Optimization Model of Express–Local Train Schedules Under Cross-Line Operation of Suburban Railway

Abstract

Cross-line operation and express–local train coordination are both crucial for enhancing the efficiency of multi-level urban rail transit systems. Most studies address suburban railway operations in isolation, overlooking coordination and inducing supply–demand mismatches that weaken system efficiency. This study addresses the joint optimization of cross-line operation and express–local scheduling by proposing a novel train timetable model. The model determines train service plans and departure times to minimize total system cost, including train operating and passenger travel costs. A space–time network represents integrated train–passenger interactions, and an extended adaptive large neighborhood search (E-ALNS) algorithm is developed to solve the model efficiently. Numerical experiments verify the effectiveness of the proposed approach. The E-ALNS achieves near-optimal solutions with less than 4% deviation from Gurobi. Comparative analysis shows that the proposed hybrid operation mode reduces total passenger travel cost by 6% and improves the cost efficiency ratio by 13% compared to independent operations. Sensitivity analyses further confirm the model’s robustness to variations in transfer walking time, passenger penalties, and waiting thresholds. This study provides a practical and scalable framework for optimizing train timetables in complex cross-line transit systems, offering insights for enhancing system coordination and passenger service quality.

1. Introduction

As metropolitan areas continue to expand, the demand for commuting between suburban and central urban regions has increased significantly. This has resulted in extended commuting distances and has positioned suburban railway as a critical mode for medium- and long-distance commuting. The passenger flow characteristics of suburban railways exhibit pronounced spatial and temporal imbalances, typically manifesting as tidal commuting patterns and substantial differences across line segments. In particular, major commuting nodes experience considerably higher passenger densities than small- and medium-sized stations along the line. In traditional multi-tier rail systems operated independently, a large number of suburban railway passengers are required to transfer to urban rail lines at hub stations. This leads to substantial transfer pressure, increased commuting time, and a decline in overall service efficiency and passenger experience. To address these challenges, several cities have implemented cross-line operation strategies between suburban railway and urban rail systems, aiming to enhance direct accessibility and alleviate the burden on transfer hubs.

Suburban railway commonly adopts express–local train organization in its operations. Under a cross-line operation framework, the integration of express–local services can improve the alignment between service supply and demand, thus reducing overall travel time and enhancing system responsiveness. However, such integration also substantially increases the complexity of timetable design and train service planning. As a result, there is a growing need for scientifically sound methods that can optimize train timetables in the combined context of cross-line operation and express–local coordination to ensure operational feasibility and improve overall system efficiency. Despite growing interest in suburban railway planning, most existing studies focus on optimizing train schedules under isolated operation modes, such as either cross-line service or express–local coordination alone. Few have explored the joint optimization of these two dimensions within an integrated framework. This lack of synergy limits the capacity of current models to cope with real-world complexities and provide coordinated solutions for multi-tier transit systems. To fill this gap, this study proposes a train timetable optimization model that jointly determines train service plans and departure times under a hybrid operation framework. A space–time network is constructed to capture interactions between trains and passenger flows, and an extended adaptive large neighborhood search (E-ALNS) algorithm is developed to solve the model efficiently. The proposed approach is evaluated through numerical experiments and sensitivity analyses, aiming to verify its performance and provide insights for the design of effective suburban–urban integrated rail operations.

Figure 1 presents the overall research flow of this study. The process begins with the identification of practical challenges and literature gaps in suburban railway scheduling, particularly under the combined context of cross-line operation and express–local service coordination. This is followed by the formulation of a timetable optimization model based on a space–time network, including an objective function and operational constraints. To solve the model, an extended adaptive large neighborhood search (E-ALNS) algorithm is designed with tailored destruction and repair operators. The algorithm is iteratively applied and refined to enhance solution quality. Computational experiments are then conducted, involving comparative analysis of different operation modes and sensitivity analyses on key parameters. The final stage synthesizes the findings and discusses implications for improving coordinated scheduling and operational performance in suburban railway systems.

2. Literature Review

In recent years, extensive research has been conducted on the optimization of urban rail operation organization, focusing on multiple aspects, including multi-transfer coordination, express–local train services, and cross-line operations. These studies aim to improve the passenger travel experience and better match transport supply with dynamic demand. For instance, Ding et al. [1] proposed a scheduling optimization method for large and small crossings based on passenger flow distribution to alleviate peak-hour congestion on specific segments. With the advancement of passenger flow data acquisition techniques, the research focus has gradually shifted from infrastructure constraints to passenger-oriented scheduling optimization. Niu et al. [2] proposed a train service optimization model based on accurate demand inputs, expressing passenger travelling behavior through 0–1 integer planning and designing genetic algorithms to solve it. Niu et al. [3] considered the time-varying passenger flow demand and constructed a train hopping scheme and schedule optimization model with the objective of reducing the waiting time of passengers. Shang et al. [4] and Li et al. [5] constructed an optimization model for express–local train schedules from the perspective of fairness and crossing behavior. Wong et al. [6] proposed a mixed integer planning model that minimizes the total waiting time while guaranteeing that all passengers reach their destinations, and verified through an arithmetic example that it was able to significantly improve synchronization as compared to the prevailing practice of using a fixed number of schedules and travel times. Yin et al. [7] proposed an optimization model for train scheduling and operation, aiming to minimize operational costs and passenger waiting time, and solved it using a heuristic algorithm based on Lagrangian relaxation. Different from most existing studies that treat these two problems separately, this paper proposes an integrated approach for the train scheduling problem on a bi-directional urban metro line in order to minimize the operational costs and passenger waiting time. Barrena et al. [8] proposed an adaptive large neighborhood search algorithm to efficiently optimize train headways for non-periodic railway lines. Dong et al. [9] proposed an integrated optimization model combining train stop plans and timetables under time-dependent passenger demand for commuter railways. The model, formulated as a mixed-integer nonlinear program, accounted for realistic factors such as no predefined schedules, variable train counts, and oversaturation. It aimed to minimize total passenger waiting time, stop-induced train delay, and train running time. An extended adaptive large neighborhood search (ALNS) algorithm was developed and validated through experiments and a real-world case, demonstrating improved scheduling efficiency and passenger service. Corman et al. [10] proposed an optimization model that accounts for the effect of boarding and alighting passenger volumes on dwell times, aiming to minimize passenger dissatisfaction, and solved it using a heuristic algorithm. Veelenturf et al. [11] developed a real-time scheduling method that responds to dynamic passenger demand by adding temporary stops at key stations. Using a heuristic algorithm, they demonstrated in Dutch railway cases that demand-responsive adjustments can substantially reduce total passenger delays. Gao et al. [12] proposed an optimization model for metro rescheduling under time-dependent and overcrowded passenger flow after a disruption. To ensure computational efficiency, the model was decomposed and solved using an iterative algorithm. Numerical experiments on the Beijing Metro demonstrated the method’s effectiveness in accelerating recovery by strategically skipping stations and reducing passenger accumulation.

Furthermore, other works have explored cross-line scheduling under system interoperability. Sun et al. [13] optimized the intersections and frequency of cross-line trains from the perspectives of saving travel time and enterprise cost. Zhu et al. [14] proposed a rescheduling model integrating flexible stopping and short-turning to minimize passenger delays, using time-dependent demand weights. Case studies on Dutch railways showed that flexible stopping significantly reduced delays and improved operational adaptability during disruptions. Zhou et al. [15] constructed an operation scheme based on infrastructure constraints. From a comparative perspective, Kurosaki [16] highlighted the institutional differences between Japan and Europe in organizing through-train services, noting Japan’s integrated management and operator collaboration versus Europe’s vertically separated and open-access framework, which presents important implications for the design of cross-line rail systems. Yang et al. [17] evaluated the impact of cross-line trains on line throughput capacity, and constructed a cross-line capacity allocation model with the objective of minimizing the cost of vehicle use, train operation cost, and passenger travel cost based on the analysis of the consumption of station capacity by cross-line train operation, which provided a quantitative optimization method for cross-line operation. Zhao et al. [18] proposed an optimization model to balance passenger cost and train load during peak cross-line operations. Introducing section load rate balance, the model used turn-back stations and train frequencies as decision variables to minimize travel cost, load imbalance, and operating cost. Case studies on Beijing’s Changping Line and Line 8 showed improved capacity balance and reduced overcrowding.

