Optimizing transfer operations for air-rail intermodal shipments: A case study of Zhengzhou aviation hub

Abstract

Zhengzhou, with its unique “*” shaped high-speed rail network and strong advantages as an aviation hub, is leading a new trend in air-rail intermodal freight transportation, continuously enhancing its competitiveness in the global logistics. This paper focuses on the transfer operations of intermodal freight at the air-rail hub, investigating the integrated optimization of the cargo transfer scheme and the train template to ensure the most effective connection between flights and trains. Considering constraints such as the operational capacity of airports and high-speed rail stations, as well as the availability of facilities and resources, we developed a mixed-integer programming model. This model aims to minimize both cargo transfer time and the volume of remaining cargo. In the experimental section, we validated the model’s performance under large-scale scenarios using a model solver and quantitatively analyzed the relationship between the two objectives. These two objectives are integrated into a single objective function via a linear weighted sum, achieving a quantifiable trade-off between transfer efficiency and distribution levels. Finally, through sensitivity analysis, we discussed the impact of train loading rates and service time windows on the efficiency of intermodal cargo transfers. The results indicate that higher loading rate requirements tend to extend the average transfer time for cargo, while longer service time windows reduce the average transfer time. Both scenarios facilitate trains serving a greater number of flights.

Introduction

With the continuous development of the socio-economic landscape and trade, traditional single-mode transportation methods are increasingly inadequate in meeting the demands for rapid, efficient, and cost-effective logistics. ¹ Railway freight technology plays a pivotal role in the facilitation of goods and resource allocation both domestically and internationally. ² The emergence of high-speed rail express services has demonstrated the potential for swift, high-capacity, and punctual delivery of parcels, representing a new form of productive capacity within the logistics industry.^3,4 On the other hand, air cargo plays an important role in the transport of high-value cargo in global trade. ⁵ Building on this foundation, the integration of the international accessibility advantages of air transport with hub resources to organize an air-rail intermodal transportation system for parcels can lead to complementary benefits. ⁶ This approach has the potential to significantly enhance the capacity for logistics collection and distribution, and better meet the diverse transportation needs of customers.

Currently, Zhengzhou has taken the lead in implementing air-rail intermodal transportation practices. In December 2024, a batch of cargo from Europe arrived at Zhengzhou Xinzheng International Airport, marking the successful launch of a new transportation mode. By leveraging the city’s integrated aviation and high-speed rail hubs, this mode combines international air imports with domestic high-speed rail distribution. This mode has now entered a phase of regular operation.

In terms of the aviation network, Zhengzhou Xinzheng International Airport has established a comprehensive route network connecting major global economies and has been approved as the first airport-based national logistics hub. ⁷ Currently, the airport operates 214 passenger routes and 44 cargo routes, including 36 international routes. Zhengzhou ranks among the top in the nation for international cargo capacity, the number of freighter routes, flight frequency, and the number of cities served, positioning it within the top 40 global cargo airports.

In terms of the high-speed rail network, Zhengzhou Hangkonggang Railway Station is the closest dual-use passenger and cargo high-speed rail station to an airport in the country, featuring the largest construction scale and the most convenient connections. It houses the nation’s first comprehensive cargo distribution center for ‘air-rail intermodal transport’. Additionally, this station serves as a central hub within the *-shaped network, offering unique advantages for air-rail intermodal transport and the large-scale, networked operation of high-speed rail logistics, ⁸ as illustrated in Figure 1.

Figure 1.

The cross-shaped and X-shaped overlapping high-speed rail network.

The Zhengzhou Airport Economy Zone facilitates the integration of aviation and high-speed rail networks. This zone integrates transport, logistics, and manufacturing to attract various companies through its superior infrastructure.^9,10 The logistics process begins with international cargo arriving at the airport, which is then moved by road to the railway station. From there, it is distributed via the high-speed rail network. Specifically, since air transport involves palletized units, arriving cargo is first de-palletized and cleared through customs. During de-palletization, items are sorted by destination. Customs clearance is completed with the cargo remaining at the airport for a short duration, rather than prolonged storage. Subsequently, the cargo is transported by road to the high-speed rail cargo handling area at the station for unloading. Finally, after passing through security checks, the cargo is loaded onto high-speed rail trains via a conveyor belt, as illustrated in Figure 2.

Figure 2.

The layout of Zhengzhou airport economy zone.

This process reveals that the ‘aviation + high-speed rail’ transportation mode is not merely a combination of different modes of transport. Its essence lies in the efficient interconnectivity of infrastructure, specifically the integration of aviation hubs and high-speed rail terminals, as well as the precise temporal coordination between flight schedules and high-speed rail services. The Zhengzhou Airport Economy Zone has, to some extent, achieved physical connectivity; however, there is a pressing need to further optimize the temporal coordination of various operations during the cargo transfer process to prevent prolonged delays in the handling of cargo.

Specifically, for the airport, it is essential to consider factors such as the unloading time, the sorting capacity for cargo, customs clearance times. These considerations will help determine the direction and volume of cargo transfers during different time periods. For high-speed rail stations, an effective coordination method involves leveraging operational resources and capabilities, such as track capacity, and loading capabilities, to determine the transport direction and service times for trains. This approach enables the train scheduling template to align with the arrival of cargo flights and the subsequent transfer of cargo at the airport.

Notably, the decision-making in the distribution process is predicated on flight arrival data to synchronize outbound rail services. Conversely, the collection process is governed by pre-existing flight schedules to which rail arrivals must align. This study adopts the perspective of railway decision-makers, specifically investigating how constrained rail resources—such as track allocation and train capacity—can be optimized to match incoming air cargo. Unlike the collection process, where flight schedules often render the railway a ‘passive responder’, the distribution process requires the railway terminal to proactively manage complex dispersal operations. Consequently, this research prioritizes the distribution process as it presents more significant operational challenges and optimization opportunities for railway operators.

In summary, this paper takes into account the resource capacity constraints and transportation organization requirements at each stage to optimize the cargo transfer scheme, thereby enhancing the distribution efficiency of cargo in the air-high-speed rail intermodal system. The key decisions involved include: (a) the arrival and departure times of trains; (b) the direction of trains;(c) the volume of cargo loaded onto the trains; and (d) the transfer time and the quantity of cargo transported from the airport to the high-speed rail station.

The main contribution of this paper is as follows.

• (1) This study focuses on the practical optimization of air-rail intermodal operations within the cargo transportation sector. By constructing a mixed-integer programming (MIP) model, it provides a quantitative tool to enhance collaboration between air transport and high-speed rail in freight organization.

• (2) From the perspective of spatial and temporal coordination, an integrated optimization approach is proposed for both intermodal cargo transfer schemes and distribution train templates. This integration effectively resolves the matching and coordination challenges between flights and trains, thereby supporting the development of comprehensive air-rail intermodal service plans.

• (3) By balancing the minimization of transfer time from air to high-speed rail against the minimization of residual cargo volume, a multi-objective optimization framework is introduced, and the trade-offs between these objectives are quantitatively analyzed. Furthermore, practical operational constraints—such as train loading rates and service time windows—are incorporated into the experiments, providing actionable management insights for practitioners in air-rail intermodal hub operations.

The remainder of this paper is organized as follows. Section 2 reviews the relevant research findings and summarizes the existing gaps in the literature. Section 3 provides a detailed description of the problem, and Section 4 formulates the mathematical model. Computational experiments are conducted in Section 5 to validate the proposed approach, following with concluding remarks in Section 6.

Literature review

So far, optimization on multimodal transport of goods has been extensively explored, particularly in the context of road-rail and sea-rail intermodal transport. However, in the case of air-rail intermodal transport, researchers have primarily focused on competitive analyses between the two modes or on optimizing passenger transport services, with little attention given to the combined transportation of goods via air and high-speed rail. In this section, we will review the literature related to air-rail intermodal transportation and the transfer operations of intermodal cargo, thereby providing a comprehensive overview of topics closely related to this study.

