The rapid growth in the number of motor vehicles in cities exacerbates the issue of traffic congestion and pollution. Meanwhile, growing e-commerce has significantly increased the demand for urban freight transport. For example, the daily volume of the received and delivered express has reached approximately 15 million pieces in Beijing. All along, urban freight transport has always been road-dominated [1] and mostly relies on trucks, which account for 40% of the pollutants within cities [2,3] and increase the risk of accidents. According to statistics [4], global freight demand is growing by approximately 3% per year, which implies that by 2050, freight demand will triple than that of the current demand today. And almost half of the freight transport takes place in urban areas in terms of vehicle-kilometers [5].

Developing the underground space of a city for freight transportation become a potential and superior plan for coping with a series of problems caused by limited urban road resources. Considering the complexity of building an underground logistics system, which includes city planning, transportation safety, infrastructure construction, and logistics management, building a specialized underground freight transportation and distribution system is difficult in the short term [6]. As a punctual, convenient, large-capacity, and low-emission transportation system, the metro system provides a more realistic possibility as a new mode of urban freight transport. In 2007, Nuzzolo et al. [7] explored the potential of utilizing passenger train infrastructure for freight transportation between Naples and Sorrento during off-peak hours, proposing modifications to passenger trains to accommodate additional freight capacity. In Newcastle, Motraghi and Marinov [8] assessed the viability of incorporating the metro system into urban freight transport and the feasibility of a metro-based baggage transportation service from the city center to the airport was also verified in practice by Brice et al. [9]. In particular, some innovative transportation concepts have been proposed to integrate freight and passenger transports within shared systems [[10], [11], [12]]. For instance, Chebbi and Chaouachi [13] introduced a two-tier shared transportation framework for the simultaneous movement of goods and passengers using the same personal rapid transit (PRT) vehicles. Bruzzone et al. [14] proposed the use of an integrated system for passenger and cargo transportation, instead of the traditional approach of separate transport for passengers and freight delivery. These studies highlighted the growing interest in integrated transport and underscored the potential of metro systems in freight transportation.

However, integrating freight transport into an urban passenger metro network is a complex and systemic decision [15]. Existing metro systems were initially engineered for passenger conveyance and are not suitable for direct adaptation to logistic operations. Therefore, the first step is to retrofit the station and train (e.g., storage space for freight in the station, carriage of trains), which is a long-term strategic decision. This study mainly focuses on the operational level, specifically during off-peak hours, and explores the feasibility of passenger and freight co-transportation (PFCT) in urban metro systems from the following aspects.

(1) Determining the optimal train timetable (TT) for the operation period. PFCT means that the passenger and freight share both infrastructure and operating time. Notably, passenger transport remains the first priority, and its operational constraints must be ensured (e.g., train dwell times, skip-stop strategies). Nevertheless, to provide quality service, the TT should be designed to accommodate both time-varying passenger and freight demands.

(2) Formulating an appropriate train composition scheme. Sharing carriages between passengers and freight inevitably affects passenger transport, particularly in terms of conflicts between cargo loading/unloading and passenger boarding/alighting at stations. In some regions, the mixing of freight with passenger carriages is prohibited. A more suitable approach is to employ separate passenger and freight carriages that can be coupled or uncoupled based on the dynamic passenger and freight demands, thereby ensuring a precise match between service provision and demand. Furthermore, different train composition schemes significantly affect the operating costs, and the feasibility of these schemes is constrained by the availability of rolling stocks. Therefore, an optimal train composition scheme requires balancing the service capacity with operating costs under the support of rolling stock circulation plan (RSCP).

(3) Coordinating passenger assignment and freight dispatch. For both passenger and freight, it is essential to design appropriate allocation rules and make decisions regarding specific trains and boarding times. In this process, both temporal (i.e., arrival times of passengers and freight) and spatial (i.e., train capacity, skip-stop strategies) constraints must be considered. Moreover, considering the priority of passenger and freight services, one party (that is, operators, passengers, or freight agents) may need to compromise to maximize the overall system benefits.

The issues mentioned above are closely connected. To achieve high-quality train operation and passenger/freight allocation plans, this study develops a joint optimization approach for TT and RSCP by considering flexible composition and skip-stop strategies to promote a better match between the service capacity and transport demand, thereby reducing the operating costs and further verifying the possibility of PFCT in urban metro system.

As an environmentally friendly alternative to urban freight transport, metro-based urban freight transport has significant potential, along with substantial economic and environmental benefits [[15], [16], [17]]. This motivated the investigation on the joint optimization problem for PFCT in an urban metro system. Considering the two key issues of TT and RSCP in operating an urban metro system, this study is dedicated to optimizing both train operations and passenger/freight allocation plans. The related literature review is organized into the following three parts.

For urban metro systems, train stop and TT are two key components in operation and management, and have attracted considerable interest. As an effective technology for enhancing service and efficiency, train skip-stop strategies refer to situations where the train may skip some stations along the metro line. For instance, to determine the optimal skip-stop strategy, a nonlinear integer programming (IP) model was formulated by Abdelhafiez et al. [18]. The results obtained by the heuristic algorithm were compared with the results of the all-stop strategy, which saved 10% of passenger travel time. Focusing on enhancing the fairness performance in an overcrowded urban network, Shang et al. [19] constructed discrete states according to the times that the passenger misses a train to characterize the fairness performance of the whole system, and developed a multi-commodity flow model to optimize the train skip-stop schedule. Jiang et al. [20] introduced a synchronized optimization strategy for managing passenger inflows and optimizing the train skip-stop operations. A nonlinear model was constructed to minimize the detained passenger penalties, which was solved using Q-learning.

The TT problem, which is a key problem in urban metro systems, determines when each train arrives at and departs from each station. Considering the time-dependent characteristics of passenger demand in urban metro systems [21], the formulation of a demand-sensitive TT is particularly important for satisfying passenger demands and improving service quality [22,23]. Specifically, with the dynamic passenger flow, Niu and Zhou [22] optimized the TT for an oversaturated metro corridor. A model was constructed to reduce passenger waiting time and the number of detained passengers due to limited train capacity, which was solved using a genetic algorithm. To optimize the TT, Barrena et al. [24] presented three formulations to minimize the passenger waiting time and performed numerical experiments using the branch-and-cut algorithm, which highlighted the benefits of designing a flexible timetable that aligns with fluctuating demand. To meet the passenger travel demand with the required service, Shi et al. [23] developed a specific maximum headway function to optimize the TT. Facing the phenomenon of oversaturation, Shi et al. [25] presented an efficient approach to simultaneously optimize the TT and precise passenger flow control tactics, which was also applied in metro networks [26]. Considering the dynamics and uncertainty of passenger demand, Gong et al. [27] constructed a model aimed at jointly optimizing the overall number of trains, determining departure intervals and making decisions regarding speed profiles. The model was linearized and solved using an improved variable neighborhood search (VNS) algorithm. Based on an innovative flexible composition mode, Zhou et al. [28] developed a synergistic optimization model of TT and RSCP to better cope with the tidal traffic phenomenon in some metro lines. However, few studies exist on timetable optimization for time-varying passenger and freight demands. Behiri et al. [15] addressed the issue of timetabling at the operational level in their discussion on freight transport in an urban metro system. However, they did not provide specific methods for solving this problem.

