In recent years, shared autonomous electric cars have emerged as a promising solution for urban mobility, combining the benefits of electric propulsion, automation, and shared use to enhance road efficiency and reduce environmental impact. However, traditional single ride-hailing modes often underutilize vehicles, leading to low order fulfillment rates and economic inefficiencies. To address this, we propose integrating ridesplitting and relay strategies into the dynamic operation of shared autonomous electric car systems. This approach allows multiple passengers to share a single electric car and enables vehicle relays to extend service range despite battery limitations. In this paper, we formulate the problem using a spatiotemporal network model, simplify it for computational tractability, and develop a rolling horizon optimization framework to handle dynamic demand. Our case study, based on real-world data, demonstrates significant improvements in operational profit and service quality.
The operation of shared autonomous electric cars involves managing a fleet of electric cars across a network of stations to serve time-varying travel demands. We consider a system where electric cars are deployed in a urban area divided into regions, each with a station. The operational timeline is discretized into time intervals, and the battery levels of electric cars are represented as discrete energy layers. Key decisions include assigning electric cars to serve orders, allowing ridesplitting (where multiple passengers share an electric car), and enabling relays (where two electric cars接力 complete a long trip via an intermediate station). The goal is to maximize operational profit while respecting constraints such as battery safety, charging station capacity, and vehicle availability.
To model this, we construct a four-dimensional spatiotemporal network based on time, space, battery level, and passenger count. Nodes represent the state of an electric car at a specific time, location, battery level, and passenger load, while arcs represent actions like pickup, drop-off, movement, charging, and relay. However, this four-dimensional network is computationally complex due to its exponential growth in size. Therefore, we simplify it by merging arcs related to passenger services, resulting in a three-dimensional network that aggregates passenger counts into service types. This reduction maintains model accuracy while improving solvability.
The mathematical model is formulated as a pure integer linear program with the objective of maximizing operational profit. The profit components include revenue from direct rides, ridesplitting services (with discounts for shared rides), and relay services, minus charging costs. Constraints ensure flow conservation, limit the number of electric cars based on fleet size and parking availability, and cap service orders by demand. Let $I$ be the set of stations, $T$ the set of time intervals, $L$ the set of battery levels, and $N$ the set of passenger counts. The decision variables $f_s$ represent the flow on arc $s$ in the simplified network. The objective function is:
$$ \text{Maximize} \quad \sum_{n \in N} \sum_{s \in \zeta^{(Z)}} n P_n P_z f_s + \sum_{n \in N} \sum_{w \in \zeta^{(W)}} n P_n P_w f_w – \sum_{e \in \zeta^{(C)}} P_c f_e $$
where $P_n$ is the discount factor for ridesplitting with $n$ passengers, $P_z$ is the profit per direct ride, $P_w$ is the profit per relay ride, and $P_c$ is the charging cost. The constraints are:
$$ \sum_{s \in \zeta^{(SC)}} f_s \leq FS \quad \text{(Fleet size)} $$
$$ \sum_{s \in \zeta^{(SC)} \cup \zeta^{(C)}} f_s \leq PA_{i,t} \quad \forall i \in I, t \in T \quad \text{(Parking capacity)} $$
$$ \sum_{s \in \zeta^{-}(o)} f_s = \sum_{s \in \zeta^{+}(o)} f_s \quad \forall o \in O \quad \text{(Flow balance)} $$
$$ \sum_{s \in \zeta^{(Z)}} n f_s + \sum_{w \in \zeta^{(W)}} n f_w \leq D_{i,j,t} \quad \forall i,j \in I, t \in T \quad \text{(Demand fulfillment)} $$
$$ f_s \in \mathbb{N} \quad \forall s \in \zeta \quad \text{(Integer flow)} $$
To handle dynamic demand, we employ a rolling horizon optimization approach. The operational period is divided into overlapping time windows. In each window, we solve a subproblem that optimizes decisions for a forward-looking period, while fixing decisions from previous windows. This allows the system to adapt to real-time demand updates. We set different window sizes for direct rides and relay services to balance solution quality and computational efficiency. For instance, direct rides have a longer allocation window than relays to account for their higher resource consumption and impact on future states.
