High-Level Implicit Enumerations for Quadratic Periodic Train Timetabling with Prioritized Cross-Line Operations

Abstract

Periodic train timetables provide passengers with easily accessible rail transport services. However, in networked railway operations, some long-distance cross-line trains have high operational priority and pose difficulties for scheduling local services. In this paper, we address the minimal-cycle-length periodic train timetabling problem with high-priority cross-line operations and complex local train types. We propose a special set of constraints to accommodate the prespecified operational times of cross-line operations with regard to system robustness. As the cycle length is regarded as a decision variable, the formulation is nonlinear. To solve the problem, we exploit the connection between cycle length and consumed capacity of periodic timetables and propose high-level cycle-capacity and binary search-guided iterative solution frameworks, which implicitly enumerate the periodic train timetabling problems. Using the real-world operational data of the Guangzhou–Zhuhai Intercity Rail Line, we explore the solution performance of the proposed solution approaches and the straight linearization of the problem, and we also compare the practices of fixing prespecified operational times and our proposed constraints for the cross-line services. The results demonstrate that our proposed method can efficiently achieve flexible while recoverable operational times for the cross-line services and the proposed implicit enumeration algorithms significantly outperform the direct linearization, which increases the search space significantly due to the considerable dimensionality of the periodic decision variables involved. Numerical computations also suggest that our proposed constraints provide a type of approach for balancing the operational convenience and stability margins available in the periodic timetable with the presence of cross-line operations.

1. Introduction

Periodic train timetables define the arrival and departure times of all passenger trains in cyclic terms. They are highly passenger-friendly and are mostly adopted in European operating environments. In other parts of the world, travelers do not enjoy such convenience. To ensure the general accessibility of rail transport services in complex railway networks, long-distance trains are in widespread operation. For example, in China, the train “G337” originates from Beijing and terminates in Zhuhai, covering a distance of 2261 km (according to Google Maps^® route planner). It serves the passenger demands between major cities in China, such as Beijing, Shijiazhuang, Zhengzhou, Wuhan, Changsha, and Guangzhou. Such a long-distance train traverses multiple railway lines, each of which serve a relatively dedicated local zone, including our case study example of the Guangzhou–Zhuhai Intercity Rail. For this reason, it is usually called a cross-line (or interline) service. In the current timetabling system setting, cross-line train paths usually have high priority and are planned at a national scale and before the local services are scheduled. In this scenario, service planning with the existing cross-line trains occupying part of the temporal resources of the infrastructure poses challenge for implementing periodic timetabling for scheduling local trains in such a mixed environment. It is the local operator’s responsibility to reasonably coordinate the operations of these cross-line and local services, particularly within the framework of periodic timetables.

In existing periodic timetabling studies, most researchers focus on efficient formulations and solution algorithms for local service designs [1,2], but few look at the real hindrance of cross-line operations for the applicability of periodic timetables in the realistic situation of networked railway operations. In highly complex railway networks, operators usually want train lines to cover extensive distances while maintaining short journey times and adhering to strict operational priorities. Failure to effectively coordinate cross-line services can lead to significant delays, reduced network capacity, and suboptimal passenger experiences. This makes the implementation of periodic timetabling rather difficult. As a matter of fact, this problem can be generalized as the coordination of train services. In networks of lesser complexity, such coordination can be performed by mixing cross-line trains within the set of periodic train lines and optimizing the schedule. However, this does not necessarily satisfy the operational requirements of the cross-line services in other sections and may incur extra passenger waiting times. In this context, we take a step to address this gap by developing a novel modeling approach that features recoverable cross-line operations and a high-level implicit enumeration solution framework. By tackling this problem, our work aims to enhance operational efficiency, reduce delays, and improve service reliability, ultimately benefiting railway operators and passengers who rely on these networks daily.

1.1. Related Literature

We provide a concise review of the characteristics of the optimization problem, the modeling approaches, and the solution algorithms, with a focus on general research on periodic timetable optimization, the stability of periodic timetables, cycle length, and cross-line operation.

The Periodic Event Scheduling Problem (PESP), introduced by [3], serves as a fundamental modeling framework for most studies on periodic timetables. Nachtigall employed the Periodic Event Activity Network (PEAN) to model train timetables [4], where event nodes represent train arrivals and departures, and arcs denote activities such as dwell, travel, or transfer. By fixing dwell and travel times and setting the objective to minimize travel time, their study significantly enhanced the performance of the branch-and-bound algorithm using the Hermite normal form. To ensure that related events occur within the same cycle, the PESP model applies modular variables to map two connected events in the PEAN to periodic intervals [5]. Building on the Constrained and Differential Algebraic Nonlinear System Solver (CADANS) developed by [6], a constraint programming system called Designer Of Network Schedules (DONS) was proposed and implemented on the Dutch railway network [7,8]. The PESP framework has been extended in multiple directions. Ref. [9] expanded the PESP model by introducing buffer times as decision variables and leveraged the graph structure of the network to reduce the number of variables. Lindner incorporated operational costs into the objective function, further enriching the PESP framework [10]. Kroon and Peters extended PESP by allowing variable running times, achieved through the introduction of additional virtual nodes where necessary [11]. Other studies have focused on optimizing periodic timetables with variable cycle lengths. Several studies have treated cycle length as a decision variable and aimed to minimize its value [1,12,13,14,15]. Bortoletto et al. proposed a PESP extension that considers infrastructure constraints [16], optimizing timetables by fixing cyclic sequences for specific infrastructure elements.

The complexity of the PESP model has been proven to be NP-complete [17], and two variants of the PESP model have also been shown to exhibit MAXSNP-hardness [18]. Various solution algorithms have been proposed for addressing the PESP, including constraint generation algorithms [17], genetic algorithms [19], branch-and-bound methods [19,20], the modulo simplex method [21,22,23], SAT solver applications [24,25,26], and customized iterative algorithms [1,14,15]. Herrigel et al. proposed a heuristic sequential decomposition method, which divides all train lines into multiple priority groups and gradually adds them to the PESP model, thereby fixing the previously planned train schedule to a certain time margin [27]. Yan et al. presented a multi-objective periodic railway timetabling model formulated as a mixed-integer linear program (MILP) with four objectives—minimizing journey time, regularity deviation, vulnerability, and overtakings—and designed algorithms to generate the Pareto frontier [28]. Matos et al. proposed a method based on reinforcement learning and multi-agent systems, combined with an SAT solver (for the Boolean Satisfiability Problem) [29], to optimize the travel time of periodic timetables. Martin-Iradi and Ropke modeled periodic timetables using a time–space graph formalism based on a symmetric timetable strategy and fixed train-running-time assumptions, employing a column generation-based heuristic approach [30]. Sartor et al. introduced the concept of quasi-periodic timetables, which allow certain train groups to have slightly varying starting times at each cycle, and formulated an MILP model, solved using a hybrid optimization method combining column generation and Benders decomposition [31]. Bortoletto et al. studied novel connections between periodic timetabling and discrete geometry, representing the feasible periodic timetable space as a disjointed union of polytropes, and leveraged their neighborhood relationships to propose a new heuristic algorithm for PESP [32]. Furthermore, heuristic algorithms like DOSA-PIO for TSP [33] and IBM-Dy-SGA [34] for high-speed rail control systems demonstrate their effectiveness in solving complex optimization problems.

