Dynamic Occupancy Rate for Shared Taxi Mobility-on-Demand Services through LSTM and PER-DQN

Abstract

As an effective public transportation system, a Shared Taxi Mobility-on-Demand (STMoD) provides passengers with door-to-door shared taxi service. This study proposes a dynamic occupancy rate rebalancing approach with centralised dispatching for STMoD systems to equalise taxi supply in response to passengers’ demands in a city. The occupancy rate changes dynamically since the passengers’ demand varies during the time, as predicted using a Long Short-Term Memory (LSTM) machine learning algorithm. The zone, weekday, time, and holidays are used as effective parameters to train the LSTM model. The occupancy rate increases in peak hours and decreases in off-peak hours to balance the number of passengers and the number of idle taxis in the corresponding zones. Then, the taxi transferring procedure applies to the remaining imbalanced zones, balancing the request and response in the whole city. The proposed approach adjusts the drivers’ incomes to increase the number of taxis earning money and decrease the idle taxis without income. Also, it reduces passenger waiting time. Taxis learn to follow the shortest paths to pick up and drop off passengers using the Prioritised Experience-Deep Q Network (PER-DQN) reinforcement learning algorithm. Using the New York City passenger demand data in Manhattan, we simulated and compared the STMoD performance with the classic shared taxi system in an agent-based simulation environment. The evaluation results showed a a 28.18% improvement in the balance ofmoney earned by taxis compared to the classic shared taxi scenario. Also, the number of idle taxis decreased by 38%, and the passenger waiting time significantly reduced by 22.69%.

Introduction

Taxis are a vital part of public transport, filling gaps left by buses and trains [1, 2]. They offer a more comfortable option due to limited or no sharing compared to buses or trains [3]. However, inefficiencies in passenger pick-up and drop-off lead to taxi congestion in hotspot areas, causing traffic, excessive fuel use, and air pollution [4, 5–6]. Classic taxi services assign one passenger per taxi, which increases the number of taxis during peak hours, contributing to congestion and noise pollution during off-peak times [7, 8]. Enhancing efficiency through taxi reuse and ride-sharing can help reduce these issues.

Shared taxis offer a solution during peak hours by transporting multiple passengers with nearby destinations [9]. Drivers plan the shortest routes to minimize travel and waiting times, and shared taxis are more affordable as costs are divided among passengers. This system also reduces the number of taxis needed in high-demand areas. In 2015, 8 million people used taxi-sharing globally, with projections reaching 36 million by 2025 [10].

The Shared Mobility-on-Demand (SMoD) system offers door-to-door service [11], allowing passengers to select their origins and destinations while interacting with drivers via mobile/web apps [12]. Several researchers have investigated the possibility of improving MoD (Mobility-on-Demand) and SMoD systems [13, 14, 15, 16–17]. A branch of these investigations has focused on balancing the number of vehicles available in different parts of the city (zones) and passenger requests [4, 18, 19, 20–21]. In a balanced SMoD system, where taxis are properly distributed in zones, all passengers can access taxis around the city at any time. In reality, taxis tend to go to high-demand zones to find more passengers, and low-demand zones become barren of idle taxis. In other words, idle taxis and passengers are in different zones because there is a lack of information about passenger demand and the number of idle taxis in each zone. In this circumstance, we need to rebalance the number of taxis in different zones by transferring taxis from zones with idle taxis to high-demand zones to increase passenger service in low-demand zones, prevent congestion in high-demand zones [22], enhance drivers’ income, and decrease passenger waiting time all over the zones.

Problem Description

Figure 1 shows the distribution of passengers and taxis on Monday at 5 PM across three different zones: Zone A (industrial) and Zones B and C (residential). The figure illustrates that at 5 PM, there is a high concentration of passengers in Zone A, the industrial area, waiting to travel to their homes in Zones B and C. Consequently, Zone A becomes a high-demand zone with an accumulation of taxis.

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Fig. 1

Illustration of an imbalanced situation

While there are idle taxis in Zone A unable to earn money, some passengers in Zones B and C are still waiting for taxis. This situation reveals a mismatch between passenger demand and taxi availability. The shortage of taxis in Zones B and C increases wait times for passengers, while the excess in Zone A reduces drivers’ chances of finding fares and lowers their income. Consequently, drivers must cruise between zones to find passengers, making the situation worse.

Background

Researchers have proposed three different approaches to tackle the imbalanced situation. Some studies delegated the responsibility of supply and demand balancing to a group of taxi drivers or extra staff, which were particularly employed to relocate vehicles to high-demand zones and answer the high request for taxis [4, 23, 24]. In some cases, the extra staff is supposed to use public transportation or a folding bicycle to reach the vehicles and drive to high-demand zones [25].

In another direction, some researchers devised incentive mechanisms to urge passengers to choose their destination in high-demand areas or ask drivers to bring them back to the origin station [26, 27–28].

In the third line of research, autonomous vehicles are used as transportation means to transfer empty robotic vehicles between stations [4, 14, 23, 29, 30, 31–32].

Although previous studies have addressed the rebalancing problem to some extent, there are still several unsolved issues that need further research. In particular, studies that addressed rebalancing through transferring taxis to other zones [20, 25] cause congestion, noise pollution, and greenhouse gas emissions. Some other researchers limited themselves to stations so that passengers had to walk to and drop off at the stations, which is not desirable when the origin or destination is far from the stations [24, 28, 32]. We have also seen cases where rebalancing was addressed by taxis cruising among zones regardless of the passengers’ demand, e.g [26]. , . In addition, each line of the above-mentioned research has its drawbacks. In the first line of approach, extra staff or a group of drivers is hired to relocate the vehicles between the zones. The incentive mechanisms are usually costly and need to be adjusted continuously to be effective. Furthermore, an entirely new set of rules to communicate with other non-autonomous vehicles on the same road is required, and building new road infrastructure is necessary to use robotic vehicles [33].