However, most existing studies treat cross-line operation and express–local train scheduling as independent systems. Skip-stop operation, a well-known acceleration method in urban rail transit, assigns each train a subset of stations to stop at. Abdelhafiez et al. [19] propose a nonlinear integer optimization model—later relaxed and linearized—to minimize passengers’ average travel time. The model accounts for five station arrangement scenarios, a safety collision constraint, and accommodates transfers and backtracking. Calculations show that this mode saves about 10 percent in passenger travel time compared to the all-stop strategy. Gao et al. [20] investigated the fast and slow train scheduling problem under the fixed overrun station mode and quantitatively analyzed the relationship between energy consumption, travel time, and timetable to reduce the operational energy consumption by optimizing the train overrun points and at the same time reduce the passenger travel time. Parbo et al. [21] formulated skip-stop pattern optimization as a bi-level model to minimize passengers’ generalized travel cost without increasing operational cost. The upper level is a mixed-integer skip-stop optimization model, while the lower level is a schedule-based transit assignment that captures passenger route choices. A heuristic solution method was developed to tackle the large-scale urban setting. Tested by example, the method reduced passenger travel time and lowered generalized cost. Huang et al. [22] introduced a path scale Logit model with fare factors to analyze the distribution of passenger flows under the cross-line hopping stations of national railroads and suburban railroads.

A unified modeling approach that simultaneously considers both aspects remains largely unexplored. An enhanced adaptive large neighborhood search (ALNS) algorithm [23] is developed to efficiently solve the proposed model. Canca et al. [24] proposed an integrated optimization model for railway rapid transit network design and line planning, which incorporated construction, operational, rolling stock, and personnel costs. Due to the problem’s computational complexity at realistic scales, the authors developed an adaptive large neighborhood search (ALNS) algorithm. They benchmarked the method against commercial solvers on a synthetic test instance and further applied it to the rapid transit network design in Seville, Spain. Moritz et al. [25] proposed an integrated planning model for classification yard operations, covering tasks such as train cut generation, car classification, and service scheduling. Due to its computational complexity, a tailored adaptive large neighborhood search heuristic was developed. For realistic instances involving up to 20 inbound and outbound trains, the method achieved high-quality solutions within 20 min on average, with an average optimality gap of only 0.5% when benchmarked against known optimal solutions.

3. Problem Statement

Consider a Y-shaped line composed of a single suburban railway line and a single urban rail transit line. Figure 2 shows the schematic diagram of urban rail transit and suburban railway operation. Assume that the total number of stations of the Y-shaped line is 12, the set of stations of the suburban railway line is $S_{urb}=\left\{s_1, s_2, \dots , s_5\right\}$, the set of stations of the urban rail transit is $S_{sub}=\left\{s_6, s_7, \dots , s_{12}\right\}$, suburban railway lines and urban rail transit connect at station $s_5/s_9$, and the station has cross-line conditions. In this paper, only the upward direction of the train is considered. The suburban railway train runs from station $s_1$ to station $s_5$ and then crosses the line to the urban rail transit to continue to run. This train is a cross-line operation train, and the set of urban rail transit stations covered by the cross-line operation is a co-lined section, i.e., station $s_9$ to station $s_{12}$. Stations $s_1, s_3, s_5$ are identified as major stations with high demand, and stations $s_2{, s}_4$ are identified as minor stations with low demand.

The research problem focuses on the optimization of train operation organization in the context of cross-line operation between suburban railways and urban rail transit. It adopts the joint decision-making of the train service plan and train schedule, with the set of train service plans $P=\left\{p_1, p_2, p_3, p_4\right\}$ representing different train operation strategies, covering the two dimensions of whether the train carries out cross-line as well as whether it adopts the express-local train mode or not, in which

(1)

$p_1$ indicates that the suburban train adopts stop-and-go mode and operates only within the suburban line;

(2)

$p_2$ indicates that the suburban train adopts the station stop mode based on cross-line to the common line section of the urban rail transit line;

(3)

$p_3$ denotes suburban trains using a large stop-and-go mode, skipping some small stops and operating only within the suburban line;

(4)

$p_4$ indicates that the suburban train adopts the large station express train mode based on the cross-line to the common line section of the urban rail transit line.

To effectively simulate the dynamic changes of the rail transit system during the operation period, this chapter discretizes the study period $\left|T\right|$ in equal-interval steps into a set of time nodes $T$. The optimization objective is to comprehensively optimize the cost structure of the operator and passengers, and the objective function is to minimize the sum of the train operation cost and the total passenger travel cost. The passenger travel cost components include the waiting cost at the originating station, the travel time cost of riding the train, the transfer walk cost in the rail network, and the penalty cost corresponding to passengers who fail to make a successful trip due to service capacity constraints.

3.1. Basic Assumption

The following assumptions are adopted to facilitate model formulation:

Assumption 1: The running time of the suburban railway train in each interval is known; the information related to the urban rail transit system is used as the input of the problem, including the passenger flow of the urban rail transit system and the planned schedule of the urban rail transit train;

Assumption 2: The cross-line operation conditions are satisfied between the suburban railway and urban rail transit lines, and the undercarriage utilization problem is not considered;

Assumption 3: All OD pairs of passenger flows are known.

3.2. Symbol Definition

We define the following sets, parameters, and variables:

Parameters and sets:

$T$: Set of time nodes where $T=\left\{t|t=1,\dots ,\left|T\right|\right\}$;

$P$: Set of train service plans for suburban railway trains, $P=\left\{p|p=p_1,p_2,p_3,p_4\right\}$;

$P^{*}$: Set of cross-line train service plans for suburban railway trains, $P^{*}=\left\{p|p=p_2,p_4\right\}$, $P^{*}\subseteq P$;

$S$: Set of all stations on suburban railway and urban rail transit lines,$\forall s\in S$, $S=\left\{s_1,s_2,\dots ,s_n\right\}$, where $n$ is the total number of all stations on suburban railway and urban transit lines;

$S_{urb}$: Set of all stations on the suburban railway line, $S_{urb}=\left\{s|s=s_1,s_2,\dots ,s_r\right\}$, where $r$ is the total number of all stations on the suburban railway line, $S_{urb}\subseteq S$;

$S_{sub}$: Set of all stations on the urban rail transit line, $S_{sub}=\left\{s|s=s_{r+1},{\dots ,s}_{r+g},\dots ,s_n\right\}$, $S_{sub}\subseteq S$;

$S*:$ Set of urban rail stations to be covered by suburban railway trains after cross-line operation, i.e., the set of stations in the co-located section, $S*\subseteq S$, $s_{r+g}$ is the set of urban rail transit stations where the suburban railway line meets the urban rail transit line;

$L_p:$ Set of station intervals covered by the suburban railway train service program $p$, $(s,s^{\prime })\in L_p$, $\forall p\in P$;

$S_p:$ Set of station covered by the suburban railway train service program $p$, $\forall s\in S_p,\forall p\in P$;

$W:$ OD pairs set, $\forall w\in W$;

$U_{wt}$: Number of passengers departing at time $t$ for OD pair $w$, $\forall w\in W,t\in T$;