Furthermore, time–space networks have been employed in modeling and optimizing periodic timetable problems. Caprara et al. fixed the cycle length to one day, allowing train schedules to cross day boundaries, thereby enabling precise calculations of the scheduled time differences between events [35]. Bešinović et al. proposed a path-based integer programming model and solved it using a randomized multi-start greedy heuristic algorithm [36]. Zhang et al. developed an extended time–space network structure to model periodic timetables as a multi-commodity network flow problem, obtaining high-quality solutions through Lagrangian relaxation and the Alternating Direction Method of Multipliers (ADMM) [2]. Zhan et al. addressed the train rescheduling problem by considering the simultaneous rescheduling of bidirectional trains based on a space–time network, formulating an integer linear programming (ILP) model to minimize train deviation costs, and solving it using ADMM [37]. Yao et al. developed a periodic time–space network for integrated train-stop planning and passenger routing in periodic timetabling, formulating a multi-commodity network flow model and solving it with an ADMM-based algorithm [38].

Robust and stable periodic timetables, which improve delay resilience, are a key focus of research in periodic timetabling. Fischetti et al. compared four different approaches [39], including the lightweight robustness method proposed by [40], aimed at improving timetable robustness. Cacchiani and Toth proposed a Lagrangian optimization-based method capable of generating a Pareto set of solutions with varying robustness weights [41]. Liebchen et al. introduced the concept of recoverable robustness, focusing on disruption scenarios and their recovery strategies, which involve not only shortening buffer times but also severing connections if necessary [42]. Yan and Goverde studied the robustness optimization of periodic timetables for multiple train lines operating at different frequencies [43]. Grafe and Schöbel formulated the recoverable robust periodic timetable problem and proposed three equivalent mixed-integer programming formulations, providing detailed analyses of their properties [44].

Several studies have focused on cycle time, which is a critical component in the design of periodic timetables. Goverde evaluated the feasibility and stability of timetables through max-plus analysis and introduced the concept of the minimum cycle time, representing the shortest time required to maintain a periodic timetable [45]. Heydar et al. [12] developed a mixed-integer programming model to minimize cycle time for a single-track, single-direction railway (with passing loops), considering a fixed number of express and regular trains and allowing variable dwell times at intermediate stations, building on [46]. Sparing and Goverde aimed to maximize the stability of periodic timetables and transformed the nonlinear constraints caused by variable cycle time into linear constraints [1,15]. Zhang and Nie constructed non-conflict constraints and a series of flexible overtaking constraints based on the original binary variables in the PESP model, targeting the minimization of cycle time and proposing an iterative approximation method [14]. For flexible overtaking, they both introduced constraints allowing at most one overtaking per dwell activity in the optimization models. Yan et al. [28] conducted an in-depth analysis and modeling of scenarios involving multiple overtakes, building upon the overtaking constraints in [14]. Zheng et al. [47] proposed a multi-cycle train timetable optimization model that comprehensively considered variables such as train spatio-temporal paths, cycle time, and operating lines to improve passenger satisfaction and optimize energy consumption, and designed a hybrid heuristic Lagrangian decomposition algorithm.

Cross-line trains are also known as through trains in railway systems, and their technical requirements, such as infrastructure, signaling, communications, power supply, vehicle design, station configuration, and track layout, were analyzed in [48,49,50,51]. Yang et al. formulated an allocation model and described the necessary conditions to successfully deploy cross-line operation [52]. Yang et al. proposed a mixed-integer nonlinear programming (MINLP) model to systematically analyze the benefits of cross-line express trains in reducing passenger travel time and operational costs, and developed a genetic algorithm (GA) to solve the model [53]. Wang et al. addressed the integrated optimization of cross-line train operations and timetables for non-periodic schedules, constructing a mixed-integer programming model aiming to minimize deviations from ideal schedules for main-line trains while maximizing direct service frequency for cross-line passengers, based on event-activity network modeling and a genetic algorithm [54].

A summary of the related prior research is presented in

Table 1. A brief summary of the literature related to periodic timetabling.

Reference	Cyclic	Cross-line	Objective(s)	Modeling	Solution
[17]	no	no	Feasible timetable	PESP	ConGen
[19]	no	no	Passenger waiting	PESP	GenAlg
[20]	no	no	Total cost	PESP	B&B
[11]	no	no	Multiple	PESP	CADANS
[18]	no	no	Train idle time	PESP	CPLEX + Heuristic
[23]	no	no	Train idle time	PESP	Modulo simplex
[12]	yes	no	Cycle, dwell time	PESP	CPLEX
[36]	no	no	Total cost	TSN	Heuristics
[14]	yes	no	Cycle time	PESP	CPLEX + Heursitc
[1]	yes	no	Cycle time	PESP	CPLEX + Heuristic
[27]	no	no	Total cost	PESP	CPLEX + Heuristic
[2]	no	no	Travel time	TSN	LR + ADMM
[28]	no	no	Multiple	PESP	$ϵ$-constaint + Gurobi
[29]	no	no	Journey time	PESP	SAT solver + ML
[30]	no	no	Travel time	TSN	ColGen + Heuristic
[38]	no	no	Journey time, tickets	TSN	LR + ADMM
[32]	no	no	Journey time	PESP	Geometric heuristic
[54]	no	yes	Adjustment time, OD frequency	MILP	Heuristic GA
This paper	yes	yes	Cycle, journey time	PESP	Implicit enumeration + CPLEX

Cyclic: Cycle length as variable? Cross: Cross-line operation included? Modeling: Modeling framework. ConGen: Constraint generation. GenAlg: Genetic algorithm. B&B: Branch-and-bound. LR: Lagrangian relaxation. ADMM: Alternating direction method of multipliers. SAT: Satisfiability problem. ML: Machine learning. ColGen: Column generation.

.

There exists convincing research on cycle-time minimization in periodic train timetabling that also works to maintain the operating efficiency of the system. In these studies, the main focus is on coordinating the speed levels, rather than addressing prioritization differences across train classes. As a result, between the state-of-the-art research and real practices is a gap that limits the implementation of periodic timetabling in local search planning due to the prespecified operation times of prioritized cross-line trains. We note the distinction between the scheduling of mixed-speed trains and the coordination of cross-line and local services. The former is just the problem of applying different running or dwell times, which is present in the methodological structure of periodic timetabling. The latter involves the treatment of prefixed arrival and departure times at stations. One can either stick to these times in scheduling the local trains, or deviate from them. As noted earlier, research on periodic timetable optimization models with the objective of minimizing cycle time is relatively limited. Studies such as those by [1,12,15] have explored this objective. However, there has been no comprehensive consideration of both minimizing cycle time and cross-line operations. In this paper, we apply them as a set of constraints and we argue that this returns a better timetable. Adopting the practice of minimizing the cycle length for allowing maximum buffer time insertion, the optimal (and compressed) timetable should also allow the cross-line trains to retain their original arrival and departure times. An introduction to these two approaches is given in Section 2.

1.2. Contribution

This paper contributes to the railway operations research community in the following ways:

• Theoretically, our paper aims to fill the gap between periodic train timetabling and the reality of networked railway operations, where long-distance trains are scheduled before local services. In the case of China railway, high-priority cross-line train services are scheduled at a nationwide scale and usually have prespecified operation times when scheduling the local services. To avoid the shortcoming of fixing the operation times of those services, we set their operational times as decision variables subject to a special set of constraints, making it possible to retain their original operation times while allowing for abundant room for stability adjustment. In this way, we successfully coordinate the cross-line and local services in the framework of periodic timetabling.
• From a solution perspective, cycle-capacity and binary search-guided iterative solution frameworks are proposed as high-level implicit enumeration strategies for addressing the nonlinearity caused by taking the minimal cycle length as a decision variable. Given that a periodic timetable of minimal cycle length is “compressed”, in the sense of capacity analysis, its cycle length is equal to that of its consumed capacity, according to the definition of consumed capacity. Otherwise, the cycle length is at least that of its consumed capacity. The cycle length decision variable is noticeably one-dimensional, which can be used for enumerating the periodic timetabling problem fixed cycle lengths. These features of the nonlinear periodic timetable with prioritized cross-line services for minimizing the cycle length can be effectively enumerated.
• To verify our approach with a practical case, we take as an example a typical real-world railway line with high-priority cross-line services, the Guangzhou–Zhuhai Intercity Rail Line, to conduct a set of numerical experiments to demonstrate the efficacy of our proposed approach. The results demonstrate positive performance in treating the first and cross-line services. In particular, our proposed implicit enumeration methods significantly outperform straight linearization, which can be explained by considerably restricting the search space in variable-based enumeration rather than introducing additional logic continuous variables (for replacing the product of binary and continuous decision variables), which expands the problem search space along the way.
• Furthermore, our study has considerable application potential in justifying stability measures for periodic timetabling in the presence of cross-line operations. On the one hand, one can directly compute and use the periodic prespecified operational times of the cross-line services, resulting in a timetable favorable for inserting running-time margins within stations and distributing buffer times between trains scheduled later than the cross-line services in a cycle. Otherwise, it would be impossible. On the other hand, it derives an alternative timetabling strategy, where the trains contained between the fixed and the cross-line trains receive the appropriate strategic space for robustness. Therefore, our research suggests a new direction for balancing the treatment of cross-line operations with respect to the robustness of network periodic timetables.

