Optimizing Scheduled Train Service for Seaport-Hinterland Corridors: A Time-Space-State Network Approach

Abstract

Effective cooperation between railways and seaports is crucial for enhancing the efficiency of seaport-hinterland corridors (SHC) . However, existing challenges stem from fragmented decision-making across seaports, rail operators, and inland cities, leading to asynchronous routing and scheduling, suboptimal service coverage, and delays. Addressing these issues requires a comprehensive approach to scheduled train service design from a network-based perspective. To tackle the challenges in SHCs, we propose a targeted networked solution that integrates multimodal coordination and resource optimization. The proposed framework is built upon a time-space-state network model, incorporating service selection, timing, and frequency decisions. Furthermore, an improved adaptive large neighborhood search (ALNS) algorithm is developed to enhance computational efficiency and solution quality. The proposed solution is applied to a representative land–sea transport corridor to assess its effectiveness. Compared to traditional operational strategies, our optimized approach yields a 7.6% reduction in transportation costs and a 56.6% decrease in average cargo collection time, highlighting the advantages of networked service coordination. The findings underscore the potential of network-based operational strategies in reducing costs and enhancing efficiency, particularly under unbalanced demand distributions. Additionally, effective demand management policies and targeted infrastructure capacity enhancements at bottleneck points may play a crucial role in practical implementations.

1. Introduction

Seaport-hinterland corridors (SHC) play a vital role in regional economic development by enhancing connectivity between seaports and inland cities [1]. A typical SHC consists of seaports, inland cities, and an integrated transport network, where rail transport serves as a crucial backbone for long-distance freight movement. These corridors are vital not only for the competitiveness of seaports but also for inland cities seeking seamless integration into the global trade network.

To improve logistics efficiency along these corridors, previous research has mainly focused on infrastructure enhancements, such as the development of dry ports [2] and the expansion of inland transport facilities [3,4]. However, these efforts have often overlooked the importance of optimizing transportation services [5]. With inland transport, particularly rail, becoming an increasingly competitive domain for seaport operators [6,7,8,9,10], it has become imperative to address inefficiencies in service coordination. Scheduled train services offer reliable departure and arrival times, stable operating frequencies, and higher capacity, making them a cost-effective solution for long-distance freight transport. These features make scheduled trains vital for SHCs, providing a structured approach to reduce service disruptions and enhance overall corridor efficiency [11,12,13]. Despite their importance, the optimization of scheduled train services in SHCs has received limited research attention. This lack of focus can be attributed to the predominant seaport-centric perspective in existing studies, which often overlooks optimizing rail transport. As a result, inland cities’ needs are underrepresented, leading to conflicts among stakeholders and limiting the overall efficiency of these corridors [9,14,15].

This paper addresses these gaps by focusing on the New Western Land–Sea Corridor (NWLSC) in Western China. Based on an actual project investigation, we identify challenges in the NWLSC, such as limited service coverage, poor timeliness, and unpredictable schedules, which stem from disjointed city-level decision-making and a lack of coordination. As a collaborative initiative between the Association of Southeast Asian Nations (ASEAN) countries and western Chinese provinces, the NWLSC is a vital international SHC and a key trade route. However, operational inefficiencies hinder its potential. The rail service provider is tasked with designing scheduled train services to transport export container cargo from various inland cities to the export seaport, spanning a defined planning horizon. Traditionally, a point-to-point operational approach is adopted, where only local demand and capacity are considered. However, the imbalance in development between western provinces and the uneven distribution of cargo along the NWLSC present significant challenges. In this context, individual transportation for smaller cities becomes uneconomical due to low demand compared to the available rail transport capacity. Conversely, cities with sufficient demand volumes may benefit from demand consolidation across multiple locations, but over-consolidation could harm the interests of other cities. Thus, the lack of coordination in routing and scheduling exacerbates inefficiencies, making it critical to address these challenges through more integrated, corridor-wide planning. These practical challenges provide the motivation for our study, which aims to address these inefficiencies by proposing a more coordinated networked approach.

In this study, we formalize the problem as a Scheduled Service Network Design with Synchromodal Transportation (SSND-ST), a variant of the scheduled service network design problem, where decisions on train scheduling, routing, and frequency must be optimized to ensure efficient and cost-effective operations. We propose a networked operational approach to balance transport cost-efficiency with customer interests by reasonably consolidating demand across the network. Since the destination is the export seaport, the problem inherently exhibits multimodal transport characteristics. Therefore, when planning rail services, it is necessary to consider synchronized planning (synchromodal) between rail services and port operations, specifically incorporating the time windows of port operations.

By incorporating the principles of networked service coordination, our approach offers insight with both theoretical and practical implications, enhancing the efficiency and reliability of scheduled train services. Our key contributions are as follows:

• We introduce a Networked Scheduled Train Planning (NSTP) model that employs a time-space-state network to formulate the SSND-ST problem. By extending the state network to incorporate stop patterns, load volume, and frequency, our approach provides a more comprehensive representation of real-world operational constraints. Compared to conventional modeling methods, this formulation significantly reduces the number of decision variables and complexity, enhancing both efficiency and scalability. To solve the model efficiently, we develop a tailored adaptive large neighborhood search algorithm (ALNS) to ensure computational efficiency.
• Demonstrating the application of our solution to the New Western Land–Sea Corridor (NWLSC), we validate its practicality and effectiveness. Our approach, which includes both the TSS framework and the ALNS algorithm, has led to substantial improvements in operational efficiency, reductions in transportation costs, and optimization of travel times, directly benefiting the involved stakeholders. Although our study is based primarily on the NWLSC, the principles and challenges addressed may resonate in similar contexts worldwide [14,16,17,18].

This paper is organized as follows. Section 2 reviews the relevant literature on SHCs, service network design of scheduled trains, and time-space-state network applications. Section 3 defines the research problem and details the NWLSC case study. Section 4 introduces our NSTP model based on a TSS network. Section 5 discusses the findings of our experiments. Finally, Section 6 concludes the paper by summarizing the key outcomes and their broader implications.

2. Literature Review

2.1. Seaport-Hinterland Corridor

The evolution of the trade industry since the 1960s has been profoundly shaped by the advent of container transportation, initiating a significant shift in seaport networks [19]. This era saw the emergence of corridor-shaped hinterlands, increasingly recognized as a key strategy to enhance the global competitiveness and market positioning of seaports [20]. Although the development of dry ports to improve SHCs has received extensive attention, encompassing conceptual models [21], location selection [22], and capacity optimization [23], the effectiveness of transport services, particularly for dry ports along these corridors, remains a critical yet underexplored aspect. Without reliable transport services, the full potential of these corridor infrastructures cannot be realized, leading to inefficiencies and increased costs.