Air-rail intermodal transportation

Both air transport and high-speed rail are characterized by their rapid speeds, making them suitable for long-distance transportation. Consequently, many scholars have focused on the competitive relationship between these two modes and have engaged in discussions regarding future development. In terms of potential collaboration, some researchers have also explored the design of integrated transport schemes for passengers utilizing both air and high-speed rail services.

In the field of relationship analysis and evaluation, Sato and Chen ¹¹ discussed the feasibility of multimodal transport involving air travel and high-speed rail. In the work of Li et al., ¹² the authors demonstrated that the distance between airports and high-speed rail stations is a significant factor influencing the collaboration between these two modes of transport. Yao et al. ¹³ examined the carbon reduction potential of air-high-speed rail express networks compared to traditional aviation networks. Lu et al. ¹⁴ established an evaluation framework to measure the service capacity and connectivity of the integrated aviation and high-speed rail network in China. Furthermore, Hou et al. ¹⁵ employed a game-theoretic model to analyze the competitive substitution relationship between air-high-speed rail intermodal networks and aviation networks, based on various multi-airport management strategies. Lastly, Avenali et al. ¹⁶ explored the interaction between cooperation in aviation and high-speed rail and the congestion levels at high-speed rail hubs in the presence of sunk costs.

In the field of network optimization and service design for passenger transport, Ke et al. ¹⁷ successfully coordinated flights and trains by adjusting train schedules to maximize the number of matched flights and their coverage while minimizing passenger transfer costs. Similarly, Tan et al. ¹⁸ focused on the temporal coordination between aviation and high-speed rail services. In their study, they allowed for adjustments to the existing high-speed rail timetable, but aimed to minimize the extent of these changes. On the other hand, Huan et al. ¹⁹ sought to improve the connection between the two modes of transport by adjusting flight schedules, while also considering ticket pricing. Sun et al. ²⁰ proposed a collaborative transport solution for air and high-speed rail in scenarios where capacity shortages lead to airport disruptions. Their approach involved decision-making regarding which flights to cancel, which to replace with high-speed rail, and how to manage delayed flights to minimize overall losses. Lu et al. ²¹ investigated the network optimization problem for air-high-speed rail intermodal transportation, employing a mixed-integer linear programming model to hierarchically determine routes, frequencies, and timing connections, with the objective of minimizing passenger costs, emissions, and operational expenses. Taking Zhengzhou Xinzheng International Airport as an example, Zhang et al. ²² investigated the pricing issues of air–rail intermodal passenger transport under different scenarios, including early, on-time, and delayed flight arrivals. Guitart and Buire ²³ considered passengers’ personalized needs and, without relying on existing timetables, proposed a solution framework to generate integrated train and flight schedules from scratch, which was tested on the French transportation network.

From the aforementioned literature, it is evident that while existing research on air-high-speed rail intermodal transportation has yielded significant findings in areas such as competitive analysis, network evaluation, and service optimization, it predominantly focuses on passenger transportation. There is a notable lack of exploration into the optimization of cargo air-high-speed rail intermodal transportation. The transfer process for freight involves more complex procedures and is constrained by resources and operational capacities. Additionally, considerations for the consolidation and distribution of cargo moving in the same direction introduce further constraints and decision-making elements. Therefore, there is an urgent need to conduct optimization research in this area.

Transfer operations of intermodal cargo

Multimodal transport of goods has proven to be an efficient and cost-effective logistics solution. Numerous scholars have conducted research on service network design, transportation schemes, and terminal operations in the contexts of sea-rail and road-rail intermodal transportation. There is a wealth of research on crane scheduling optimization for transshipment operations within terminals; interested readers can refer to the review by Boysen et al. ²⁴ for more information. Given the relevance to this study, this section will summarize the optimization issues related to transfer operations at hub nodes or ports within these two scenarios.

Regarding container transfer operations in sea-rail intermodal ports, Xie and Song ²⁵ considered a scenario where the railway operation area includes a buffer area. They optimized the transfer plan for containers moving between the yard, buffer area, and trains to reduce logistics costs. Rusca et al. ²⁶ employed simulation methods to explore train shunting strategies within the port. Yan et al. ²⁷ investigated an integrated train schedule template and container transfer scheme for import operations, where containers could either be processed in the yard or directly transported from the vessel to the railway operation area. Grishin et al. ²⁸ studied the direct transshipment process between ships and trains, optimizing berth allocation on the quayside and the distribution of container groups according to their destinations on landside. Focusing on export containers, Wang and Sun ²⁹ examined transfer plans under a direct transshipment mode, aiming to maximize the number of connecting trains, minimize the number of vessels involved, and reduce schedule adjustments. Building on a novel port layout that extends the railway line to the quayside, Ji et al. ³⁰ integrated train scheduling and container transfer operations to enhance the temporal and spatial coordination between trains and vessels.

Regarding container transfer operations at road-rail intermodal terminals, Schulz et al. ³¹ primarily focused on resource and capacity assignment, optimizing train scheduling sequences and track allocation issues. Basallo-Triana et al. ³² developed a method to measure the operational performance of intermodal transfer yards based on terminal layout, assessing factors such as yard capacity, crane operational cycles, and truck connections. Essghaier et al. ³³ addressed the uncertainty of truck arrival times and studied the truck scheduling problem in road-rail Physical Internet hubs. Xia et al. ³⁴ conducted an integrated study on train-track allocation and crane capacity allocation issues in a road-rail intermodal terminal with multiple yards. Chen et al. ³⁵ constructed a model based on a multi-commodity flow approach to optimize transfer station selection and train timetables. Their optimization was conducted from the perspective of the entire intermodal network, without specifically considering the internal operations of the transfer stations.

Scholars have recognized the impact of operations at transfer nodes on the efficiency of multimodal transportation, leading to various research initiatives. Their focus areas differ, encompassing equipment scheduling, resource allocation, and transshipment plans, among others. However, there has been limited research integrating time matching between different transport modes with cargo transshipment operations, despite the close relationship between the two. In light of this, this paper considers the air-high-speed rail intermodal transport of cargo and aims to achieve coordinated optimization of transfer plan and high-speed train scheduling template, thereby facilitating efficient spatial and temporal connections for the cargo.

Problem description

Given a time period T, there are $\left|I\right|$ international flights arriving at the airport carrying imported cargo. The expected arrival time of flight $i,i\in I$ is denoted as $a_i$ , and each flight carries various cargos destined for multiple locations, with the quantity of cargo for each direction $d$ represented as $v_i^d$ . These cargos can be unloaded within given time requirement $T_{\mathrm{unload}}$ , but they still require de-palletization and sorting operations, which can be conducted simultaneously.

For the airport, the de-palletization and sorting capacity is limited to $C_{\mathrm{sort}}$ . We assume that the sorting time for each batch of cargo is directly proportional to the quantity of cargo, with a picking time of $p$ per unit of cargo. Additionally, we assume that the customs clearance time for each batch of cargo is constant, denoted as $T_{\mathrm{clear}}$ , and that the customs clearance procedures can be conducted simultaneously with the sorting operations. The airport is equipped with a temporary storage area for short-term storage of intermodal cargo, with sufficient capacity.

The cargo arriving at the airport can either be temporarily stored at the airport or directly transported via road to the high-speed rail station. It is important to note that all cargo from the same flight and destined for the same transport direction must either be entirely stored at the airport or entirely transferred to the high-speed rail station; partial splitting of the shipment is not permitted. In this study, we assume that the road transfer capacity is sufficient, with a transportation time denoted as $T_{\mathrm{road}}$ .