1.3. Research Methodology Formulation

To tackle the problem of merging the prioritized cross-line services into periodic timetables, we propose the following methodologies in our paper. Section 2 introduces the periodic timetabling problem in the case of prioritized cross-line operations. Section 3 formulates the mathematical model for the concerned problem and introduces the constraints for implementing recoverable cross-line services, in which we show that the proposed constraints achieve what we want. Section 4 develops the cycle-capacity iterative solution framework and a variant using binary-guided search based on the problem structure. A set of numerical experiments based on real-world operating data are reported in Section 5. The conclusions are briefly summarized in Section 6, where we also identify future research opportunities.

2. Problem Description

2.1. Periodic Timetable

According to Nachtigall [4] and Sparing et al. [1], we model the railway system as a periodic event–activity network $\mathcal{G}=(\mathcal{E},\mathcal{A},T)$. $\mathcal{E}$ is the set of all events, $\mathcal{A}$ is the set of all activities (arcs connecting respective events), and T stands for the common cycle time of events, as shown in Figure 1.

An event $i\in \mathcal{E}$ can be expressed by a tuple $i:=(s,k,\epsilon ,l_i,u_i)$.

$s\in S_{all}$ refers to the train, with the set of all trains $S_{all}=S_{first}\cup S_{cross}\cup S_{other}$, where $S_{first}$ represents the set of the first train (which is by definition a singleton), $S_{cross}$ represents the set of cross-line trains, and $S_{other}$ represents other local trains.

$k\in K$ refers to a station in the railway network, consisting of a set of stations K.

$\epsilon \in \{arr,dep\}$, where $dep$ represents a departure event and $arr$ represents an arrival event.

The occurrence time of event $i\in \mathcal{E}$ is represented by a continuous variable ${\pi }_i\in [0,T-1)$ in periodic train timetabling, which is also constrained in a certain time window. In realistic applications, $\pi$ can just be assumed to represent the integer number of seconds due to the accuracy of train control and operations.

An arc $ij\in \mathcal{A}$ is represented as a tuple $ij:=(i,j,\varsigma ,l_{ij},u_{ij})$, where $l_{ij}$ and $u_{ij}$ represent the lower and upper bounds of the occurrence time of activity $ij$, ensuring that the activity is scheduled within a specified time window to meet operational requirements and practical constraints.

$\varsigma$ takes a value from $\{dwell,run,pass,over,safe,reg\}$, representing different styles of arcs, including dwelling, running, passing, overtaking, train separation, and service regularity of the same-line concept. According to the types of arcs, they can be written as $\mathcal{A}={\mathcal{A}}_{run}\cup {\mathcal{A}}_{pass}\cup {\mathcal{A}}_{dwell}\cup {\mathcal{A}}_{over}\cup {\mathcal{A}}_{safe}\cup {\mathcal{A}}_{reg}$. Among them, running arcs connect events to arrival events, passing arcs connect arrival events to departure events, dwelling arcs connect arrival events to departure events, overtaking arcs connect an arrival event event through a passing arc to a departure event, safety arcs connect arrivals (or departures) to adjacent arrivals (or departures), and regularity arcs connect departure events to departure events.

We have $i\in {\mathcal{E}}_s$ if event i is incurred by train s, where ${\mathcal{E}}_s$ represents the set of all events incurred by the train $s\in S_{other}$. Likewise, we write $ij\in \mathcal{A}$ for the arc $ij$ in activity set $\mathcal{A}$. In particular, if we define ${\mathcal{A}}_{run,s}$ to be the set of running activities incurred by train s, we may write ${\mathcal{A}}_{run}={\cup }_{s\in S_{all}}{\mathcal{A}}_{run,s}$. Such notation and set algebra are used in this paper without further detailed explanation. The lists of sets, parameters, and variables are provided in Appendix A.

$[l_{ij},u_{ij}]$ represents the lower and upper bounds of the duration for different types of arcs and can be expressed as

$$l_{ij}\le ({\pi }_j-{\pi }_i)modT\le u_{ij}.$$

We introduce a binary variable ${\rho }_{ij}$, which indicates whether certain activities or transitions occur within the cycle, to eliminate the modulus operation, and thus this constraint can be rewritten as

$$l_{ij}\le {\pi }_j-{\pi }_i)+T·{\rho }_{ij}\le u_{ij}.$$

2.2. Prioritized Cross-Line Operation

Cross-line trains are given priority over local trains in networked train timetabling. For the given cross-line train $s\in S_{cross}$, its event time in the periodic timetable ${\mathcal{T}}_P$ of nominal cycle time P is denoted as ${\widehat{\pi }}_i$. If the cross-line operation also assumes its current operation times by taking ${\pi }_i={\widehat{\pi }}_i,\forall i\in {\mathcal{E}}_s,s\in S_{cross}$, since we wish to generate a periodic timetable with maximal possible stability, the local trains would be circling around the cross-line service. This is acceptable, because in the final nominal cycle length timetable it assumes its initial operation times and the other trains can be translated to their own favorable positions (in terms of service provision perhaps).

However, in the increasingly marketing-oriented rail service environment, some services have fixed favorable times of operation. The Shanghai Airport Linking Line operates such a service, with the first train having stop-skipping patterns. As illustrated in Figure 2a, the blue line represents the first train, the orange line represents the cross-line operation scheduled at ${\pi }_{L_c,a}$, the dashed line represents the replica of the first train $L_1$ and indicates the minimal cycle length $T_1$, and the dash-dot-dotted line represents the time span of the nominal cycle length. In Figure 2b, the orange line marks the cross-line operation scheduled at ${\pi }_{L_c,a}^{\prime }$, the dash-dotted line stands for the previous position of the cross-line service in the time axis, and the blue dashed line indicates the minimal cycle length $T_2$. If we stick to the idea of taking ${\pi }_i={\widehat{\pi }}_i,\forall i\in {\mathcal{E}}_s,s\in S_{cross}$ in the case of first-train operations, a fixed parallel-like region is formed between the first train and the cross-line train as the first train has fixed zero-time departure in a cycle. Since the objective of the model is to minimize the cycle time (i.e., maximize the available buffer time), the optimization will try to arrange other trains within this fixed region. This may lead to some drawbacks. The train paths situated at this region may not fully utilize the included infrastructure times, which do not align with the optimization goal of the model. On the other hand, the resulting minimal-cycle-length timetable needs to be recovered into one of the nominal cycle lengths. However, due to the fixed region of the first and cross-line trains, it is impossible to further insert buffer or marginal running times for the trains situated there. This challenges the robustness of the timetable. Therefore, corresponding solutions are required to address such scenarios, which frequently present themselves in network operations.