In response, this study proposes a dynamic occupancy rate rebalancing approach for a Shared Taxi Mobility-on-Demand (STMoD) system with centralised dispatching to equalise supply and demand, adjust drivers’ incomes, decrease passengers’ waiting time, and optimise route planning.

In the proposed dynamic occupancy rate rebalancing approach, we adjust the taxi occupancy rate to balance passengers and taxis across different zones. This adjustment responds to fluctuating demand using machine learning predictions. During peak hours, the occupancy rate increases, reducing the number of taxis needed. In contrast, off-peak hours see a decrease in occupancy, allowing more taxis to serve passengers within their zones. This strategy optimizes the system, increases driver incomes, and reduces idle taxis. In zones with a taxi shortage, a targeted transfer procedure guided by centralized dispatching ensures efficient relocation. This approach minimizes unnecessary movement by avoiding widespread transfers across all zones [22, 25, 29].

The STMoD was tested using the agent-based model on NYC Open Data [34]. We ran two scenarios to compare the viability of the proposed dynamic occupancy rate rebalancing approach against the classic, well-established shared taxi system.

Our proposed approach presents a comprehensive analysis by concurrently addressing key parameters such as pick-up and drop-off algorithms, determining the shortest path, and pricing strategies for drivers and passengers. Notably, synthesizing these three critical elements in a singular study distinguishes our research from others in the field, where investigations often tend to focus on just one or two of these dimensions.

To summarize the main contributions of this paper are directly aligned with the innovative strategies and technologies discussed above. They include:

• Balancing Benefits: It systematically considers the benefits for both drivers and passengers, ensuring a more equitable service delivery.

• Income Adjustment: By reducing idle vehicle times, the paper effectively adjusts drivers’ incomes, leading to improved economic efficiency and job satisfaction within the taxi industry.

The paper begins with an Introduction, followed by a Methods section detailing the notations, STMoD system mechanics, agent-based model using RL, and simulation setup. Section 3: Results presents the findings, Section 4: Discussion interprets these findings in the context of existing research, and Section 5: Conclusion and future work summarizes the study and outlines directions for further research.

Method

Figure 2 presents the three main components of this study. Firstly, the STMoD component forecasts passenger demand by analysing historical taxi usage data, including pick-up/drop-off times, locations, and passenger counts. This component relies on a predictive analytics model utilising an LSTM neural network to forecast passenger demand based on historical taxi usage data. Subsequently, a centralised dispatch system calculates the required number of taxis for each zone, optimising occupancy rates and ensuring an efficient assignment of passengers to taxis. Secondly, an agent-based model is designed to test the occupancy rates calculated by STMoD component. In this model, taxis pick up and drop off passengers with different origins and destinations along the shortest possible path. Each taxi acts as an agent following defined policies in the environment and receives rewards to optimise the path between origins and destinations. Finally, the third component ran two simulations to compare the performance of the STMoD system with classic shared taxi systems.

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Fig. 2

STMoD system research methodology

The proposed system represents a shared taxi system that offers passengers door-to-door service. This system benefits driver income, passenger waiting times, fare cost-effectiveness, and route planning.

Notations

Table 1 shows the notation used throughout the article.

Table 1. Notations in this article

Symbol	Description
${\stackrel{\vee }{T}}_{\mathit{zt}}$	The total idle taxis in zone $\text{Z}$and timeslot $\text{t}$.
$T_{\mathit{zt}}$	The total number of available taxis in zone $\text{Z}$ and timeslot $\text{t}$.
$N_z$	The neighbouring zones of zone $\text{Z}$ .
$Trn(\stackrel{_{}^{\prime }}{z},z)$	The number of taxis transfer from zone ${\text{Z}}^{\prime }$ to zone $\text{Z}$
$S_{\mathit{zt}}$	The shortage of taxi in zone $\text{Z}$ and timeslot $\text{t}$.
$O_{\mathit{zt}}$	The occupancy rate in zone $\text{Z}$ and timeslot $\text{t}$.
$r_{\mathit{zto}}$	The ratio of taxis that work in zone $\text{Z}$ and timeslot $\text{t}$ with a fixed occupancy rate $O$.
$D_z$	The total number of passenger demand in zone $\text{Z}$ .
$D_{\mathit{zt}}$	The total passenger demand in zone $\text{Z}$ and timeslot $\text{t}$ .
$d_k$	The day number of the week, $K\in \{1,2,3,\cdots ,7\}$
$H$	Holiday, $\text{H}\in \left\{\text{0,1}\right\}$
$a$	The action in the agent-based model, $a\in \left\{a_1,,,,a_2,,,,a_3,,,,\cdots ,,,,,a_8\right\}$
$s$	The state in the agent-based model.
$s^{\prime }$	The next state in the agent-based model.
$R$	The reward in the agent-based model.
$Q(s,a)$	The expected value of doing action $a$ in state $S$.
$P\left(j\right)$	The probability of choosing experience $j$.

The notation defines the symbols used in the modelling while representing city zones, timeslots of the day, taxis, passengers, and occupancy rate per zone and timeslot.

A city is divided into m zones $\text{z}\in \{\text{1,2}\cdots ,\text{m}\}$, usually defined based on the municipal and neighbourhood boundaries. The taxis’ zones, passengers’ requests, and the destinations they want to go to are known in our system. Each day is split into eight 3-hour timeslots $t\in \{\text{1,2},\cdots ,8\}$ so that the first timeslot is from 6 AM to 9 AM and the last timeslot is from 3 AM to 6 AM. A city has N taxis, represented as $t\_i$, where $i\in \{\text{1,2},\cdots ,N\}$. Each taxi has four seats to pick up passengers, so the minimum occupancy rate is one, and the maximum occupancy rate is four.