We conduct a case study based on the road network of Chengdu, China, with 30 stations and a 5-hour operational period from 16:00 to 21:00. The electric car fleet size is 600, with initial battery levels at 70%. Each electric car has an average speed of 30 km/h, a battery capacity of 50 kWh, and a power consumption rate of 20 kW while driving. Charging stations have a power of 20 kW, and we discretize battery levels into 30 layers. Travel demand is generated using spatiotemporal attractiveness factors, with scenarios representing uniform and non-uniform distributions. Key parameters include a direct ride profit of $0.3 per minute, a relay discount of 10%, and ridesplitting discounts of 40% for two passengers and 45% for three passengers. The maximum acceptable delay for relay services is 10 minutes.
We test the rolling horizon algorithm with various time window sizes to find a balance between profit and computation time. The results show that a direct allocation window of 45 minutes and a relay allocation window of 5 minutes yield high profit while keeping solution times under 60 seconds, suitable for real-time operations. For example, with a demand pressure factor of 0.8, the profit reaches 36,840.87 units, with an order fulfillment rate of 66.35% and a relay rate of 11.87%. The table below summarizes the impact of window sizes on performance:
| Direct Window (min) | Relay Window (min) | Profit (units) | Order Fulfillment Rate (%) | Max Solve Time (s) |
|---|---|---|---|---|
| 30 | 5 | 33,678.20 | 60.00 | 10.3 |
| 45 | 5 | 36,840.87 | 66.35 | 21.20 |
| 60 | 5 | — | — | >300 |
Comparing the strategies, we find that integrating ridesplitting and relay significantly improves profit over single ride-hailing. In a non-uniform demand scenario, the profit increases by 11.60%, while in a uniform scenario, it rises by 13.85%. The table below contrasts the two strategies under different demand distributions:
| Demand Scenario | Strategy | Profit (units) | Profit Increase (%) | Order Fulfillment Rate (%) | Relay Rate (%) | Ridesplitting Rate (%) |
|---|---|---|---|---|---|---|
| Non-uniform | Single Ride-hailing | 33,011.20 | — | 58.47 | 0.00 | 0.00 |
| Non-uniform | Ridesplitting + Relay | 36,840.87 | 11.60 | 66.35 | 11.87 | 18.52 |
| Uniform | Single Ride-hailing | 33,504.70 | — | 59.86 | 0.00 | 0.00 |
| Uniform | Ridesplitting + Relay | 38,147.60 | 13.85 | 68.44 | 11.23 | 17.21 |
Furthermore, we investigate the effect of passenger capacity on ridesplitting. As shown in the table below, higher passenger limits improve order fulfillment and profit under high demand pressure, but may reduce average profit per order under low demand due to discounts:
| Max Passengers | Demand Pressure | Profit (units) | Relay Rate (%) | Ridesplitting Rate (%) | Order Fulfillment Rate (%) |
|---|---|---|---|---|---|
| 1 | 0.6 | 28,685.87 | 10.35 | 0.00 | 65.94 |
| 2 | 0.6 | 27,855.12 | 10.13 | 13.46 | 66.48 |
| 3 | 0.6 | 29,502.62 | 9.95 | 9.85 | 70.00 |
| 1 | 0.8 | 36,009.14 | 11.46 | 0.00 | 61.73 |
| 2 | 0.8 | 36,840.87 | 11.87 | 18.53 | 66.35 |
| 3 | 0.8 | 37,615.78 | 10.98 | 14.52 | 67.67 |
| 1 | 1.0 | 39,363.82 | 13.24 | 0.00 | 54.23 |
| 2 | 1.0 | 40,328.53 | 12.96 | 24.26 | 59.30 |
| 3 | 1.0 | 41,488.04 | 11.86 | 16.99 | 61.11 |
In conclusion, our approach demonstrates that dynamic operation of shared autonomous electric cars with ridesplitting and relay strategies can significantly enhance operational efficiency and profitability. The three-dimensional network simplification and rolling horizon optimization enable practical implementation for large-scale systems. Future work could extend the model to handle group travel demands, where multiple passengers in a single order require sophisticated matching with other electric car services. Overall, the integration of these strategies promises a more sustainable and efficient urban mobility system centered around electric cars.