In response to this problem, we propose recoverable operation times for cross-line operations, where the operation times of cross-line trains in the objective minimal-cycle-time periodic timetable are treated as variables; see Section 3.3. By applying some valid inequalities, we wish to first optimize a minimal-cycle periodic timetable, which can then have the prespecified operation times reinstated in the nominal cycle length timetable. Readers are referred to Figure 3 and its accompanying explanations for better intuition. The advantage of this approach is that it avoids the shortcoming that the train scheduled between the first and cross-line services cannot be altered for the purpose of robustness enhancement. In this way, the overall robustness of the nominal cycle length timetable can be ensured.

3. Mathematical Formulations

3.1. The Formulation

The general model for a periodic train timetable is given as follows:

$$\begin{array}{c}{l}_{ij}\le ({\pi }_j-{\pi }_i)+T·{\rho }_{ij}\le u_{ij},\forall ij\in {\mathcal{A}}_{dwell}\cup {\mathcal{A}}_{run}\cup {\mathcal{A}}_{safe},\hfill \end{array}$$

(1)

$$\begin{array}{c}{\pi }_i\in [0,T),\forall i\in {\mathcal{E}}_s,s\in S_{other},\hfill \end{array}$$

(2)

$$\begin{array}{c}{\rho }_{ij}\in \{0,1\},\forall ij\in \mathcal{A}.\hfill \end{array}$$

(3)

Limiting overtakes. To locally reduce the journey time of a train, the times of being overtaken should be regulated. We define a special type of overtaking decision variable ${\chi }_{i{i}^{\prime }j{j}^{\prime }}$, which indicates whether the dwelling arc $i{i}^{\prime }$ of train s is overtaken by the passing arc $j{j}^{\prime }$ of a high-priority train $s^{\prime }$. If the arc $i{i}^{\prime }$ does not conflict with the arc $j{j}^{\prime }$, then ${\chi }_{i{i}^{\prime }j{j}^{\prime }}=0$. If the arc $i{i}^{\prime }$ is overtaken by the arc $j{j}^{\prime }$, then ${\chi }_{i{i}^{\prime }j{j}^{\prime }}=1$, indicating that an overtaking arc $i{i}^{\prime }j{j}^{\prime }$ is formed between trains s and s’. In this case, the arc $i{i}^{\prime }j{j}^{\prime }$ completely encompasses the arc $i{i}^{\prime }$, allowing the arc $i{i}^{\prime }$ to be replaced by the arc $i{i}^{\prime }j{j}^{\prime }$.

Figure 4 presents schematic diagrams depicting the two possible interaction scenarios between trains: non-overtaking and overtaking. The line styles follow the conventions used in Figure 1.

For the major class of trains on this line, we have

$$\begin{array}{cc}\hfill & \sum _{s^{\prime }\in S_{all}\setminus \left\{s\right\}}\sum _{i{i}^{\prime }j{j}^{\prime }\in {\mathcal{A}}_{over,s{s}^{\prime }}}{\chi }_{i{i}^{\prime }j{j}^{\prime }}\le R_1,\forall i{i}^{\prime }\in {\mathcal{A}}_{dwell,s},s\in S_{other},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill & \sum _{s^{\prime }\in S_{all}\setminus \left\{s\right\}}\sum _{i{i}^{\prime }j{j}^{\prime }\in {\mathcal{A}}_{over,s{s}^{\prime }}}\sum _{i{i}^{\prime }\in {\mathcal{A}}_{dwell,s}}{\chi }_{i{i}^{\prime }j{j}^{\prime }}\le R_2,\forall s\in S_{other}.\hfill \end{array}$$

(5)

For any overtake to be successful, the following constraints are defined for dwell activity, $\forall i{i}^{\prime }\in {\mathcal{A}}_{dwell,s},s\in S_{other}$:

$$\begin{array}{cc}\hfill & \sum _{i{i}^{\prime }j{j}^{\prime }\in {\mathcal{A}}_{over,s{s}^{\prime }}}{\chi }_{i{i}^{\prime }j{j}^{\prime }}·(l_{i{i}^{\prime }j{j}^{\prime }}^{over}-l_{i{i}^{\prime }}^{dwell})-({\pi }_{i^{\prime }}-{\pi }_i+T·{\rho }_{i{i}^{\prime }})\le -l_{i{i}^{\prime }}^{dwell},\hfill \end{array}$$

(6)

$$\begin{array}{cc}\hfill & \sum _{i{i}^{\prime }j{j}^{\prime }\in {\mathcal{A}}_{over,s{s}^{\prime }}}{\chi }_{i{i}^{\prime }j{j}^{\prime }}·(u_{i{i}^{\prime }}^{dwell}-u_{i{i}^{\prime }j{j}^{\prime }}^{over})+({\pi }_{i^{\prime }}-{\pi }_i+T·{\rho }_{i{i}^{\prime }})\le u_{i{i}^{\prime }}^{dwell}.\hfill \end{array}$$

(7)

There are two types of constraint to consider: when ${\chi }_{i{i}^{\prime }j{j}^{\prime }}=1$, the length of the overtaking arc $i{i}^{\prime }j{j}^{\prime }\in {\mathcal{A}}_{over,s{s}^{\prime }}$ is bounded by $[l_{i{i}^{\prime }j{j}^{\prime }}^{over},u_{i{i}^{\prime }j{j}^{\prime }}^{over}]$; when ${\chi }_{i{i}^{\prime }j{j}^{\prime }}=0$, the length of the dwelling arc $i{i}^{\prime }\in {\mathcal{A}}_{dwell,s}$ is bounded by $[l_{i{i}^{\prime }}^{dwell},u_{i{i}^{\prime }}^{dwell}]$. Usually, $u_{i{i}^{\prime }}^{dwell}\le l_{i{i}^{\prime }j{j}^{\prime }}^{over}$.

Cycle length is usually regulated within a certain bound, namely,

$$Q\le T\le P.$$

(8)

Here, we use P to represent the nominal cycle length, which is eventually applied in the timetable, and Q stands for an initial estimate of the minimal cycle length and usually assumes the shortest time length required to separate the number of train paths in the periodic timetable.

3.2. First Train

Real-world operations often have a strong focus on marketing demands, where a type of skip-stop pattern is required in order to serve fast travelers as a so-called first train. Given the operation times ${\widehat{\pi }}_i,i\in {\mathcal{E}}_s,s\in S_{first}$ of the first train, the following constraints must be satisfied:

$$\begin{array}{cc}\hfill & {\pi }_{i^{★}}={\widehat{\pi }}_{i^{★}}=0,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill & {\pi }_j-{\pi }_i+T·{\rho }_{ij}={\widehat{\pi }}_j-{\widehat{\pi }}_i,\forall ij\in {\mathcal{A}}_{dwell,s}\cup {\mathcal{A}}_{run,s},s\in S_{first}.\hfill \end{array}$$

(10)

Here, $i^{★}$ represents the initial operation time, which is zero, of the first train at the first station of the examined rail line. Note that when considering the operation on the studied line we simply repeat the departure time at the first station as the corresponding arrival time, which does not impact the feasibility of the whole timetable. Consider the fact that cross-line operations often have prefixed operation times that may conflict with that of the first train in Equation (9). However, it is almost immediately known whether such conflicts occur with the cross-line trains, which can be manually resolved by changing (perhaps) the initial operation times of the first train. Therefore, we generally assume that no such conflict occurs due to the explanation above, and thus the “first” train is but a class of marketing-oriented services.