$N:$ Set of points in space–time network, $\forall i\in N$;

$A:$ Set of arcs in space–time networks,$\forall \left(i,j\right)\in A$;

$o_w/d_w$: Origin and destination of OD pair $w$, $\forall w\in W$;

${\tau }_p^{ope}$: Operating hours for suburban railway train service plan $p$, $\forall p\in P$;

${\tau }_{ps}^{arr}/{\tau }_{ps}^{dep}$: Suburban railway train service plan $p$ arrival time from first station to station $s$, departure time, $\forall p\in P,s\in S$;

${\tau }_s^{tra}$: Transfer walk time for passengers transferring at station $s$, $\forall s\in S$;

$\Delta t$: Maximum single waiting time tolerated by passengers;

${\delta }_{s,1}^{arr}/{\delta }_{s,1}^{dep}/{\delta }_{\mathrm{s},1}^{da}$: Minimum arrival headway, minimum departure headway, and minimum arrival–departure headway at station $s$ after both suburban railway trains operate in cross-line mode, $\forall s\in S$;

${\delta }_{s,2}^{arr}/{\delta }_{s,2}^{dep}/{\delta }_{\mathrm{s},2}^{da}$: Minimum arrival headway, minimum departure headway, and minimum arrival–departure headway at station s between suburban railway trains after cross-line operation and urban rail transit trains, $\forall s\in S$;

$C_{ij}$: Cost of arc$(i,j)$, $\forall (i,j)\in A$;

$c_{train}^1/c_{train}^2$: Passenger cost per unit travel time on suburban railway trains, passenger cost per unit travel time on urban rail transit trains;

$c_{wait}/c_{walk}/c_{penalty}/c_{ope}$: Passenger cost per unit waiting time, passenger cost per unit transfer walking time, unit penalty cost for unserved passengers, and cost per unit operating time of suburban railway trains;

$Q$: Capacity of suburban railway trains;

$K$: Passenger flow thresholds for triggering cross-line trains.

Variables:

$x_{pt}$: Binary variable, where

• $x_{pt}=1$ if a suburban railway train that takes service timetable $p$ and has first departure time $t$ is put into service;
• $x_{pt}=0$ otherwise, $\forall p\in P,t\in T$;

${\pi }_{pt}^{ij}$: Binary variable where

• ${\pi }_{pt}^{ij}=1$ if the suburban railway train travel arc$(i,j)$ is covered by a suburban railway train that adopts service timetable $p$ and has first station departure time $t$;
• ${\pi }_{pt}^{ij}=0$ otherwise, $\forall p\in P,t\in T,(i,j)\in A$;

${\epsilon }_{s,p}$: Binary variable, where

• ${\epsilon }_{s,p}=1$ if a train stops at station $s$ in a suburban railway train service plan $p$;
• ${\epsilon }_{s,p}=0$ otherwise, $\forall p\in P,s\in S$;

$f_{ij}$: Non-negative integer variable assigned to the arc$(i,j)$ of the passenger flow, $\forall (i,j)\in A$;

${Dep}_{pts}$: The departure time at station $s$ for a suburban railway train operating under service plan $p$ with an initial departure time $t$, $\forall p\in P,t\in T,s\in S$;

${Arr}_{pts}$: The arrival time at station $s$ for a suburban railway train operating under service plan $p$ with an initial departure time $t$, $\forall p\in P,t\in T,s\in S$.

4. Space–Time Network Construction

Rail transit train scheduling optimization involves complex space–time constraints, and traditional methods make it difficult to dynamically portray train and passenger behaviors. For this reason, a space–time network is constructed to uniformly model train operation, passenger flow distribution, and system constraints. The method clearly describes the space–time paths of trains and passengers through a multi-commodity flow model, which is suitable for large-scale problems such as operating map optimization. Space–time network is constructed as shown in Figure 3.

The space–time network comprises four categories of points, the set of train arrival points $N_{arr}$, the set of train departure points $N_{dep}$, the set of passenger access points $N_{start}$, and the set of passenger termination points $N_{end}$, of which the set of train arrival points $N_{arr}=N_{arr}^1\cup N_{arr}^2$, $N_{arr}^1$, and $N_{arr}^2$ are, respectively, the set of suburban railway train arrivals and the set of urban railway train arrivals.

The set of train arrivals is the set of suburban railway train arrivals and the set of urban railway train arrivals. Additionally, there is a set of train departures and an urban rail train arrival point set; regarding the train departure point set, $N_{dep}=N_{dep}^1\cup N_{dep}^2$, $N_{dep}^1$, and $N_{dep}^2$ are the suburban railway train departure point set and urban rail train departure point set, respectively. In space–time, each point $i\in N$ is represented as a tuple containing four elements $\left({\eta }_1^i,{\eta }_2^i,{\eta }_3^i,{\eta }_4^i\right)$ that are associated with the type, train service plan or OD pair, station, and time node corresponding to that point, respectively. Specifically, when $i$ is a train departure point or a train arrival point, the value of ${\eta }_1^i$ is 1 or 2, and the value of ${\eta }_2^i$ is the suburban railway train service plan corresponding to point $i$. (If point $i$ is an urban rail train arrival point or an urban rail train departure point, ${\eta }_2^i=0$.) When point $i$ is a passenger access point or a passenger termination point, the value of ${\eta }_1^i$ is 3 or 4, and the value of ${\eta }_2^i$ is the number of the OD pair corresponding to point $i$. The set $\mathit{A}$ of arcs in the space–time network consists of six parts: the train traveling arc set ${\mathit{A}}_{\mathit{r}\mathit{u}\mathit{n}}$, the train stopping arc set ${\mathit{A}}_{\mathit{d}\mathit{w}\mathit{e}\mathit{l}\mathit{l}}$, the passenger transferring arc set ${\mathit{A}}_{\mathit{t}\mathit{r}\mathit{a}}$, the passenger accessing arc set ${\mathit{A}}_{\mathit{s}\mathit{t}\mathit{a}\mathit{r}\mathit{t}}$, the passenger end arc set ${\mathit{A}}_{\mathit{e}\mathit{n}\mathit{d}}$, and the passenger penalizing arc ${\mathit{A}}_{\mathit{p}\mathit{e}\mathit{n}\mathit{a}\mathit{l}\mathit{t}\mathit{y}}$. Similarly, the train traveling arc set ${\mathit{A}}_{\mathit{r}\mathit{u}\mathit{n}}$ can be divided into the suburban railway traveling arc set ${\mathit{A}}_{\mathit{r}\mathit{u}\mathit{n}}^1$ and the urban rail train traveling arc set ${\mathit{A}}_{\mathit{r}\mathit{u}\mathit{n}}^2$. The set of train stopping arcs ${\mathit{A}}_{\mathit{d}\mathit{w}\mathit{e}\mathit{l}\mathit{l}}$ can be divided into the set of suburban railway train stopping arcs ${\mathit{A}}_{\mathit{d}\mathit{w}\mathit{e}\mathit{l}\mathit{l}}^1$ and the set of urban rail train stopping arcs ${\mathit{A}}_{\mathit{d}\mathit{w}\mathit{e}\mathit{l}\mathit{l}}^2$. Each arc connects two space–time points in the space–time network, and the specific construction process of the space–time network is as follows.