At the high-speed rail station, there are $\left|K\right|$ available tracks for the loading of intermodal cargo, and there is no storage space within the station for cargo. Therefore, upon arrival at the high-speed rail station, the cargo must be unloaded and undergo security checks before being directly loaded onto the train. We assume that the unloading time is proportional to the cargo volume, with a rate of u per unit of cargo, while the security check time is fixed at $T_{\mathrm{safe}}$ . In practice, continuous loading operations commence only after all designated cargo is staged. The rationale for this is twofold: first, it prevents the unproductive occupancy of limited track resources; second, it helps maintain an uninterrupted workflow, avoiding the labor inefficiencies associated with intermittent loading processes. The high-speed rail station utilizes conveyor belts for the loading operations, with a capacity of q per unit of time. Consequently, the duration of the train’s stay at the station depends on the loading volume and operational capacity. However, due to the constraints of the train schedule, each train has an earliest arrival time $E_j$ and a latest departure time $L_j$ at the high-speed rail station. In this study, we assume that each train has a maximum carrying capacity of $Q_j$ . Dispatchers need to determine the optimal scheduling sequence for the trains and their directions based on flight arrival times and cargo loads. The scheduling of each train’s arrival at and departure from the operational tracks, along with its designated service direction, collectively constitute the train scheduling template. Figure 3 illustrates an example involving four trains, three directions, and two operational tracks. Specifically, Trains 1 and 2 initially enter Tracks 1 and 2 at times $t1$ and $t2$ , respectively. As Train 1 departs for direction D1 at $t4$ , Track 1 becomes available and is subsequently allocated to Train 3. Similarly, once Train 2 completes its operations and departs for direction D2 at $t5$ , Track 2 is then released for the assignment of Train 4.

Figure 3.

An example of train scheduling.

Given the information above, the cargo unloading and transfer process can be integrated with the high-speed rail distribution plan. This integration involves joint decision-making on both the cargo transfer scheme and the train scheduling template to ensure effective temporal coordination, as illustrated in Figure 4. Specifically, arriving air cargo must undergo unloading, depalletization and sorting, and customs clearance before being temporarily stored or directly transshipped. Notably, depalletization and sorting and customs clearance can be executed concurrently. Upon arriving at the HSR station via road drayage, the cargo undergoes unloading and security check before being loaded onto trains. Regarding railway operations, a train may enter the handling track directly if an available track exists; otherwise, it must wait. Furthermore, the destination of the transshipped cargo must align with the direction of the departing train, and the volume of cargo must meet the capacity requirements of the train. Therefore, the organization of intermodal distribution requires comprehensive synchronization across the dimensions of time, volume, and transport direction to maximize both transfer efficiency and throughput.

Figure 4.

Schematic diagram of air-rail intermodal cargo transfer operations.

In this paper, the time period T will be divided into multiple discrete sub-periods, $t\in T$ . The cargo transfer scheme determines which flights and directions of cargo will be transported to the high-speed rail station during each time period, while the train scheduling template specifies the operational times and directions of the trains, and the volume of cargo to be loaded. Given the constraints of available resources and capacity limitations, our objective is to minimize the dwell time of cargo and reduce the volume of remaining cargo as much as possible.

Mathematical formulation

Assumptions

To formulate a practical model for the problem at hand, we make the following assumptions:

• (1) The arrival schedule of flights, the cargo volume carried by each flight, and the destination of each cargo shipment are known in advance.

This assumption reflects standard practice in air cargo operations, where flight schedules and cargo manifests are typically determined prior to flight departure. Moreover, international flights usually have longer durations, providing sufficient time for information exchange with domestic airports.

• (2) The storage yard has sufficient capacity to accommodate all incoming cargo.

In practice, terminals are designed to handle peak volumes, and capacity shortages are rare under normal operating conditions. If necessary, this assumption can be relaxed in future studies to address scenarios with limited storage space.

• (3) The fleet capacity for road transfer is assumed to be adequate, with the transfer time treated as a constant parameter.

This assumption is motivated by the following factors. For the case study in this paper, truck transfer occurs within the Zhengzhou Airport Economy Zone, where transfer distances are short and road conditions are straightforward. Moreover, the airport is capable of pre-scheduling a sufficient truck fleet based on operational plans to ensure a stable service level. Furthermore, a constant transfer time represents a stable, benchmark service level, which is a common assumption in optimization studies for intermodal terminals involving truck drayage, such as sea-rail and rail-rail transshipment.^34,36 On the other hand, incorporating truck quantity constraints would necessitate auxiliary sub-models—such as vehicle scheduling or queuing theory—which could divert the focus from the core air-rail coordination logic. By isolating these variables, the model can more effectively optimize the primary decisions involving key stakeholders in the aviation and railway sectors. For future increases in operational scale or more complex transfer scenarios, while this assumption simplifies real-world traffic stochasticity, this research can be extended by employing uncertainty optimization to explore the impacts of truck shortages and travel time fluctuations.

• (4) Each cargo shipment is indivisible and must be transferred as a whole unit.

Many types of air cargo, such as containers or palletized shipments, are handled as single units for efficiency and security reasons. This assumption aligns with standard cargo handling practices. Additionally, if a shipment does need to be split, it can be represented in the input data as multiple demands with the same arrival time and destination.

Notations

The following notations are used in the model formulation (Table 1).

Table 1.

Notations.

Notations	Description
Sets
I	The set of flights.
J	The set of trains.
T	The set of time slots
K	The set of handling tracks
D	The set of directions
Indices
i	The index of flights, $i\in I$
j	The index of trains, $j\in J$
t	The index of time slots, $t\in T$
k	The index of tracks, $k\in K$
d	The index of directions, $d\in D$
Parameter
$a_i$	The arrival time of flight i.
$v_i^d$	The volume of cargo from flight i to direction d.
$T_{\mathrm{unload}}$	The unloading time of cargo from flights.
$T_{\mathrm{clear}}$	The customs clearance time for each batch of cargo.
$T_{\mathrm{road}}$	The connection transportation time by road.
$T_{\mathrm{safe}}$	The security check time at high-speed rail station.
$C_{\mathrm{sort}}$	The maximum de-palletization and sorting capacity of the airport.
p	The picking time per unit of cargo.
u	The unloading time per unit of cargo from trucks.
q	The loading capacity per unit of time.
$Q_j$	The carrying capacity of train j.
$[E_j,L_j]$	The feasible service time window of train j.
Decision variables
$x_j^d$	=1 if the direction of train j is d; =0, otherwise.
$y_{\mathrm{ij}}^d$	=1 if the cargo to direction d from flight i is transported by train j; =0, otherwise.
$z_{\mathrm{jk}}$	=1 if train j is assigned to track k; =0, otherwise.
$s_j$	The time when train j enters the track for operations.
$f_j$	The time when train j completes its operations and leaves the track.
$f_i^d$	The time when the cargo to direction d from flight i completes its operations.
$h_i^d$	The time when the cargo to direction d from flight i leaves the airport.
$r_{\mathrm{it}}$	=1 if flight i is being sorted at time slot t; =0, otherwise.
$b_i$	The time when the cargo of flight i begins the sorting operation.
$e_i$	The time when the cargo of flight i ends the sorting operation.
$w_{\mathrm{jj}\prime }$	$\in \{0,1\}$ , an auxiliary variable.
${\mu }_{\mathrm{jj}\prime }$	$\in \{0,1\}$ , an auxiliary variable.

Model formulation

The mathematical formulation is as follows:

(1) Objectives

In the intermodal transportation system for cargo, the efficiency of cargo transfer and the assurance of service quality are core demands. From the perspective of transportation organizers, the total transfer time of cargo is directly linked to operational costs and resource utilization efficiency. By precisely coordinating the timing and spatial connections between flights and trains, it is possible to reduce the dwell time of cargo at hub nodes, thereby lowering storage costs and enhancing the overall operational efficiency of the transportation network.