Since the first train is designed to travel fast, it should not be overtaken by any other train, and thus is defined by the following constraints:

$$\sum _{s^{\prime }\in S_{all}\setminus \left\{s\right\}}\sum _{i{i}^{\prime }j{j}^{\prime }\in {\mathcal{A}}_{over,s{s}^{\prime }}}\sum _{i{i}^{\prime }\in {\mathcal{A}}_{dwell,s}}{\chi }_{i{i}^{\prime }j{j}^{\prime }}=0,\forall s\in S_{first}.$$

Any overtake of the first train can be sufficiently eliminated by keeping dwell times lower than the time needed to perform an overtake—recall the requirement that $u_{i{i}^{\prime }}^{dwell}\le l_{i{i}^{\prime }j{j}^{\prime }}^{over}$ for successful overtaking. Such a constraint is unnecessary for the first train, and this is similar for the cross-line operations.

3.3. Cross-Line Operations

To avoid the shortcomings mentioned in Section 2, we intend to mobilize the operation times of the cross-line trains to two ends. In the minimal-cycle-length timetable, their operation times may be different than the original ones; meanwhile, they can be reset as the original ones without significantly “damaging” the current train timetable structure. Due to the fact that the minimal cycle length is shorter than the practical full cycle in general, these timetables are smaller than the original ones. Here, the term timetable structure can be interpreted as “train-following relations”, which should be familiar to any railway operations expert. However, interested readers are referred to [55] for a definition of this concept in graph form [56].

Define ${\pi }_i,i\in {\mathcal{E}}_{cross}$ as the decision variable for the event time of a cross-line train. Analogously, denote the prespecified operation time as ${\widehat{\pi }}_i,i\in {\mathcal{E}}_{cross}$. The scheduling parameters of cross-line trains cannot be altered in our computation, which implements the following constraints in the periodic timetable of nominal cycle length P:

$${\pi }_j-{\pi }_i+P·{\rho }_{ij}={\widehat{\pi }}_j-{\widehat{\pi }}_i,\forall ij\in {\mathcal{A}}_{dwell,s}\cup {\mathcal{A}}_{run,s},s\in S_{cross}.$$

(11)

To express the operation of the cross-line services in the periodic timetable of cycle length variable T, we first translate their initial operation times in cycle time T using the following equations:

$${\pi }_j^{\prime }-{\pi }_i^{\prime }+T·{\rho }_{ij}={\widehat{\pi }}_j-{\widehat{\pi }}_i,\forall ij\in {\mathcal{A}}_{dwell,s}\cup {\mathcal{A}}_{run,s},s\in S_{cross}.$$

(12)

Here, ${\pi }_i^{\prime },i\in {\mathcal{E}}_s,s\in S_{cross}$ represent the translated operation times of the cross-line services.

Then, the following set of constraints ensures that the cross-line trains in the minimal-cycle periodic timetable can be restored to the original operation times.

$$0\le {\pi }_{i^{★}}-{\pi }_{i^{★}}^{\prime }\le P-T,$$

(13)

where $i^{★}\in {\mathcal{E}}_s$ represents the initial departure event of train $s\in S_{cross}$. Inequality (13) expresses that, for any prioritized cross-line service, the difference between the original operation time of any event in the timetable of the nominal cycle length and its corresponding operation time in the timetable of minimal cycle length cannot exceed $P-T$, otherwise this cross-line service cannot be restored to its prespecified times in the timetable of the nominal cycle length.

To further illustrate the practical implication of constraint (13), we provide the following numerical example.

Assume the nominal cycle length is $P=120$ min, and the minimal cycle length is $T=90$ min. Consider a cross-line train $s\in S_{\mathrm{cross}}$, whose initial departure event $i^{★}$ occurs at ${\pi }_{i^{★}}=75$ min in the nominal cycle timetable. The corresponding translated event time in the minimal-cycle timetable is ${\pi }_{i^{★}}^{\prime }=60$ min. The timing deviation can thus be calculated as

$${\pi }_{i^{★}}-{\pi }_{i^{★}}^{\prime }=75-60=15\min ,$$

which clearly satisfies the restoration threshold $P-T=30$ min as prescribed by constraint (13). Therefore, the operation schedule of this train in the minimal-cycle timetable can be effectively mapped back to its original timing in the nominal cycle timetable. In contrast, if the deviation were 35 min, it would exceed the maximum allowed restoration tolerance of 30 min, and thus the consistency of this train’s operation times between the two timetables could not be guaranteed.

Let ${\mathcal{T}}_T$ be a cyclic timetable in T, and let $\mathcal{T}$ be its equivalent timetable with acyclic times. We first give an introductory result for an acyclic timetable in the following lemma.

Lemma 1.

In an acyclic timetable $\mathcal{T}$, the cross-line train path $s\in S_{cross}$ can be translated back to the times ${\left\{{\widehat{\pi }}_i\right\}}_{i\in {\mathcal{E}}_s,s\in S_{cross}}$ within a maximal shift of ${\Delta }_s\ge 0$, $s\in S_{cross}$, without changing the train-following relations if and only if

$$\begin{array}{cc}\hfill & {\pi }_i+{\Delta }_s-{\widehat{\pi }}_i\ge 0,\forall i\in {\mathcal{E}}_s,s\in S_{cross},\hfill \end{array}$$

(14)

$$\begin{array}{cc}\hfill & {\pi }_j-{\pi }_i={\widehat{\pi }}_j-{\widehat{\pi }}_i,\forall ij\in {\mathcal{A}}_{dwell,s}\cup {\mathcal{A}}_{run,s},s\in S_{cross}.\hfill \end{array}$$

(15)

Proof. 

First of all, cross-line train paths retain their structure during translation by maintaining Equation (15), which indicates the preservation of the stopping times and running times within station sections of these train paths.

We modify Equations (14) and (15) as

$$\begin{array}{cc}\hfill & {\widehat{\pi }}_i-{\pi }_i\le {\Delta }_s,\forall i\in {\mathcal{E}}_s,s\in S_{cross},\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & {\widehat{\pi }}_j-{\pi }_j={\widehat{\pi }}_i-{\pi }_i,\forall ij\in {\mathcal{A}}_{dwell,s}\cup {\mathcal{A}}_{run,s},s\in S_{cross}.\hfill \end{array}$$

(17)

The left-hand side of Equation (17) stands for the distance between the original arrival time and the current one, and the right-hand side represents the maximal possible shift allowed for cross-line service s. As illustrated in Figure 2b, the operational times of cross-line operations are contained in the reachable region between the solid blue line and the dash-dotted line, which shows that such a translation is possible. □

Now, combining Proposition 1 of [2], we can show that these constraints achieve what we want, as in the following proposition.

Proposition 1.

The constraints presented in Equations (11)–(13) are sufficient to characterize the required cross-line train paths in the minimal-cycle timetable.

Proof. 

Given an acyclic timetable $\mathcal{T}$ subject to Lemma 1, we can simply compute its cyclic times in any cycle time T, which results in a cyclic timetable in T, ${\mathcal{T}}_T$. Proposition 1 of [2] states that a cyclic timetable is equivalent to that of an extended one spanning the necessary number of cycles, in the modeling framework of the time–space network. This in turn tells us that if the cross-line train paths of $\mathcal{T}$ can be translated in a way that meets our requirements, the same process can be performed for ${\mathcal{T}}_T$. Computationally, we just need to take the modular operation of $\mathcal{T}$ in T, which is exactly ${\mathcal{T}}_T$. Thus, the result of Lemma 1 for an acyclic timetable can be carried to the case with its cyclic equivalent, and taking ${\Delta }_s:=P-T,\forall s\in S_{cross}$ provides what is required. □

As illustrated in Figure 5, the black line represents the first train, the red line represents the cross-line operation scheduled within $[{\pi }_i,{\pi }_j]$ in the minimal-cycle timetable, the red dashed line stands for the previous position of the cross-line service in the time axis, the green line represents the replica of the first train and indicates the minimal cycle length T, and the green dashed line represents the time span of nominal cycle length P. The red rectangle indicates the deviation range $[{\pi }_i^{\prime },{\pi }_i]$ during the recovery of a cross-line train from the minimal to the nominal timetable, and the green rectangle marks the time gap $(P-T)$, which defines the upper limit for achievable restoration. Even if the timetable is compressed to the minimal cycle length T to optimize the cycle length, as long as the event time offsets for each cross-line service do not exceed $(P-T)$, the original running times can still be restored within the nominal length P without disrupting the relative operational structure among trains.