For a suburban railway train with a train service plan of $p$ and a first station departure time of $t,N_{arr}^1,N_{dep}^1$ are, respectively, used to denote the train arrival and train departure points involved. The train traveling arc $A_{run}^1$ represents the running process of the train from the train departure point to the train arrival point, and the train stopping arc $A_{dwell}^1$ represents the stopping process of the train at the station, i.e., the running process from the train arrival point at the station to the train departure point at the station. The cost of train traveling arc and stopping arc $C_{ij}=c_{train}^1\left({\eta }_4^j-{\eta }_4^i\right)$. Similarly, a collection of urban rail train arrival points, a collection of urban rail train departure points, a collection of urban rail train traveling arcs, and a collection of urban rail train stopping arcs are constructed based on known urban rail train timetables, with the cost of the arcs $C_{ij}=c_{train}^2\left({\eta }_4^j-{\eta }_4^i\right)$. The construction of points and arcs is shown in Equations (1)–(4):

$$N_{pt}^{1,arr}=\left\{i=\left(1,p,s,t+{\tau }_{ps}^{arr}\right),\forall t\in T,p\in P,\left(s,s^{\prime }\right)\in L_p\right\}.$$

(1)

$$N_{pt}^{1,dep}=\left\{i=\left(2,p,s^{\prime },t+{\tau }_{p{s}^{\prime }}^{dep}\right),\forall t\in T,p\in P,\left(s,s^{\prime }\right)\in L_p\right\}.$$

(2)

$$A_{pt}^{1,run}=\left\{\left(i,j\right),\forall t\in T,p\in P,\left(s,s^{\prime }\right)\in L_p|\begin{array}{c}i=\left(2,p,s,t+{\tau }_{ps}^{dep}\right),\\ j=\left(1,p,s^{\prime },t+{\tau }_{p{s}^{\prime }}^{arr}\right)\end{array}\right\}.$$

(3)

$$A_{pt}^{1,dwell}=\left\{\left(i,j\right),\forall t\in T,p\in P,\left(s,s^{\prime }\right)\in L_p|\begin{array}{c}i=\left(1,p,s,t+{\tau }_{ps}^{arr}\right),\\ j=\left(2,p,s,t+{\tau }_{ps}^{dep}\right)\end{array}\right\}.$$

(4)

Each OD pair and departure time combination corresponds to a passenger access point $N_{start}$ and a passenger termination point $N_{end}$, which, respectively, denote the passenger’s start and end points in the space–time network under the combination. Each passenger access arc$(i,j)\in A_{start}$ denotes a passenger waiting for a train at the starting point, and the cost of the arc $C_{ij}=c_{wait}\left({\eta }_4^j-{\eta }_4^i\right)$. Each termination arc$(i,j)\in A_{end}$ denotes the process of passengers getting off the train after reaching the endpoint, and the cost of the arc $C_{ij}=0$. Each penalty arc$(i,j)\in A_{penalty}$ denotes the scenario in which a passenger is not served, and the cost of the arc $C_{ij}=c_{penalty}$. The specific point and arc construction process is described in Equations (5)–(9):

$$N_{wt}^{start}=\left\{i=\left(3,w,o_w,t\right),\forall t\in T,w\in W\right\}.$$

(5)

$$N_{wt}^{end}=\left\{i=\left(4,w,d_w,t\right),\forall t\in T,w\in W\right\}.$$

(6)

$$A_{wt}^{start}=\left\{\left(i,j\right),\forall j\in N_{pt}^{dep},w\in W,t\in T|\begin{array}{c}i=\left(3,w,o_w,t\right),\\ {\eta }_3^j=o_w,0\le {\eta }_4^j-t\le \Delta t\end{array}\right\}.$$

(7)

$$A_{wt}^{end}=\left\{\left(i,j\right),\forall i\in N_{pt}^{arr},w\in W,t\in T|j=\left(4,w,d_w,t\right),{\eta }_3^j=d_w,t\le {\eta }_4^j\right\}.$$

(8)

$$A_{wt}^{penalty}=\left\{\left(i,j\right),\forall w\in W,t\in T|i=\left(3,w,o_w,t\right),j=\left(4,w,d_w,t\right)\right\}.$$

(9)

The passenger transfer arc $A_{tra}$ considers both passenger transfers from suburban railway trains to urban rail trains and transfers between suburban railway trains and does not consider passenger transfers from urban rail trains to suburban railway trains. The total transfer time is the sum of the transferring walk time and the waiting time for a passenger to transfer from the transfer station from the current train to a train on another line, and the cost of the arc $C_{ij}=c_{walk}{\tau }_{{\eta }_3^i}^{tra}+c_{wait}({\eta }_4^j-{\eta }_4^i-{\tau }_{{\eta }_3^i}^{tra})$. The transfer arc construction is shown in Equation (10):

$$A_{ptp\prime t\prime }^{\mathrm{t}\mathrm{r}\mathrm{a}}=\left\{\left(i,j\right),\forall i\in N_{pt}^{1,arr},j\in N_{p\prime t\prime }^{1,dep}\cup N_{p\prime t\prime }^{2,dep}|\begin{array}{c}{\eta }_2^i\in \left\{p_1,p_3\right\},{\eta }_2^j\in \left\{p_0,p_2,p_4\right\},\\ {\eta }_3^i={\eta }_3^j=s_{r+g},0\le {\eta }_4^j-{\eta }_4^i-{\tau }_{{\eta }_3^i}^{tra}\le \Delta t\end{array}\right\}.$$

(10)

In addition to the above arcs used by trains and passengers, this paper also proposes conflict arcs that consider the train spacing case. For any pair of suburban railway train traveling arcs $(i,j)\in A_{run}^1$ and $(i\prime ,j\prime )\in A_{run}^1$, if the arrival and departure times of two trains at the same station do not meet the minimum tracking interval, they are said to be a pair of conflicting suburban railway train traveling arcs denoted by $(i,j,i\prime ,j\prime )$ and encapsulated by the set ${\mathit{\Theta }}_{pt}$, as shown in Equation (11):

$$\begin{gathered} {\mathit{\Theta }}_{pt} \\ =\left\{\left(i,j\right),\left(i^{\prime },j^{\prime }\right),\forall \left(i,j\right),\left(i^{\prime },j^{\prime }\right)\in A_{pt}^{1,run}|\begin{array}{c}\left(i,j\right)\ne \left(i^{\prime },j^{\prime }\right),{\eta }_3^i={\eta }_3^{i^{\prime }},\\ {\eta }_3^i,{\eta }_3^j\in S_{urb},\left|{\eta }_4^{i^{\prime }}-{\eta }_4^i\right|\le {\delta }_{{\eta }_3^i,1}^{dep}\end{array}\right\} \\ \cup \left\{\left(i,j\right),\left(i^{\prime },j^{\prime }\right),\forall \left(i,j\right),\left(i^{\prime },j^{\prime }\right)\in A_{pt}^{1,run}|\begin{array}{c}\left(i,j\right)\ne \left(i^{\prime },j^{\prime }\right),{\eta }_3^j={\eta }_3^{j^{\prime }},\\ {\eta }_3^i,{\eta }_3^j\in S_{urb},\left|{\eta }_4^{j^{\prime }}-{\eta }_4^j\right|\le {\delta }_{{\eta }_3^j,1}^{arr}\end{array}\right\} \\ \cup \left\{\left(i,j\right),\left(i^{\prime },j^{\prime }\right),\forall \left(i,j\right),\left(i^{\prime },j^{\prime }\right)\in A_{pt}^{1,run}|\begin{array}{c}\left(i,j\right)\ne \left(i^{\prime },j^{\prime }\right),{\eta }_3^i={\eta }_3^{j^{\prime }},\\ {\eta }_3^{i^{\prime }},{\eta }_3^{j^{\prime }}\in S_{urb},\left|{\eta }_4^{i^{\prime }}-{\eta }_4^{j^{\prime }}\right|<{\delta }_{{\eta }_3^j,1}^{da}\end{array}\right\}. \end{gathered}$$

(11)

