Semi-on-Demand Off-Peak Transit Services with Shared Autonomous Vehicles — Service Planning, Simulation, and Analysis in Munich, Germany

The advent of shared autonomous vehicles (SAVs) brings new perspectives to enhancing multimodal public transit services, particularly as feeders in less dense areas (Ng et al., 2024b). With lower operating costs, they can address the first-mile-last-mile problem and bring passengers to other transit modes more efficiently than conventional ride-sharing. While transit agencies retain some drivers due to union contracts or operational reasons, the lower operating costs of SAVs also allow for a larger fleet of smaller vehicles and thereby more frequent services (Hatzenbühler et al., 2020). Transit agencies can plan their fleet (vehicle size and type) according to peak-hour fixed-route service, which leaves redundant capacity at off-peak times for more flexible services and expand their catchment areas. For example, smaller but more SAVs can pick up and drop off some passengers at their homes, bringing more convenience and stimulating more demand (Frei et al., 2017).

In semi-on-demand (SoD) hybrid-route services investigated in previous studies (Ng and Mahmassani, 2023; Ng et al., 2024a) (Figure 1), SAVs first serve all fixed-route stops based on a schedule, then drop off and pick up passengers in the flexible-route portion (pre-determined with length $x_{f}$), and return to fixed-route scheduled stops and the terminus. It is shown to combine the cost efficiency of fixed-route buses in denser areas with the flexibility of on-demand services in less populated regions.

This study considers improving current transit services with SAVs by enhancing peak-hour fixed-route service (in the existing format but more frequent due to SAV cost savings) and implementing off-peak SoD hybrid-route services in the same route catchment area. We address four primary research questions:

1. 1.

   At peak hours, what are the optimal fleet and vehicle sizes for current-day transit lines with the lower operating costs of SAVs, when we minimize waiting time and operator’s cost subject to capacity and budget constraints? ¹¹1Determining fleet and vehicle size with fixed routes at peak hours ensures sufficient capacity to satisfy demand. This also leads to more conservative estimates of the benefits of using smaller vehicles to provide flexible services at off-peak.

2. 2.

   During a transition phase from drivers to full automation, how would the operating budget affect the previous optimal results?

3. 3.

   At off-peak hours, what is the optimal service planning (schedule and route form) for SoD hybrid-route services, considering stochastic requests and balance between schedule time, request rejection, and operational performance?

4. 4.

   What are the use cases of off-peak SoD services considering benefits and impacts to passengers and the operator?

For SoD, too long flexible routes would lead to excessive detours, longer journey times, and higher operating costs. Meanwhile, smaller headways (e.g., more smaller vehicles) favor longer flexible routes by reducing the detours of each trip (Ng et al., 2024a). Therefore, optimizing SAV fleet and vehicle sizes for peak-hour fixed routes may also benefit off-peak SoD hybrid-route services.

The problem scope is defined with some key assumptions: (i) continuous operations during the study hours; (ii) inelastic demand (yet sensitive to walking distance); (iii) requests made when passengers are ready for boarding, with no reservations considered; and (iv) individual travelers, i.e. no group bookings are considered.

Building upon the theoretical formulations of costs and benefits derived in previous work (Ng et al., 2024a), this study focuses on the service planning, simulation, and analysis of the SoD hybrid-route transit feeders using SAVs. It presents conceptual, theoretical, and methodological contributions:

1. 1.

   We formulate the service planning problem of peak-hour fixed-route services in terms of optimal fleet size and vehicle size, considering changes in costs and retention of drivers. The study further conceptualizes the off-peak SoD hybrid-route service.

2. 2.

   We derive analytical results of convex optimization for the peak-hour fleet and vehicle sizes (in full-SAV and partial-driver scenarios) and off-peak schedules considering detours, service guarantee, and peak/off-peak service level. This is a timely addition to the literature for planning the gradual introduction of SAVs in transit service.

3. 3.

   By analyzing the agent-based simulation results on ten existing bus routes in the city of Munich, Germany, we optimize the flexible-route portion $x_{f}$ and headway $h$, and examine the use cases and benefits of hybrid routes (in terms of access, waiting, and riding times for users and vehicle distances and requirements for operators). We contrast theoretical predictions with simulation results and study variations in user experience.

Figure 2 summarizes the flow of the methodology. The remainder of the paper is structured as follows. Section 2 presents an overview of relevant literature on demand-responsive transit, SAVs in public transit, and ride-sharing simulation. Section 3 follows with the methodology, illustrated with an experiment described in Section 4 and results presented in Section 5. Section 6 concludes the paper.