Remark 1.

Algebraically, from Equation (13), we deduce

$${\pi }_i^{★}\ge {\widehat{\pi }}_i^{★}+T-P.$$

(18)

In combination with Equation (11), this is sufficient to characterize the recoverable region for the cross-line trains in any periodic timetable of cycle length $T\le P$.

3.4. Objective Functions

The primary objective of this problem is to minimize the cycle length, in exchange for maximal leftover time within the nominal cycle length for marginal running time and buffer time distribution.

$$min{z}_1:=T$$

(19)

The secondary objective is to minimize the total journey time for all trains.

$$min\sum _{s\in S_{other}}\sum _{{\mathcal{A}}_{dwell,s}\cup {\mathcal{A}}_{run,s}}{z}_2:=({\pi }_j-{\pi }_i+T·{\rho }_{ij}).$$

(20)

We should also note that our study aims to maximize the robustness of the periodic timetable by minimizing the minimal cycle length, as depicted in [1]. Therefore, the cycle length is taken as the dominant goal and the total journey time is considered less important. This approach resembles assigning a weight to the first objective significantly larger than that assigned to the second objective, such as the case of a single node in the Pareto frontier.

4. Solution Approaches

A noticeable feature of the minimal-cycle model for periodic timetabling is that the precedence constraints are accompanied by nonlinearity due to the product of cycle length and the periodicity decision variables. A traditional linearization technique exists for this, for example, in [1]. However, such linearization usually relies on the introduction of several additional decision variables and could increase the search space by a considerable extent. In the meantime, we notice that such nonlinearity is due to a single-dimensional decision variable for the cycle length. Suppose, given sufficient computational power, we perform enumeration on the cycle length decision variable t using constraint (8). If one can solve all instances of our problem with T subject to constraint (8), the smallest feasible value of T would be the exact value of our primary objective. Realistic timetables only assume integral values of operational times in seconds. Even if we limit T among the integer values of seconds within its bound, such a search is exhaustive (thousands of instances for a one-hour cycle length) and therefore computationally inefficient.

To tackle the nonlinearity of the model and solve it efficiently, we propose a cycle-capacity iterative framework for controlling the solution process, which resembles the technique of implicit enumeration. We only examine the values of cycle lengths as the consumed capacity of a previous timetable. In each iteration, we fix the cycle length and solve the periodic timetabling problem with the secondary objective function and then compute the consumed capacity of the resulting timetable as the new cycle length. Within each iteration, we employ the linear programming relaxation as the lower bound.

4.1. Cycle-Capacity Iterative Solution

Consumed capacity measures the minimum time required to operate a given set of trains over certain railway infrastructure. The definition of UIC 406 is usually adopted in its computation. The first train is repeated in the last position as a dummy train, meaning that the consumed capacity can be computed at any station. This is similar to the periodicity of a train timetable. For more detailed discussions on the computation of railway line capacity and its technological improvements, readers are referred to [57,58].

We start with a technical lemma on the basic relationships between the cycle length and the consumed capacity of a periodic timetable.

Lemma 2.

If a periodic timetable ${\mathcal{T}}_T$ of cycle length T has consumed capacity C, then $T\ge C$.

Proof. 

We prove this by contradiction.

Suppose $T<C$, that is a set of trains I repeat themselves in the cycle length T, and their operation requires a time of at least C. Denote the first train of I in the first cycle as ${\alpha }^1$, which repeats itself after time duration T and is denoted as ${\alpha }^2$. From the capacity side, the minimum time (further compression is impossible) required for scheduling I in this periodic timetable is C, from the departure of train ${\alpha }^1$ from a certain station to the departure of ${\alpha }^2$ at the same station. This is clearly a contradiction because, according to the definition of consumed capacity, I requires a minimal duration of C and cannot be feasibly scheduled in cycle length T, which is smaller than C. □

The implication of Lemma 2 is that the consumed capacity of a periodic timetable is not necessarily equal to its cycle length. Lemma 2 sets the consumed capacity of a periodic timetable as the new cycle length of a new periodic timetable that is isomorphic to the original one. In fact, the value of the consumed capacity is always a feasible cycle length for such a periodic timetable. We furnish this idea in the following lemma.

Lemma 3.

If a periodic timetable ${\mathcal{T}}_T$ has consumed capacity C, then C is a feasible cycle length for another periodic timetable ${\mathcal{T}}_C$.

Proof. 

We only need to construct a periodic timetable of cycle C given the periodic timetable ${\mathcal{T}}_T$ of consumed capacity C.

Recall that in calculating the consumed capacity of timetable ${\mathcal{T}}_T$, all buffer times are eliminated to create a compressed timetable ${\mathcal{T}}^{\prime }$. This in fact resembles the procedure of railway line capacity analysis. We briefly describe this here.

First, compute the contraction minor G of timetable ${\mathcal{T}}_T$, which can be achieved by applying a labeling algorithm [55,57]. Denote trains $\alpha ,\beta$ corresponding to the vertex with no preceding or following vertex as the first and last trains of ${\mathcal{T}}_T$, respectively. For each train i in timetable ${\mathcal{T}}_T$, denote its arrival and departure times as the vector $A_i$. Then, we preserve the arrival times of the first train $\alpha$ and label vertex $\alpha$ of minor G by $A_{\alpha }$ as its arrival times. The rest of these vertices in G can be labeled in a way that maintains the stopping times, running times within station sections, and train separation constraints. In this way, we arrive at the compressed timetable ${\mathcal{T}}^{\prime }$. Denote the label of train $\beta$ as $D_{\beta }^{\prime }$. According to the definition of consumed capacity, the difference $D_{\beta }^{\prime }-D_{\alpha }$ is a singleton consisting of the consumed capacity value of timetable ${\mathcal{T}}_T$ since train $\beta$ is a replica of $\alpha$. The train-following relations of ${\mathcal{T}}_T$ are preserved in ${\mathcal{T}}^{\prime }$ by maintaining its train contraction minor during our procedure, ${\mathcal{T}}^{\prime }$ is cyclic and has a cycle length of C. As a result, ${\mathcal{T}}_C:={\mathcal{T}}^{\prime }$. □

A brief restatement of Lemma 3 is that given a periodic timetable ${\mathcal{T}}_T$ of cycle length T and consumed capacity C, we can generate a compressed timetable ${\mathcal{T}}_C^{\prime }$, which has consumed capacity C as its cycle length. With this theoretical guarantee that $C\le T$ must be a feasible cycle length for a new periodic timetable, we may use it to start another iteration for updating the current periodic timetable, which potentially improves the compressed timetable that can be constructed. This provides us with the main idea of devising the so-called cycle-capacity iterative solution framework.

Now, we formally propose a solution framework equivalent to the original mixed-integer nonlinear program. The complete problem is referred as the Minimal-Cycle-Length Periodic Timetabling Problem with Cross-line Operations, and is denoted as $MPTwC\left(\pi \right)$. Fixing $T\in [Q,P]$, the original $MPTwC\left(\pi \right)$ can be regarded as a traditional mixed-integer linear program, which can be addressed as $PTwC$, with the cycle parameter t, denoted as $PTwC(\pi ,T)$. Before presenting the complete solution framework, we present its necessary optimality conditions.

Theorem 1.

If, fixing $T\in [Q,P]$, ${\mathcal{T}}_T$ is an optimal solution to $PTwC(\pi ,T)$, then ${\mathcal{T}}_T$ is an optimal solution to $MPTwC\left(\pi \right)$ only if periodic timetable ${\mathcal{T}}_T$ has consumed capacity $C=T$.