Furthermore, the volume of remaining cargo reflects the service assurance capability of the transportation system and is closely related to customer satisfaction. This is particularly critical in scenarios involving high-value cargo, where time sensitivity is even more stringent. Therefore, the model proposed in this paper aims to minimize both the total transfer time of cargo and the volume of remaining cargo. The synergistic optimization of these two objectives can lead to improvements in transportation efficiency and service quality.

$$min{F}_1=\sum _{i\in I}\sum _{d\in D}(f_i^d-a_i)$$

(1)

$$min{F}_2=\sum _{i\in I}\sum _{d\in D}(1-\sum _{j\in J}{y}_{\mathrm{ij}}^d)·v_i^d$$

(2)

The objective function (1) is to minimize the air-to-rail transfer time of all cargo. The objective function (2) is to minimize the volume of remaining cargo during the planning period.

Due to the significant difference in magnitude between the two objective functions, it is essential to normalize them to ensure the rationality and scientific validity of the calculations. Let $F_1\_\min$ and $F_1\_\max$ represent the minimum and maximum values of the transfer time of all cargo, respectively, and let $F_2\_\min$ and $F_2\_\max$ denote the minimum and maximum values of the volume of remaining cargo. The normalized objective functions can then be expressed as follows:

$$F_1\_\mathrm{norm}=(F_1-F_1\_\min )/(F_1\_\max -F_1\_\min )$$

(3)

$$F_2\_\mathrm{norm}=(F_2-F_2\_\min )/(F_2\_\max -F_2\_\min )$$

(4)

(2) Constraints

• Assignment constraints

$$\sum _{d\in D}{x}_j^d=1,\forall j\in J$$

(5)

$$\sum _{k\in K}{z}_{\mathrm{jk}}=1,\forall j\in J$$

(6)

$$\sum _{j\in J}{y}_{\mathrm{ij}}^d\le 1,\forall i\in I,d\in D$$

(7)

$$y_{\mathrm{ij}}^d\le x_j^d,\forall i\in I,j\in J,d\in D$$

(8)

$$y_{\mathrm{ij}}^d\le v_i^d,\forall i\in I,j\in J,d\in D$$

(9)

Constraint (5) indicates that each train has a unique transportation direction. Constraint (6) states that each train must be assigned to only one track. Constraint (7) specifies that the cargo from the same flight in a particular direction can only be loaded onto one train, meaning the cargo cannot be split. Constraint (8) requires that the destination of the cargo loaded onto the train must align with the train’s operational direction. Constraint (9) indicates that non-existent cargo cannot be transported.

• Time related constraints

$$h_d^i\le M·\sum _{j\in J}{y}_{\mathrm{ij}}^d,\forall i\in I,d\in D$$

(10)

$$\begin{array}{c}\multicolumn{1}{c}{h_i^d+M·(1-\sum _{j\in J}{y}_{\mathrm{ij}}^d)\ge a_i+T_{\mathrm{unload}}+T_{\mathrm{clear}},}\\ \multicolumn{1}{c}{\forall i\in I,d\in D}\end{array}$$

(11)

$$h_i^d+M·(1-\sum _{j\in J}{y}_{\mathrm{ij}}^d)\ge e_i,\forall i\in I,d\in D$$

(12)

$$\begin{array}{c}\multicolumn{1}{c}{h_i^d+T_{\mathrm{road}}+v_i^d·u+T_{\mathrm{safe}}\le s_j+M·(1-y_{\mathrm{ij}}^d),}\\ \multicolumn{1}{c}{\forall i\in I,j\in J,d\in D}\end{array}$$

(13)

$$\begin{array}{c}\multicolumn{1}{c}{h_i^d+T_{\mathrm{road}}+v_i^d·u+T_{\mathrm{safe}}\ge s_j-1-M·(1-y_{\mathrm{ij}}^d),}\\ \multicolumn{1}{c}{\forall i\in I,j\in J,d\in D}\end{array}$$

(14)

$$\begin{array}{c}\multicolumn{1}{c}{f_j\ge s_j+(\sum _{i\in I}{y}_{\mathrm{ij}}^d·v_i^d)/q-M·(1-x_j^d),}\\ \multicolumn{1}{c}{\forall j\in J,d\in D}\end{array}$$

(15)

$$f_i^d\ge f_j-M·(1-y_{\mathrm{ij}}^d),\forall i\in I,j\in J,d\in D$$

(16)

$$f_i^d·v_i^d\ge \left|T\right|·(1-\sum _{j\in J}{y}_{\mathrm{ij}}^d)·v_i^d,\forall i\in I,d\in D$$

(17)

$$f_i^d\ge a_i-M·v_i^d,\forall i\in I,d\in D$$

(18)

$$s_j\ge E_j,\forall j\in J$$

(19)

$$f_j\le L_j,\forall j\in J$$

(20)

Constraint (10) restricts the leaving time of cargo that has not been transferred from the airport. Constraint (11) ensures that the leaving time of the cargo from the airport must be later than the time when customs clearance procedures are completed. Constraint (12) guarantees that the leaving time of the cargo from the airport must be later than the time when sorting is completed. Constraints (13) and (14) indicate that the time when the transferred cargo is ready for loading onto trains must be earlier than the time when the train enters the track to begin operations. Constraint (15) calculates the time when the train completes the loading operation. Constraints (16)–(18) specify the completion times for the transfer operations of cargo with various directions from each flight, where the end time for transferred cargo cannot be earlier than the train’s departure time, while the end time for remaining cargo is set to the end of the planning period. Meanwhile, the completion time involving non-existent cargo is equal to their arrival time, meaning that no transfer duration will be incurred in this case. Constraints (19) and (20) ensure that the start and end times of the train’s operations must meet the specified operational time window requirements.

• Capacity constraints

$$\sum _{i\in I}{y}_{\mathrm{ij}}^d·v_i^d\le Q_j,\forall j\in J,d\in D$$

(21)

$$\begin{array}{c}\multicolumn{1}{c}{s_j-f_{j\prime }\ge -M·(3-w_{\mathrm{jj}\prime }-z_{\mathrm{jk}}-z_{j\prime k}),}\\ \multicolumn{1}{c}{\forall j,j\prime \in J,j\ne j\prime ,k\in K}\end{array}$$

(22)

$$\begin{array}{c}\multicolumn{1}{c}{s_{j\prime }-f_j\ge -M·(3-{\mu }_{\mathrm{jj}\prime }-z_{\mathrm{jk}}-z_{j\prime k}),}\\ \multicolumn{1}{c}{\forall j,j\prime \in J,j\ne j\prime ,k\in K}\end{array}$$

(23)

$$w_{\mathrm{jj}\prime }+{\mu }_{\mathrm{jj}\prime }\ge 1,\forall j,j\prime \in J,j\ne j\prime$$

(24)

$$\sum _{t\in T}{r}_{\mathrm{it}}\ge v_i·p,\forall i\in I$$

(25)

$$\sum _{t\in T}{r}_{\mathrm{it}}\le v_i·p+1,\forall i\in I$$

(26)

$$b_i\ge a_i+T_{\mathrm{unload}},\forall i\in I$$

(27)

$$b_i\le r_{\mathrm{it}}·t+M·(1-r_{\mathrm{it}}),\forall i\in I,t\in T$$

(28)

$$e_i\ge r_{\mathrm{it}}·t,\forall i\in I,t\in T$$

(29)

$$b_i+\sum _{t\in T}{r}_{\mathrm{it}}=e_i+1,\forall i\in I$$

(30)

$$\sum _{i\in I}{r}_{\mathrm{it}}\le C_{\mathrm{sort}},\forall t\in T$$

(31)

Constraint (21) states that the total amount of cargo loaded onto the train cannot exceed its maximum capacity. Constraints (22) to (24) indicate that the operational times of trains assigned to the same track cannot overlap. Constraints (25) and (26) specify the duration required for sorting all cargo from a flight. Constraint (27) ensures that the sorting of a flight’s cargo does not begin earlier than its arrival time plus the unloading time. Constraints (28) to (30) calculate the start and end times for the sorting of cargo, and ensure that the sorting operations for the same flight must be conducted consecutively. Constraint (31) requires that the number of flights undergoing sorting operations at any given time cannot exceed the airport’s maximum sorting capacity.