Similarly, based on the known urban rail train timetable and the minimum tracking interval between a suburban railway train and an urban rail train after crossing the line, the set of suburban railway train traveling arcs conflicting with the known urban rail train timetable is constructed, and the set is denoted by $A_{run,pt}^{conflict}$, as shown in Equation (12):

$$\begin{gathered} A_{run,pt}^{conflict}=\left\{\left(i,j\right),\forall \left(i,j\right)\in A_{pt}^{run},\left(i^{\prime },j^{\prime }\right)\in A_{pt}^{2,run}|{\eta }_3^{i^{\prime }}={\eta }_3^i,|{\eta }_4^{i^{\prime }}-{\eta }_4^i|\le {\delta }_{{\eta }_3^i,2}^{dep}\right\}\cup \\ \left\{\left(i,j\right),\forall \left(i,j\right)\in A_{pt}^{run},\left(i^{\prime },j^{\prime }\right)\in A_{pt}^{2,run}|{\eta }_3^{j^{\prime }}={\eta }_3^j,|{\eta }_4^{j^{\prime }}-{\eta }_4^j|\le {\delta }_{{\eta }_3^j,2}^{arr}\right\}\cup \\ \left\{\left(i,j\right),\forall \left(i,j\right)\in A_{pt}^{run},\left(i^{\prime },j^{\prime }\right)\in A_{pt}^{2,run}|{\eta }_3^i={\eta }_3^{j^{\prime }},|{\eta }_4^{j^{\prime }}-{\eta }_4^i|\le {\delta }_{{\eta }_3^j,2}^{da}\right\}. \end{gathered}$$

(12)

5. Mathematical Model

5.1. Objective Function

This model takes minimizing train operation cost and total passenger travel cost as the optimization objective, in which the total passenger travel cost mainly consists of four parts: passenger travel cost (including running and stopping), passenger waiting cost, passenger penalty cost, and passenger transfer cost, as shown in Equations (13)–(15):

$$minobj=\sum _{t\in T} \sum _{p\in P} c_{ope}{\tau }_p^{ope}{x}_{pt}+\sum _{(i,j)\in A} C_{ij}{f}_{ij}.$$

(13)

$$x_{pt}\in \left\{\mathrm{0,1}\right\},\forall t\in T,p\in P.$$

(14)

$$f_{ij}\in N,\forall \left(i,j\right)\in A_{run},A_{dwell},A_{acc},A_{penalty},A_{tra}.$$

(15)

5.2. Constraints

To ensure the safety of the train in the process of traveling and to ensure that the train has enough time to be able to carry out the necessary technical operations at the departure station, the departure time of any two trains of the suburban railway line must meet the given minimum departure interval and the minimum departure interval constraints of the suburban railway train, as shown in Equation (16):

$$\sum _{p\in P} x_{pt}+\sum _{p\in P} x_{p{t}^{\prime }}\le 1,\forall t,t^{\prime }\in T|1\le t^{\prime }-t\le {\delta }_{s,1}^{dep}-1.$$

(16)

The passenger assignment conservation constraints ensure that each OD pair and departure time combination, respectively, corresponds to the passenger access point and passenger termination point passenger flow assignment conservation, and each train departure point and train arrival point passenger flow assignment conservation, as shown in Equations (17)–(19):

$$\sum _{(i,j)\in A|i=\left(3,w,o_w,t\right),j\in \mathrm{N}} f_{ij}=U_{wt},\forall w\in W,t\in T.$$

(17)

$${\sum }_{(i,j)\in A|j=\left(4,w,d_w,t\right),,i\in \mathrm{N}} f_{ij}=U_{wt},\forall w\in W,t\in T.$$

(18)

$${\sum }_{(i,j)\in A} f_{ij}={\sum }_{(j,i)\in A} f_{ji},\forall i\in N_{arr}\cup N_{dep}.$$

(19)

The train capacity constraint ensures that the number of passengers assigned to each traveling arc is less than the given suburban railway train capacity, as shown in Equation (20):

$$f_{ij}\le \sum _{t\in T} \sum _{p\in P} {\pi }_{pt}^{ij}{x}_{pt}Q,\forall (i,j)\in A_{run}.$$

(20)

Conflict arc constraints ensure that trains are safe to travel and that minimum tracking intervals are maintained between neighboring trains in the same zone, as shown in Equations (21) and (22):

$$\sum _{t\in T} \sum _{p\in P}\sum _{(i,j)\in A_{run,pt}^{conflict}} {\pi }_{pt}^{ij}=0.$$

(21)

$$\begin{gathered} {\sum }_{t\in T} {\sum }_{p\in P} {\pi }_{pt}^{ij}{x}_{pt}+{\sum }_{t\in T} {\sum }_{p\in P} {\pi }_{pt}^{i^{\prime }{j}^{\prime }}{x}_{pt}\le 1,\forall \left(i,j\right),\left(i^{\prime },j^{\prime }\right)\in \\ {A}_{run},\left\{\left(i,j\right),\left(i^{\prime },j^{\prime }\right)\right\}\in {\mathit{\Theta }}_{pt}. \end{gathered}$$

(22)

Equations (23) and (24) are used to decide whether a train should cross-line or not based on the cross-section passenger demand. Equations (25) and (26) are calculated for the arrival and departure time of the train at station s.

$$\sum _{o_w=s,d_w\in S_{sub}}{U}_{w\overline{t}}-K\ge -M·(1-{\mu }_{s\overline{t}}),\forall s\in S_{sub},\overline{t}\in T.$$

(23)

$$\sum _{t\in T} \sum _{p\in P^{*}}{\epsilon }_{ps}{x}_{pt}\ge {\mu }_{s\overline{t}},\left.\forall t,\overline{t}\in T\right|{\tau }_{ps}^{arr}\le \overline{t}-t,s\in S_{sub}.$$

(24)

$${Dep}_{pts}=(t+{\tau }_{ps}^{dep})·x_{pt},\forall p\in P,t\in T,s\in S.$$

(25)

$${Arr}_{pts}=(t+{\tau }_{ps}^{arr})·x_{pt},\forall p\in P,t\in T,s\in S.$$

(26)

6. Algorithmic Design

The extended adaptive large neighborhood search (E-ALNS) algorithm is used to solve the train timetable model. E-ALNS uses different destruction and repair operators to find the current solution neighborhood, evaluates the operators, and then selects them with a roulette strategy, which prevents the algorithm from falling into the local optimum too early, improves the global search ability, and is highly efficient in dealing with large-scale combinatorial optimization problems. This paper follows this classical algorithm, and the algorithm design covers the design of destruction and repair operators, operator selection, weight update rules, and termination conditions.

The purpose of the destruction operators is to perturb the current solution by deleting or modifying the existing train operation arrangements, thereby encouraging the exploration of new solution spaces. In this study, two categories of destruction operators are designed:

1.

Train Removal-Based Destruction Operators:

(a)

Randomly select a train from the current timetable and remove it.

(b)

Calculate the departure intervals between neighboring trains at each station, and remove the train whose deletion results in the smallest overall interval during operating hours.

(c)

For each train with service plan p and first-station departure time t, evaluate the number of passenger boardings at each station s. Remove the train with the lowest total number of passenger boardings.

2.

Service Plan-Based Destruction Operators:

(a)

Randomly change a train’s service pattern from a local train to an express train.

(b)

For each express train, count the total number of boardings at small stations. Identify the local train with the lowest small-station boardings and change it to an express train.

(c)

Randomly change a cross-line train into a non-cross-line train.

(d)

Identify the cross-line train with the lowest number of passengers traveling to segments shared with urban rail lines and convert it into a non-cross-line train.

The purpose of the repair operators is to restore feasibility by reinserting or adjusting trains within the solution to compensate for missing or unreasonable configurations caused by the destruction phase. Two categories of repair operators are developed:

1.