In addition, previous literature viewed the role of SHCs primarily as a solution to seaport congestion [24], but the demands of the rail industry or inland cities are not being met. It was also noted that the lack of high degrees of logistic collaboration limits efficiency gains [15], and few studies have explicitly considered this issue [25].

Numerous studies have highlighted the key role of rail transport in cost savings and time efficiency [7], environmental protection [2], and market share enhancement [17] within these corridors. However, research on the scheduling of rail transport services in SHCs is still insufficient. Yıldırım et al. [26] provided a scheduling solution for freight train shuttles on a shared rail corridor connecting a seaport and a dry port. Furthermore, the design of a comprehensive freight train service network that includes multiple seaports and dry ports remains an area that has not been adequately explored. These research gaps highlight the need for more focused research on optimizing rail transport services to improve corridor performance.

2.2. Service Network Design for Scheduled Trains

The problem of scheduled train service network design forms a crucial aspect of tactical freight planning. This complex process is often conceptualized as a time-dependent service network design problem, widely known as the SSND problem. For a comprehensive understanding of general SSND issues, one can refer to the review by [27]. However, this section focuses on literature specifically related to scheduled trains.

Most SSND models address a wide range of planning challenges, extending beyond mere service selection, which categorizes them as integrated planning methods. These methods involve not only choosing services but also optimizing the schedules of these services to align with time-dependent demands. The integrated optimization approach in SSND for trains, which includes service selection, path planning, flow distribution, terminal policies, and scheduling optimization, is well illustrated in studies by [28,29,30]. The primary aim is to fulfill the demand utilizing available resources while minimizing the system’s total cost.

When it comes to SHCs, the service design problem is multifaceted, addressing several optimization needs like cost reduction and efficiency improvement. SSND emerges as a potent method to improve rail transport operations in SHCs, leveraging its inherent integrated and systemic perspective.

Most SSND models offer comprehensive planning methods; yet, there is insufficient research on their application specifically to SHCs. This indicates an opportunity for our research to contribute to the field by applying the SSND model in a new context, particularly focusing on the complexities of these corridors.

2.3. Time-Space-State Networks in the SSND Problem

The concept of time plays a pivotal role in the SSND problem. A prevalent approach to incorporating time into network design issues is the use of time-expanded networks, as outlined by [31]. These networks are based on static networks, with a defined planning horizon and a specific time discretization strategy [32].

Time-space (TS) networks, which represent the timing of objects, events, and decisions at precise moments, are often employed to address time-dependent factors in network design. A fundamental illustration of a TS network can be seen in Figure 1. Here, the space dimension signifies the physical nodes, whereas time is discretized into specific intervals. The TS network framework facilitates the pre-definition of demand and resource constraints within the network.

In Figure 1, the green line represents the customer demand entering the network, the blue line represents the correlation between demands and services, the purple line indicates the service path, and the black line reflects actual rail lines in the Resource Layer. When designing a service, it is necessary to take into account both the quantity and preferences from the demand layer, as well as the capabilities and constraints imposed by the resource layer, as illustrated by the red arrows in the figure. The rectangles in the resource layer represent the stages during which a service remains stationary in physical space—for example, when a train is at a stop. In such cases, the spatial dimension remains unchanged, while the temporal dimension continues to evolve.

Recent research has proposed various TS networks to model rail transport service network design problems. Yao et al. [33] developed a time-space network based on railway infrastructure to optimize train stop patterns and schedules from a passenger-centric perspective. Yin et al. [34] constructed a time-space network to jointly optimize rolling stock allocation and train timetabling in urban rail transit systems. Ambrosino and Asta [35] tackled a logistics issue in a rail-sea transfer node using an operation-time-space network. Wang et al. [36] utilized a space-time-speed network methodology to generate train trajectories that encapsulate both driving strategies and timetables. Zhu et al. [30] introduced a modeling framework that employs a three-layer cyclic space-time network to represent operations, decisions, their interrelations, and temporal dimensions to address the challenge of scheduled service network design. Zhang et al. [37] innovated an approach to reformulate the integer programming model for the problem of cyclic train timetabling using an extended TS network, demonstrating potential applicability to a wider range of scheduling and planning issues with regular activities.

To further streamline modeling, some studies have extended the traditional time-space network by introducing additional dimensions. Mahmoudi and Zhou [38] developed a state-indexed time-discretized time-space-state (TSS) network. Guan et al. [39] proposed a bi-level TSS network-based model that explicitly incorporates passenger load states. The model optimizes routing plans for customized bus services while considering multi-trip operations, time windows, vehicle capacity, and mixed passenger flows.

The application of TSS networks in solving scheduled service network problems is a growing area of research. However, there is a need to adapt and expand these frameworks to the specific challenges of SHCs, which is the focus of our study.

2.4. Research Gaps and Focus

This review of the relevant literature underscores several key challenges in improving rail transport services for SHCs. Firstly, the design of the service network for scheduled trains in these corridors is a largely unresolved area. Secondly, the multi-stakeholder nature of these corridors, involving seaports, rail services, shippers, and local governments, calls for a more integrated approach, which is currently lacking. Finally, the potential of TSS networks to address these challenges is evident but requires further exploration and adaptation. Our research aims to bridge these gaps by developing a comprehensive model that addresses these challenges and diverse stakeholder needs within SHCs.

3. Problem Definition and Notation

3.1. General Problem Statement

A SHC consists of a combination of inland cities, a seaport, rail stations, and potential transport routes. Each city has a different number of containerized cargoes that need to be sent to the seaport and then delivered to foreign destinations, and every city is equipped with a rail station to consolidate these cargoes. Railway operators must decide how to set up different types of trains to meet the demands of the entire corridor within a given planning horizon.

This paper focuses on a problem characterized by multi-objective planning, regular service, and rail-sea intermodal integration. First, since the SHC involves many participants, the design of rail transport services must coordinate the interests of all stakeholders. Second, because the scheduled trains must operate on a regular timetable, regular service is an inherent feature of the problem. Moreover, since the application context is SHC, rail-sea intermodal integration is another critical factor that needs to be considered.

The external inputs of the problem include:

• Rail network: A rail network connects inland rail stations and the seaport terminal along the corridor, with capacity limits specified for each segment.
• Potential train service patterns: Each train service pattern is defined by its origin/destination (OD) stations, capacity, mode (stopping patterns), and earliest and last departure times. In this context, the term mode refers to a series of stations and segments, as well as to the station at which a train must stop. To ensure service efficiency, two modes—direct train and step train—are predetermined in this study. Direct trains [40] do not stop between the origin and destination, whereas step trains include one intermediate stop.
• Transport demands: This dataset includes the initial volume of transport demand for each inland city, together with the corresponding OD stations.
• Seaport’s collection time window: As rail-sea intermodal integration is a key feature of the corridor, a time window constraint is imposed at the seaport. All demands terminate at the seaport and must meet this constraint. On one hand, the port provides a certain amount of free storage time, typically related to the shipping schedule [41]. On the other hand, the railway arrival times must be coordinated as closely as possible with the shipping schedule [7]. As shown in Figure 2, if a train arrives within the designated time window, no storage cost is incurred; otherwise, additional storage waiting time is required. This constraint is unique to intermodal-based train operations in SHCs.