Research on demand-responsive transit (DRT) spans a few decades (see surveys by Errico et al. (2013); Vansteenwegen et al. (2022)). Several studies have explored flexible transit route designs, suggesting conditions to use demand-responsive over fixed-route services (Li and Quadrifoglio, 2010) and cost advantages for low-to-moderate passenger demand (Nourbakhsh and Ouyang, 2012). Recent research has focused on integrating DRT with traditional fixed services, e.g., supplementary DRT improvements to fixed-route bus services (Chen and Nie, 2017; Fielbaum et al., 2024) and combination of fixed-schedule and DRT services during off-peak hours and in the suburbs (Calabrò et al., 2023).

Various optimization strategies have been employed in semi-flexible transit system research, including analytical models (Qiu et al., 2014), metaheuristics (Galarza Montenegro et al., 2021), stochastic approaches with Markov decision processes (Rambha et al., 2016; Li et al., 2023), and agent-based simulation (Rich et al., 2023).

However, most studies focus on the performance of flexible-route services or the decision to deploy either flexible-route or fixed-route services. In contrast, this study builds on previous work (Ng et al., 2024a) to design and simulate hybrid-route service, representing a continuum between these two service modes.

SAVs have shown the potential in expanding transit coverage and enhancing service attractiveness (Salazar et al., 2020; Shen et al., 2018) (also see a survey (Narayanan et al., 2020)). Several studies have explored the synergistic relationship between SAVs and public transit systems (Shen et al., 2018; Cortina et al., 2023; Gurumurthy et al., 2020a), sometimes by integrating SAVs into traditional transit (Levin et al., 2019). Conversely, some considered SAVs as competitors (Sieber et al., 2020), with performance improvements by replacing fixed bus routes with flexible routes in suburbs in Chicago, USA (Ng and Mahmassani, 2023; Volakakis et al., 2023). The joint design of transit and SAVs has been another area of focus, e.g., optimizing vehicle size (Alonso-Mora et al., 2017), fleet size (Pinto et al., 2020; Dandl et al., 2021), network (Ng et al., 2024b), and timetabling and vehicle scheduling (Cao and Ceder, 2019).

Automation can also benefit line-based buses, with a general shift towards higher frequencies and smaller vehicles (Bösch et al., 2018) found to attract passengers through improved service (Hatzenbühler et al., 2020) and SAVs as feeders to improve efficiency (Badia and Jenelius, 2021).

In summary, the integration of SAVs in public transit systems presents opportunities for improved service quality, expanded coverage, and enhanced operational efficiency. However, challenges remain in optimizing the balance between traditional fixed-route services and more flexible autonomous options across urban contexts and demand patterns.

Agent-based simulation has been widely used to represent the dynamic and stochastic nature of underlying vehicle routing problems and the interaction between operators and riders. Studies focused on solving the online assignment problem (Alonso-Mora et al., 2017; Simonetto et al., 2019; Wang et al., 2023) or repositioning idle vehicles to accommodate future expected demand (Zhang and Pavone, 2016; Tafreshian et al., 2021; Engelhardt et al., 2024). Apart from door-to-door services, short walking legs for riders (Engelhardt and Bogenberger, 2021; Fielbaum et al., 2021) or transfers between vehicles (Masoud and Jayakrishnan, 2017; Namdarpour et al., 2024) have been proposed to increase efficiency and sharing rate for the service.

On the operational aspects of demand-responsive services, models like SUMO (Armellini, 2021), MATSim (Horni et al., 2016), and Polaris (Gurumurthy et al., 2020b; Cokyasar et al., 2022) have been extended to simulate both fixed- and flexible-route services. Other studies have developed new agent-based simulation models focusing on operational aspects or computational performance (Alonso-Mora et al., 2017; Fagnant and Kockelman, 2018; Dandl et al., 2019; Engelhardt et al., 2022). These studies focused on the supply side, i.e., how to serve a fixed set of riders as efficiently as possible. Meanwhile, agent-based demand models (Ruch et al., 2018; Gurumurthy et al., 2020c; Wilkes et al., 2021) have been developed to estimate the demand attracted to the new modes.

This study adapts the open-source Python package Fleetpy (Engelhardt et al., 2022) to model the hybrid-route SoD service and the interaction between riders, operator, and vehicles.

We expand the FleetPy simulation framework (Engelhardt et al., 2022) with a scheduled SoD module (Figure 2). It models the interaction among three agents: riders $r\in\mathcal{R}$, a service operator, and vehicles $v\in\mathcal{V}$ within a street network $\mathcal{G}=(\mathcal{N},\mathcal{E})$ with nodes $\mathcal{N}$ and edges $\mathcal{E}$ in discrete time steps. Customers request trips (with demand as a function of access distance), where each rider request $r$ is defined by the request time $t_{r}$, origin $o_{r}\in\mathcal{N}$, and destination $d_{r}\in\mathcal{N}$. The operator evaluates requests, considers service quality and operational constraints (e.g., maximum waiting and travel times and vehicle capacity), and assigns requests to individual vehicles operating on SoD schedules $z$. A schedule is an ordered list of stops that are processed by the vehicle one after another, whereas a stop $i\in\mathcal{I}$ is characterized by a location $Y_{i}\in\mathcal{N}$, the latest start time $T^{l}_{i}$, the earliest start time $E_{i}$, a duration $T^{s}_{i}$, and sets of requests boarding and alighting the vehicles.