STMoD System

The STMoD system is composed of four procedures.

1. Prediction of passenger demand.

2. Calculation of dynamic occupancy rate.

3. Taxi transferring procedure.

4. Centralised dispatching system.

In the following sections, each procedure is explained in detail.

Predicting Passenger Demand

Passenger demand in a zone is the number of passengers requesting taxis in that zone. The time of the day, day of the week, land-use, weather conditions [30], transport system conditions [35], and public holidays [36] affect passenger demand. This study considers zone, public holiday (non-working days due to an event), time of the day, and the day of the week (both weekdays and weekends) as the affecting parameters in calculating the passenger demand [37].

Previous studies have focused on several linear [38, 39, 40–41] and non-linear algorithms [42, 43–44] for the passenger demand prediction. Linear Regression (LR) was chosen as a baseline model because it is widely used in initial comparative studies [45, 46–47]. Its simplicity serves as an essential benchmark, allowing us to demonstrate the incremental value provided by more complex models.

Recently, learning-based methods have been employed to predict traffic accidents [48], passenger flow [49], travel time [50], pedestrian trajectory [51], and vehicle trajectory [52] Neural Networks (NN) have the ability to handle complex data structures, as highlighted in recent literature [53, 54–55]. However, NNs do not inherently support sequential data, which is crucial for our application. This limitation underscores the need for models that can effectively handle sequential data. Recurrent Neural Networks (RNNs) [56] are popular networks that can effectively model sequential data. Among RNNs, Gated Recurrent Units (GRU) are competitive model for processing sequential data, as supported by the literature review [57, 58, 59–60]. Another powerful RNN model for supporting sequential data is the LSTM network, which is frequently cited [61, 62, 63–64] and widely utilised in studies involving passenger demand data.

In this study, we apply LSTM to estimate passenger demand per zone ($D_z$) in the STMoD system. Since passenger demand per zone ($D_z$) has a repeated pattern, we exploit the LSTM model as a powerful RNN method with the ability to capture repetitive patterns in long sequences [65]. The choice of LSTM over traditional Recurrent Neural Networks (RNNs) is guided by the specific advantages of LSTM in handling sequential data. Unlike standard RNNs, which struggle with long-range dependencies due to issues like vanishing gradients. The LSTM’s unique architecture, featuring forget, input, and output gates, allows it to selectively remember patterns over long-time intervals, making it more suitable for predicting passenger demand in the STMoD system than conventional RNNs, which might overlook such extended temporal correlations.

Figure 3 shows the structure of the proposed LSTM model. The LSTM receives zone id $\left(z\right)$, day of the week $\left(d_k\right)$, number of passenger demand in a zone, time $\left(D_{\mathit{zt}}\right)$ and whether the day is a holiday or not $\left(H\right)$ for a week-long sequence as input variables. The LSTM generates an output representing the sequential passenger demand pattern for 8 timeslots per day. To process sequential data using the LSTM, eleven weeks of data from the dataset were utilised for training, and its predictive accuracy was assessed on a separate, subsequent two-week test period. A comprehensive random search approach was employed to ascertain the hyperparameters, resulting in a configuration with two hidden layers, each containing 32 neurons. The model adopted a learning rate of 0.1, with the Mean Squared Error (MSE) as the loss function and Adam as the optimisation algorithm, selected for its adaptive learning rate capabilities.

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Fig. 3

The proposed LSTM model

Further enhancing model reliability, the LSTM model utilised L2 regularization to penalize complexity, dropout to reduce interdependencies among neurons, and early stopping to halt training when the validation score ceased to improve, thereby preventing overfitting. These strategies collectively strengthened the LSTM model, enhancing its ability to detect complex patterns within the passenger demand data.

Calculation of Dynamic Occupancy Rate

In classic shared taxi services, it is customary to allow up to four passengers to share a taxi, mirroring the real-world practice of taxi sharing [9, 66, 67–68]. During peak hours, a four-occupancy rate efficiently serves many passengers with fewer taxis. However, during off-peak hours, passenger demand drops, but the number of taxis and occupancy rate remain fixed. This study introduces the dynamic occupancy rate rebalancing approach to solve this issue. By increasing the occupancy rate during peak hours and by lowering it in off-peak hours.

The dynamic occupancy rate is calculated from $D_{\mathit{zt}}$ and the number of taxis available in each zone and timeslot $\left(T_{\mathit{zt}}\right)$ with the constrain of minimising the number of idle taxis per zone and time $\left({\stackrel{\vee }{T}}_{\mathit{zt}}\right)$ every three hours.

Equation 1 calculates the ratio of taxis working in each zone and timeslot with a particular occupancy rate $\left(r_{\text{zto}}\right)$, where $D_{\mathit{zt}}$ and $T_{\text{zt}}$ are fixed in each zone and timeslot; the only variable is the occupancy rate $\left(O_{\text{zt}}\right)$. We need to find the optimum $\left(O_{\mathit{zt}}\right)$, while accounting for the constraint of having minimum idle taxis per zone and time.