The problem consists of three sets of decisions:

• Selecting a service from the set of potential train service patterns for each city.
• Defining each train’s arrival and departure timings, as well as its route, in accordance with its service pattern. Note that the departure time in this study is focused on coordinating operations between rail and sea transport; it serves as a macro-level outline rather than a detailed timetable.
• Specifying the frequency of each train service within the planning horizon. Due to this frequency requirement, identical trains must be scheduled at evenly spaced intervals throughout the planning horizon.

3.2. Notation and Definition

The sets, parameters, and variables used in TSS formulation and modeling are listed in

Table 1. Notation and definition.

Type	Symbol	Definition
Sets	N	Set of nodes in the physic network
	R	Set of links in the physic network
	T	Set of time horizon in the network
	S	Set of states corresponds to demands and trains in the network
	V	Set of TSS vertexes
	E	Set of arcs in the TSS network
	A	Set of trains
	P	Set of demands, denoted by origin and destination nodes, in terms of OD pairs
	L	Set of links in space dimension
	$E_a$	Set of arcs correspond to train a, $E_a=\left\{(i,i^{\prime },s,s^{\prime },t,t^{\prime })\right|i,i^{\prime }\in N\}$
	${\psi }_{p,a}$	Set of arcs denoted train a is selected by OD pair p, ${\psi }_{p,a}\in E_a$
Indexes	i	Index of nodes, $i,i^{\prime }\in N$
	t	Index of times, $t,t^{\prime }\in T$
	$\tau$	Index of times in subsequent periods, $\tau ,{\tau }^{\prime }\in T$
	s	Index of state, $s,s^{\prime }\in S$
	a	Index of trains, $a,a^{\prime }\in A$
	p	Index of OD pairs
	$q_p$	Initial volume of OD pair p
	$(i,s,t)$	Index of time-space-state vertex
	$(i,i^{\prime },s,s^{\prime },t,t^{\prime })$	Index of time-space-state arcs, $(i,i^{\prime },s,s^{\prime },t,t^{\prime })\in E$
Parameters	$o_p$	Origin station of OD pair p
	$d_p$	Destination station of OD pair p
	$t_{i,a}$	Arrival time of train a at node i
	$t_{i,a}^{\prime }$	Departure time of train a at node i
	$t_{d,a}^w$	Storage time of train a at destination
	$s_{i,a}$	State of train a when arriving at node i
	$s_{i,a}^{\prime }$	State of train a when departing from node i
	$wa$	The first train a in the initial period
	${\beta }_a$	The order of train a with the same schedule
	$c^{(i,i^{\prime })}$	Cost of link $(i,i^{\prime })$, $\forall t,t^{\prime },s,s^{\prime }$
	$h^{(i,i^{\prime })}$	Transport time of link $(i,i^{\prime })$, $\forall t,t^{\prime },s,s^{\prime }$
	$[{\min }_a,{\max }_a]$	Capacity of train a
	$[E{T}_d,L{T}_d]$	Time windows of the destination node
	$T_p$	Length of the planning horizon
	$T_i$	Length of the initial period
	$T_s$	Length of the subsequent period
	H	Number of plan periods
	$\alpha$	Order of subsequent period, $\alpha \in \{0,\dots ,H\}$
Variables	$x_a^{(i,i^{\prime },s,s^{\prime },t,t^{\prime })}$	=1 if arc $(i,i^{\prime },s,s^{\prime },t,t^{\prime })$ is selected, =0 otherwise
	$y_a^{(i,i^{\prime },s,s^{\prime },\tau ,{\tau }^{\prime })}$	=1 if arc $(i,i^{\prime },s,s^{\prime },\tau ,{\tau }^{\prime })$ is selected, =0 otherwise
	$f_a$	Frequency of train a
	$t_a$	The departure time of train a

.

3.3. Time-Space-State Network Formulation

We propose the SSND-ST formulation on a TSS network. The TSS network encompasses three dimensions: time, space, and state. The space dimension represents the physical nodes N and transport arcs R within the railway network. The nodes include rail stations and the seaport terminal, which are crucial components of the SHC. For each inland city, the demand P can be represented by the nodes of origin $o_p$ and destination $d_p$ along with the volume $q_p$, where p refers to the pair of OD. In this context, because the demand is aggregated at the local rail station, all destinations $d_p$ are seaports, and both origins $o_p$ and destinations $d_p$ belong to the set of nodes N.

In the time dimension, time mainly includes transportation and transshipment times. The transportation time, denoted as $T_{(i,i^{\prime },t)}$, represents the time it takes to travel between two nodes on a transportation arc R starting at time t, where $(i,i^{\prime })$ indicates the specific arc between node i and node $i^{\prime }$ within the railway network. The transportation time can be calculated based on the distance between nodes and the average operating speed of the vehicle. The transshipment time, denoted as $T_{(i,t)}$, refers to the time required for handling and transferring goods at rail station i starting at time t. This includes the time needed for loading, unloading, and preparing goods for further transportation, ensuring a seamless transition in the movement of goods along the SHC.

A state $s\in S$ in the network is defined by a set of components $s_{(i,a)}$, where $s_{(i,a)}=1$ indicates that the OD pair $p_i$ is being transported in a particular train $a\in A$, and $s_{(i,a)}=0$ otherwise. This representation helps to track the cumulative volume of demands on a train. For example, consider three pairs of customer OD: $p_1,p_2,p_3$. If $p_1$ and $p_3$ are transported by train a, the state is denoted as $[1,0,1]$. The state dimension is then integrated with the time and space dimensions to form a three-dimensional vertex $(i,s,t)$.

The set of arcs corresponding to the train a, denoted as $E_a=\{(i,i^{\prime },s,s^{\prime },t,t^{\prime })\mid i,i^{\prime }\in N\}$, describes the transitions between different nodes, state changes, and temporal variations for the train a. These arcs represent the movement of trains through the rail network and the actions performed at the stations. For each train a, there are fixed starting points o and destination points d. The train route a consists of connected transport arcs. With the inclusion of the time dimension, the arrival time at node i can be represented as $t_{(i,a)}$, the departure time from node i as $t_{(i,a)}^{\prime }$, and the waiting time at the destination as $t_{(d,a)}^w$.