In between stops, vehicles travel in the network on the fastest path. If they arrive at the next stop $i$ before the scheduled time $E_{i}$, they wait there until $E_{i}$. In the next step, they perform the boarding task associated with this stop for a duration of $T^{s}_{i}$ and then go to the next stop.

Requests are assigned by sequential insertion into currently assigned vehicle schedules. The feasibility of each resulting schedule is then evaluated based on the following constraints: (i) each stop $i$ in the schedule is served no later than $T^{l}_{i}$; (ii) each rider waits less than the maximum waiting time $T^{w,max}$; (iii) the maximum in-vehicle travel time of each passenger is within the direct travel time by a factor $\phi^{p}$; and (iv) the number of passengers never exceeds the vehicle capacity $C_{v}$.

The objective $\phi(z)$ of all feasible schedules is then calculated in Eq. (19) that balances vehicle distance, rider travel time, and requests served in the fixed-route portion. $\gamma^{o}_{v}$, $\gamma^{t}$, $\gamma^{r}$, and $\gamma^{s}$ are the cost/reward coefficients for vehicle distance, traveler time, requests satisfied, and requests served in a fixed route, respectively. ³³3$\gamma^{s}$ prioritizes the assignment of riders to fixed stops if possible. $d_{v,z}$, $a_{r,z}$, $n^{r}_{z}$, and $n^{s}_{z}$ are respectively the distance traveled by vehicle $v$, arrival time of rider $r$, number of requests satisfied, and number of requests served in a fixed route.

The feasible schedule with the minimum objective is then selected and assigned. The process is summarized as follows:

1. 1.

   If a rider request $r$ is picked up or dropped off in the fixed-route portion, $o_{r}$ or $d_{r}$ is shifted to $o_{r}^{f}$ or $d_{r}^{f}$ corresponding to the nearest bus stop.

2. 2.

   A request is assigned to the vehicle schedule by: (i) adding the request to the boarding or alighting list of a stop if the corresponding locations match; or (ii) inserting a new stop to pick up or drop off the request. All feasible insertions for a request and each vehicle are computed by an exhaustive search, and the schedule $z$ that minimizes $\phi(z)$ in Eq. (19) is assigned. If no feasible insertion is found, the rider’s request is rejected.

3. 3.

   Steps 1 and 2 are repeated until all current requests are processed. Then, vehicles move and perform boarding processes according to their assigned schedules.

The simulation records user and vehicle data on trip level and at the end of the simulation, metrics in aggregate terms are computed.

The simulation results of each scenario (each flexible-route length $x_{f}$) are analyzed and compared to identify the suitable use cases for SoD. Generalized costs are used as metrics that include both user’s and operator’s costs.

The user’s cost $c^{u}_{r}$ of rider $r$ in Eq. (20) is composed of access (walking), waiting, and riding costs, where $t^{a}_{r}$, $t^{w}_{r}$, and $t^{t}_{r}$ are the access, waiting, and riding times and $\gamma^{a}$, $\gamma^{w}$, and $\gamma^{r}$ are the respective cost coefficients.

The vehicle cost $c_{v}^{o}$ for vehicle $v$ in Eq. (21) consists of distance-based operating cost and vehicle marginal cost, where $d_{v}$ and $t^{v}_{v}$ are the distance traveled and time deployed with $\gamma^{o}$, and $\gamma^{v}$ as the respective cost coefficients.

The total generalized cost $c^{g}$ in Eq. (22) is the sum of all users’ costs and operator’s costs.

The optimal SoD route form may be fixed route ($x_{f}=0$), hybrid route ($0\leq x_{f}\leq L^{x}$), or flexible route ($x_{f}=L^{x}$), depending on the minimum generalized cost $c^{g}$ and number of passengers served. We identify such cases and also analyze the more fine-grained effects on journey time and operational costs, in comparison with the theoretical values (Ng et al., 2024a).

A simulated SoD run of Route 7 is illustrated in Figure 5 as an example. After departing from the Trudering terminus and serving the fixed-route stops (black line), the SAV detours from the original bus route to pick up and drop off passengers in the flexible-route part (blue line). It then returns to the fixed route and terminus.

This subsection discusses the simulation results for different routes under varying flexible-route length $x_{f}$. All statistics shown are median values unless otherwise specified.