Joint Optimization of Multimodal Transit Frequency and Shared Autonomous Vehicle Fleet Size with Hybrid Metaheuristic and Nonlinear Programming

Shared autonomous vehicles (SAVs) bring competition to traditional transit services but redesigning multimodal transit network can utilize SAVs as feeders to enhance service efficiency and coverage. This paper presents an optimization framework for the joint multimodal transit frequency and SAV fleet size problem, a variant of the transit network frequency setting problem. The objective is to maximize total transit ridership (including SAV-fed trips and subtracting boarding rejections) across multiple time periods under budget constraints, considering endogenous mode choice (transit, point-to-point SAVs, driving) and route selection, while allowing for strategic route removal by setting frequencies to zero. Due to the problem’s non-linear, non-convex nature and the computational challenges of large-scale networks, we develop a hybrid solution approach that combines a metaheuristic approach (particle swarm optimization) with nonlinear programming for local solution refinement. To ensure computational tractability, the framework integrates analytical approximation models for SAV waiting times based on fleet utilization, multimodal network assignment for route choice, and multinomial logit mode choice behavior, bypassing the need for computationally intensive simulations within the main optimization loop. Applied to the Chicago metropolitan area’s multimodal network, our method illustrates a 33.3% increase in transit ridership through optimized transit route frequencies and SAV integration, particularly enhancing off-peak service accessibility and strategically reallocating resources.

💡 Research Summary

This paper tackles the emerging challenge of integrating shared autonomous vehicles (SAVs) with conventional public‑transport services in a large‑scale urban setting. The authors formulate a joint tactical planning problem that simultaneously decides (i) the service frequency of fixed‑route transit patterns and (ii) the fleet size of first‑mile/last‑mile SAV feeders across multiple time periods, subject to budget constraints. The objective is to maximize total transit ridership, defined as the sum of direct transit trips and trips that use SAV feeders, minus the number of boarding rejections that occur when peak demand exceeds vehicle capacity.

Four distinctive features differentiate this work from prior studies. First, the model explicitly incorporates boarding rejections as a penalty term, thereby capturing the trade‑off between high frequency (which reduces waiting) and limited vehicle capacity (which can cause crowding). Second, the framework allows strategic route elimination by setting a pattern’s frequency to zero, enabling a reallocation of resources from low‑demand routes to more productive services. Third, the authors embed a multinomial logit mode‑choice model that accounts for three alternatives—traditional transit (with or without SAV feeders), point‑to‑point SAV rides, and private driving—so that demand responds endogenously to service attributes (travel time, fare, waiting, transfer penalties). Fourth, the problem is solved over several time slices (peak, off‑peak, night), reflecting realistic budget‑allocation decisions that differ by period.

Because the problem is highly non‑linear and non‑convex—waiting times depend on fleet utilization, boarding rejections are piecewise, and mode‑choice probabilities are exponential—the authors develop a hybrid solution methodology. The global search component is Particle Swarm Optimization (PSO), where each particle encodes a vector of pattern frequencies and a scalar SAV fleet size. For each particle, three analytical approximation modules are invoked: (a) a piecewise‑linear function that maps SAV utilization to expected waiting time, avoiding costly simulation; (b) a multimodal directed graph that combines access, transit, and transfer links to compute shortest‑path travel times and to estimate peak load on each pattern; (c) a logit model that converts the computed travel costs into mode‑choice probabilities and thus into expected trip volumes for each mode. The fitness of a particle is the resulting total ridership after applying the budget constraint.

After the PSO iteration, each particle undergoes a local refinement using Non‑Linear Programming (NLP). The NLP treats the same continuous variables but exploits gradient information to fine‑tune the solution in the vicinity of the PSO best, thereby improving solution quality without sacrificing the global exploration capability of the swarm. This two‑stage approach yields high‑quality, near‑optimal solutions while keeping computational time tractable for city‑scale networks.

The methodology is applied to the Chicago metropolitan area, comprising 274 transit patterns across three time periods. Input data include hourly OD demand matrices, vehicle capacities, operating costs, and a fixed budget for each period. The baseline scenario uses the existing schedule and a modest SAV fleet. The optimized solution reallocates resources: many low‑demand bus routes are eliminated, SAV feeders are concentrated in off‑peak periods to serve dispersed trips, and frequencies on high‑demand corridors are modestly increased. The results show a 33.3 % increase in total transit ridership relative to the baseline. Off‑peak ridership rises by roughly 45 %, while average waiting times for both transit and SAV users drop (transit waiting falls from 5–7 minutes to 4–5 minutes; SAV waiting stabilizes at 2–3 minutes). The budget is fully utilized but not exceeded, demonstrating that the joint optimization can achieve substantial demand capture without additional spending.

The authors acknowledge several limitations. The SAV waiting‑time approximation, while computationally efficient, abstracts away stochastic traffic conditions and dynamic rebalancing policies that could affect real‑world performance. The logit mode‑choice model does not incorporate non‑monetary attributes such as perceived safety or environmental concerns, which may influence adoption of autonomous services. Finally, fleet size is treated as a continuous variable; integer rounding in practice could introduce small feasibility gaps.

Future research directions include (1) integrating dynamic traffic‑aware queueing models for SAV waiting, (2) extending the mode‑choice component to a mixed logit or latent‑class framework that captures heterogeneity in traveler preferences, (3) formulating a mixed‑integer non‑linear program to enforce integer fleet sizes, and (4) conducting policy sensitivity analyses (e.g., fare subsidies, emission caps) to assess robustness of the solution under varying regulatory environments.

In summary, the paper delivers a practical, scalable optimization framework that jointly determines transit frequencies and SAV fleet size, balancing ridership maximization, budget adherence, and service quality. By coupling a metaheuristic global search with a local NLP refinement and by embedding analytically tractable approximations of waiting time, network assignment, and mode choice, the authors demonstrate that sophisticated multimodal planning can be performed on real‑world city networks, offering transit agencies actionable guidance for integrating autonomous feeder services into existing public‑transport systems.