Considering the interaction between the train stop pattern and timetable, many studies have focused on devising efficient strategies to simultaneously optimize these two problems from the perspective of system optimization. The investigation by Yang et al. [29] focused on establishing a modeling framework for the joint optimization of train stop and TT problems. From the perspective of energy-efficient subway operations, Yang et al. [30] constructed a multi-objective model to jointly optimize the TT and stopping pattern. Based on a time–space network, Tian and Niu [31] constructed a linear IP model to optimize TT and skip-stop decisions under dynamic passenger demand, which was solved using a column generation framework. Wu et al. [32] sought to address the issue of unequal waiting times by refining the train skip-stopping scheme and TT. They introduced a mixed-integer linear programming (MILP) model combined with a VNS algorithm to achieve acceptable solutions within a reasonable computational time.

Most previous studies on train operations in urban metro systems have assumed that the train composition is fixed. However, it is difficult for the fixed composition mode to satisfy the time and space imbalance distribution of passenger demand. The flexible composition mode, which can change the train composition according to variations in passenger demand, has received tremendous attention. For instance, Liu et al. [33] defined four types of train compositions based on existing train composition techniques, which implied that trains with different compositions could be operated to accommodate passenger demand. With considering multiple routes and train sizes, Li et al. [34] constructed an optimization model for train operations, which was directly solved by commercial solvers. Xu et al. [35] established a model for optimizing a train operating scheme considering full-length and short-distance routings, and different types of train compositions, which was solved using a genetic algorithm.

In addition to the given multiple types of train compositions, many studies have considered the RSCP when examining the flexible composition mode. To accommodate the tidal passenger demand, Zhou et al. [28] constructed a collaborative model to co-optimize the TT and RSCP by permitting the adjustment of train compositions through coupling or decoupling operations at the terminals of metro lines. Pan et al. [36] proposed two models for this problem using time–space networks by incorporating flexible coupling/decoupling activities, which was solved using a heuristic algorithm based on a column generation framework. Both considering the passenger service quality and operator costs, Zhao et al. [37] proposed a multi-objective mixed-integer nonlinear programming model (MINLP) model aimed at reducing passenger waiting time, minimizing the rolling stock requirements, and lowering the frequency of coupling and decoupling operations.

Moreover, the RSCP is typically formulated subsequent to the TT [38,39]. Since the sequential decision-making approach usually falls short in achieving optimal system performance, many studies have focused on collaboratively optimizing the TT and RSCP problems [28,36,[40], [41], [42]]. For example, considering energy consumption, Mo et al. [43] developed a collaborative optimization framework for the TT and RSCP, which can be directly addressed using the CPLEX solver. With a similar problem background, Wang et al. [44] extended their analysis to incorporate various train operation parameters (running and dwelling times). Yang et al. [45] concentrated on integrated optimization of the TT and RSCP with heterogeneous rolling stocks, which was solved using an iterative programming method.

The development of underground logistics systems provides an effective method for addressing sustainable urban freight transport in the future [5,[46], [47], [48]]. In the short term, developing urban freight transportation based on the existing metro network is an alternative method for alleviating the pressure on urban freight transport, which can also help make full use of the surplus capacity of metro lines during off-peak periods.

At the macro level, several studies have been performed on issues such as logistics network planning and infrastructure upgrades. Regarding the coordinated planning of land use and policy for infrastructure development, Horl et al. [49] explored the potential for the enhanced utilization of railway infrastructure in freight transportation, specifically focusing on the first and last urban miles. Zhao et al. [6] divided an urban metro network into several subnetworks and formulated a facility location model to identify the final metro distribution hubs by evaluating the importance of each station. Considering service capacity, freight demand, and regional connectivity, Dong et al. [50] introduced a metro-based logistic system network planning strategy, which constructed an MILP model, and developed a hybrid algorithm for this problem. Zheng et al. [51] developed a location optimization approach for a metro-based underground logistics system (MULS) to facilitate freight transportation using a metro network during off-peak hours, which adopted a Voronoi diagram to determine station locations.

At the operational level, Motraghi and Marinov [8] and Dampier and Marinov [52] demonstrated the conceptual feasibility of incorporating urban metro systems into freight transport through an examination of the Newcastle metro network. As a further application, Brice et al. [9] investigated a baggage transportation service from the city center to Newcastle airport using the metro system and showcased the possibility of this innovative system. Behiri et al. [15] centered their research on the rail freight transport scheduling problem. They developed a decision support tool to assist decision makers in modeling and assessing dynamic changes in the system under various scenarios. Ozturk and Patrick [2] introduced a decision support system for the metro-based freight transport problem together with mathematical methods for the optimal allocation of freight. Li et al. [53] formulated an optimization model to determine the TT and freight allocation plan, where freight transport can be realized using dedicated freight trains or the unused capacity of passenger carriages. Hörsting and Cleophas [54] introduced a model to optimize the TT and cargo allocation plans to reduce waiting passengers and container delays. To advance sustainable urban freight transportation at the operational level, Di et al. [55] developed an optimization model with aim of minimizing the operational costs and total delay penalties, and an improved Benders decomposition algorithm was developed to solve the proposed model. Based on the traditional optimization model of passenger transportation under the fixed composition mode, Qi et al. [56] developed a PFCT model based on the flexible composition mode and all-stop strategy.

As discussed above, most literature has studied the metro-based urban freight transport problem with a pre-given timetable or without considering the RSCP. In face of the diversity of passenger and freight demands, the separately optimized method is difficult to achieve a system-optimized operational plan that balances the tripartite interests of operators, passengers, and freight agents.

To utilize the excess capacity of urban metro systems during off-peak periods and realize the sustainable development of urban freight transport, this study develops a joint optimization method of TT and RSCP for PFCT, in which the flexible composition mode and the skip-stop strategies are particularly considered. Table 1 [2,15,[53], [54], [55],[57], [58], [59], [60]] lists several closely related studies. It can be observed that: ① Few studies considered the tripartite interests of operators, passengers, and freight agents. Typically, a single perspective on service quality or operating cost is not the optimal decision for the system. Therefore, the key issue that needs to be addressed is a method to enhance the freight transport services quality with the most economical train operating plan while ensuring the first priority of passenger transport; ② the time-dependent passenger demand is an important piece of information for designing train operation plans, whereas in some metro-based freight transport problems, only the freight transport process was modeled, and very few studies have included both time-dependent passenger and freight demand in their specific modeling process; and ③ the train composition scheme, stopping plan, TT, and passenger/freight allocation plan are typically intertwined and greatly affected by each other, and previous studies generally considered only a partial of them and have not included all these closely connected elements in a unified modeling framework.