Figure 4 represents the optimisation process of dynamic $O_{\mathit{zt}}$. The process starts with an initial value of one for $O_{\mathit{zt}}$ and calculate $r_{\text{zto}}$ through Eq. 1 for each zone. For each zone, if $r_{\text{zto}}$is less than one, it means that some taxis will not receive passengers, so there will be idle taxis in the zone. The number of idle taxis is calculated afterwards for the current zone using Eq. 2. In case $r_{\text{zto}}$is greater than one, it means the occupancy rate does not meet the passenger demand; therefore, the process increases $O_{\mathit{zt}}$ one by one and recalculate $r_{\text{zto}}$till $r_{\text{zto}}\le 1$. The maximum value for $O_{\mathit{zt}}$ is four, because the maximum occupancy rate for taxis is four. The minimum $O_{\mathit{zt}}$ which meets the passenger demand and keep $r_{\text{zto}}<1$ will be the proper value of $O_{\mathit{zt}}$ for the current zone $z$.

1

$$r_{\mathit{zto}}=\frac{D_{\mathit{zt}}}{O_{\mathit{zt}}\text{*}{T}_{\mathit{zt}}}{O}_{\mathit{zt}}\in \left\{1,,,,2,,,,3,,,,4\right\}$$

2

$${\stackrel{\vee }{T}}_{\mathit{zt}}=\left(1,-,r_{\mathit{zto}}\right)\text{*}{T}_{\mathit{zt}}if{r}_{\mathit{zto}}<1$$

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Fig. 4

This is a process of calculating of dynamic occupancy rate

Suppose we increase $O_{\text{zt}}$ to four for a zone, but it is still insufficient to address passenger requests in the high-demand zone. In this case, there is a shortage of taxis, indicated as $S_{\text{zt}}$, which can be calculated through Eq. 3.

3

$$S_{\mathit{zt}}=\left(1,-,r_{\mathit{zto}}\right)\text{*}{T}_{\mathit{zt}}if{r}_{\mathit{zto}}>1$$

Note that if $r_{\text{zto}}$is equal to one, the number of taxis is sufficient for the passenger demand, and there is no need to calculate neither ${\stackrel{\vee }{T}}_{\mathit{zt}}$ nor $S_{\text{zt}}$, because both are equal to zero.

Taxi Transferring Procedure

Adjusting the occupancy rate based on passenger numbers balances most zones. However, when the occupancy rate reaches its limit and taxi shortages persist in some zones, we use a transfer procedure (Fig. 5) to move idle taxis from neighboring zones. Zones with touching boundaries are considered neighbors.

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Fig. 5

This is a process of taxis transferring

The taxi transferring procedure checks whether the total idle taxis in the neighbouring zones are sufficient for those passengers or not using Eq. 4.

4

$$\sum _{N_z}^{}{\stackrel{\vee }{T}}_{\mathit{zt}}\ge S_{\mathit{zt}}$$

In the condition presented in Eq. 4, $\sum _{N_z}^{}{\stackrel{\vee }{T}}_{\mathit{zt}}$ determines the total number of idle taxis in the neighbouring zones of a zone of interest. If the condition is met, the neighbouring zones’ taxis are sufficient to address the shortage in the zone of interest. Thus, we calculate how many taxis should be transferred from each neighbouring zone to the zone of interest with a shortage of taxis ($Trn\left(\stackrel{_{}^{\prime }}{z},,,z\right)$ Eq. 5). Regarding the constraint of having minimum idle taxis per zone and time, the zone with more idle taxis transfers more taxis, and the zone with fewer idle taxis transfers fewer taxis.

5

$$f{O}_{\mathit{zt}}<4Trn\left(\stackrel{_{}^{\prime }}{z},,,z\right)=\frac{\stackrel{\vee }{T}_{z_{}^{_{}^{\prime }}t}^{}}{\sum _{N_z}^{}{\stackrel{\vee }{T}}_{\mathit{zt}}}\text{*}{S}_{\mathit{zt}}z,z_{}^{_{}^{\prime }}\in {\text{N}}_{\text{z}}$$

In contrast, if Eq. 4 is not met, we increase the $O_{\mathit{zt}}$ of the neighbouring zones to increase the ${\stackrel{\vee }{T}}_{\mathit{zt}}$. We recalculate $r_{\mathit{zto}}$, and ${\stackrel{\vee }{T}}_{\mathit{zt}}$ according to Eqs. 1 and 2, respectively. To apply the constrain, we increase the $O_{\text{zt}}$ for the neighbour which has the minimum ${\stackrel{\vee }{T}}_{\mathit{zt}}$ among neighbours. This will be addressed by reducing the number of idle taxis in every zone. This process continues through the neighbouring zones till Eq. 4 is met. If the nearest neighbours cannot provide the required taxies, the idle taxis in the second and third-level neighbour zones can be transferred to the zone of interest. Then we calculate the number of taxis that should be transferred from each neighbouring zone (Eq. 5).

Note that the taxi transferring procedure only deals with the zones which remained imbalanced after occupancy rate calculation. Also, in the taxi transferring procedure, we transfer taxis from the zones with the $O_{\mathit{zt}}<4$ because the zones with the $O_{\mathit{zt}}=4$ either are imbalanced and have a shortage of taxis or are balanced zone without extra idle taxi capacity. Therefore, balancing the remaining imbalanced zones with extra taxis coming from zones with $O_{\mathit{zt}}<4$ does not affect the balance of previously balanced zones.

Centralised Dispatching System

Figure 6 shows the centralised dispatching system calculates the dynamic occupancy rate rebalancing using the LSTM model and determines the number of taxis that should be transferred through the taxi transferring procedure eight times a day (every 3 h). Subsequently, the system assigns passengers to the taxis every five minutes, thereby ensuring minimal waiting times.

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Fig. 6

Dispatch system workflow

Passengers are assigned to the same taxis when they have close origin and destination if they satisfy two conditions. First, their origins are in the same zones. Second, they must commute to the same or neighbouring zones.