The state dimension can further describe the operations of the train a at node i. When the train arrives at node i, its state is denoted as $s_{(i,a)}$. If any intermediate operations occur, $s_{(i,a)}^{\prime }$ differs from $s_{(i,a)}$; otherwise, it remains unchanged. Through the TSS network, we can model train services with varying starting points, departure times, and patterns.

Figure 3 exemplifies a sample TSS network for train a with 5 nodes from time $t_0$ to $t_6$. In Figure 3a, a simple physical space path of a train is shown, where the train is loaded with cargo $p_1$ at the station $n_1$ and $p_4$ at station $n_4$. Consequently, the state of the link $(n_1,n_4)$ is set to $s_1$, while the state of the link $(n_4,n_5)$ is set to $s_2$. The corresponding TSS network path, as shown in Figure 3b, follows the sequence of the TSS nodes $(n_1,s_0,t_0)\to (n_1,s_1,t_1)\to (n_4,s_1,t_4)\to (n_4,s_2,t_5)\to (n_5,s_2,t_6)$.

Capacity limits $[{\min }_a,{\max }_a]$ and time window constraints $[E{T}_d,L{T}_d]$ are incorporated into the TSS network using time and state dimensions. The decision-making process involves selecting suitable train services for each city, defining the departure time $t_a$ of each train and the route according to its service pattern, and specifying the frequency $f_a$ of each service within the planning horizon $T_p$. In the network, all scheduled train operations can be represented.

The SSND-ST problem can then be solved by solving a network flow problem, aiming to select the optimal itinerary for each demand, as well as the associated service pattern and schedule.

4. Model Formulation

4.1. Modeling Approach

Due to the inherent cyclic nature of scheduling, conventional modeling often necessitates the use of modulo variables, which can complicate the model construction and solution process. Inspired by the research of [37], we have incorporated the concepts of expansion and replication to model this scenario. This approach offers several advantages: It simplifies the modeling process and effectively averts potential conflicts that may arise in periodic services without modulo variables. Our study extends the state dimension, enriching the practical application of this modeling approach in real-world scenarios. The original problem could be extended to include an initial planning period and some subsequent periods. Figure 4 shows the relationship between the main concepts of the study. Then, the SSND-ST problem of scheduled trains is to schedule a set of train services repeatedly for every period.

To ensure conflict-free scheduling throughout the planning horizon, we incorporate the concept of replicating the initial period into several subsequent periods within the TSS network framework. The temporal dimension is uniformly segmented into distinct periods for easier representation of scheduled planning. Subsequent periods are identical replicas of the initial period, with the only difference being the adjusted time for each train a, retrogressed by $\alpha T_i$ (where $\alpha$ denotes the order of the period and $T_i$ is the duration of the period). The frequency $f_a$ determines the regularity of the train service a per period. The initial period is structured to resemble non-cyclic service planning, while subsequent periods focus on resolving train conflicts over the planning horizon. Each train a is characterized by its unique frequency $f_a$, stopping pattern, and departure time from the origin. As illustrated in Figure 5, Train 1 (red) operates with a frequency of 2 per period, while Train 2 (blue) operates with a frequency of 1 per period. The duration $T_i$ extends beyond the maximum arrival times of all trains’ first departures at their respective destinations, serving as an initial period. During this period, the TSS arcs progress forward in time. The planning horizon $T_p$, determined by the operator requirements, is given by $T_p=\alpha ·H·T_i$, where trains in the initial period are repeated $\alpha$ times. This formulation ensures conflict-free operations throughout the planning horizon, a conclusion that has been formally validated and proven by [37].

4.2. Modeling Assumptions

Considering practical aspects and without loss of generality, the model includes the following 6 assumptions:

• Assume that the time, space, and quantity of demands are known. In this context, demands refer to customer cargo transport requirements, calculated by containers. This is a common assumption in deterministic rail service planning studies, as these data represent key parameters.
• Assume that each OD pair is assumed to be indivisible; hence, cargo from the same OD pair must be transported on trains of the same mode. This assumption is made for management simplicity and is commonly used in some studies [42,43].
• To maintain competitive advantage, it is assumed that each train service makes no more than one stop during its journey, as additional stops increase travel time [40].
• To simplify the work of rail operators, we assume that the planning horizon is a week and the time step is an hour. When demand fluctuates weekly or seasonally, this fixed planning horizon can be handled using a rolling or periodic approach, where the model is updated regularly to reflect the latest demand patterns.
• The scheduled train services are planned to operate at regular intervals [44].
• Given the ongoing infrastructure improvements, we assume that the capacity of the rail track, station, and port infrastructure will meet the operational requirements of the proposed train services.

4.3. Objective Function

The decision factors in the model, based on the network mentioned above, are as follows:

• $x_a^{(i,i^{\prime },s,s^{\prime },t,t^{\prime })}=1$ if arc $(i,i^{\prime },s,s^{\prime },t,t^{\prime })$ is selected by train a in the initial period; otherwise, $x_a^{(i,i^{\prime },s,s^{\prime },t,t^{\prime })}=0$.
• $y_a^{(i,i^{\prime },s,s^{\prime },\tau ,{\tau }^{\prime })}=1$ if arc $(i,i^{\prime },s,s^{\prime },\tau ,{\tau }^{\prime })$ is selected by train a in subsequent periods; otherwise, $y_a^{(i,i^{\prime },s,s^{\prime },\tau ,{\tau }^{\prime })}=0$.
• $f_a$ represents the frequency of the train a, indicating the number of times the train a operates within the planning horizon.
• $t_a$ denotes the departure time of the train a, specifying when the train a begins its journey on the schedule.

It is worth noting that time intervals in the initial period are indexed by t and $t^{\prime }$, whereas time intervals in the subsequent periods are indexed by $\tau$ and ${\tau }^{\prime }$. To simplify the formulation of the model, let $e=(i,i^{\prime },s,s^{\prime },t,t^{\prime })\in E^{\prime }$, and $e^{\prime }=(i,i^{\prime },s,s^{\prime },\tau ,{\tau }^{\prime })\in E^{″}$.

In our interviews with stakeholders from the NWLSC Railway Express, it was evident that constructing a service plan based on user feedback is vital to customer attraction. In addition, protecting customer interests is instrumental in the advancement of rail development. To align the objectives of inland cities, rail operators, and seaports, our objective function focuses on minimizing both total transport costs and total journey times.