Based on the above analysis, the main contributions of this paper are as follows:

(1) Focusing on metro-based PFCT during off-peak hours, this paper proposes a joint optimization of TT and RSCP on a bi-directional metro line, with special consideration of flexible composition mode and skip-stop strategies. In this method, different train composition modes can be selected to cope with unbalanced and time-varying passenger and freight demands, and train skip-stop strategies are considered to improve transportation efficiency. In comparison with the conventional fixed composition mode and all-stop pattern, this approach enables a more accurate match between passenger and freight demands, enhances the quality of transport services, and significantly minimizes the operating costs for metro enterprises.

(2) An MINLP model is constructed for the system optimization problem. The model considers the practical train operation constraints as well as passenger and freight demand constraints, in which the ideas of flexible train composition and skip-stop strategies are embedded into the model to capture the dynamic change process of passenger and cargo group boarding, on-boarding, and alighting. In particular, based on the idea of cargo grouping, multiple constraints (i.e., two first in, first out (FIFO) rules) are used in the model to portray the inherent priorities in cargo group allocation to ensure both efficiency and fairness of freight transport. The objective function of the model is to reduce operating costs, passenger detention, and cargo group delays, which reflects a synergistic consideration of the interests of operators, passengers, and freight agents.

(3) Given the complexity of the model, a customized VNS algorithm is applied to generate high-quality solutions within a reasonable computational time. To evaluate the performance of the proposed methods, several sets of numerical experiments are performed with a simplified example and an empirical example of the Beijing Metro Yizhuang Line. Furthermore, we also compare our approaches with the conventional operating modes (i.e., fixed composition mode and all-stop strategy), which further validates the effectiveness of the proposed approaches for meeting time-dependent passenger and freight demands.

The remaining sections are organized as follows. Section 2 presents a detailed problem statement and modeling assumptions. An MINLP model is formulated for the considered problem in Section 3. To solve the complex model, in Section 4, a heuristic algorithm based on the VNS algorithm is developed. In Section 5, a simplified example and an empirical case with historical operational data of the Beijing Metro Yizhuang Line, are carried out to verify the proposed methods. Section 6 ends with some conclusions and directions for further research.

The problem of interest considers the PFCT on a bi-directional urban metro line, as shown in Fig. 1. Two sets of trains represented by $\overline{K}$ and $\underset{\_}{K}$, are respectively scheduled in different directions and there are two depots at each end of the line.

Regarding transportation carriers, there are generally two basic modes for metro-based freight transport problems. One is the shared train mode, in which a train serves both passengers and freight [55], and the other is the dedicated train mode, in which a train exclusively carries either passengers or freight [2]. Some studies have also proposed a mixed mode of utilizing extra space in passenger train carriages for freight transportation in addition to dedicated freight trains [53]. However, legislation in some countries prohibits the mixing of freight with passenger transportation in the same carriage for various reasons [15]. Therefore, we particularly take into account the flexible train composition mode, where trains can couple or decouple in the depot at the end of the line. By this way, multiple train composition types with different lengths (i.e., different numbers of passenger and freight carriages in each type of train composition mode) can be considered by giving a set of different composition types. That means the train can be a single passenger or freight train composition, or a mixed passenger and freight train composition, to address different scales of passenger and freight demand, as shown in Fig. 2. Nevertheless, the effective implementation of a flexible train composition scheme relies heavily on the quality of the RSCP. In addition, the initial number of each type of rolling stock in the depot is usually given in advance, and the formulation of the composition scheme needs to strictly meet the constraints of the available rolling stocks (i.e., resources are limited).

Typically, considering the large passenger flow during peak hours, utilization of the metro system for freight transportation services usually occurs during off-peak hours. For ease of modeling, the continuous scheduling horizon $[0,T]$ is discretized into a sum of $T$ time intervals with the length of each interval as $\delta$. With passenger data collected from the Auto Fare Collection (AFC) system, we use the time-dependent origin–destination matrix to describe each passenger’s itinerary and arrival time.

The inclusion of freight transport makes management more difficult for operators. To guarantee the safety, efficiency, and fairness of the process of freight aggregation, handling, and transportation, and to achieve large-scale freight transport while minimizing the inconvenience to passengers, the idea of cargo grouping is applied, which has been employed for passenger transport by Shang et al. [19] and Wu et al. [32]. Specifically, if cargos successively arrive at the origin station during the same time period and have the same destination, they can be integrated into a cargo group. The completion of the cargo group is subject to any of the following conditions: ① The cumulative number of cargos is up to the number limitation of the cargo group, and ② the duration of the grouping process reaches the upper limit of the setting time. The cargo grouping process serves as a preprocessing step to enhance train resource utilization while ensuring high-quality freight transportation services. In this context, grouped cargo can include some high-value, lightweight goods such as express parcels. In this paper, it is assumed that the information of the cargo groups is given in advance. Cargo groups are transported in standard freight containers, as assumed and applied in several previous studies [2,15]. Furthermore, from a fairness perspective, the cargo group allocation process should follow a spatial–temporal prioritization, which can be expanded into the following sub-rules [19,32]: ① For each station, cargo groups that arrive earlier have priority in terms of the temporal sequence for boarding trains, and ② cargo groups waiting at upstream stations have spatial priority for boarding trains. Hence, the cargo group allocation process is jointly constrained by the train capacity (i.e., train composition type), the train skip-stop strategies, and the two FIFO rules.

From the above analysis, this study aims to develop a reasonable train operating scheme (i.e., TT, skip-stop strategies, and RSCP) to better match the time-dependent passenger and freight demands by considering flexible train operation modes and establishing strict rules for passenger and freight allocation. And some assumptions are made as follows:

Assumption 1. This paper addresses the passenger and freight co-transportation under the general conditions, of which the uncertain disturbances of various events are not considered. The section travelling time and the required dwell times for passenger and freight transportation at each stop station are assumed to be fixed values.

Assumption 2. We consider a bi-directional metro line with multiple types of train compositions being operated in each direction. The coupling and splitting operations are assumed to be implemented only in the depot and the corresponding duration times of these operations are pre-given.

Assumption 3. Since this paper studies the PFCT during the off-peak hours with sufficient train capacity, the passenger and freight demands are assumed be given in advance and all transportation requirements are guaranteed to be satisfied.

Assumption 4. Taking into account the interference between passengers and freight, in this study, only some of the designated stations are allowed for carrying out freight transport. And all cargos are pre-grouped and loaded into standard freight containers in accordance with the grouping rules.

All the symbols and parameters involved in the formulation of the model are listed in Table 2.

Depending on the characteristics of the problem studied, the variables can be divided into two main parts. The first part contains variables related to train operations, including the TT, skip-stop strategies, and RSCP. The second includes variables related to passenger and freight allocations. Then, the first part of variables is listed as follows.