Proof. 

Suppose ${\mathcal{T}}_T$ is an optimal solution to $MPTwC\left(\pi \right)$ with consumed capacity $C\ne T$. According to Lemma 2, we have $C<T$, in which case we can always use ${\mathcal{T}}_T$ to construct a compressed timetable ${\mathcal{T}}_T^{\prime }\ne {\mathcal{T}}_T$, which is also an optimal periodic timetable up to $PTwC(\pi ,C)$ (as per Lemma 3), with a smaller cycle length $T^{\prime }<C$, as in the proof of Lemma 3. This contradicts the hypothesis of ${\mathcal{T}}_T$ being an optimal solution to $PTwC(\pi ,T)$. □

The formal description of the cycle-capacity iterative solution framework is presented in Algorithm 1.

Algorithm 1 Cycle-capacity iterative solution

Input: $\mathcal{G}=(\mathcal{E},\mathcal{A})$, the periodic event-activity network for $MPTwC\left(\pi \right)$.       The train operational parameters fo $MPTwC\left(\pi \right)$. Output: $({\pi }^{*},T^{*})$, which solves $MPTwC\left(\pi \right)$.       Cycle parameter $T\leftarrow P$.       Consumed capacity $C\leftarrow Q$.       Iteration $k\leftarrow 0$.       Solution $\mathcal{X}\leftarrow NULL$.       while  $C\ne T$ do           $T\leftarrow C^{\left(k\right)}$.           Solve $PTwC(\pi ,T)$.           if $PTwC(\pi ,T)$ solved with ${\pi }^{\left(k\right)}$ then               $\mathcal{X}\leftarrow {\pi }^{\left(k\right)}$.               Compute $C^{\left(k\right)}$ based on timetable ${\pi }^{\left(k\right)}$.               Consumed capacity $C\leftarrow C^{\left(k\right)}$.               $k++$.           end if       end while       Return $\mathcal{X}$.

The cycle-capacity iteration is initialized with cycle parameter $T\leftarrow P$ and consumed capacity $C\leftarrow Q$. Solving $PTwC(\pi ,T)$ results in a periodic timetable ${\pi }^{\left(k\right)}$, which has consumed capacity $C^{\left(k\right)}$. If this value is equal to the current cycle parameter T, this solution stops. Otherwise, we reiterate. Below, we analyze the working efficiency of Algorithm 1.

Lemma 4.

Let $\underline{T}$ be the minimal possible cycle length. If periodic timetable ${\mathcal{T}}_{n·T}$, with $T\ge \underline{T}$ and $n\in \mathbb{N}$, is a feasible solution to $PTwC(\pi ,n·T)$, then ${\mathcal{T}}_T$ is also a feasible solution to $PTwC(\pi ,n·T)$.

Proof. 

Clearly, periodic timetable ${\mathcal{T}}_T$ can be constructed out of ${\mathcal{T}}_{n·T}$ of cycle length $n·T$, with $T\ge \underline{T},n\in \mathbb{N}$. Since the minimal cycle is $\underline{T}\le T$, cycle T is a feasible cycle length and then ${\mathcal{T}}_T$ is a periodic timetable of cycle length T, according to Lemma 3. The optimality condition satisfying the second objective (20) is implicit in ${\mathcal{T}}_{n·T}$ being the optimal solution to $PTwC(\pi ,n·T)$. □

As a consequence of Lemma 4, the cycle-capacity solution control framework of Algorithm 1 is iteratively efficient in the sense that it simply ignores those periodic timetables that share the cycle lengths of the least common multiples. Suppose we have obtained the periodic timetable ${\mathcal{T}}_{n·T}$ of cycle length $n·T$, then we compute its consumed capacity, which is C. According to the definition of consumed capacity, timetables ${\left\{{\mathcal{T}}_{n·T}\right\}}_{n=1,2,\dots }$ have the same consumed capacity $C\le n·T$ for $n=1,2,\dots$. Algorithm 1 directly uses C for computing the new timetable in line 6, avoids examining the intermediate cycle lengths, and thus spares the computational resources. This is perhaps more theoretically than practically interesting.

4.2. Binary Search-Guided Solution

We first furnish the idea of performing decomposition and exploiting enumeration on the cycle-length decision variable t, and then we move on to delve into another solution technique based on the cycle-length variable t. Since $t\in [Q,P]$ is bounded and the operation times and train parameters are in seconds, which suffices for actual operations, we need only consider integer numbers of cycle lengths in seconds. Then, if we solve $PTwC(\pi ,t)$ for all integral values of t in $[Q,P]$, then the complete solution set of $MPTwC\left(\pi \right)$ is enumerated by computing ${\left\{PTwC(\pi ,t)\right\}}_{t\in {[Q,P]}_1}$, where ${★}_1$ stands for the set of integers in set ★. In this sense, ${\left\{PTwC(\pi ,t)\right\}}_{t\in {[Q,P]}_1}$ is also a decomposition of $MPTwC\left(\pi \right)$, and thus the solution found is exact. On top of this, our proposed Algorithm 1 avoids some of the cycle lengths in $[Q,P]$, in particular those in $(T^{\left(k\right)},C^{\left(k\right)})$ for iteration k. Such an ideal process is illustrated in Figure 6.

The horizontal and vertical axes represent the number of iterations and the time length, respectively. The black line represents the set of problem instances ${\left\{PTwC(\pi ,t)\right\}}_{t\in {[Q,P]}_1}$, which enumerates the original problem $MPTwC$. The red line represents the set of corresponding consumed capacities ${\left\{C\left(t\right)\right\}}_{t\in {[Q,P]}_1}$. Given $T^{\left(k\right)}$ as the cycle-length parameter, we solve $PTwC(\pi ,T^{\left(k\right)})$ in iteration k and obtain ${\mathcal{T}}_{T^{\left(k\right)}}$. The cycle lengths eliminated from the search space in iteration k are given by ${[C^{\left(k\right)},T^{\left(k\right)}]}_1$, expressed by the interval between the lines of cycle length and capacity. Then, $C^{\left(k\right)}$ is used as the cycle length (by taking $T^{(k+1)}:=C^{\left(k\right)}$) for generating the periodic timetable of the next iteration $k+1$. Unfortunately, we do not have sufficient information to assert that these two lines collide with each other only at the horizontal line of minimal cycle length. This actually corresponds to the missing sufficiency condition of Theorem 1. Therefore, we can only state that this cycle-capacity iterative approach is not in an exact sense.

To further look at our problem at a higher level, we aim to solve the original problem with as few iterations as possible, where each iteration is a PESP and can be effectively treated given that it is of a moderate input scale. As we have described previously, t is just an enumerable one-dimensional decision variable. A trivial and underlying assertion regarding cycle length in periodic timetabling is that a positive number T must be a feasible cycle length for a periodic timetable if $T^{\prime }\le T$ spans one of the same parameters. This is fundamental to the exploration of the periodic timetabling problem space over the one-dimensional line segment of cycle lengths $[Q,P]$. A technique that is known to be efficient in handling such one-dimensional exploration is binary search. Using this technique, we repeatedly divide the interval, checking each midpoint until we find the smallest viable cycle length. Binary search can be quite efficient because it reduces the search space by half at each step, meaning it converges logarithmically. Given the segment [2400, 3600], it only takes ${log}_2\left(1200\right)\approx 10-11$ steps to narrow down the cycle time if each check is independent. This approach can be advantageous if evaluating each cycle time is computationally inexpensive since fewer evaluations are needed to zero in on the minimum valid cycle.

We first propose the binary search-guided solution in Algorithm 2.