The objective function (1) targets the reduction of overall transport costs for all trains throughout the planning horizon. In contrast, the objective function (2) is designed to decrease the total travel times of all trains within the same planning period. The cost of transportation services for each TSS arc is denoted as $c^e$, while the travel time for each arc is represented by $h^e$.

$$\min Z_1=\left(\sum _{a\in A}\sum _{e\in E_a^{\prime }}{c}^e·x_a^e·f_a\right)·H$$

(1)

$$\min Z_2=\left(\sum _{a\in A}\sum _{e\in E_a^{\prime \prime }}(h^e+t_{d,a}^w)·x_a^e·f_a\right)·H$$

(2)

4.4. Constraints

OD satisfaction constraint:

$$\sum _{a\in A}\sum _{e\in {\psi }_{(p,a)}}{x}_a^e=1,\forall p\in P$$

(3)

Constraint (3) ensures that each OD pair is allocated a unique train service, thereby prohibiting multiple service selections for the same OD pair. Here, ${\psi }_{(p,a)}\in E_a$ represents the set of arcs where train a is selected by OD pair p.

Flow balance constraints:

$$\sum _{e\in E_{wa}^{\prime }}{x}_{wa}^e-\sum _{e\in E_{wa}^{\prime }}{x}_{wa}^e=\left\{\begin{array}{cc}-1,\hfill & \mathrm{if}{i}^{\prime }=o_{wa},t^{\prime }=t_{(o,a)}^{\prime }\hfill \\ 1,\hfill & \mathrm{if}{i}^{\prime }=d_{wa},t^{\prime }=T_i\hfill \\ 0,\hfill & \mathrm{otherwise}\hfill \end{array}\right.,\forall a\in A$$

(4)

Constraint (4) ensures the preservation of flow balance at every vertex within the TSS network for each train a. Here, $wa$ represents the first instance of the train a in the initial period, with $o_{wa}$ and $d_{wa}$ representing the nodes of the origin and destination of the train a, respectively.

Initial period coupling constraint:

$$x_{wa}^e=x_a^{(i,i^{\prime },t+{\beta }_a·min\{\lfloor \frac{T_i}{f_a}\rfloor ,T_i-1\},t^{\prime }+{\beta }_a·min\{\lfloor \frac{T_i}{f_a}\rfloor ,T_i-1\})}\forall a\in A,e\in E^{\prime },f_a>1,{\beta }_a\in \{1,\dots ,f_a-1\}$$

(5)

Constraint (5) ensures that train a in the initial period adheres to a consistent schedule, achieved by shifting train $wa$ within the initial period at intervals of $min\{\lfloor {\displaystyle \frac{T_i}{f_a}}\rfloor ,T_i-1\}$. The parameters ${\beta }_a$ in this constraint delineate the sequence of train a operating under the same schedule, ranging within $\{1,\dots ,f_a-1\}$.

Subsequent period coupling constraint:

$$y_a^e=x_a^{(i,i^{\prime },s,s^{\prime },t+\alpha T,t^{\prime }+\alpha T)}\forall a\in A,e\in E^{\prime },e^{\prime }\in E^{″},\alpha \in \{0,\dots ,H\},\tau =t+\alpha T,{\tau }^{\prime }=t^{\prime }+\alpha T$$

(6)

Constraint (6) facilitates the replication of the TSS arcs from the initial period to the subsequent periods $H-1$. The parameter $\alpha$ indicates the sequence of subsequent periods, ranging within $\{0,\dots ,H\}$. The case where $H=0$ corresponds to the initial period.

Capacity constraint:

$$s_{(i,a)}\in [{\min }_a,{\max }_a],\forall a\in A,i\in N$$

(7)

Constraint (7) ensures that train a can reach state s if its container carrying capacity remains within the specified lower and upper limits.

Time-windows constraints:

$$t_{(o,wa)}\in [0,min\{\lfloor {\displaystyle \frac{T_i}{f_l}}\rfloor ,T_i-1\}]$$

(8)

$$t_{(d,wa)}=t_a+\sum _{e\in E_a}{x}_a^e·h^{(i,i^{\prime })}$$

(9)

$$t_{(d,wa)}^w=\left\{\begin{array}{cc}0,\hfill & \mathrm{if}{t}_{(d,wa)}\in [E{T}_d,L{T}_d]\hfill \\ E{T}_d-t_{(d,wa)},\hfill & \mathrm{if}{t}_{(d,wa)}<E{T}_d\hfill \\ E{T}_d+24-t_{(d,wa)},\hfill & \mathrm{if}{t}_{(d,wa)}>L{T}_d\hfill \end{array}\right.,\forall wa\in A$$

(10)

Constraint (8) sets the departure time window for the first train $wa$ to $[0,min\{\lfloor {\displaystyle \frac{T_i}{f_a}}\rfloor ,T_i-1\}]$. Constraint (9) specifies the calculation method for $t_{(d,wa)}$, the arrival time of the train $wa$, while Constraint (10) details the methodology to calculate the storage waiting time in the seaport.

Domain of variables:

$$x_a^e\in \{0,1\},\forall a\in A,e\in E^{\prime }$$

(11)

$$y_a^e\in \{0,1\},\forall a\in A,e\in E^{″}$$

(12)

$$f_a\in {\mathbb{N}}^{+}$$

(13)

$$t_a\le 168,\forall a\in A$$

(14)

The binary domain of the TSS flow decision variables is established by Constraints (11) and (12) for the initial and subsequent periods, respectively. The frequency and departure time are defined by Constraints (13) and (14).

5. Solution Algorithm

Within the TSS framework, the research problem is reformulated as a multi-commodity flow problem, where the generation of TSS arcs serves as the backbone of our solution representation. After linearizing the formulation, solvers such as Gurobi are typically employed to obtain an optimal solution. In most previous studies, a candidate set approach is used to pre-eliminate infeasible or suboptimal solutions before invoking Gurobi [45]. To verify the feasibility of our model, we employed conventional methods in small-scale cases. However, when addressing real-world problems, the numerous TSS arcs and constraints increase significant challenges to achieving efficient optimization.

To address these challenges, meta-heuristic algorithms—especially ALNS—have been successfully applied to relative problems [46], particularly in medium- to large-scale instances. Motivated by its scalability and effectiveness, our approach incorporates an ALNS framework that integrates dynamic neighborhood operator design, adaptive parameter adjustment, and robust constraint handling into a unified iterative process, as outlined in Algorithm 1.

Algorithm 1: Adaptive large neighborhood search for SSND-ST problem

Initially, a feasible solution is generated, and key parameters are set. Then, in each iteration, the destroy phase employs a dynamically adjusted destruction rate to remove selected OD pairs, while the repair phase uses a weighted roulette wheel mechanism to select a repair operator from a multi-strategy pool. The reconstructed solution is rigorously checked for constraint violations, with penalties applied as needed, and a hybrid acceptance criterion, inspired by simulated annealing, determines whether the new solution should be accepted based on both cost improvement and probabilistic acceptance of non-improving moves. In addition, an elite update mechanism continuously refines the best solution and adjusts the destruction rate to intensify the search in promising regions. This integrated design not only effectively navigates the complex solution space but also mitigates premature convergence, thereby enhancing the efficiency of finding an optimal schedule with minimal cost.