• Variable constraints

$$x_j^d,y_{\mathrm{ij}}^d,z_{\mathrm{jk}},r_{\mathrm{it}}\in \{0,1\},\forall i\in I,j\in J,d\in D,k\in K,t\in T$$

(32)

$$s_j,f_j,f_i^d,h_i^d,b_i,e_i\ge 0,\forall i\in I,j\in J,d\in D$$

(33)

Constraints (32)–(33) specify variable ranges.

Computational experiments

In this section, computational experiments are carried out to evaluate the proposed model on Windows 10, Intel (R) Core (TM) i7-10875H CPU @ 2.30 GHz, 16 GB RAM. The model is coded in PyCharm 2017.3.2 linked with Cplex 12.8.

Case introduction

This section conducts numerical experiments with reference to the Zhengzhou Airport Economy Zone. The air-rail intermodal cargo is transported from Zhengzhou Xinzheng International Airport to Zhengzhou Hangkonggang Railway Station via road connections. Currently, as the air-rail intermodal service is still in its early stages and the volume of goods is relatively small, the cargo transfer organization mode at this hub operates on a one-to-one basis between flights and trains. Considering the potential for future increases in volume and service coverage, this paper examines the matching of multiple flights with multiple trains. The parameter values are set with reference to relevant literature^3,21,22 and the industry report, ³⁷ and are reasonably assumed according to practical conditions, as shown in Table 2.

Table 2.

The values of parameters.

Parameters	Definition	Values
$T_{\mathrm{unload}}$	The unloading time of cargo from flights.	10 min
$T_{\mathrm{clear}}$	The customs clearance time for each batch of cargo.	20 min
$T_{\mathrm{road}}$	The connection transportation time by road.	30 min
$T_{\mathrm{safe}}$	The security check time at high-speed rail station.	5 min
$C_{\mathrm{sort}}$	The maximum de-palletization and sorting capacity of the airport.	3
p	The picking time per unit of cargo.	1 min
u	The unloading time per unit of cargo from trucks.	1 min
q	The loading capacity per unit of time.	5

The above parameters can be adjusted manually based on the increase in airport staff or upgrades to facilities, without affecting the quality and efficiency of the model’s solutions. For the high-speed railway station, we assume that there are three operational lines available for the loading and unloading of goods in the air-rail intermodal service. This means that the station can simultaneously accommodate up to three trains, which should be sufficient for the current volume and potential future capacity development.

Objective discussion

In this section, we discuss the impact of setting two objective functions on the problem-solving process through multiple sets of numerical experiments. Both objectives have been normalized during the computation, as outlined in Section 4.3. We designate $F_1\_\mathrm{norm}$ , which aims to minimize the total cargo transfer time, as the primary objective. Meanwhile, $F_2\_\mathrm{norm}$ , which seeks to minimize the remaining cargo volume, is incorporated into the model as a constraint:

$$F_2\_\mathrm{norm}\le \epsilon$$

where $\epsilon$ takes values in the range of [0, 1].

Given a planning period of $T=600\min$ , there are 10 flights arriving, carrying air-rail intermodal cargo in 3 different directions, with 4 trains available to provide distribution services. The results are presented in Table 3, where ‘ALR’ denotes the average loading rate of the trains, ‘ATT’ represents the average transfer time for cargo. Figure 5 illustrates the relationship between the two objective functions.

Table 3.

Comparison of results under different objective settings.

$\mathbf{\epsilon }\in [0,1]$ with a step size of 0.1					$\mathbf{\epsilon }\in [0,0.1]$ with a step size of 0.01
$\epsilon$	$F_1$	$F_2$	ALR	ATT	$\epsilon$	$F_1$	$F_2$	ALR	ATT
0.0	5996	145	0.93	253.1	0.0	5996	145	0.93	253.1
0.1	5884	163	0.84	232.3	0.01	5896	147	0.92	223.1
0.2	5884	165	0.83	225.6	0.02	5892	148	0.92	232.8
0.3	5884	165	0.83	225.6	0.03	5892	150	0.91	236.1
0.4	5884	165	0.83	225.6	0.04	5892	150	0.91	236.1
0.5	5884	163	0.84	232.3	0.05	5888	153	0.89	245.9
0.6	5884	163	0.84	232.3	0.06	5888	154	0.89	249.2
0.7	5884	165	0.83	225.6	0.07	5888	155	0.88	239.2
0.8	5884	165	0.83	225.6	0.08	5888	153	0.89	245.9
0.9	5884	165	0.83	225.6	0.09	5888	159	0.86	229.2
1.0	5884	165	0.83	225.6	0.1	5884	163	0.84	232.3

Figure 5.

Relationship between the two objectives.

We first evaluate the values of $\epsilon$ in the range [0, 1] with a step size of 0.1. The results of the objective functions are shown in Figure 5(a) and Table 3. At this stage, there is a noticeable trend in the values of the two objective functions only between $\epsilon =0$ and $\epsilon =0.1$ , while the results in the other intervals remain relatively stable. Furthermore, we evaluate the values of $\epsilon$ in the range [0, 0.1] with a finer step size of 0.01. The results of the objective functions are presented in Figure 5(b) and Table 3. In the figure, the green points indicate the minimum value obtained for $F_1$ , while the purple points represent the minimum value for $F_2$ .

As illustrated in Figure 5(b), varying the value of $\epsilon$ leads to observable shifts in both objective functions, revealing a complex coupling and competitive relationship between them. To further analyze the trade-off, we identified several representative solutions that reflect different operational priorities:

• Time-Optimized Solution (Point A): The model achieves the minimum cargo transfer time ( $F_1$  = 5884). However, this leads to a higher volume of remaining cargo ( $F_2$  = 163), suggesting that rapid dispatching may prevent some late-arriving cargo from being loaded in time.

• Throughput-Optimized Solution (Point B): The remaining cargo is minimized to 145. This improvement in throughput comes at the expense of a significantly longer transfer time ( $F_1$  = 5996), as a prolonged waiting period is required for additional cargo to arrive to maximize loading capacity.

• Balanced Trade-off Solutions (Points C and D): Points C ( $F_1$  = 5896, $F_2$  = 147) and D ( $F_1$  = 5892, $F_2$  = 148) offer superior compromises. Specifically, compared to Point A, Solution C achieves a nearly 10% reduction in remaining cargo while incurring only a marginal increase (approximately 0.2%) in transfer time.

In practice, this provides decision-makers with a flexible framework to accommodate different operational preferences, allowing them to select a plan that best aligns with current station congestion and cargo urgency. When $\epsilon$ is set to 0.01 or 0.02, it effectively achieves a good balance between the two objectives, helping to avoid excessive losses for either one. For computational convenience, in the subsequent experiments, we consider the fairness between the two objective functions by defining the overall objective function as a linear combination of the normalized values of both.

Additionally, as shown in Table 3, the trade-off between cargo transfer time and remaining cargo amount also affects the average loading rate of the trains. In practice, the train loading rate is a crucial criterion that influences whether a train operates and its operational efficiency. We will conduct a detailed analysis of this in Section 5.5.

Performance of the proposed model

Performance under different objective settings

In this section, we validate the solution efficiency and quality of the model under different objective settings. The case studies involve 10–15 flights and 4–6 trains over a planning period of 1 days ( $T=1440$  min). There are a total of 3–5 serviceable cargo directions. Due to the multi-objective nature of this study, different approaches to handling these objectives can significantly impact the efficiency and quality of the model’s solutions. We compared two methods: the first method optimizes the normalized $F_1\_\mathrm{norm}$ as the primary objective while incorporating $F_2\_\mathrm{norm}$ as a constraint, with $\epsilon =0.01$ and $\epsilon =0.02$ , respectively. The second method involves a linear weighted sum of the two objective functions. To ensure fairness and avoid scale-dominance, we use the normalized objective values derived from equations (3) and (4) in Section 4.3. Accordingly, equal weights are assigned to both objectives to treat them with equivalent priority. The results are presented in Table 4, where ‘OF1’ indicates the optimal value for minimizing total cargo transfer time, ‘OF2’ represents the optimal value for minimizing the remaining cargo quantity, and ‘CT’ denotes computational time. The shaded areas identify the results that are closest to the optimal values in each case. Figures 6 and 7 show the deviations of $F_1$ and $F_2$ from the optimal value obtained during single-objective optimization under different bi-objective processing strategies, respectively.