The location of taxis and passengers, and drivers’ income are known to the centralised dispatching system. In the real world, this information is collected from the taxis’ GPS sensors and the mobile application installed on the handle device of the user.

The decision to divide the day into eight timeslots was intentional. Fewer timeslots might oversimplify demand fluctuations, while more could create redundant data. Therefore, choosing 8 timeslots offers an optimal balance between granularity and redundancy in passenger demand data, allowing for effective prediction of passenger demands.

Agent-based Model Using RL

To evaluate the calculated occupancy rates and test different scenarios, we developed an agent-based simulation where passengers and taxis interact. In the model, drivers pick up passengers from various origins and drop them at different destinations, requiring intelligent route planning. To handle this complexity, we implemented reinforcement learning [22], which adapts to the dynamic nature of taxi sharing, unlike traditional routing algorithms that rely on fixed rules for the shortest path.

RL methods have three main elements: states, actions, and rewards. States represent the current situation of the agent in the environment. Actions are the activities that change the states, and rewards are the environmental response to the agent’s action. Each action changes the agent’s state, and the environment rewards the agent based on the outcome of the action. The agent learns to select actions that maximise its total rewards [69].

To address the complexity of route planning in our agent-based model, we employed deep Q-learning (DQN), a popular RL method [70, 71].

DQN uses two deep learning networks to stabilize training and selects actions using an $ϵ$-greedy policy for balancing exploration and exploitation. Instead of random experience replay, we used Prioritized Experience Replay [45], which focuses on more impactful experiences. This helps optimize route planning, reducing travel distances and improving the order of passenger pick-ups and drop-offs.

The PER-DQN operates in the following settings:

Agent

Each taxi in the environment is an agent that operates independently. The agent interacts with the environment and receives rewards. Notably, there is no direct interaction between agents (taxis), as they are managed by a centralised dispatch system responsible for assigning passengers and directing them to their destinations.

Environment

The agent learns in the grided study area. The study area is two-dimensional, and each cell is identified with its row and column.

State

The state of the taxi includes the location of the taxi and the assigned passengers’ origins and destinations.

Actions

Moving to the eight possible neighbouring cells (four primary directions and four secondary directions) is considered as the eight actions of a taxi agent.

Rewards

The taxi receives a negative reward for every state, except when picking up or dropping off passengers, which earns a positive reward. This encourages the taxi to follow the shortest path to passengers and their destinations.

In Algorithm 1, PER-DQN method, the initialization phase sets up the Q-network and the target Q-network with random weights, denoted as $\theta$ and $\widehat{\theta }$, respectively. Both networks are initialized identically to establish a stable learning baseline. The replay buffer, crucial for storing experiences, is empty at the onset of training (Line 2).

As the simulation progresses, taxis and passengers are placed within the environment at random starting locations, with passengers assigned random origins and destinations. This randomness introduces variability into the simulation, reflecting the unpredictable nature of real-world scenarios (Line 3 to 6).

During training, the agent interacts with the environment by selecting actions based on its current state. After each action, the taxi transitions to a new state, and a reward is calculated based on the efficacy of the action taken by the $ϵ$-greedy policy. In every state, the reward is negative unless the taxi picks up or drops off the passengers (Line 8 - 9).

Critical to the PER-DQN approach is the prioritized replay buffer mechanism. Experiences, represented as tuples ${(s}_t,a,r,s_{t+1})$, are stored in the buffer. Here, $s_{t+1}$ represents the new state following action $a$. These experiences are not treated uniformly; rather, they are assigned probabilities $P\left(j\right)$ for being sampled, are stored in a sum tree [72], that reflect their importance, prioritising those that are deemed more informative for learning (Line 10–11). The experiences with the highest $P\left(j\right)$ are more important for updating the network, so they are selected more frequently.

When the number of experiences reaches the length of the experience reply’s batch, it is time to select the effective bunch of experience replies (Line 12–13).

Training updates are conducted in batches selected from the replay buffer based on prioritization. These updates are then applied to the target Q-network, achieving smoother learning updates over time.

The loss function used during these updates is critical for adjusting the parameters $\widehat{\theta }$ of the target network. It gradually improves the policy the $Q$-network follows, every $M$ step (Line 14 to 19).

Simulation

We simulated two scenarios on the agent-based model using RL with real-world NYC data to test and evaluate the STMoD compared to the classic shared taxi system. The classic shared taxi system scenario, representing the status quo in real-world settings, is a solid baseline for comparing our developed approach.

The STMoD scenario was implemented with the dynamic occupancy rate rebalancing, whereas the classic shared taxi scenario has a fixed rate. We compared these two scenarios to demonstrate the efficiency of our system in terms of the number of taxis earning money and decreasing passenger waiting time.

Case Study and Dataset

The study area is located in Manhattan, New York City, as illustrated in Fig. 7. It encompasses 63 zones, as defined in the publicly available dataset provided by the NYC Taxi & Limousine Commission. These predefined taxi zones are designed to accommodate traffic flow, urban planning considerations, and the operational needs of taxi and ride-sharing services in the city, as shown in Fig. 8(A).

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Fig. 7

Study area

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Fig. 8

(A) the 63 Manhattan Zones – (B) the number of taxis in each zone

We used a dataset including 22 million records of yellow taxis from July to September 2018. The selection of these three consecutive months was influenced by the high density and continuity of observations. This extensive dataset was provided by the Taxi & Limousine Commission [34], includes several key fields crucial for our study. The fields utilised are as follows:

• VendorID: Identifies the provider of the trip record.

• Tpep_pickup_datetime and tpep_dropoff_datetime: Timestamps marking the start and end of each trip, essential for determining trip duration and the exact timing of service demand.