Based on the problem characteristics, we have developed a set of specialized operators to enhance the search process. In the destruction phase, the dynamic destruction operator employs an adaptive removal rate, which leverages the cost contributions of individual trains to selectively eliminate candidate OD pairs, thereby perturbing the solution space effectively. In the repair phase, our multi-strategy pool comprises three distinct strategies: (1) the capacity-prioritized insertion operator, which is invoked when residual capacity is available; (2) the temporal-relaxed insertion operator, designed to address time window violations; and (3) the new train generation operator, which constructs a new service route when the other strategies fail to restore feasibility. These operators are tailored to the specific challenges of the seaport-hinterland scheduling problem and work adaptively to improve the algorithm’s efficiency and robustness.

Due to the large-scale solution space, Gurobi 11.0.3 failed to solve our practical problem within the 3600 s time limit. To provide a comparative evaluation of algorithm performance, we designed three small-scale test cases. As shown in

Table 2. Comparison of GUROBI and ALNS Performance.

Instances	GUROBI			ALNS
	Weighted Obj.	CPU (s)	Gap	Weighted Obj.	CPU (s)	Gap
3 cities	1654	52.06	0.00%	1654	23.35	0.00%
4 cities	4335	368.08	0.00%	4335	56.88	0.00%
5 cities	25,145	919.86	0.00%	25,159	68.56	0.06%

, our ALNS algorithm achieved results with a very small gap while maintaining a relatively low CPU time. This demonstrates the effectiveness of the ALNS algorithm in solving the problem efficiently.

6. Results and Discussion

This section begins by introducing the background case that motivated the study, followed by the experimental results. Finally, we discuss the key findings and their implications.

6.1. Case Study

Our research is driven by a significant challenge in western China, focusing on the development of the NWLSC. This corridor, crucial for regional development, is shown in Figure 6. The railway departments in China are tasked with harnessing green and low-carbon initiatives to enhance the efficiency of the NWLSC’s freight network system. Since its official launch in 2019, the NWLSC Railway Express has been characterized by its regular scheduling (mainly referring to freight trains operating on a fixed schedule), speed, and rail-sea intermodality. Despite these features, the current service mode fails to meet the comprehensive needs of the network. As the NWLSC network continues to expand, there is an urgent need to refine the operations of the NWLSC Railway Express, particularly in terms of service scheduling and network integration.

To facilitate our analysis, the cities within the NWLSC network have been numerically designated as shown in

Table 3. Number of Cities in the NWLSC Network.

City	Number	City	Number	City	Number
Ürümqi	1	Chengdu	6	Nanning	11
Xining	2	Chongqing	7	Qinzhou Port	12
Lanzhou	3	Huaihua	8
Yinchuan	4	Guiyang	9
Xi’an	5	Kunming	10

. It is important to note that Qinzhou Port serves as the destination node in this network setup.

According to the projections by the National Development and Reform Commission in their NWLSC planning, container shipping volume through NWLSC is expected to reach 500,000 equivalent units of twenty feet (TEU) by 2025 [47]. For this study, the weekly container shipping volumes for each typical city in 2025 have been estimated based on the proportion of each province’s total exports to ASEAN countries in 2024, sourced from China Customs Statistics.

Table 4. Each City’s Weekly Container Volume.

City Number (O–D)	Origin City	Weekly Container Shipping Volume (TEU)
1→12	Ürümqi	58
2→12	Xining	20
3→12	Lanzhou	60
4→12	Yinchuan	35
5→12	Xi’an	580
6→12	Chengdu	2657
7→12	Chongqing	1017
8→12	Huaihua	936
9→12	Guiyang	133
10→12	Kunming	218
11→12	Nanning	3886

presents these projected weekly container volumes for each city. Additionally,

Table 5. Distances Between Cities.

City Number (O–D)	City (OD)	Distance (km)	City Number (O–D)	City (OD)	Distance (km)
1–2	Ürümqi–Lanzhou	1926	5–8	Xi’an–Huaihua	837
2–3	Xining–Lanzhou	216	5–6	Xi’an–Chengdu	842
3–6	Lanzhou–Chengdu	1172	6–10	Chengdu–Kunming	1100
3–7	Lanzhou–Chongqing	1676	6–7	Chengdu–Chongqing	504
4–3	Yinchuan–Lanzhou	468	7–8	Chongqing–Huaihua	708
4–5	Yinchuan–Xi’an	1144	9–11	Guiyang–Nanning	865
5–7	Xi’an–Chongqing	1346	11–12	Nanning–Qinzhou Port	173

provides the railway distances between these cities, sourced from China Railway Express. The transportation cost per container is obtained from the railway tariff table. By combining the segment distance, the corresponding transportation cost $c^{(i,i^{\prime })}$ is determined.

6.2. Result

To align with operational norms, our model assumes a 7-day planning horizon T, discretized into hourly intervals. This approach enables a comprehensive weekly overview of transport activities. Additional key parameters required for the model are detailed in

Table 6. Other Parameters Setting.

Parameter	${\mathit{T}}_{\mathit{p}}$	[ET_d, LT_d]	[${\text{Min}}_{\mathit{a}}$, ${\text{Max}}_{\mathit{a}}$]
Unit	hour	/	TEU
Value	168	12:00–16:00 everyday	[20, 100]

, with data sourced from the survey conducted.

The computational experiments were conducted on a desktop computer equipped with an Intel i7-10710U CPU @ 1.61 GHz and 16 GB RAM, and Python 3.8. To address the bi-objective function, we employed the weighted sum method, with the impact of weight selection on the results discussed in Section 6.3.

Utilizing the case data, we successfully formulated a service plan for the NWLSC Railway Express. This plan involves selecting six cities to operate direct train services and five cities for step train services.

Table 7. Service Plan of Scheduled Trains.

Type	Original	Departure Day of the First Train	Departure Time of the First Train	Stop City	Frequency
Direct trains	Xining	Saturday	10:00	—	1
	Xi’an	Monday	14:00	—	12
	Chengdu	Monday	06:00	—	28
	Kunming	Monday	04:00	—	10
	Huaihua	Monday	01:00	—	45
	Nanning	Monday	00:00	—	47
Step trains	Ürümqi	Thursday	05:00	Lanzhou	2
	Yinchuan	Thursday	02:00	Lanzhou	1
	Chongqing	Monday	15:00	Guiyang	11

provides detailed information on the service planning for each city, including the type of service, the first train’s departure time, journey duration, and service frequency. The optimal transport routes derived from this plan are illustrated in Figure 7.