Table 4.

Solution performance under different objective settings.

Case	$\left|I\right|$	$\left|J\right|$	OF1	OF2	F2 with e = 0.01			F2 with e = 0.02			Linear combination
					$F_1$	$F_2$	CT	$F_1$	$F_2$	CT	$F_1$	$F_2$	CT
Directions = 3
1	10	4	12,758	92	12,780	93	21.3	12,780	96	17.1	12,780	92	3.8
2		5	11,074	65	11,569	66	6.9	11,565	69	11.9	11,082	71	10.4
3		6	9930	38	9949	40	23.5	9939	43	28.5	9949	38	4.5
4	15	4	15,325	211	15,342	211	102.3	15,342	212	31.5	15,342	211	6.4
5		5	13,639	184	14,359	186	39.5	13,644	188	144.3	13,644	187	11.9
6		6	12,496	149	12,509	151	390.2	12,509	151	103.6	12,515	149	7.4
Directions = 4
7	10	4	19,890	166	20,442	168	42.4	20,442	167	36.6	19,907	172	5.2
8		5	18,233	142	18,739	144	44.5	18,739	143	38.1	18,739	143	24.3
9		6	16,241	100	16,663	102	45.2	16,658	105	67.8	16,663	100	25.1
10	15	4	22,907	306	23,454	308	586.6	23,454	308	1211.9	23,454	306	31.6
11		5	21,255	282	21,751	284	190.1	21,751	283	54.6	21,751	283	42.5
12		6	19,251	236	19,268	237	2189.1	19,263	241	608.3	19,268	236	6.7
Directions = 5
13	10	4	22,045	192	22,062	194	62.6	22,062	196	42.3	22,062	194	19.6
14		5	20,835	169	20,853	171	57.5	20,853	172	72.9	20,853	170	5.2
15		6	18,479	127	18,484	127	39.8	18,479	132	59.3	18,484	127	49.2
16	15	4	25,164	336	25,164	338	35.4	25,164	338	38.7	25,164	338	7.8
17		5	23,954	312	23,955	314	84.7	23,955	314	116.1	24,205	312	54.8
18		6	21,781	269	21,786	269	109.3	21,781	274	51.8	21,786	269	26.3

Note: Shading denotes results nearest to the optimal values.

Figure 6.

The deviation of $F_1$ from the optimal value.

Figure 7.

The deviation of $F_2$ from the optimal value.

As indicated in Table 4, for a given planning period of T = 1440 min, the model is capable of obtaining effective optimal solutions across different scales of cases. In terms of solution efficiency, the computational time when linearly combining the two normalized objective functions is generally significantly faster than the time required to solve the problem using $\epsilon$ -constraint method for multi-objective optimization. For the latter approach, the solution times for most cases remain relatively consistent when $\epsilon$  = 0.01 and $\epsilon$  = 0.02. Furthermore, it can be observed that as the number of service directions and flights increases, the computational time correspondingly increases. However, the variation in the number of trains does not exhibit a clear pattern in its impact on computational time.

In terms of solution quality, the optimization of $F_2$ is more effective when employing the multi-objective linear combination approach, as most cases under this strategy are closer to the optimal solution obtained through the individual optimization of $F_2$ . Conversely, when using the $\epsilon$ -constraint method for multi-objective optimization, it is more conducive to achieving better results for $F_1$ . Specifically, when $\epsilon$  = 0.02, $F_1$ performs optimally, with most cases under this strategy being closest to the optimal solution obtained through the individual optimization of $F_1$ .

Overall, when considering the solution performance, there are three instances where both $F_1$ and $F_2$ perform best among three scenarios at $\epsilon$  = 0.01, and similarly, three instances at $\epsilon$  = 0.02. In contrast, when using the linear combination approach, there are eight cases where both $F_1$ and $F_2$ achieve optimal performance. Therefore, in subsequent experiments, we will adopt the method of linearly summing the two normalized objective functions to solve the various cases.

Performance on large-scale instances

In this section, we evaluate the performance of the CPLEX solver through large-scale computational experiments. If the planning horizon is excessively long, data uncertainty and variability will significantly increase, thereby affecting the feasibility and optimization effectiveness of the proposed solutions. Therefore, designing the scheme on a daily basis allows for better integration with actual operational data and enhances the executability of the plan.

According to actual data, Zhengzhou Xinzheng International Airport handles approximately 250 inbound and outbound international cargo flights per week (HPPG, 2025). It can be reasonably inferred that there are, on average, about 17–18 inbound flights per day. Therefore, the large-scale instances in this section involve 20–30 inbound flights and 10–20 trains over a planning period of 1 day (T = 1440 min). There are a total of 5–10 serviceable cargo directions. The experimental results are presented in Table 5.

Table 5.

Results of large-scale instances.

$\left|D\right|$	$\left|I\right|$	$\left|J\right|$	F1	F2	CT
5	20	10	23,346	140	259.8
		15	17,866	111	276.5
		20	14,822	11	1059.4
	25	10	32,143	162	344
		15	23,376	100	328.2
		20	18,556	31	1363.3
	30	10	38,670	252	315.4
		15	27,844	139	458.5
		20	24,635	86	1664.7
10	20	10	69,463	706	359.4
		15	57,401	507	502.2
		20	53,237	246	1785.1
	25	10	79,778	798	554.7
		15	64,763	553	676.4
		20	59,182	250	2034.8
	30	10	101,582	1071	613.6
		15	82,798	745	767.3
		20	72,552	436	2456.4

The data scale set in this section not only reflects current operational realities but also accommodates potential future increases in cargo volume. Experimental results demonstrate that CPLEX is capable of obtaining optimal solutions within an acceptable time frame. Specifically, for single-day instances involving 30 flights, 20 trains and 10 directions, CPLEX successfully finds the optimal solution within 2500 s. For long-term intermodal transport schemes, the planning horizon can be divided into multiple daily segments for separate optimization, thereby ensuring both flexibility and practical operability of the solutions. Therefore, CPLEX can be directly applied to solve most scenarios of the current scale.

However, considering the theoretical maximum applicability of the approach, if the data scale increases exponentially, the computational time required by exact algorithms such as CPLEX may rise significantly, presenting certain limitations. Consequently, future research may consider designing heuristic strategies or employ metaheuristic algorithms—such as genetic algorithms or large neighborhood search algorithms—as potential alternative solution approaches, based on the specific characteristics of the problem.

Value of integrated optimization

This study integrates the cargo transfer scheme with the train template into a unified decision-making framework that is, integrated optimization, to achieve globally optimal cargo distribution organization. In this section, to evaluate the value of integrated optimization, we experimentally compare the results obtained by the integrated optimization approach (IOA) with those from the sequential optimization approach (SOA). In the SOA, the transfer scheme is optimized first, and then the train template problem is optimized based on the results of the transfer scheme; the two are not optimized simultaneously in a coordinated manner. The comparison is conducted from two perspectives: the transfer time of all cargo ( $F_1$ ), the volume of remaining cargo ( $F_2$ ), and the average loading rate of trains (ALR). We define the percentage gap between $F_1$ under IOA and $F_1$ under SOA as $GAP=\left(\frac{F_1\_SOA-F_1\_IOA}{F_1\_IOA}\right)·100\%$ . The results are presented in Table 6.

Table 6.

Comparison of integrated optimization and sequential optimization approach.