(1) Variables related to TT.

$t_{k,i}^{\left(\mathrm{a}\right)}$: integer variable, the arrival time of train $k$ at station $i$.

$t_{k,i}^{\left(\mathrm{d}\right)}$: integer variable, the departure time of train $k$ from station $i$.

The above variables are the primary decision variables in the TT formulation.

(2) Variables related to skip-stop strategies.

$x_{k,n}$: binary variable, if train $k$ chooses the passenger stopping mode $n$, $x_{k,n}=1$; otherwise, $x_{k,n}=0$.

${\psi }_{k,v}$: binary variable, if train $k$ chooses the freight stopping mode $v$, ${\psi }_{k,v}=1$; otherwise, ${\psi }_{k,v}=0$.

These two variables are related to train skip-stop strategies for passenger and freight transport, respectively, which together determine the stopping mode of each train. Typically, as mentioned in many studies, these variables are also closely related to the TT problem (i.e., when a train decides to skip a station, it reduces its dwell time at that station, and the TT is changed accordingly). For clarity, the following variables are introduced to denote the actual dwell time of each train at each station, which is determined by the selected stopping mode for passenger and freight transport.

$t_{k,i}^{\left(\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\right)}$: integer variable, denoting the dwell time of train $k$ at station $i$, taking either $0$ or a fixed value depending on the chosen stopping modes.

(3) Variables related to rolling stock circulation plan.

$y_{k,m}$: binary variable, if train $k$ chooses the train composition $m$, $y_{k,m}=1$; otherwise, $y_{k,m}=0$.

For the determination of the RSCP, the above variables are first defined to select a specific composition for each train in accordance with passenger/freight demand, which is limited by the number of available rolling stocks in each depot. However, the number of available rolling stocks in each depot usually changes with the train departures or arrivals at the depot. For the successful execution of train composition scheme, the following variables are introduced as the indicator of train departure by calculating the number of passenger/freight rolling stocks available at each depot (see Section 3.2.2 for more details).

${\rho }_{k,i}\left(t\right)$: binary variable, if train $k$ departs from station $i$ at or before time t, ${\rho }_{k,i}\left(t\right)=1$; otherwise, ${\rho }_{k,i}\left(t\right)=0$.

The second part contains variables related to the allocation of passengers and freight, including those representing their distribution on each train and those capturing variations in passenger and freight flows, listed as follows.

(1) Variables related to passenger allocation.

${\lambda }_{k,i,j}$: integer variable, the number of passengers arriving at station $i$ with destination station $j$ that successfully board train $k$ between trains $k$ and $k-1$.

$s_{k,i,j}$: integer variable, the number of passengers arriving at station $i$ going to station $j$ between trains $k$ and $k-1$ that can not board train $k$ and need to wait for train $k+1$ or train $k+2$.

${\varrho }_{k,i}\left(t\right)$: binary variable, if timestamp $t$ falls between the departure time of train $k-1$ from station $i$ and the departure time of train $k$ from station $i$ (i.e., [$t_{k-1,i}^{\left(\mathrm{d}\right)}$, $t_{k,i}^{\left(\mathrm{d}\right)}$]), ${\varrho }_{k,i}\left(t\right)=1$; otherwise, ${\varrho }_{k,i}\left(t\right)=0$.

$P_{k,i}^{\left(\mathrm{b}\mathrm{o}\mathrm{a}\mathrm{r}\mathrm{d}\right)}$: integer variable, the number of passengers boarding train $k$ at station $i$.

$P_{k,i}^{\left(\mathrm{a}\mathrm{l}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}\right)}$: integer variable, the number of passengers alighting from train $k$ at station $i$.

$P_{k,i}^{\left(\mathrm{i}\mathrm{n}\right)}$: integer variable, the number of passengers on train $k$ between station $i$ and station $i+1$.

(2) Variables related to freight allocation.

${\gamma }_{k,f}$: binary variable, if $f$ group of cargos are assigned to train $k$, ${\gamma }_{k,f}=1$; otherwise, ${\gamma }_{k,f}=0$.

$z_{k,f}$: binary variable, if $f$ group of cargos arrive at their departure station earlier than the departure time of train $k$, $z_{k,f}=1$; otherwise, $z_{k,f}=0$.

${\mathrm{Cap}}_{k,i}$: integer variable, the available cargo capacity of train $k$ at station $i$.

$w_f$: integer variable, the delay time of $f$ group of cargo.

This subsection systematically introduces various constraints related to the TT, RSCP, and the dynamic evolution of passengers and cargo groups, ensuring the feasibility of the solutions. Additionally, the concepts of flexible train composition and skip-stop strategies are incorporated into these constraints.

According to assumption 1, the section travelling time and train dwell time are pre-determined. Consequently, the departure and arrival times of train $k$ at each station $i$ should comply with the following constraints:

In particular, this study takes the train skip-stop strategies into consideration, which are embedded into the constraints of TT. The decision variable $x_{k,n}/{\psi }_{k,v}$ indicates the choice of stopping mode by train service $k$ for passenger/freight transport, where $x_{k,n}=1/{\psi }_{k,v}=1$ indicates that train service $k$ chooses the stopping mode $n/v$, and $x_{k,n}=0/{\psi }_{k,v}=0$ otherwise. Constraints in Eqs. (1), (2) imply that train $k$ can only choose one passenger stopping mode $n$ and one freight stopping mode $v$. The specific information of the stopping modes $n/v$ is given in advance, where ${\theta }_{n,i}=1/{∊}_{v,i}=1$ indicates that the stopping mode $n/v$ stops at station $i$, and ${\theta }_{n,i}=0/{∊}_{v,i}=0$ otherwise. With the given required dwell time (i.e., $t_{k,i}^{\left(\mathrm{P}\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\right)},t_{k,i}^{\left(\mathrm{F}\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\right)}$) and the selected stopping modes, the actual dwell time of train $k$ at station $i$ can be calculated by constraint in Eq. (3). Thus, given the departure time of the first train from the original station in the upstream and downstream direction, that is, $t_{1,1}^{\left(\mathrm{d}\right)}$, $t_{1,\left|S\right|+1}^{\left(\mathrm{d}\right)}$, if the departure headway is determined, the time of each train departing from/arriving at each station can be calculated using constraints in Eqs. (4), (5).