Train Reinsertion-Based Repair Operators:

(a)

Randomly select a train and reinsert it into the current timetable.

(b)

Compute the departure intervals at each station, identify the time slot with the largest interval across all stations, and insert a train at that point.

(c)

Identify stations with high passenger backlogs and insert a train into the timetable, with the specific train selected randomly.

2.

Service Plan-Based Repair Operators:

(a)

Randomly convert an express train back to a local train.

(b)

For each local train, identify small stations with severe passenger backlogs, and convert the associated train to an express train if appropriate.

(c)

Randomly convert a non-cross-line train into a cross-line train.

(d)

Identify the non-cross-line train with the highest passenger flow destined for segments shared with urban rail lines and convert it into a cross-line train.

In each iteration, a roulette selection mechanism is adopted to select the currently applied operator from the set of destruction and repair operators. To measure the contribution of each operator $\phi$ in the search process, a score ${\pi }_{\phi }$ is used to evaluate its performance in successive iterations, i.e., after each selection of operators, the corresponding score is increased according to the quality of the obtained solution $S\prime$. The logical framework of the E-ALNS algorithm is schematized as shown in the following Algorithm 1.

Algorithm 1. E-ALNS framework.

Algorithmic Process

Step 1 Input:    ${\mathit{S}}_{\mathit{c}\mathit{u}\mathit{r}}, {\mathit{S}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}\leftarrow$ Randomly generated initial solutions;    Destruction operator ${\mathit{\phi }}_{\mathit{d}}\in {\mathit{\varphi }}_{\mathit{d}}$ and its weight ${\mathit{\omega }}_{\mathit{\phi }\mathit{d}}$, $\mathrm{repair} \mathrm{operator}{\mathit{\phi }}_{\mathit{r}}\in {\mathit{\varphi }}_{\mathit{r}}$ and its weight ${\mathit{\omega }}_{\mathit{\phi }\mathit{r}}$, initial temperature ${\mathit{T}}_0$, cooling rate $\partial$, maximum iteration rounds ${\mathit{\alpha }}_{\mathit{m}\mathit{a}\mathit{x}}$, starting iteration rounds into the unimproved solution statistics ${\mathit{\alpha }}_{\mathit{t}\mathit{h}\mathit{r}\mathit{e}\mathit{s}\mathit{h}\mathit{o}\mathit{l}\mathit{d}}$, maximum number of iteration rounds for unimproved solutions ${\mathit{\alpha }}_{\mathit{e}\mathit{n}\mathit{d}}$. Step 2 Extended adaptive large neighborhood search:    ${\mathit{w}\mathit{h}\mathit{i}\mathit{l}\mathit{e} \mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{x}<\mathit{\alpha }}_{\mathit{m}\mathit{a}\mathit{x}}$:        $\mathit{w}\mathit{h}\mathit{i}\mathit{l}\mathit{e} \mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{x}\left(\mathit{n}\mathit{o} \mathit{b}\mathit{e}\mathit{t}\mathit{t}\mathit{e}\mathit{r}\right)<{\mathit{\alpha }}_{\mathit{e}\mathit{n}\mathit{d}}:$           $\mathrm{Select} \mathrm{a} \mathrm{destruction} \mathrm{operator} {\mathit{\phi }}_{\mathit{d}}\in {\mathit{\varphi }}_{\mathit{d}}$ with roulette;           ${\mathit{S}}_{\mathit{c}\mathit{u}\mathit{r}}$$\leftarrow \mathrm{destroy}({\mathit{S}}_{\mathit{c}\mathit{u}\mathit{r}}$);           $\mathrm{Select} \mathrm{a} \mathrm{repair} \mathrm{operator} {\mathit{\phi }}_{\mathit{r}}\in {\mathit{\varphi }}_{\mathit{r}}$ with roulette;           ${\mathit{S}}_{new}$$\leftarrow \mathrm{repair}({\mathit{S}}_{\mathit{c}\mathit{u}\mathit{r}}$);           Compare the objective values of the solutions:              If F(${\mathit{S}}_{\mathit{n}\mathit{e}\mathit{w}})<\mathit{F}\left({\mathit{S}}_{\mathit{c}\mathit{u}\mathit{r}}\right):$                 ${\mathit{S}}_{\mathit{c}\mathit{u}\mathit{r}}\leftarrow {\mathit{S}}_{\mathit{n}\mathit{e}\mathit{w}}$;                 If $\mathit{F}\left({\mathit{S}}_{\mathit{n}\mathit{e}\mathit{w}}\right)<\mathit{F}\left({\mathit{S}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}\right):$                    ${\mathit{S}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}\leftarrow {\mathit{S}}_{\mathit{n}\mathit{e}\mathit{w}}$;              Else evaluate ${\mathit{S}}_{\mathit{n}\mathit{e}\mathit{w}}$ using simulated annealing:                 $\mathit{r}\mathit{a}\mathit{n}\mathit{d}\leftarrow \mathit{U}[0,1]$;                 If $\mathit{r}\mathit{a}\mathit{n}\mathit{d}<{\mathit{e}}^{-\left(\mathit{F}\left({\mathit{S}}^{\mathit{\prime }}\right)-\mathit{F}\left({\mathit{S}}_{\mathit{c}\mathit{u}\mathit{r}}\right)\right)/\mathit{T}}$:                    ${\mathit{S}}_{\mathit{c}\mathit{u}\mathit{r}}\leftarrow {\mathit{S}}_{\mathit{n}\mathit{e}\mathit{w}}$;           Update operator scores:              If $\mathit{F}\left({\mathit{S}}_{\mathit{n}\mathit{e}\mathit{w}}\right)<\mathit{F}\left({\mathit{S}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}\right)$ then ${\mathit{\sigma }}_1;$              Else $\mathit{F}\left({\mathit{S}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}\right)\le$ $\mathit{F}\left({\mathit{S}}_{\mathit{n}\mathit{e}\mathit{w}}\right)\le$ $\mathit{F}\left({\mathit{S}}_{\mathit{c}\mathit{u}\mathit{r}}\right)$ then ${\mathit{\sigma }}_2$;              Else $\mathit{F}\left({\mathit{S}}_{\mathit{c}\mathit{u}\mathit{r}}\right)<\mathit{F}\left({\mathit{S}}_{\mathit{n}\mathit{e}\mathit{w}}\right)\mathrm{b}\mathrm{u}\mathrm{t}\mathrm{a}\mathrm{c}\mathrm{c}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{e}\mathrm{d} \mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}{\mathit{\sigma }}_3$;              Else no change;              Update the weights based on the operator scores;           ${\mathit{T}}^{\mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{x}}={\mathit{T}}^{\mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{x}-1}\mathit{\ast }\mathsf{\partial }$;        If $\mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{x}\ge {\mathit{\alpha }}_{\mathit{t}\mathit{h}\mathit{r}\mathit{e}\mathit{s}\mathit{h}\mathit{o}\mathit{l}\mathit{d}}$:           $\mathit{F}\left({\mathit{S}}_{\mathit{n}\mathit{e}\mathit{w}}\right)\ge \mathit{F}\left({\mathit{S}}_{\mathit{b}\mathit{e}\mathit{s}\mathit{t}}\right)$:              $\mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{x}\left(\mathit{n}\mathit{o} \mathit{b}\mathit{e}\mathit{t}\mathit{t}\mathit{e}\mathit{r}\right)=\mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{x}\left(\mathit{n}\mathit{o} \mathit{b}\mathit{e}\mathit{t}\mathit{t}\mathit{e}\mathit{r}\right)+1$;        $\mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{x}=\mathit{i}\mathit{t}\mathit{e}\mathit{r}\mathit{x}+1$; Step 3 Output: train_timetable, stop_plan, stoptime_plan

7. Computational Results

All computational experiments in this study were implemented in Python 3.9 on an Apple MacBook equipped with an 8-core M2 processor, 16 GB of unified memory, and macOS Ventura 13.5. The mathematical model was solved using the commercial optimization solver Gurobi, and an extended adaptive large neighborhood search (E-ALNS) algorithm was developed for instances. Python has been widely adopted in transportation and logistics research due to its flexibility and compatibility with optimization solvers such as Gurobi. Zheng et al. [26] proposed a matrix-based ALNS heuristic for air cargo network scheduling problems, demonstrating the effectiveness of Python and Gurobi in solving complex planning models. The train timetable optimization model is solved and analyzed in the upward direction of a Y-shaped line formed by a suburban railway and an urban rail line as an example. To verify the effectiveness of the proposed method, the solver Gurobi and the extended adaptive large neighborhood search algorithm proposed in this paper are used to solve the model based on the above demands, respectively, and the original passenger flow is multiplied by different fluctuation coefficients, which are used to simulate the fluctuation scenarios of the passenger flow and recorded as the cases of different demands, respectively.

