. This fraction decreases further if drivers are allowed to pool into shared customer rebalancing trips (multi-driver taxi rides).
• Vehicle utilization: The fraction of drivers performing rebalancing-only trips (i.e., moving empty cars) remains low—improving overall system efficiency.
• Scalability: As the number of stations grows, these efficiency ratios stabilize, supporting structural insights for system designers.

Implications, Limitations, and Future Directions

The theoretical structure and numerical evidence jointly enable quantitative trade-off analysis for MOD system operators. These insights are directly applicable to current urban car-sharing operations using human drivers for rebalancing, providing benchmarks for staff sizing and fleet management. Furthermore, the results quantify the operational gap between today's driver-based systems and future autonomous (self-rebalancing) vehicle systems.

The analysis assumes a deterministic, fluid approximation. The paper acknowledges the need for extending the results to stochastic queueing models to capture arrival variability, for handling time-varying or seasonal demand, and to account for uncertainties in travel time and road capacities. Dynamic pricing and real-time incentive mechanisms for customer-based rebalancing also represent promising extensions.

Conclusion

This work formulates and solves the bi-level vehicle and driver rebalancing problem for MOD systems via coupled but decouplable linear program optimizations. It provides explicit stability conditions and numerical prescriptions for driver and fleet sizing as functions of demand patterns and network geometry. These results yield structural guidelines for practical MOD system design and highlight future research directions, particularly toward stochastic, time-varying, and incentive-compatible system operation.