• Passenger_count: Indicates the number of passengers per trip, relevant for assessing demand intensity.

• PULocationID and DOLocationID: Represent taxi zone codes for pickup and drop-off locations, crucial for analysing the spatial distribution of taxi demand.

These fields were selected to accurately model demand dynamics and optimise the dispatching system within the urban context of New York City. Detailed metadata for these fields is available [73], providing further clarity on data definitions and usage.

Scenarios

In our study, we designed the simulation to replicate real-world scenarios of taxi dispatch and mobility patterns, ensuring a robust evaluation of STMoD. The simulation was conducted using the agent-based model where taxis and passengers interact within a defined urban environment. Below, the important aspects of the simulation are highlighted.

Initialisation of Taxis

The initial distribution of taxis across the urban zones was calibrated based on historical demand data to ensure realistic representation of typical urban mobility patterns. We set the number of taxis in each zone to align with the peak passenger demand observed in that zone, ensuring that the supply of taxis closely matched the potential demand. Figure 8(B) illustrates the initial assignment of taxis in each zone, providing a visual distribution of taxis across the city in response to expected demand patterns.

Time Frame

The simulation progressed in discrete time intervals. Passengers were introduced into the system every 5 min, reflecting a continuous influx of demand. This interval was selected to simulate the dynamic nature of transport where passenger requests vary frequently.

Fare Calculation

Each taxi fare was calculated based on the number of states traversed from pickup to drop-off locations. A unit fare was charged per state, making the fare calculation straightforward.

Evaluation Metrics

The effectiveness of each scenario was evaluated based on several metrics, including the average waiting time for passengers, the average idle time for taxis, and the income generated per taxi. These metrics provided a comprehensive evaluation of the performance and efficiency of the proposed STMoD system compared to the classic shared taxi scenario.

Result

Predicting Passenger Demand Results

To assess the effectiveness of the LSTM model in predicting passenger demand, we employed the same LSTM setup for both STMoD and classic shared taxi scenarios. We compared the LSTM with LR, NN, and GRU algorithms. The aim of this comparison is to determine the model structure that best meets the specific requirements and complexities of the dataset used in this study. Mean Square Error (MSE) (Eq. 6) and Mean Absolute Error (MAE) (Eq. 7) metrics were exploited to compare the performance of the models for predicting passenger demand in two weeks test data.

6

$$MSE=\frac{1}{n}\sum _{i=1}^n{\left({\widehat{y}}_{\text{i}},-,y_i\right)}^2$$

7

$$MAE=\frac{\sum _{i=1}^n\left|{\widehat{y}}_{\text{i}},-,y_i\right|}{n}$$

In Eq. 6 and Eq. 7, $y_i$ is the observed passenger demand and ${\widehat{y}}_{\text{i}}$is the predicted passenger demand for n test records. Table 2 presents the MSE and MAE of each algorithm.

Table 2. Test results of the LSTM, LR, NN, and GRU models for predicting passenger demand

	Metrics
Methods	MSE	MAE
LR	0.92	0.73
NN	0.76	0.69
GRU	0.074	0.19
LSTM	0.070	0.17

LSTM and GRU models achieved competitive results on the test dataset, showing significantly lower MSE and MAE values compared to other methods. Additionally, NN performed better than linear regression (LR), as indicated by its lower MSE and MAE. However, NN is not as effective as the sequential models (LSTM and GRU).

For both LR and NN, the MSE and MAE values are relatively close because the errors made by these models are more uniformly distributed without significant outliers. However, LSTM and GRU have much lower MSE values, reflecting their high accuracy in most predictions. In contrast, their MAE values are comparatively higher because these models occasionally make larger errors on certain data points. MSE amplifies these few larger errors due to the squaring effect, leading to a noticeable difference between MSE and MAE.

Figure 9 illustrates the training loss of LSTM and GRU models over 200 epochs, offering a visual representation of their performance during training. Both models star with a similar loss around 1 and experience a rapid decline in the initial epochs, indicating effective learning from the training data. However, the LSTM model shows more decrease in loss compared to the GRU model as training progresses.

[See PDF for image]

Fig. 9

Training Loss for LSTM and GRU

In the first 50 epochs, the GRU model performs slightly better, with the losses of both models intersecting around this point. Beyond epoch 50, the LSTM model’s loss continues to decrease steadily, surpassing the GRU model’s performance. This suggests that the LSTM model is learning the data representation more effectively over time.

By epoch 200, the LSTM model has achieved a lower training loss compared to the GRU model. It’s important to note that the LSTM model was stopped early to prevent overfitting, as evidenced by its continued decrease in training loss.

Additionally, one of the findings concerned to the impact of the holiday parameter on demand prediction accuracy. Incorporating the holiday parameter significantly improved the model’s performance, as evidenced by a lower Mean Squared Error (MSE) of 0.070 in the test data. In contrast, excluding the holiday parameter led to a higher MSE of 0.085, underscoring its beneficial influence on enhancing the precision of passenger demand forecasts.

Calculation of Dynamic Occupancy Rate Results

In this section, we discuss the distinctions between the proposed methodology and the classic shared taxi model in terms of occupancy rate calculations. Specifically, the STMoD system employs a dynamic occupancy rate that adjusts based on passenger demand and taxi availability, derived from the LSTM predictions. Conversely, the classic shared taxi scenario utilises a fixed occupancy rate, as previously described in the 2.3.2 section.