During the solution process, the maximum runtime of Gurobi 10.0.1 is set to 3600 s, and the relative optimization gap is set to 0.001, i.e., the solution is terminated when the relative error between the current optimal solution and the lower bound of optimality is less than 0.1%. All other Gurobi parameters are kept at their default settings.

To enable comparison with an exact solver, we reformulate the original model into a tractable mixed-integer programming (MIP) form solvable by Gurobi 10.0.1. Specifically, a nonlinear constraint involving the product of two binary variables—used to represent whether a running arc is covered by an activated train—is moderately linearized. The original constraint, expressed as Equation (27), is transformed into a linear form:

$$f_{ij}\le \sum _{t\in T} \sum _{p\in P} {\pi }_{pt}^{ij}Q,\forall (i,j)\in A_{run}.$$

(27)

An auxiliary constraint ${\mathit{\pi }}_{\mathit{p}\mathit{t}}^{\mathit{i}\mathit{j}}\le {\mathit{x}}_{\mathit{p}\mathit{t}}$ is added to ensure logical consistency, indicating that a train arc can only be covered if the corresponding train is operated. This linearization simplifies the model while preserving key decision relationships. To further support solver compatibility, some additional relaxations are applied to reduce combinatorial complexity. These include decoupling certain binary interactions and simplifying selected constraints without altering the model’s core structure. The resulting relaxed model facilitates fair computational comparison with our proposed metaheuristic algorithm while retaining the integrity of the original decision-making logic.

The results of the objective function values obtained by the proposed algorithm and the exact solver are summarized in

Table 1. The solution of example.

$\mathbf{Coefficient} \mathbf{of} \mathbf{Fluctuation} \mathit{\theta }$	The Value of the Objective Function		Gap
	Gurobi	E-ALNS
0.8	2,558,611	2,640,264	3.19%
1.0	3,192,764	3,289,185	3.02%
1.5	4,758,355	4,940,115	3.82%

. Across the three test instances, the relative deviation between the objective values produced by the proposed method and those obtained by Gurobi is maintained within around 3%. Notably, when the passenger demand fluctuation coefficient is set to 1, the gap is reduced to only 3.02%, indicating that the proposed extended adaptive large neighborhood search (E-ALNS) algorithm achieves high solution accuracy in this case.

Therefore, the subsequent analysis of different operation modes is conducted based on the solution results for this representative case. In addition to delivering high-quality solutions, the proposed E-ALNS algorithm provides substantial modeling flexibility and adaptability. Its heuristic framework allows for the integration of complex real-world features such as flexible train service patterns, multi-scenario demand settings, and detailed passenger assignment logic, which are difficult handle efficiently using exact solvers. These advantages make E-ALNS suitable for solving large-scale or practically constrained scheduling problems under cross-line operation.

As shown in Figure 4, which demonstrates the curve of the objective function value with the number of iterations in the case of a fluctuation coefficient of 1, the objective curve decreases rapidly and steadily during the iteration process.

The algorithm demonstrates significant optimization performance, with a rapid decline in the objective function value observed within the first 150 iterations. In accordance with the predefined stopping criterion, the algorithm terminates at the 300th iteration, after 50 consecutive iterations without any improvement in the best-found solution.

7.1. Comparative Analysis of Different Models

In independent operation mode, no cross-line trains are operated, while in full cross-line operation mode, no small-crossing trains are included. To evaluate the effectiveness of the proposed hybrid cross-line operation mode that integrates both express and local trains, a comparative analysis with the two aforementioned modes is conducted.

Table 2. Comparison of key indicators under different operation modes.

	Full Cross-Line Operation/Independent Operation	Article Cross-Line Opera-Tion/Independent Operation
Ratio of objective function values	1.02	0.95
Ratio of train operating	1.04	1.08
Ratio of total passenger travel cost	1.02	0.94
Total passenger travel cost/train operating cost	0.97	0.87

Note: The values in the table are the ratios of the corresponding indicators under different operating modes to the corresponding indicators under independent operating modes; a value greater than 1 indicates an increase; a value greater than 1 indicates a decrease.

shows the train timetable objective function values and the ratio of each metric for this article’s cross-line, full cross-line, and independent modes of operation. As can be seen from the table regarding the implementation of this article’s cross-line mode of operation compared to the independent operation of the case, the objective function value decreases by 5%, the train operating costs increases by 8%, the total cost of passenger trips decreases by 6%, and the magnitude of passenger trip cost is greater than the train operating cost, indicating that this article’s cross-line mode of operation, although increasing train operating costs, does so in exchange for a significant reduction in the total cost of passenger trips. The ratio of total passenger travel cost to train operating cost is reduced by 13%, indicating that the article cross-line operation mode can effectively reduce the target value and improve the efficiency of passenger transportation per unit of train operating cost. Combining the results of the comparison between full cross-line operation and independent operation, where a 2% increase in the objective function value resulted in only a 3% reduction in the total passenger travel cost versus train operating cost, it is found that the article cross-line operation model is more effective.

As illustrated in Figure 5a,b, the full cross-line operation mode has the highest objective function value, total passenger travel cost, and passenger penalty cost among the three modes. Although it reduces the passenger transfer cost, in order to control the total train operating cost when the unit cost of train operation becomes higher, it reduces the number of trains, resulting in an increase in passenger penalty cost. It shows that the cost structure of this model is unbalanced, making it difficult to balance the contradiction between supply and demand, and it should be used when the demand for cross-line direct passenger flow is oversized and the short-distance passenger flow on this line is extremely low. At the same time, it shows that on the basis of cross-line operation, an appropriate increase in the number of small-crossing trains can further reduce passenger travel costs and access more passengers.

As shown in Figure 6, this article’s cross-line operation has the lowest number of unsuccessful trips and the shortest waiting time per capita, indicating that the effective integration of line resources through train cross-line operation significantly optimizes the efficiency of passenger trips, can provide direct service for cross-line passengers, and reduces the transfer walk time, transfer waiting time, and on-board travel time for some passengers. It also shows that this article’s cross-line operations are more advantageous than stand-alone operations in terms of increasing the number of successful passenger trips and reducing passenger transfers. It shows that the addition of crossover trains to independent operations can effectively reduce passenger travel costs and achieve efficient travel for more passengers.

The value of passenger transfer walking time ${\tau }_s^{tra}$ has a greater correlation with the total travel time of passengers, and its value will affect the passenger transfer behavior. Therefore, in order to further evaluate the influence of the passenger transfer walking time parameter on the results of train timetable optimization under the hybrid cross-line operation and the combination of express-–local trains, the passenger transfer walking time ${\mathit{\tau }}_{\mathit{s}}^{\mathit{t}\mathit{r}\mathit{a}}$ is set to be 2 min, 3 min, 5 min, 7 min, and 10 min for the comparison of the solutions, and the value of ${\tau }_s^{tra}$ = 5 min is taken as the baseline value.