$\left|D\right|$	$\left|I\right|$	$\left|J\right|$	IOA			SOA			GAP
			F1	F2	ALR	F1	F2	ALR
3	10	4	12,780	92	0.86	13,202	92	0.86	3.3%
		5	11,082	71	0.76	11,991	72	0.75	8.2%
		6	9949	38	0.72	10,397	39	0.72	4.5%
	15	4	15,342	211	0.86	16,044	211	0.86	4.6%
		5	13,644	187	0.77	14,397	187	0.77	5.5%
		6	12,515	149	0.74	13,189	150	0.74	5.4%
4	10	4	19,907	172	0.83	21,063	175	0.82	5.8%
		5	18,739	143	0.76	19,462	143	0.76	3.9%
		6	16,663	100	0.76	17,284	105	0.75	3.7%
	15	4	23,454	306	0.86	24,658	306	0.86	5.1%
		5	21,751	283	0.76	22,871	289	0.75	5.1%
		6	19,268	236	0.77	20,435	236	0.77	6.1%
5	10	4	22,062	194	0.85	23,380	196	0.85	6.0%
		5	20,853	170	0.76	22,153	170	0.76	6.2%
		6	18,484	127	0.75	19,814	130	0.74	7.2%
	15	4	25,164	338	0.85	27,345	338	0.85	8.7%
		5	24,205	312	0.77	26,148	314	0.76	8.0%
		6	21,786	269	0.76	23,979	272	0.75	10.1%

As shown in Table 6, the transfer time ( $F_1$ ) obtained by SOA is consistently higher than that of IOA, with a maximum increase of 10.1%. In addition, the remaining cargo volume ( $F_2$ ) also increases in some instances, and the average train loading rate decreases slightly. These experimental results indicate that IOA can comprehensively consider the coordination between transfer time and train scheduling, thus achieving the global optimum for the system and the highest overall transportation efficiency. In contrast, in SOA, the result of the first-stage optimization constrains the solution space of the subsequent stage, leading to efficiency losses and inferior performance compared to the IOA.

Impact of train operational characteristics

Loading rate of trains

In practice, the loading rate of trains significantly impacts their operation, often necessitating that a certain loading rate be met before a train is allowed to run, thereby ensuring operational efficiency. This section conducts experiments to assess the influence of train loading rates on the solution to this problem. Specifically, we introduce the variable $v_j$ to determine whether a train will operate. Constraints are implemented to ensure that the loading rate meets a specified target value LLR. The relevant constraints are as follows.

$$v_j\le \sum _{i\in I}\sum _{d\in D}{y}_{\mathrm{ij}}^d·v_i^d,\forall j\in J$$

(34)

$$v_j\ge \sum _{i\in I}\sum _{d\in D}{y}_{\mathrm{ij}}^d·v_i^d/M,\forall j\in J$$

(35)

$$\sum _{i\in I}\sum _{d\in D}{y}_{\mathrm{ij}}^d·v_i^d/Q_j+M·(1-v_j)\ge \mathrm{LLR}$$

(36)

In the planning period of $T=1440$  min, the case study involves 10–15 flights and 4–6 trains serving 3 directions. The experimental results are presented in Table 7, where ‘LLR’ denotes the lowest loading rate that trains must achieve, ‘NRT’ indicates the number of running trains, ‘ALR’ represents the average loading rate of running trains, ‘MAXLR’ indicates the maximum loading rate of running trains, ‘MINLR’ denotes the minimum loading rate of running trains, ‘ATT’ signifies the average transfer time for cargo, ‘MAXT’ indicates the maximum transfer time for cargo, ‘MINT’ denotes the minimum transfer time for cargo, ‘NFS’ represents the total number of flights served, and ‘AFS’ represents the average number of flights served by a train. Figure 8 shows the variation in cargo transfer time under different train loading rates for each group of examples.

Table 7.

Comparative results of different train loading rate requirements.

$\left|J\right|$	$\left|I\right|$	LLR	NRT	ALR	MAXLR	MINLR	ATT	MAXT	MINT	NFS	AFS
4	10	0	4	0.86	1.00	0.47	455.6	852	126	9	4.0
		0.5	3	0.99	1.00	0.97	451.3	852	102	9	4.7
		0.6	3	0.99	1.00	0.97	560.6	1032	102	8	4.7
		0.7	3	0.99	1.00	0.97	496.3	852	162	9	4.7
		0.8	3	0.99	1.00	0.97	470.6	852	162	10	4.7
		0.9	3	0.99	1.00	0.97	534.9	1032	102	8	4.7
	15	0	4	0.86	1.00	0.47	415.4	852	126	10	4.3
		0.5	3	0.99	1.00	0.97	424.0	852	132	10	5.0
		0.6	3	0.99	1.00	0.97	478.0	942	102	9	5.0
		0.7	3	0.99	1.00	0.97	466.0	1122	132	10	5.0
		0.8	3	0.99	1.00	0.97	466.0	1122	132	10	5.0
		0.9	3	0.99	1.00	0.97	466.0	1122	132	10	5.0
5	10	0	5	0.76	1.00	0.25	385.7	762	96	9	3.6
		0.5	4	0.93	1.00	0.70	434.0	872	96	10	4.3
		0.6	4	0.93	1.00	0.70	434.0	872	96	10	4.3
		0.7	4	0.93	1.00	0.70	434.0	872	96	10	4.3
		0.8	3	1.00	1.00	1.00	516.7	1122	96	9	5.0
		0.9	3	1.00	1.00	1.00	516.7	1122	96	9	5.0
	15	0	5	0.77	1.00	0.25	348.6	762	96	11	3.8
		0.5	4	0.92	1.00	0.70	397.2	872	96	12	4.5
		0.6	4	0.92	1.00	0.70	397.2	872	96	12	4.5
		0.7	4	0.92	1.00	0.70	397.2	872	96	12	4.5
		0.8	3	1.00	1.00	1.00	458.9	1122	96	11	5.3
		0.9	3	1.00	1.00	1.00	425.1	1122	96	11	5.3
6	10	0	6	0.72	1.00	0.22	335.5	670	92	10	3.3
		0.5	4	0.95	1.00	0.82	441.6	922	92	10	4.5
		0.6	4	0.95	1.00	0.82	391.6	670	92	10	4.5
		0.7	4	0.95	1.00	0.82	411.6	670	112	9	4.5
		0.8	4	0.95	1.00	0.82	431.6	922	112	10	4.5
		0.9	3	0.99	1.00	0.97	414.9	922	112	9	4.7
	15	0	6	0.74	1.00	0.25	334.5	672	112	12	3.5
		0.5	4	0.98	1.00	0.97	444.6	1122	112	12	4.8
		0.6	4	0.98	1.00	0.97	425.7	922	92	12	4.8
		0.7	4	0.98	1.00	0.97	444.6	922	112	12	4.8
		0.8	4	0.98	1.00	0.97	425.7	922	92	12	4.8
		0.9	4	0.98	1.00	0.97	425.7	1122	92	12	4.8

Figure 8.

The cargo transfer time under different train loading rates.

From Table 7, it is evident that the required loading rate of the trains significantly impacts the results of this study. Specifically, when there is no restriction on the minimum loading rate, all trains can operate. In this scenario, the loading rates of the trains vary considerably, resulting in a lower average loading rate, with some trains even having loading rates below 30%. However, in this case, the average transfer time for all transported cargo is minimized, and the maximum transfer time for any single shipment is also at its lowest. Additionally, each train services the fewest number of flights on average.