Taking Fig. 3 as an example, the stations marked in red (i.e., stations 1, 3, 6) are those capable of handling freight operations. The blue (denoted as I) and red (denoted as II) solid lines represent the selected stopping modes for passenger and freight, respectively, where solid dots indicate stops and hollow dots indicate skipping operations. The final stopping pattern is obtained by integrating these two stopping modes, as shown by the black solid line (denoted as III). For instance, the final dwell times of Train A at the first three stations can be given by $t_1=\mathrm{m}\mathrm{a}\mathrm{x}\left\{t_{\mathrm{A},1}^{\left(\mathrm{P}\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\right)},t_{\mathrm{A},1}^{\left(\mathrm{F}\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\right)}\right\}$, $t_2=t_{\mathrm{A},2}^{\left(\mathrm{P}\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\right)}$, and $t_3=t_{\mathrm{A},3}^{\left(\mathrm{F}\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\right)}$. This approach allows for more flexibly adjustment of dwell time to accommodate both passenger and freight loading/unloading requirements, while also considering the operational efficiency of the train. Furthermore, under specific spatial and temporal conditions, such as lower passenger demand during off-peak hours, as shown with Train B, dedicated freight trains (e.g., S-F, L-F) can be utilized to transport cargo groups to reduce unnecessary stops, thereby improving operational efficiency and service level.

Constraint in Eq. (6) is established to ensure that any two adjacent train services $k+1$ and $k$ are kept within a safe and reasonable range, which are the commonly used constraints.

In order to ensure an optimal alignment between the available train capacity and the passenger/freight demand, and to avoid capacity wastage or overcrowding, it is assumed that different passenger and freight rolling stocks can be coupled and decoupled in the depot. Thus, multiple train composition types will be applied in the operations. However, owing to certain realistic conditions (e.g., train length constraints and station capacity constraints), the types of train composition are usually limited. In this study, the feasibility of the generated train composition is ensured by providing alternative sets of train composition types and only one type of composition can be applied to each train, leading to constraint in Eq. (7).

In practice, successful implementation of flexible train composition scheme is based on an appropriate RSCP. With the given initial number of rolling stocks in the depot, the number of rolling stocks in the depot will dynamically change during operations with different types of train arrivals after finishing previous services or departures for executing new services. For a more explicit expression, an auxiliary binary variable ${\rho }_{k,i}\left(t\right)$ has been used in many studies [22,28] to denote the departure status of trains, where ${\rho }_{k,i}\left(t\right)=1$ signifies that train $k$ departs from station $i$ before or at timestamp $t$, and ${\rho }_{k,i}\left(t\right)=0$ otherwise. Constraint in Eq. (8) expresses the relationship between $t_{k,i}^{\left(\mathrm{d}\right)}$ and ${\rho }_{k,i}\left(t\right)$ and constraint in Eq. (9) ensures that ${\rho }_{k,i}\left(t\right)$ does not decrease over time. Furthermore, to construct linear constraints on passenger allocation (i.e., passenger–train coupling), we introduce the binary variable ${\varrho }_{k,i}\left(t\right)$ to calculate the timestamps between trains $k-1$ and $k$ at station $i$, leading to constraint in Eq. (10).

Using the definition of the auxiliary variable ${\rho }_{k,i}\left(t\right)$, the number of different types of rolling stocks in the depot at any given time can be calculated separately. The following constraints ensure sufficient rolling stocks for the subsequent services at any time while maintaining the feasibility of train operation for the metro systems.

Here, $\overline{\mathrm{I}\mathrm{N}{\mathrm{V}}_{\mathrm{P}}}$ and $\overline{\mathrm{I}\mathrm{N}{\mathrm{V}}_{\mathrm{F}}}$ represent the initial number of passenger and freight rolling stocks in depot $1$, which increases with the train in the downstream direction entering the depot and decreases with train in the upstream direction departing from the depot. Specifically, $t^{\left(\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{h}\right)}$ indicates the time of the train from the terminal station to the depot or the time from the depot to the original station. $t^{\left(\mathrm{f}\mathrm{i}\mathrm{x}\right)}$ indicates the time for the train to complete the coupling and uncoupling operations at the depot (i.e., the time for the train to prepare for the next services after it arrives at the depot). Therefore, if train $k$ has left the terminal station $2\left|S\right|$ in the downstream direction at or before moment $t-t^{\left(\mathrm{switch}\right)}-t^{\left(\mathrm{f}\mathrm{i}\mathrm{x}\right)}$, that is, ${\rho }_{k,2\left|S\right|}(t-t^{\left(\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{h}\right)}-t^{\left(\mathrm{f}\mathrm{i}\mathrm{x}\right)})=1$, it indicates that at moment $t$, all rolling stocks in the composition mode selected by train $k$ are already available for implementing the services in the upstream direction. Similarly, if train $k$ in the upstream direction has left the original station $1$ at or before the $t+t^{\left(\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{h}\right)}+t_{k,1}^{\left(\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\right)}$ moment, that is, ${\rho }_{k,1}(t+t^{\left(\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{h}\right)}+t_{k,1}^{\left(\mathrm{d}\mathrm{w}\mathrm{e}\mathrm{l}\mathrm{l}\right)})=1$, all rolling stocks in the composition mode selected by train $k$ are no longer available at $t$. Since $v_m^{\left(\mathrm{P}\right)}$ and $v_m^{\left(\mathrm{F}\right)}$ represent the number of passenger and freight rolling stocks in composition type $m$ selected by train $k$, constraints in Eqs. (11), (12) can ensure that there always exists at least one passenger and freight rolling stock at any time $t$ in depot $1$. In addition, because the required preparation time (i.e., $t^{\left(\mathrm{s}\mathrm{w}\mathrm{i}\mathrm{t}\mathrm{c}\mathrm{h}\right)}$, $t^{\left(\mathrm{f}\mathrm{i}\mathrm{x}\right)}$) has already been considered in the above constraints, the specific linkage between different rolling stocks can be easily determined according to the generated flexible train composition plan.

For the number of passenger and freight rolling stocks in depot $\mathrm{2}$, similar constraints in Eqs. (13), (14) can be formulated as follows.

Considering the background of the problem of PFCT in urban metro systems based on flexible train composition and skip-stop strategies, it is hoped that the operating costs of metro lines can be minimized under the premise of guaranteeing the quality of the transport service. This study allows a small number of passengers to at most wait for the following two services when they miss the first service at the station, leading to constraint in Eq. (15).

In these constraints, ${\lambda }_{k,i,j}$ denotes the number of passengers arriving at station $i$ with destination station $j$ that successfully board train $k$ between service trains $k$ and $k-1$. ${\varphi }_{i,j,t}$ denotes the number of passengers arriving at station $i$ at time $t$ with destination station $j$. In particular, $s_{k,i,j}$ is introduced to denote passengers that can not be serviced by the first train $k$ when they arrive at station $i$ and have to wait for the following train $k+1$ or $k+2$, which will be denoted as detained passengers in the following description. Clearly, the following three situations will be occurred: ① For the first train, the number of passengers arriving at station $i$ before the first train departing from station $i$ is exactly the passengers boarding train $1$ (i.e., ${\lambda }_{1,i,j}$) and the detained passengers (i.e., $s_{1,i,j}$). ② For the second train, the number of passengers arriving at station $i$ during the time between trains $2$ and $1$ departing from station $i$ plus the number of remaining passengers when train $1$ departs from station $i$ (i.e., $s_{1,i,j}$) equals to the number of passengers boarding train $2$ (i.e., ${\lambda }_{2,i,j}$) plus the number of detained passengers (i.e., $s_{2,i,j}$). ③ For the other trains, the number of passengers arriving at station $i$ during the time between service trains $k$ and $k-1$ departing from station $i$ plus the number of remaining passengers when train $k-1$ and train $k-2$ depart from station $i$ equals to the number of passengers boarding train $k$ plus the number of detained passengers (i.e., $s_{k,i,j}$).