Table 3. Results for different transfer walking time.

${\mathit{\tau }}_{\mathit{s}}^{\mathit{t}\mathit{r}\mathit{a}}$	Ratio of Objective Function Value to Baseline	Ratio of Train Operating Cost to Baseline	Ratio of Total Passenger Travel Cost to Baseline
2	0.99	0.97	0.99
3	0.99	1.04	1
5	1	1	1
7	0.98	1.06	0.98
10	0.98	1.08	0.97

lists the ratio changes of each index under different transfer walking time settings.

7.2. Sensitivity Analysis

(1) Transfer walking time

As the transfer walking time increases, the model cost structure changes significantly. From a ratio of 0.97 for a transfer walking time of 2 min to 1.08 for a transfer walking time of 10 min, the train operating costs show an upward trend, indicating that the model needs to mitigate the loss of accessibility due to the decrease in interchange efficiency by increasing the number of train runs. Too long or too short transfer walking time brings about an increase in the objective function value. The most optimal effect is achieved when the transfer walking time is 5 min.

As shown in Figure 7b, at 2 min transfer walking time, the number of transfers rises to 1.55 times that of the 5 min transfer walking time scenario, indicating that riders tend to choose the faster interline transfer path. As shown in Figure 7a, When the transfer time rises to 10 min, the number of transfers drops to 0, and all passengers abandon the transfer option altogether, resulting in a complete disappearance of both the transfer cost and the transfer path. This phenomenon reflects a clear threshold of transfer acceptance, indicating that there is a psychological tolerance threshold for transfer walking time, beyond which passengers will completely switch to non-transfer paths.

(2) The unit penalty cost

Penalty costs are generated for passengers due to not being served, and the results of the metrics are shown in

Table 4. Results for different unit penalty cost for unserved passengers.

${\mathit{c}}_{\mathit{p}\mathit{e}\mathit{n}\mathit{a}\mathit{l}\mathit{t}\mathit{y}}$	Ratio of Objective Function Value to Baseline	Ratio of Train Operating Cost to Baseline	Ratio of Total Passenger Travel Cost to Baseline
300	0.93	0.92	0.93
500	0.96	1.02	0.95
700	1	1	1
900	1.04	0.98	1.05
1100	1.07	1.01	1.07

when the penalty costs are changed. The unit penalty cost is set to 300, 500, 700, 900, and 1100, respectively, and the penalty cost = 700 is set as the baseline value. From the table, it can be seen that the unit penalty cost will lead to the objective function value and the total travel cost of passengers. As can be seen in Figure 8, when the unit penalty cost is elevated, the direction of optimization at this time is to reduce the number of unserved passengers, but at the same time, due to the increase in the unit penalty cost, even if the number of passengers is reduced, the total penalty cost still shows a continuous upward trend.

(3) The unit passenger waiting cost

Passengers waiting for a train at a station incur waiting costs, and when the unit waiting cost increases, the passenger waiting cost and waiting time are shown in Figure 9. By adjusting the train schedule, the waiting time is reduced, but the passenger waiting cost shows an increasing trend, indicating that the unit waiting cost has a significant impact on the passenger waiting cost.

(4) Maximum passenger waiting time

The maximum passenger waiting time $\Delta t$ has a close relationship with whether passengers can travel successfully, i.e., its value directly affects the number of unsuccessful passenger trips. Therefore, to further evaluate the impact of the maximum waiting time parameter on the optimization results of train schedules under the combination of cross-line operation and express–local train in this chapter, the maximum waiting time of passengers is set to be 3 min, 6 min, 10 min, and 14 min for the comparison of the solutions, and the value of $\Delta t=10\mathrm{m}\mathrm{i}\mathrm{n}$ is taken as the benchmark value.

As shown in

Table 5. Results for different maximum passenger waiting time.

$\Delta \mathit{t}$	$\Delta \mathit{t}=3$	$\Delta \mathit{t}=6$	$\Delta \mathit{t}=10$	$\Delta \mathit{t}=14$
Objective function value	3,725,174	3,315,659	3,289,185	3,222,923
Ratio to baseline	1.12	0.99	1	1.02

, the objective function value is highest at $\Delta t=3\mathrm{m}\mathrm{i}\mathrm{n}$, which is 12% higher than the $\Delta t=10\mathrm{m}\mathrm{i}\mathrm{n}$ scenario, indicating that when passenger tolerance is low, the system runs more trains to avoid the penalty cost increase, and therefore the overall cost increases. When $\Delta t=10\mathrm{m}\mathrm{i}\mathrm{n}$, continuing to increase the maximum passenger waiting time also causes the overall cost to increase, indicating that the system balances between service capability and cost at $\Delta t=10\mathrm{m}\mathrm{i}\mathrm{n}$. Combined with Figure 10, it can be seen that as the maximum passenger waiting time increases, the number of unsuccessful trips significantly decreases and then flattens out, and the per capita waiting time rises slightly, suggesting that significantly alleviating the problem of successful passenger trips may result in a slight compromise in the waiting experience. This indicates that setting a reasonable passenger maximum waiting time threshold can avoid resource waste while safeguarding the service level to determine the optimal service strategy.

8. Conclusions

This study investigates the train scheduling problem involving express–local trains under a cross-line operation framework between suburban railways and urban rail transit. By adopting a space–time network modeling approach, an integrated physical network of the suburban railway and urban rail systems is constructed. Based on this, various categories of passenger space–time arcs and train operation arcs are further defined. A mathematical optimization model is formulated to minimize the total system cost, including both the total passenger travel cost and operator cost, while accounting for constraints such as passenger flow distribution and minimum departure intervals.

To solve the model, an extended adaptive large neighborhood search (E-ALNS) algorithm is developed, incorporating customized destroy and repair operators tailored for express–local train combinations and cross-line timetable design. The model is reformulated into a linearized form to enable comparison with exact optimization results from Gurobi. Numerical experiments demonstrate that the proposed algorithm produces high-quality solutions with a relative gap below 4% in all tested instances, and 3.02% in the base demand scenario. Further comparisons among independent, full cross-line, and hybrid cross-line modes show that the hybrid mode achieves the best performance, reducing passenger travel costs by 6% and improving overall cost efficiency despite a slight increase in operating cost. These results indicate that integrating long and short cross-line trains enables a more balanced trade-off between supply and demand. Sensitivity analysis on key parameters reveals that a 5- min transfer walking time yields the most efficient outcome, while excessively short or long walking times degrade system performance. Higher penalty and waiting costs encourage operational adjustments but increase total cost due to value accumulation. A 10- min passenger maximum waiting time achieves a good balance between service coverage and waiting time, minimizing unserved passengers without excessive delay.

Future research can be extended in several directions. First, the passenger demand in cross-line operations may exhibit temporal fluctuations; incorporating dynamic demand into the timetable optimization framework would enhance model applicability. Second, this study focuses on mixed operation with express–local trains capable of flexible stop-skipping; further analysis can be conducted by comparing scenarios where only express or local trains are deployed. Third, although this study adopts a single-objective cost-based formulation to ensure computational tractability and consistency in evaluating operational and passenger-related costs, future research could explore multi-objective formulations to better balance trade-offs among service quality, equity, and cost-efficiency, especially in contexts involving conflicting interests between operators and passengers. Finally, the current model assumes fixed train formation sizes. Future work may explore flexible train formations in conjunction with demand-responsive passenger assignments to achieve more refined optimization results.