The formulated service plan for the NWLSC Railway Express reveals two notable patterns: (1) There is a clear inclination towards utilizing direct trains over step trains among the cities, provided the capacity constraints are met. This preference is mirrored in real-world scenarios, as direct trains offer reduced loading and unloading times and are simpler to manage. (2) Cities with lower cargo volumes can strategically opt for step trains to optimize transport costs and journey times. This includes collaboration between cities with varying cargo volumes to improve transportation efficiency. For instance, Guiyang can collaborate with Chongqing, a city with sufficient cargo volume. In addition, cities such as Yinchuan, Ürümqi, and Lanzhou, which have relatively lower cargo volumes, can effectively combine their resources to form step trains.

6.3. Discussion

This section primarily discusses the comparison between the proposed networked operation strategy and current point-to-point operation practice. In addition, we examine the impact of objective function weights and key constraints.

6.3.1. Results Improvement

To enable a clear comparison, our optimized solution is referred to as the “networked” service, while the existing operational mode is termed the “point-to-point” service. This analysis examines key aspects such as transport cost, service frequency, cargo collection time, and storage duration at the seaport. In the current “point-to-point” mode, the rail network primarily relies on direct trains. As a baseline for comparison, we calculated the transportation costs required to meet network demand in this direct train mode. This serves as a reference for assessing the benefits of the “networked” service over the existing system.

Based on the comparison in

Table 8. Transport cost, travel time, and service frequency comparison.

City	Transport Cost (USD)		Travel Time (h)		Service Frequency
	Point-to-Point	Networked	Point-to-Point	Networked	Point-to-Point	Networked
Ürümqi	91,347	74,900	156	91	0.5	2
Xining	54,668	14,025	300	103	0.3	1
Lanzhou	50,909	47,321	132	57	1	3
Yinchuan	59,053	29,148	183	109	0.5	1
Xi’an	502,216	309,844	60	27	9	12
Chengdu	1,351,377	1,372,181	36	33	44	28
Chongqing	486,364	460,744	36	26	16	11
Huaihua	302,337	309,666	16	13	15	45
Guiyang	49,479	50,520	60	27	2	11
Kunming	72,945	72,945	36	20	3	10
Nanning	501,294	501,294	12	10	64	47
Average	319,933	294,781	93	47	14	16

, the point-to-point solution in practice requires each train to meet a load restriction of 60 TEU, which leads some cities to have a service frequency of less than 1 per week due to insufficient demand. In contrast, the proposed networked solution achieves dual optimization in terms of cost and time. Specifically, the average transportation cost is reduced by approximately 7.9%, and the total transportation time for each train (including cargo collection time, transportation time, and port waiting time) is reduced by approximately 49.76%. In addition, the networked solution results in an improvement in the average service frequency, which can help attract more cargo to choose rail transport. Next, we provide a detailed comparison of these metrics.

(1) Transportation costs

Figure 8 provides a comparison of transportation costs between the “point-to-point” and “networked” services, demonstrating the cost-saving potential of our solution. Notable reductions are evident in cities such as Xining and Yinchuan, where the networked service shows cost savings of 74.34% and 50.64%, respectively (as indicated by the teal bars), against the cost of point-to-point service. For cities such as Chengdu and Huaihua, a slight increase in costs under the networked service is observed, mainly due to variable volumes of train loading. When the total OD demand is not perfectly divisible by the optimized train capacity, additional trips with partial loads may incur marginal empty container costs. This localized effect is an expected trade-off in network-based optimization and does not compromise the overall cost efficiency of the proposed solution. The aggregated data show an overall reduction in transportation costs of 7.9% (shown on the far right), supporting the feasibility of applying the networked system on a larger scale.

In addition to these savings, the cost analysis per TEU in Figure 9 further confirms the cost-effectiveness of the solution. The blue dotted line represents the average cost per TEU for point-to-point services, serving as a benchmark for comparing the networked service costs, shown by the orange dots. The orange dashed line is consistently lower than the blue dashed line, indicating cost savings across cities. In particular, Xining experiences a significant cost reduction, with a 74.34% decrease in TEU costs under the networked model. This metric is important for freight transport clients, highlighting the economic benefits of adopting the networked service within the NWLSC to support trade with ASEAN nations.

(2) Frequency

Frequency in transportation networks is a critical measure of service availability and regularity. According to Figure 10, the blue and orange dots represent the respective frequencies of the two modes, with the dotted lines indicating the mean frequency for each. In most cities, the transition to a networked service, as shown by the orange dots, leads to a heightened frequency of services, enhancing service availability and regularity. For instance, Yinchuan and Guiyang experienced significant frequency increases of 233.3% and 194.9%, respectively. In contrast, in Chengdu, Chongqing and Nanning, the frequency is reduced by 36.8%, 35.3%, and 35.2%, which, given their high volumes of freight, can contribute to alleviating congestion and logistical challenges. This strategic frequency adjustment reflects the ability of the NSTP model to tailor services to city-specific needs, improving the overall efficiency of the transportation network.

(3) Cargo collection time

The consolidation benefits of our solution also extend to cargo collection times. In Figure 11, the teal bars, representing the point-to-point service, are consistently higher than the orange bars of the networked service, indicating longer collection times. In contrast, the networked service achieves notable reductions in cargo collection times in most cities. For instance, Lanzhou and Yinchuan see reductions of 66.7% each, while Guiyang and Kunming experience decreases of 63.2% and 65.2%, respectively. On average, cargo collection time is reduced by 56.6%, highlighting the focus of the NSTP model on efficiency and its potential to improve trade logistics, particularly with ASEAN countries where timely delivery is essential [48]. These results indicate that our solution improves cargo handling efficiency, contributing to a more competitive international trade logistics system. Based on our research on the types and values of cargo of trains currently operating in NWSLC, the value of time is estimated at $15 per TEU per hour. This estimate is made by applying a 20% benchmark return rate. With our networked solution, the cargo collection time is reduced by 56.6%, saving approximately $101.88 million annually, based on an annual transportation volume of 500,000 TEUs. This reduction in cargo collection time reduces customer capital costs, allowing for reduced warehouse space requirements and shorter lead times. Improved efficiency also supports more predictable delivery schedules, helping customers optimize operations and minimize last-minute adjustments. Ultimately, these efficiencies make the service more attractive to customers, increasing cargo volume and contributing to regional import/export growth by improving logistics efficiency.

(4) Storage time in the seaport

An analysis of storage waiting times at the seaport further demonstrates the impact of the networked service. This comparison focuses on cities with scheduled train services, namely Chengdu, Chongqing, and Kunming, to contrast the networked service with the point-to-point service. Other cities were not included due to irregular train schedules, making their storage times less comparable.

The results, summarized in

Table 9. Comparison of storage time at the seaport between “point-to-point” service and “networked” service.