When a lowest loading rate is imposed, it can be observed that some trains do not operate due to failing to meet the requirements. Specifically, as the loading rate threshold is introduced or increased, the number of running trains (NRT) decreases, dropping from a maximum of six trains to as few as three. However, to accommodate the collection of cargo for the trains, the average transfer time(ATT) for consolidation increases, with the maximum increase reaching 34% compared to scenarios without a loading rate constraint. On the other hand, as the loading rate requirement rises, the average number of flights served (AFS) per train also increases. Overall, to avoid capacity waste, it is preferable to operate fewer trains with higher cargo loads, thereby reducing operational costs. However, in some scenarios, when the lowest loading rate is set to 0.9, the cargo consolidation time increases significantly—compared to a threshold of 0.7 or 0.8, ATT increases by up to 19.1% and the maximum transfer time (MAXT) increases by up to 37.6%. Therefore, we recommend setting the lowest loading rate requirement at 0.7 or 0.8, as this effectively balances train operational efficiency with cargo transfer efficiency.

Time windows of trains

The transfer of cargo in the air-rail intermodal transport system depends on the availability of high-speed trains that can provide distribution services. According to the requirements of railway transportation organization, trains typically operate within specified time windows. The duration of these time windows affects the flights that can be connected and the volume of cargo that can be loaded on trains. This section discusses the impact of different train time window configurations on the problem through sensitivity analysis. In the planning period of $T=1440$  min, the example involves 10–15 flights and 4–6 trains serving 3 directions. The experimental results are presented in Table 8 and Figure 9, where ‘TW’ denotes the length of the train time window.

Table 8.

Comparative results of different train time windows.

$\left|J\right|$	$\left|I\right|$	TW	F1	F2	ALR	ATT	MAXT	MINT	AFS
4	10	20	12,780	92	0.86	455.6	852	126	4
		40	12,780	92	0.86	455.6	852	126	4
		60	12,916	90	0.87	436	852	106	4
		90	12,916	90	0.87	413.5	852	106	4
	15	20	15,342	211	0.86	415.4	852	126	4.25
		40	15,342	211	0.86	415.4	852	126	4.25
		60	15,478	209	0.87	375.8	852	106	4.25
		90	15,478	209	0.87	370.5	852	106	4.25
5	10	20	11,512	71	0.76	394.6	872	97	3.6
		40	11,082	71	0.76	385.7	762	96	3.6
		60	11,261	63	0.78	375.6	762	96	3.6
		90	10,166	48	0.83	364.5	762	94	3.8
	15	20	14,302	187	0.77	369.6	782	97	3.6
		40	13,644	187	0.77	348.6	762	96	3.8
		60	13,644	187	0.77	343.9	672	96	3.8
		90	12,956	167	0.83	364.5	762	94	3.8
6	10	20	9949	38	0.72	335.5	670	92	3.33
		40	9949	38	0.72	335.5	670	92	3.33
		60	9067	28	0.75	324.6	670	92	3.5
		90	7964	21	0.77	308.8	670	90	3.67
	15	20	12,515	149	0.74	334.5	672	112	3.5
		40	12,515	149	0.74	334.5	672	112	3.5
		60	11,541	132	0.79	329.6	852	96	3.83
		90	10,990	128	0.8	301.7	826	90	3.83

Figure 9.

Illustration of the impact of time windows on indicators.

Table 8 indicates that the length of the available time window for trains significantly affects the transfer operations of air-rail intermodal cargo. Specifically, as the length of the time window increases, the volume of cargo that can be transferred also rises, resulting in a decrease in the amount of remaining cargo. Additionally, the average loading rate of the trains gradually improves, and the average number of flights served per train also increases. This suggests that when train capacity is sufficient, a longer service time window allows for the consolidation of more cargo without the need to wait for subsequent trains. However, this extended time window also impacts the transfer times of the cargo. While some cargo may experience longer transfer times, others may benefit from shorter transfer times. Overall, the average transfer duration for cargo decreases, indicating that the extended time windows are beneficial for enhancing the efficiency of cargo transfers.

When a train reaches its maximum loading capacity or when all demand has been met, it is necessary for the train to operate regardless of the extension of the service time window, as this extension provides no additional benefits in these scenarios. Furthermore, the time window for trains also affects the utilization of track resources, which in turn impacts the operations of subsequent trains. Therefore, in practice, the service time window for trains should be established with consideration of various factors, including adherence to train scheduling rules, the availability of track resources at the station, the maximum carrying capacity of the trains, and the volume of freight demand, to ensure optimal efficiency in cargo transfer operations and resource utilization.

Managerial insights

To enhance the operational efficiency of air-rail intermodal transport in the Zhengzhou Airport Economy Zone, the following managerial recommendations are proposed:

• (1) Drawing on the findings of this study, it is advisable to organize multiple flights connecting with trains, rather than adopting a simple one-to-one matching mode. This strategy helps facilitate efficient cargo consolidation and improves the overall efficiency and market competitiveness of air-rail intermodal transport.

• (2) Based on the optimization methods presented in this paper, it is recommended to arrange connecting scheme in a rational manner. By scheduling truck transfers according to the timetables of flights and high-speed trains, seamless connections can be achieved, thereby minimizing cargo dwell time during transshipment and enhancing overall timeliness.

• (3) Establishing a dynamic minimum loading rate threshold is crucial for balancing operational efficiency with service quality. Based on our specific experimental scenarios (e.g. Tables 7 and 8), a threshold range of 0.7–0.8 was found to effectively prevent the excessive pursuit of high loading rates, which could otherwise lead to cargo waiting. However, this optimal range is not fixed; it should be adjusted according to the scale of demand and train capacity. For instance, when train capacity is abundant relative to cargo volume, a lower threshold may be acceptable to ensure timely delivery.

• (4) The service time window should be carefully calibrated to reflect the trade-off between cargo consolidation and system throughput. While a 60 min window proved effective in our test cases by allowing sufficient cargo accumulation, this value serves primarily as an illustrative benchmark. Decision-makers should adapt the window based on real-time factors such as track capacity and freight urgency. Notably, as the scale of demand or flight frequency increases, the service time window should typically be shortened to accelerate cargo turnover and prevent bottlenecks at the handling area.

• (5) It is essential to strengthen resource coordination and information sharing among airports, railways, and logistics enterprises. Timely sharing of data such as schedules, cargo status, and handling arrangements ensures the rational allocation of track and operational time slots, thereby improving coordination efficiency across all operational segments.

Conclusion

In recent years, the intermodal transportation of express cargo using air and high-speed rail has begun to be explored and implemented. This paper takes the Zhengzhou Airport Economy Zone as a reference to analyze the transfer operation process of air-rail intermodal cargo. Based on this analysis, we propose a collaborative optimization problem for cargo transfer schemes and train templates. This approach aims to achieve efficient spatial and temporal coordination of cargo, thereby ensuring high-quality air-rail intermodal service. To this end, a mathematical model is constructed under several resource constraints, including airport sorting capacity, train time windows, track capacity, and loading limits. The model determines the transfer times and volumes for various cargo batches across multiple flights. Additionally, it optimizes train operations by defining their start and end times, directions, and loading quantities.

Next, this paper employs CPLEX to solve the model, demonstrating good performance even with large-scale instances. Through numerical experiments, we analyze the relationship between the two different objective functions, namely minimizing cargo transfer time and minimizing the amount of remaining cargo, as well as the impact of various strategies for addressing these objectives on the solution results. Furthermore, we conduct a sensitivity analysis to quantitatively investigate the effects of different lowest loading rates for trains and varying lengths of service time windows on the efficiency of transfer operations.

This study focuses on the distribution (‘air-to-rail’) process, where multiple flights and diverse cargo flows converge at the intermodal hub, necessitating sophisticated planning for train loading, routing, track allocation, and operation timing. Future research could further explore the optimization of bidirectional transfer operations for air-rail intermodal cargo. Furthermore, as unexpected scenarios may occur in real-world operations—such as flight delays, fluctuations in cargo volume, and traffic congestion—future studies should consider incorporating these key uncertainty factors into the optimization framework. For instance, modeling the road transfer time as a stochastic variable within a defined range would facilitate a more realistic assessment of transport time volatility. Leveraging advanced methodologies, such as stochastic optimization or robust programming, could significantly enhance the model’s reliability and resilience.