For clarity, Fig. 4 provides an example of the aforementioned situations for calculating the number of detained passengers. In this figure, different colors are used to indicate the corresponding passengers arriving between adjacent trains $k$ and $k-1$ board train $k$ or detaining to wait for the following train $k$ or $k+1$. Green represents the passenger who can successfully board the first arriving train upon arrival at station $i$. Blue and yellow represent passengers who have to wait for the following one or two trains, respectively, when they miss the first train at station $i$. In this example, assuming train $k-2$ has the passenger capacity of two, and four passengers arrive at station $i$ between the departures of trains $k-3$ and $k-2$ (i.e., ${\sum }_{t\in T}{\varphi }_{i,j,t}·{\varrho }_{k-2,i}\left(t\right)=4$), two passengers will be detained (i.e., $s_{k-2,i,j}=2$, as shown by yellow icons) owing to the train capacity limitation. And in order to make a better balance between the provided service and operating costs, if train $k-1$ is scheduled to not provide service for passengers at station $i$ (e.g., train $k-1$ is a freight train or train $k-1$ skips station $i$ as the brought benefit can cover the penalty cost of detained passengers), passengers arriving between trains $k-2$ and $k-1$ (i.e., ${\sum }_{t\in T}{\varphi }_{i,j,t}·{\varrho }_{k-1,i}\left(t\right)=1$, as shown by blue icons) and passengers detained by train $k-2$ (i.e., $s_{k-2,i,j}$, as shown by yellow icons) will continue to wait at station $i$ for the subsequent train $k$. It is worth noting that, because the detained passengers by train $k-2$ have reached the limit of the number of passengers waiting, it must be ensured that train $k$ has sufficient capacity to carry these passengers. Otherwise, the whole train operation plan needs to be regenerated. In particular, this study examines the impact of train skip-stop strategies and flexible composition mode. The constraint in Eq. (15) can easily be reformulated to impose different restrictions on the maximum number of passengers waiting, thereby enabling different levels of service to passengers.

Constraint in Eq. (16) establishes the connection between the selected stopping mode and the passenger allocation. A passenger can only be served by a train that stops at both the origin and destination (OD) stations. This implies that if the passengers for a certain OD pair are transported by train $k$ (i.e., ${\lambda }_{k,i,j}>0$), then the selected stopping mode of train $k$ must stop at both stations of this OD pair (i.e., $x_{k,n}·{\theta }_{n,i}·{\theta }_{n,j}=1$). Otherwise, if the selected stopping mode of train $k$ does not stop at either station of a certain OD pair (i.e., $x_{k,n}·{\theta }_{n,i}·{\theta }_{n,j}=0$), the passenger travelling between this OD pair cannot be assigned to this train (i.e., ${\lambda }_{k,i,j}=0$).

When a train stops at a station, some on-board passengers alight as their journey ends, while waiting passengers on the platform board, leading to a dynamic change in the number of on-board passengers.

Constraints in Eqs. (17), (18) denote the number of boarding passengers of each train at different stations in the upward and downward directions, respectively. Here, the number of passengers boarding train $k$ at station $i$, denoted as $P_{k,i}^{\left(\mathrm{b}\mathrm{o}\mathrm{a}\mathrm{r}\mathrm{d}\right)}$, is the total number of passengers travelling from station $i$ to subsequent stations in the same direction on this train.

Similarly, except for the original station in each direction where no passenger alight, the number of passengers alighting from train $k$ at station $j$ equals the total number of passengers transported by train $k$ from the preceding stations whose destination is station $j$, as captured by constraint in Eq. (19).

With the boarding and alighting passengers defined above, the dynamic passenger load on each train can be determined using constraints in Eq. (20).

Here, the number of on-board passengers when train $k$ departs from station $i$ (i.e., $P_{k,i}^{\left(\mathrm{i}\mathrm{n}\right)}$) is determined by the on-board passengers at station $i-1$, plus the number of passengers boarding at station $i$, minus those alighting at station $i$. Naturally, at the original station, the on-board passenger count equals the number of boarding passengers, while at the terminal station, no passengers remain on-board.

Finally, because the capacity of a train is determined its selected composition mode, we use parameter $Q_m^{\left(\mathrm{P}\right)}$ to denote the rated capacity of the $m$th train composition mode. Then, for passenger safety and comfort, constraint in Eq. (21) are formulated to ensure that the rated capacity of train $k$ cannot be exceeded.

In this study, metro-based freight transport services were conducted during periods of unsaturated passenger flows. To ensure service quality, spatial–temporal FIFO rules are specifically introduced to regulate the boarding process of cargo groups.

The allocation of cargo groups is determined by their arrival times and group sizes. Subsequently, a set of rigid constraints is considered to develop the freight transportation scheme.

The binary decision variable ${\gamma }_{k,f}$ is introduced to determine the transportation plan for cargo groups. ${\gamma }_{k,f}=1$ indicates that cargo group $f$ is arranged to train $k$, and ${\gamma }_{k,f}=0$ otherwise. Each cargo group can be transported only by one train service. To maintain the viability of the transportation plan, certain conditions must be satisfied. That is, if cargo group $f$ is arranged to train $k$ (i.e., ${\gamma }_{k,f}=1$), train $k$ must stop at both the origin $o_f$ and destination $e_f$ of cargo group $f$, and the arrival time of cargo group $f$ at the origin station should be earlier than the departure time of train $k$ from that station (i.e., $t_{o_f}^{\left(f\right)}<t_{k,o_f}^{\left(\mathrm{d}\right)}$). The following constraints were set for this purpose.

Here, the binary variable $z_{k,f}$ is introduced to indicate the relationship between the arrival time of cargo group $f$ and the departure time of train $k$ from station $o_f$. If cargo group $f$ arrives at its origin station earlier than the departure time of train $k$ from this station, $z_{k,f}=1$; otherwise, $z_{k,f}=0$. Consequently, constraints in Eqs. (26), (27), (28) ensure the temporal feasibility of cargo group transportation by linking the TT with the cargo group allocation. Additionally, the spatial feasibility of cargo group transportation is ensured by constraints in Eqs. (24), (25), which establish connections between the selected stopping mode and the cargo group allocation.