Figure 10 shows the impact of the proposed STMoD system on driver earnings compared to the classic shared taxi scenario. By reducing the occupancy rate during off-peak hours, fewer taxis pick up four passengers when demand is low, reducing the need to cruise between zones. As seen in the chart, more taxis earned money in the 0–60 income range, particularly in the 20–40 units, while the number of high-earning taxis slightly decreased. This demonstrates that the STMoD system improves income distribution by allowing more taxis to earn money.

[See PDF for image]

Fig. 10

The taxis income in both scenarios

In the STMoD scenario, 22,847 taxis earned income, compared to 22,435 taxis in the classic shared taxi scenario. With a total of 23,497 taxis in our environment, the number of idle taxis in the STMoD scenario is 650, and 1,062 in the classic shared taxi scenario, signifying that the STMoD decreases idle taxis by 38%.

Taxis Transferring Procedure Result

This result is derived from the Taxi Transferring Procedure in the STMoD method. It is compared with the traditional shared taxi scenario, where no transferring procedure exists, and taxis remain in the same zone after dropping off the last passengers. These two scenarios are compared to assess their impact on passenger waiting times.

Figure 11 illustrates the reduction in passengers waiting for taxis from other zones after applying the taxi transferring procedure. In the classic shared taxi scenario, 617 passengers were waiting, while in the STMoD scenario, this number dropped to 477, representing a 22.69% decrease. This shows how the STMoD system improves zone balance and reduces passenger wait times.

[See PDF for image]

Fig. 11

The number of passengers waiting for taxis

Agent-Based Model Performance

We used the PER-DQN algorithm to implement the dynamic route planning within the simulation environment for both the STMoD and the classic shared taxi scenarios. To demonstrate the enhancements brought by PER-DQN, we compared its performance with the DQN algorithm, which has been extensively used in previous shared taxi studies [18, 68, 74]. Figure 12 shows the pattern of cumulative rewards which is gained by taxi through DQN and PER-DQN algorithms where PER-DQN was more stable. The horizontal axis indicates the number of training steps, and the vertical axis presents the average rewards. This algorithm improves the basic DQN by incorporating a prioritized experience replay mechanism. This mechanism enables the network to focus more on significant experiences during training. This is expected to result in faster and more robust learning, particularly in complex scenarios with varied passenger demands. The choice of PER-DQN over DQN is based on its ability to handle the high variability and sporadic nature of demand in mobility-on-demand systems. By prioritising experiences with high prediction errors, PER-DQN adapts more effectively to the environment, improving learning efficiency.

[See PDF for image]

Fig. 12

DQN and PER-DQN loss result

Discussion

In this paper, three aspects of taxi-sharing research are examined: pickup/drop-off, route planning, and pricing. Within the STMoD system, pricing is strategically managed to balance driver income and reduce the number of idle taxis, while also allowing passengers to benefit from shared fares. Additionally, the agent-based model approach addresses not only the efficient management of passenger pickup and drop-off at their respective origins and destinations but also employs route planning to find the shortest path and reduce passenger waiting times. Although these three aspects, pickup/drop-off, route planning, and pricing, are the focus of recent research, only a limited number of studies have addressed all three comprehensively. Table 3 presents an overview of the key studies in this field. A positive sign (+) indicates the aspect was addressed in the research, while a negative sign (-) signifies it was not covered. The pickup and drop-off aspect is a central concept with substantial attention across multiple studies [17, 31, 75, 76, 77, 78, 79, 80–81]. Conversely, route planning [75, 76, 79, 80] and pricing [81, 82, 83–84] are aspects that fewer studies have explored.

Table 3. Research comparison in three aspects of (1) pick up/Drop off (2) Route planning (3) pricing

Number	research	Pick up/ drop off	Route planning	Pricing
1.	[18]	+	-	-
2.	[75]	+	+	-
3.	[76]	+	+	-
4.	[82]	-	-	+
5.	[83]	-	-	+
6.	[77]	+	-	-
7.	[78]	+	-	-
8.	[32]	+	-	-
9.	[79]	+	+	-
10.	[80]	+	+	-
11.	[84]	-	-	+
12.	[81]	+	-	+

Among the studies referenced, the research by Connor Riley et al. [80] is most relevant to ours within the current context of large-scale ride-sharing system optimisation. A notable difference between our approaches lies in predicting passenger demand, specifically addressing holiday effects [36, 37]. While both studies aim to accurately predict passenger demand, Riley et al. [80]. do not specifically incorporate the impact of holidays into their demand-predicting model. In contrast, our study systematically includes holiday effects, analysing these impacts on passenger demand prediction compared to non-holiday periods. This addition allows for a more nuanced understanding and prediction of demand fluctuations, which are critical for optimising system performance during peak and off-peak times.

Furthermore, balancing the number of vehicles in ride-sharing systems can be designed to prioritize passenger and driver benefits [85]. While both our study and that of Riley et al. consider the minimization of passenger waiting times, Riley et al.‘s focuses more on passenger benefits. Our approach, however, extends beyond this by incorporating the economic aspects of ride-sharing for drivers. We introduce a dynamic occupancy rate rebalancing approach that not only aims to enhance passenger service but also ensures fair and equitable income distribution among drivers. This dual focus on both passenger and driver benefits creates a balanced system for all stakeholders.

Before discussing the prediction results, it is essential to evaluate several key factors regarding the dataset used in this study.

• Volume of Dataset: The dataset contains 22 million yellow taxi trip records over three months, providing a strong foundation for demand forecasting.

• Temporal Coverage: It spans different days of the week and all hours of each day, capturing daily and hourly fluctuations in demand.

• Spatial Detail: With pickup and drop-off data across 63 Manhattan zones, the dataset allows for precise spatial demand analysis.

Relevance of Features

Key features like timestamps, passenger counts, and taxi zones support forecasting, though adding weather data in future studies could further enhance predictions.

