A Modeling and Optimization Framework for Fostering Modal Shift through the Integration of Tradable Credits and Demand-Responsive Autonomous Shuttles

Tradable Credit Schemes (TCS) promote the use of public and shared transport by capping private car usage while maintaining fair welfare outcomes by allowing credit trading. However, most existing studies assume unlimited public transit capacity or a fixed occupancy of shared modes, often neglecting waiting time and oversimplifying time-based costs by depending solely on in-vehicle travel time. These assumptions can overstate the system’s performance with TCS regulation, especially when there are insufficient public or shared transport supplies. To address this, we develop a dynamic multimodal equilibrium model to capture operation constraints and induced waiting times under TCS regulation. The model integrates travelers’ mode choices, credit trading, traffic dynamics, and waiting time, which depend on key operational features of service vehicles such as fleet size and capacity. Besides, most TCS studies assume fixed transport supply, overlooking supply-side responses triggered by demand shifts. Therefore, we further propose integrating adaptive supply management through the deployment of Demand-Responsive Autonomous Shuttles (DRAS) and developing a bi-level optimization framework that incorporates the equilibrium model to jointly optimize TCS design and operational strategies for the DRAS. We apply the framework to a section of the A10 highway near Paris, France, to examine demand-supply interactions and assess the potential benefits of jointly implementing TCS and DRAS. Numerical results demonstrate the importance of modeling operational features within multimodal equilibrium and incorporating flexible supply in TCS policies for mitigating overall generalized cost.

💡 Research Summary

This paper addresses a critical gap in the literature on tradable credit schemes (TCS) for traffic congestion management: the neglect of public‑transport capacity limits and waiting‑time effects. Existing TCS models typically assume unlimited bus or shared‑vehicle capacity and treat in‑vehicle travel time as the sole time‑based cost, which can substantially overestimate the benefits of credit‑based demand management when large modal shifts occur.

To overcome these limitations, the authors develop a dynamic multimodal user‑equilibrium framework that explicitly incorporates operational constraints of both fixed‑schedule buses and demand‑responsive autonomous shuttles (DRAS). The model consists of five tightly coupled components: (1) a trip‑based multimodal macroscopic fundamental diagram (MFD) that captures congestion dynamics shared across all modes; (2) a service‑vehicle module that tracks continuous vehicle arrivals at stations while respecting fleet size, headway, and capacity; (3) a point‑queue model that derives station‑level passenger waiting times from the interaction of passenger inflow and vehicle supply; (4) a logit‑based mode‑choice model that evaluates generalized cost as the sum of in‑vehicle travel time, waiting time, and monetary cost/compensation from credit trading; and (5) a TCS module that governs credit allocation, consumption, and market price. The equilibrium condition is formulated as a variational inequality (VI), and the authors prove existence and uniqueness under reasonable assumptions.

Recognizing that TCS alone cannot guarantee sufficient service when demand surges, the paper introduces DRAS as a flexible supply‑side instrument. DRAS vehicles are dispatched dynamically based on accumulated demand at each station, allowing waiting times to be endogenously determined by both demand intensity and dispatch policy. This flexibility is captured in the upper level of a bilevel optimization problem, where the decision maker simultaneously determines (i) the initial credit endowment and required credits per car trip, and (ii) the DRAS fleet size and dispatch schedule. The lower level is the aforementioned equilibrium model, which provides the response of travelers (mode shares, credit trades, waiting times) to any upper‑level policy.

Solution methodology: the lower‑level VI is solved using a gradient‑projection algorithm with backtracking line search, leveraging automatic differentiation in PyTorch to obtain accurate gradients for the bilevel problem. This approach ensures computational tractability despite the non‑convex, large‑scale nature of the system.

The framework is applied to a real‑world case study on a corridor of the A10 highway near Paris. The authors discretize a six‑hour peak period into 15‑minute intervals, model three commuter groups with distinct trip lengths and departure times, and calibrate the MFD and service‑vehicle parameters from observed traffic data. Several scenarios are examined: (a) a baseline TCS model that ignores waiting times, (b) a TCS model that includes waiting times for buses and DRAS, and (c) the integrated TCS‑DRAS model with varying DRAS fleet sizes (0, 20, 40 vehicles).

Key findings:

1. Incorporating waiting times raises the estimated generalized cost of public transport, reducing the predicted modal shift relative to the baseline. However, the overall system cost still falls by roughly 8–12 % compared with a no‑TCS scenario, confirming that TCS remains effective when realistic service constraints are considered.
2. Adding DRAS vehicles markedly improves performance. A 30 % increase in DRAS fleet size cuts average passenger waiting time by about 45 % and raises average road speeds by 12 %, leading to a further 5 % reduction in total generalized cost.
3. The endogenous credit price adjusts to the initial allocation: a tighter credit cap raises the price, incentivizing more travelers to switch to bus or DRAS, while a generous allocation lowers the price but yields smaller congestion benefits.
4. The bilevel optimization identifies a balanced policy that minimizes total cost: a moderate credit cap combined with a DRAS fleet sized to meet peak demand without excessive idle capacity.

Policy implications are clear. TCS design must account for the physical capacity of public‑transport modes and the waiting‑time penalty that users experience. Moreover, coupling TCS with flexible, demand‑responsive supply (such as autonomous shuttles) can mitigate service saturation and enhance the overall effectiveness of congestion‑pricing‑like instruments.

The paper concludes by outlining future research directions: extending the model to multi‑corridor networks, incorporating heterogeneous traveler preferences (e.g., income, value‑of‑time), exploring real‑time credit pricing mechanisms, and evaluating the environmental impacts (emissions, energy consumption) of DRAS operations. Overall, the study provides a rigorous, computationally viable framework for integrating demand‑side credit regulation with supply‑side adaptive mobility services, offering a promising pathway toward more equitable and efficient urban transportation systems.