Constraint in Eq. (29) is specifically expressed to ensure that the allocation of cargo groups follows the FIFO rule in time, which means that cargo groups arriving earlier are given boarding priority at each station. Specifically, for any two cargo groups $f$ and $f^{\prime }$ sharing the same origin station, where cargo group $f^{\prime }$ arrives earlier than the cargo group $f$. If cargo group $f$ is assigned to train $k$, then cargo group $f^{\prime }$ must be allocated to either train $k$ or an earlier train service.

Constraint in Eq. (30) is formulated to ensure that the allocation of cargo groups follows the FIFO rule in space. This rule means that available trains should not leave cargo groups from upstream stations behind, while reserving capacity for cargo groups in the downstream stations. And the so-called available train shall contain the following conditions: ① The train $k$ shall stop at the origin station of cargo group $f$, that is, ${\sum }_{v\in V}{\psi }_{k,v}·{∊}_{v,o_f}=1$; ② the arrival time of cargo group $f$ shall be earlier than the departure time of train $k$ at station $o_f$, that is, $z_{k,f}=1$ ; ③ the train shall have available capacity at the time of arrival at that station, that is, ${\mathrm{Cap}}_{k,o_f}-g_f\ge 0$. It is worth noting that the right-hand side of the inequality for constraint in Eq. (30) has an additional integer $1$ added to it, which is intended to ensure that if the first two conditions are satisfied and the available freight capacity of train $k$ at station $o_f$ is exactly equal to the amount of freight in cargo group $f$, that is, ${\mathrm{Cap}}_{k,o_f}-g_f=0$, cargo group $f$ should be transported by train $k$ or the preceding trains (i.e., make sure that the right-hand side of the inequality is greater than $0$).

As for the available capacity (i.e., ${\mathrm{Cap}}_{k,i}$) of each train at each station, it can be calculated using constraints in Eqs. (31), (32), (33). Specifically, the available capacity of the train at the original station can be calculated using constraints in Eqs. (31), (32), which is the rated freight capacity of the selected mode minus the number of cargos departing from the original station. The available capacity of train $k$ at intermediate station $i$ is calculated based on the available capacity at station $i-1$, adding the number of cargos unloaded at station $i$ and subtracting those loaded at station $i$, as specified in constraint in Eq. (33).

Finally, constraint in Eq. (34) ensures that the available capacity of train $k$ when it departs from any station $i$ does not exceed the rated cargo capacity of its composition mode.

In this study, the objective function considers both economic and service quality objectives from the perspectives of metro operators, passengers, and freight agents.

Train skip-stop strategies and flexible composition modes are considered to better align the service provision with demand. Typically, the number of train stops and the type of composition mode significantly affect the operating costs of metro lines. To capture this effect, the operating cost was represented as the product of the chosen train composition cost, the number of train stops, and stop-related coefficient $\xi$. Notably, when calculating the number of train stops, both passenger and freight stopping modes must be considered. Specifically, a train can skip a station only if both passenger and freight stopping modes allow it (i.e., ${\sum }_{n\in N}{x}_{k,n}·{\theta }_{n,i}=0$ and ${\sum }_{v\in V}{\psi }_{k,v}·{∊}_{v,i}=0$); otherwise, it must stop at this station to meet the transport demand. Accordingly, the final operating cost can be expressed as follows:

For clarity, an auxiliary binary variable ${\phi }_{k,i}$ is introduced to denote the specific stopping information of a train after integrating the two stopping modes. If train $k$ stops at station $i$, ${\phi }_{k,i}=1$; otherwise, ${\phi }_{k,i}=0$. Therefore, ${\phi }_{k,i}=\mathrm{m}\mathrm{a}\mathrm{x}\{{\sum }_{n\in N}{x}_{k,n}·{\theta }_{n,i},{\sum }_{v\in V}{\psi }_{k,v}·{∊}_{v,i}\}$ is equivalently replaced by the following linearized constraint in Eq. (35).

Additionally, another auxiliary binary variable ${\varpi }_{k,i,m}=y_{k,m}·{\phi }_{k,i}$ is introduced and the associated linear constraint in Eq. (36) are as follows:

Therefore, Eq. (37) represents the total operating cost of the metro line.

To ensure high service quality while minimizing operating costs and enhancing train efficiency, some passengers may be required to wait for the next available train, which could be the second or even third one. These passengers are referred to as detained passengers. Naturally, the number of detained passengers is a key indicator of service quality, and the total number of detained passengers can be expressed as follows:

For the freight transport on metro lines, freight agents typically focus on the arrival times of cargo groups. Therefore, minimizing the delay for cargo groups is an important aspect of transportation services. In this study, the cargo groups can be transported by only one train. If cargo group $f$ takes train $k$ for its trip (i.e, ${\gamma }_{k,f}=1$), the delay time for the cargo group is the arrival time of train $k$ minus the expected arrival time of cargo group $f$ at station $e_f$; otherwise, $w_f$ is set to zero. Therefore, the delay of the cargo group can be expressed using Eq. (39).

The total delay time for all the cargo groups on the metro line is calculated as follows:

Subsequently, by incorporating two cost coefficients, ${\sigma }_1$ and ${\sigma }_2$, which represent the costs for detained passengers and cargo delays, respectively, the objectives pertaining to economic efficiency and service quality can be converted into a single-objective framework. Additionally, introducing three weighting factors, $c_1$, $c_2$, and $c_3$, which represent the relative importance of the operational cost, number of detained passengers and delay time of cargo groups in the objective function, respectively, enables decision makers to set different combinations of these weights based on various scenarios to balance operational costs and service quality.

In summary, the studied model can be formulated as follows.

Notably, the formulated model is a general optimization method for TT and RSCP with flexible composition and skip-stop strategies for the co-transportation of passengers and freight. Decision makers can adjust the values of some key parameters, such as the initial number of rolling stocks at the depot, coupling/uncoupling time, train capacity, and so forth, to better align with real-world operations.

Table 3 reflects the size and complexity of the proposed model. The overall number of variables and key constraints are predominantly connected to the number of trains, stations, composition types, skip-stop strategies, cargo groups, and discrete timestamps. For example, in the case of 10 stations with 30 trains in each direction, five train composition types, five train stopping modes, ten cargo groups, and 120 timestamps, the proposed model contains 7 980 decision variables and 151 210 auxiliary variables. Clearly, the joint optimization model of TT and RSCP with consideration of flexible composition and skip-stop strategies for passenger and freight co-transportation is an MINLP model, and the computational complexity increases with the increase of the problem size.

It can be inferred from Section 3.4 that for a real-world problem with a large number of services, the computational scale of the model is large. This will require considerable computational time, and the commercial solver (e.g., GUROBI) even cannot find a viable solution. By analyzing the characteristics of the studied model, this paper develops a heuristic algorithm based on VNS algorithm and GUROBI solver. The VNS algorithm was proposed by Mladenović and Hansen [61], which is an efficient metaheuristic algorithm that has been successfully applied to solve some TT-related optimization problems [28,32,62]. The following subsections describes the details of the implementation of the algorithm.