City	Networked Service		Point-to-Point Service		Reduction Rate
	Arrival Time	Storage Time	Arrival Time	Storage Time
Chengdu	31.275	4.725	23.9	12.1	61.0%
Chongqing	33.775	2.225	18.77	17.23	87.1%
Kunming	15.365	0	1.28	10.72	100%

, show that our solution effectively reduces storage time. In Chengdu, storage time decreased from 12.1 h to 4.725 h, a reduction of approximately 61.0%. Similarly, Chongqing saw a decrease from 17.23 h to 2.225 h, representing an 87.1% reduction. Kunming achieved a complete elimination of storage time, down from 10.72 h, highlighting the efficiency gains enabled by the networked approach. These reductions improve cargo flow and seaport operations, supporting more efficient transport within the SHC [49,50]. Increasing timeliness and operational efficiency is crucial to maintaining competitiveness in logistics.

6.3.2. Cost and Time Object

Given that the NSTP model aims to optimize both cost and time, the allocation of weights to these objectives plays a key role in shaping the results. To evaluate the adaptability of the model, we examined the impact of different weight assignments.

Figure 12 shows how the cost and time change rates vary with different weight settings for the cost objective in the NSTP model. As the weight assigned to cost increases, both the cost change rate and the time change rate decrease. This suggests that both objective functions are sensitive to weight adjustments, with the time objective showing a consistent decline, indicating a strong response to increased cost weighting. The overall objective change rate also decreases, highlighting the influence of weighting on the results of the model and the importance of carefully balancing the cost and time considerations in the NSTP model.

Table 10. Effect of objective weights on train service patterns.

Weight of Cost	0	0.1	0.2	0.3	0.4	0.5–0.9	1.0
Ürümqi	step	step	step	step	step	direct	step
Xining	step	step	step	direct	direct	direct	step
Lanzhou	step	step	step	step	step	direct	step
Yinchuan	step	step	step	step	step	direct	step
Xi’an	direct	direct	direct	direct	direct	direct	step
Chengdu	step + direct	direct	direct	direct	direct	direct	step
Chongqing	step + direct	step + direct	step + direct	step + direct	direct	direct	step
Huaihua	direct	direct	direct	direct	direct	direct	step
Guiyang	step	step	step	step	direct	direct	step
Kunming	direct	direct	direct	direct	direct	direct	step
Nanning	direct	direct	direct	direct	direct	direct	step

Step refers to step train, direct refers to direct train.

presents the impact of the adjustments of the cost weights on the selection of service patterns. The cities that choose step trains to fulfill their transportation demand are marked in green, while those that use a combination of step trains and direct trains are marked in orange. Increasing the weight on cost objectives shifts service preferences towards direct services until cost becomes the dominant factor. This trend continues until the cost weight reaches 1, at which point step services become more favorable due to their higher cost-efficiency, achieved through demand consolidation and reduced service frequencies. A cost weight of 0.4 is identified as optimal, balancing cost efficiency and service quality. This weighting avoids extreme trade-offs, ensuring a scheduling approach that integrates both economic efficiency and reliable service.

The adaptability of the NSTP model to different weight settings demonstrates its flexibility in addressing varying operational priorities. When cost is prioritized, the model effectively reduces transportation costs and reduces cargo collection and storage times. In contrast, when time is emphasized, the model is adjusted to prioritize timely service delivery while maintaining cost efficiency. This balance between cost and time objectives enhances cargo flow and minimizes delays in seaport operations, contributing to the overall efficiency of the SHC.

6.3.3. Demand and Capacity Limits

This section presents a sensitivity analysis of demand fluctuations and capacity constraints. Given that the demand in some cities is already relatively small, we focus primarily on the impact of increasing demand. In the NSTP model, a critical capacity constraint is the 20–100 TEU train load capacity range. During the experiments, when the upper limits of the load volume were reduced to around 70 TEU, the demand could no longer be met, making the model infeasible. Therefore, our analysis focuses mainly on the effects of changes in the lower limit of the load volume.

As shown in Figure 13, there is a strong linear relationship between demand changes and the cost objective. Specifically, as demand increases, the costs increase linearly, with a correlation greater than 0.97. However, the increase in time follows a polynomial pattern, indicating congestion effects and non-linear inefficiencies in the system. The relationship between the capacity limit and two objectives shows a moderate correlation (less than 0.22). The impact of the capacity limit on time is relatively more pronounced. As the limit is relaxed, the time objective decreases; however, the extent of this reduction is influenced by the scale of demand, revealing bottleneck points within the network.

Regarding the upper limit, when the upper limit of 100 TEUs decreases to approximately 70 TEUs, the system’s ability to meet demand is severely affected, particularly in terms of time. As the upper limit shrinks, it forces the system to operate more trains to meet the same demand, leading to an increase in the time objective.

This sensitivity analysis highlights the dynamic relationship between demand and capacity constraints. The results underscore the importance of demand management strategies, such as adjusting pricing or incentivizing off-peak usage, to optimize system performance. Furthermore, lowering the minimum load volume constraint can improve service flexibility and reduce time costs, while strengthening capacity infrastructure is crucial to ensure that the network can efficiently handle increasing demand.

7. Conclusions

This study addresses the challenge of optimizing transportation in the seaport-hinterland corridor (SHC), with a focus on the importance of stable rail services. By characterizing the multi-objective nature, regular service structure, and rail-sea intermodal integration, we develop the Networked Scheduled Train Planning (NSTP) model using a time-space-state network approach. This model incorporates the needs of various stakeholders, including seaports, rail operators, and inland cities, while providing a structured framework for managing logistic complexities. To efficiently solve the problem, we designed a customized adaptive large neighborhood search algorithm (ALNS), ensuring computational efficiency and solution quality.

Applied to the New Western Land–Sea Corridor (NWLSC), our solution, based on networked operations, has shown significant improvements in transportation efficiency compared to current practices. By integrating coordinated scheduling and resource management, the results indicate a 7.9% reduction in transportation costs, a 56.6% improvement in average cargo collection efficiency, and a notable decrease in storage waiting times at seaports. These findings highlight the benefits of a networked approach in aligning stakeholder interests while balancing cost and time objectives. Furthermore, the flexibility of the model to adjust key parameters improves its strategic value for operational planning, ensuring both cost efficiency and reliable service delivery.

The proposed networked operational strategy and modeling framework provide practical solutions for complex logistical challenges in SHCs. Our findings underscore the need for effective demand management and targeted infrastructure enhancements to address bottlenecks and improve network resilience, particularly under demand fluctuations and capacity constraints. However, the framework does have some limitations, such as the exclusion of unscheduled events or disruptions, which could further impact its applicability in dynamic environments. It is also most suitable for SHCs with a solid foundation of cooperation or where a leading enterprise can organize networked cooperation. Future research should focus on extending the adaptability to large-scale instances, improving its robustness in uncertain scenarios, and addressing more complex networks.