According Table 2, the LSTM model better handles the complexities of passenger demand data than the other three algorithms. Linear Regression (LR) struggles with non-linear relationships and lacks time dependency modeling, making it unsuitable for sequential data. Neural Networks (NN) can model non-linear relationships but fail to capture temporal dynamics, resulting in poor performance when past data significantly influences future outcomes.

In contrast, the analysis presented in Fig. 9 is critical as it highlights the LSTM model’s superior ability to manage dependencies in our dataset, which is essential for accurate predicting required in our STMoD system. The faster reduction in loss indicates that LSTM can more quickly adapt to the complexities of the data, leading to more robust and reliable predictions. While the GRU model demonstrates commendable performance, the LSTM consistently outperforms it in our evaluations.

The LSTM’s ‘forget gate’ is instrumental in managing information flow, allowing it to filter out irrelevant data points and focus on meaningful patterns over longer sequences. This feature is particularly beneficial given the structure of our dataset, where each week consists of 7 days, each divided into 8 sequences. Although the LSTM architecture is inherently more complex and involves more parameters, which might suggest a higher risk of overfitting, we have implemented several strategies to mitigate this risk. These strategies include the use of dropout layers, L2 regularization, and early stopping during the training process.

Figure 10 illustrates 266 taxis reach 60 income units in the classic shared taxi scenario because they are allowed to pick up four passengers without considering other drivers during off-peak hours. Accordingly, the number of taxis has increased by 71% more than the number of taxis has decreased in the STMoD scenario. Overall, by changing the occupancy rate and adjusting the distribution of drivers’ incomes, we have improved the balance of earnings among drivers by 28.18%. This adjustment ensures that more drivers earn money during off-peak hours.

Due to the accurate prediction of the LSTM model, all the zones are balanced after applying the taxi transferring procedure; thus, the number of passengers waiting for taxis from other zones declines when simulating real-world passenger requests. According to Fig. 11, the number of passengers waiting for a taxi to be transferred from other zones in the STMoD scenario is 477, compared to 617 for the shared taxi scenario. Therefore, we decrease waiting time by about 22%.

We compared PER-DQN with base DQN in the agent-based model over 3500 epochs, using average cumulative rewards. As shown in Fig. 12, PER-DQN outperforms DQN, with a more stable and consistent improvement, indicating better agent learning progress.

Our proposed method improves traffic flow and congestion management by optimizing the number of taxis in each zone and adjusting passenger loads, reducing the need for additional taxis. Efficient route planning further minimizes traffic, emissions, and road wear. Environmentally, it cuts cruising time and lowers fuel consumption and emissions, vital for urban areas. Economically, it ensures more consistent earnings for drivers and reduces passenger wait times and costs. The principles of demand prediction and dynamic occupancy can also be applied to other public transport systems, contributing to smart city initiatives and urban planning.

Conclusion and Future Work

We proposed the STMoD system to balance the number of passengers and the number of idle taxis in corresponding zones utilising a dynamic occupancy rate rebalancing approach using a predictive LSTM model and taxi transferring procedure administered by centralised dispatching. We simulated our proposed method using real-world data and demonstrated its efficiency in adjusting the drivers’ income. This included increasing the number of taxis that earned money, reducing idle taxis, and decreasing passenger waiting times compared to the current real-world scenario. By changing the occupancy rate, we allowed taxis to make money in their zones and avoided cruising between zones, which directly affected air pollution, fuel consumption, and taxi depreciation costs. Besides, our taxis learned to find the shortest path, which reduced taxis’ carbon footprint and travel time. Generally, the STMoD scenario takes advantage of shared taxis by raising occupancy rate in peak hours and non-shared taxis by decreasing occupancy rate in off-peak hours.

Our system capitalizes on the benefits of shared taxis by increasing occupancy rates during peak hours and decreasing them during off-peak hours, aligning service provision more closely with demand variability. The determining factors in calculating occupancy rates were passenger demand and the fixed number of taxis in urban settings. While our model already shows promising results, its predictive accuracy could be enhanced by incorporating additional variables such as weather conditions and proximity to major attractions like shopping centres and recreational facilities. Extending the training dataset to cover a full year would also improve our model’s ability to capture seasonal variations in taxi demand.

However, our study has limitations. For instance, we did not account for the dynamic nature of urban traffic, which can have significant impact on taxi availability and efficiency. Future research will aim to integrate real-time traffic data into our model, thereby refining our approach to simulate the complexities of urban transport environments more realistically. Moreover, exploring the application of our model in a multi-modal transport setting, where taxis interact with other forms of public and private transport, could yield further insights. Transitioning from centralised to decentralised dispatching using multi-agent reinforcement learning methods offers a promising avenue for enhancing the scalability and adaptability of our system. In conclusion, the STMoD system represents a significant step forward in the intelligent management of urban taxi services. It holds considerable potential not only for improving economic outcomes for taxi drivers but also for enhancing the overall efficiency of urban transportation networks. By continuing to refine this approach, we can better meet the challenges of urban mobility and contribute to more sustainable and responsive transportation systems.

Data Availability

The data will be made available upon request.

Declarations

Competing Interests

The authors report there are no competing interests to declare.

Abbreviations

SMoD

Shared mobility-on-demand

STMoD

Shared taxi mobility-on-demand

NYC

New york city

LSTM

Long short-term memory

RNN

Recurrent neural network

DQN

Deep q network

PER

Prioritised experience

DL

Deep learning

R.L.

Reinforcement learning

GRU

Gated recurrent unit

NN

Neural network

LR

Linear regression

MSE

Mean squared error

MAE

Mean absolute error

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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