Generally, the VNS algorithm contains two parts, the shaking process and variable neighborhood descent (VND) process. The VND process is essentially a local search procedure that alternately searches in different neighborhood structures composed of different neighborhood actions. The basic idea is that if a better solution cannot be found in the current neighborhood, the search continues by jumping to the next neighborhood (as shown by the black dashed line in Fig. 5). When a better solution is found in the current neighborhood, the search starts again by jumping back to the first neighborhood (as shown by the red solid line in Fig. 5). The framework of the VND procedure is described as Algorithm 1.

Algorithm 1 The VND framework.

Input: Current focus solution: $x;$ neighborhood structures for VND: ${\mathcal{N}}_l$, $l=1,2,...,L;$

Output: A local optimal solution:$x^{\prime };$

1: Initialization: $l\leftarrow 1$, $x^{\prime }\leftarrow x;$

2: repeat

3:   Local search: find best neighbor of solution $x^{″}$ in its neighborhood ${\mathcal{N}}_l\left(x\right);$

4:   if $f\left(x^{″}\right)<f\left(x\right)$ then

5:    $x\leftarrow x^{″}$; $l\leftarrow 1$

6:  else

7:   $l\leftarrow l+1;$

8:   end if

9: until $l>L$

10: if $f\left(x\right)<f\left(x^{\prime }\right)$ then

11:  $x^{\prime }\leftarrow x;$

12: end if

13: return $x^{\prime }$

For the optimization problem, a local optimal solution that is effective within one neighborhood may not retain its optimality in a different neighborhood. However, the global optimal solution must maintain its optimality across all the potential neighborhoods. Therefore, it is better to explore as many different types of neighborhoods as possible if the search time allows. Consequently, the VNS algorithm extends the breadth and depth of the local search method by using the VND process alternately searching in various neighborhoods, whereas the shaking process generates a new solution with certain rules when a better solution cannot be found in the VND process. The specific process of VNS is shown in Algorithm 2.

Algorithm 2 The VNS algorithm framework.

Input: A set of neighborhood structures ${\mathcal{N}}_r$($r=1,2,...,R$) for shaking; a set of neighborhood structures ${\mathcal{N}}_l$($l=1,2,...,L$) for VND$;$ generate the initial solution: $x=x_0;$

Output: Best found solution: $x^{\ast }$;

1: Initialization: $x^{\ast }\leftarrow x_0;$

2: repeat

3:   $r\leftarrow 1$

4:   while $r\le R$ do

5:    $x^{\text{'}}\leftarrow \mathrm{S}\mathrm{h}\mathrm{a}\mathrm{k}\mathrm{i}\mathrm{n}\mathrm{g}\left(x^{*},r,{\mathcal{N}}_r\right)$: select a neighbor $x^{\prime }$ from neighborhood structure $r$ of $x^{\ast }$ randomly$;$

6:    $x^{\text{'}}\text{'}\leftarrow \mathrm{V}\mathrm{N}\mathrm{D}\left(x^{\text{'}},l,{\mathcal{N}}_l\right)$: find the local optimal solution $x^{″}$ by VND $;$

7:    if $f\left(x^{″}\right)<f\left(x^{\ast }\right)$ then

8:     $x^{\ast }\leftarrow x^{″};$ $r\leftarrow 1$

9:    else

10:    $r\leftarrow r+1;$

11:    end if

12:   end while

13: until stopping criterion14: return $x^{\ast }$

According to the complexity analysis of the model in Section 3.4, the model has a great number of decision variables that are difficult to solve. In the mathematical model, the decision variables are mainly divided into four categories: ① TT variables, that is, $t_{k,i}^{\left(\mathrm{a}\right)}$, $t_{k,i}^{\left(\mathrm{d}\right)}$; ② skip-stop strategies variables, that is, $x_{k,n},{\psi }_{k,v}$; ③ train composition variable $y_{k,m}$; and ④ passenger and freight allocation variables ${\lambda }_{k,i,j}$, $s_{k,i,j}$, and ${\gamma }_{k,f}$. If the values of $t_{k,i}^{\left(\mathrm{a}\right)}$, $t_{k,i}^{\left(\mathrm{d}\right)}$, $x_{k,n}$, and ${\psi }_{k,v}$ are fixed and the associated complex constraints are removed to simplify the problem, the remaining problem can be efficiently solved using a solver (e.g., GUROBI). Therefore, the original problem can be decomposed into two interrelated layers: The upper layer determines the TT and skip-stop strategies, whereas the lower layer focuses on selecting the composition type and allocating passengers and cargo groups. The TT and skip-stop strategies in the upper layer can then be solved and passed to the lower layer. By solving the simplified lower layer problem with eliminating the corresponding complex constraints, the obtained solution can be evaluated and updated in the upper layer. Through iterative resolution of the problem’s upper and lower layers, a solution that approaches optimality can be achieved within a reasonable computational time. Regarding the representation of the solution, because the section travelling time and train dwell time are predetermined, if the departure intervals of two consecutive trains from the origin station are known and the two stopping modes of each train service are also determined, the specific TT can be easily calculated from constraints in Eqs. (4), (5). Thus, the solution of the algorithm is denoted by $h=\left(\mathit{\partial },\mathit{\chi }\right)$, where $\mathit{\partial }$ represents the train headway vector with $\left(\right|\overline{K}|+|\underset{\_}{K}|-2)$ dimensions and $\mathit{\chi }$ represents the skip-stop strategy vector with $2\left(\right|\overline{K}|+|\underset{\_}{K}\left|\right)$ dimensions.

In the designed algorithm, considering the distribution characteristics of passenger and freight flows, several alternative stopping modes are pre-determined to prevent trains from skipping excessive stations and to ensure solution feasibility. Furthermore, during the solution process, each generated solution is evaluated for feasibility at every iteration and if an infeasible solution is found, a new solution will be regenerated. Consequently, the objective function of the model can be used directly as the evaluation function throughout the search process. For the initial solution in the first iteration, both a set of randomly generated train headways within the range [$h_{\mathrm{m}\mathrm{i}\mathrm{n}}$, $h_{\mathrm{m}\mathrm{a}\mathrm{x}}$] and the traditional all-stop strategy served as the inputs.

As mentioned above, the VNS algorithm mainly comprised two important parts: ① generating different neighbor solutions by the shaking procedure (i.e., generating different solutions for the VND procedure), aiming to increase the diversity of the solution; and ② applying the VND procedure to alternately search in different neighborhoods, to improve the optimality of the solution. The neighborhood structures are extremely important for the efficiency of the designed algorithm, and different neighborhood actions should be implemented for the shaking and VND procedures, as described below.

The solution primarily involves two types of variables that pertaining to the TT and skip-stop strategies. Thus, two types of neighborhood structures can be constructed for the shaking procedure, which are denoted by the set ${\mathcal{N}}_r$($r=1,2,...,R$), $R=2$, as described below.