Algorithm 2 Binary search-guided solution

Input: $\mathcal{G}=(\mathcal{E},\mathcal{A})$, the periodic event-activity network for $MPTwC\left(\pi \right)$.       The train operational parameters fo $MPTwC\left(\pi \right)$. Output: $({\pi }^{*},T^{*})$, which solves $MPTwC\left(\pi \right)$.   1: Upper bound $U\leftarrow P$.   2: Lower bound $L\leftarrow Q$.   3: Iteration $k\leftarrow 0$.   4: Cycle length $T^{\left(k\right)}\leftarrow U$.   5: Solution $\mathcal{X}\leftarrow NULL$.   6: while  $U-L\ge 2$ do   7:     Solve $PTwC(\pi ,T^{\left(k\right)})$, and   8:     if problem feasible then   9:         $\mathcal{X}\leftarrow {\pi }^{\left(k\right)}$. 10:         $U\leftarrow T^{\left(k\right)}$. 11:     else 12:         $L\leftarrow T^{\left(k\right)}$. 13:     end if 14:     $T^k\leftarrow \lfloor (L+U)/2\rfloor$. 15:     $k++$. 16: end while 17: Return $\mathcal{X}$.

In Algorithm 2, we initialize with upper bound P and initialize the solution with $NULL$. In line 7 of Algorithm 2, we can perform a feasibility check, namely, by solving a periodic timetabling problem of fixed cycle length, to see whether the periodic timetable exists, using a CPLEX solver, for instance. When each instance is feasible (and thus solved), we record the current solution in $\mathcal{X}$ in line 9 and set the current cycle length as the new upper bound in line 10; otherwise, we update the lower bound as the current cycle length. Then, the problem is solved with the new cycle length $\lfloor (L+U)/2\rfloor$. When the upper and lower bounds converge by 1, the solution stops and returns the optimal minimum-cycle-length periodic timetable. The lower bound L always corresponds to an infeasible timetable, while the upper bound U corresponds to a feasible one. Recall that all train scheduling parameters are in integral seconds; therefore their algebraic operations can only return integer numbers in seconds. Therefore, the final upper bound U gives the minimal cycle length and $\mathcal{X}$ the corresponding minimal-cycle periodic timetable.

An additional trick we can notice from this solution is that we only need the solution for $PTwC(\pi ,T^{*})$ and not the intermediate solutions for $PTwC(\pi ,T)$ for any $T\ne T^{*}$. Therefore, we could just define a recall function for the feasibility check and verify that a solution exists during the solution process in line 7. When the minimal feasible $T^{*}$ is identified, we solve $PTwC(\pi ,T^{*})$ for the optimal timetable ${\mathcal{T}}_{T^{*}}$. This further saves time in removing unnecessary intermediate solutions.

4.3. Hybrid

Comparatively, cycle-capacity iterations start from the nominal cycle length and use each timetable’s consumed capacity as the next cycle length, which introduces a sequential dependency. The occupancy time after each cycle may vary significantly, making it hard to predict the total number of cycles needed. However, the iterative approach can also be powerful since each previous consumed capacity is guaranteed as a feasible cycle for the next timetable. This iterative approach can minimize the unnecessary feasibility checks associated with binary search, as every step is guaranteed to build on a feasible cycle. By leveraging the feasible cycle length of the previous timetable, the search space reduces dynamically. This may also lead to early termination if the optimal cycle length is reached before the lower boundary.

Therefore, it is also plausible to combine these two methods to make use of the property of binary search for finding the minimum viable cycle according to its predictable convergence rate and the cycle-capacity iterative approach’s ability to reveal the next feasible cycle length and skip excessive checks of intermediate cycle lengths. In this way, we seek to quickly locate a relatively small feasible region of cycle lengths by excluding intermediate ones and use binary search to find the minimum through its logarithmic convergence rate.

As explained, this hybrid approach is applied due to the absence of theoretical convergence of the cycle-capacity iterative procedure. Algorithm 3 readily shows that it is just the embedding of cycle-capacity iteration within the binary search-guided method. For the first part of the computation, cycle-capacity iteration is exploited until the equivalence of cycle length and the corresponding consumed capacity, and then the solution is guided using binary search to find the global minimum. One can just think of the hybrid approach as the binary search-guided solution taking over the solution process from a potential minimal cycle length, which is no greater than the nominal cycle length due to the cycle-capacity iteration. Since the starting point of binary-guided search is always a feasible cycle length, the algorithmic convergence is due to the binary search-guided version. And due to the numerical convergence of binary search, this algorithm also converges.

Algorithm 3 Hybrid iterative solution

Input: $\mathcal{G}=(\mathcal{E},\mathcal{A})$, the periodic event-activity network for $MPTwC\left(\pi \right)$.       The train operational parameters fo $MPTwC\left(\pi \right)$. Output: $({\pi }^{*},T^{*})$, which solves $MPTwC\left(\pi \right)$.   1: Iteration $k\leftarrow 0$.   2: Solution $\mathcal{X}\leftarrow NULL$.   3: Cycle parameter $T\leftarrow P$.   4: Consumed capacity $C\leftarrow Q$.   5: while$C\ne T$ do   6:     $T\leftarrow C^{\left(k\right)}$.   7:     Solve $PTwC(\pi ,T)$.   8:     if $PTwC(\pi ,T)$ solved with ${\pi }^{\left(k\right)}$ then   9:         $\mathcal{X}\leftarrow {\pi }^{\left(k\right)}$. 10:         Compute $C^{\left(k\right)}$ based on timetable ${\pi }^{\left(k\right)}$. 11:         Consumed capacity $C\leftarrow C^{\left(k\right)}$. 12:         $k++$. 13:     end if 14: end while 15: Upper bound $U\leftarrow C$. 16: Lower bound $L\leftarrow Q$. 17: Cycle length $T^{\left(k\right)}\leftarrow \lfloor (C+Q)/2\rfloor$. 18: while  $U-L\ge 2$do 19:     Solve $PTwC(\pi ,T^{\left(k\right)})$, and 20:     if problem feasible then 21:         $\mathcal{X}\leftarrow {\pi }^{\left(k\right)}$. 22:         $U\leftarrow T^{\left(k\right)}$. 23:     else 24:         $L\leftarrow T^{\left(k\right)}$. 25:     end if 26:     $T^{\left(k\right)}\leftarrow \lfloor (L+U)/2\rfloor$. 27:     $k++$. 28: end while 29: Return $\mathcal{X}$.

5. Numerical Experiments

We take the example of the Guangzhou–Zhuhai Intercity Rail in Guangdong, China. To evaluate the effectiveness of our proposed algorithms for solving the $MPTwC\left(\pi \right)$ problem, we compare them with the straight linearization of the quadratic term in $MPTwC\left(\pi \right)$, allowing a comprehensive assessment of solution performance across different paradigms.

• Straight linearization (LNR-CPLEX): This approach employs a straight linearization of the quadratic term in $MPTwC\left(\pi \right)$ and leverages the commercial solver CPLEX. It serves as a baseline, representing a traditional optimization strategy that transforms the problem into a linear form for exact solutions, though it may face scalability challenges for large instances.
• Cycle-capacity iterative solution (CC-NMR): The cycle-capacity iterative solution adopts an iterative framework that decomposes the problem based on cycle-capacity relations. This method represents a decomposition-based strategy, which aims to identify the next feasible cycle length as the consumed capacity of a computed periodic timetable.
• Binary guided search (BS-NMR):The binary guided search navigates the solution space by one-dimensional cycle length. This method embodies a search-based strategy, prioritizing fast computational speed given reasonable solution time in the lower level of periodic timetabling problem of fixed cycle length.
• Hybrid method (HB-NMR): The hybrid method combines elements of iterative decomposition and guided search, integrating the strengths of CC-NMR and BS-NMR. This approach represents a hybrid strategy, aiming to achieve the combined advantage in the different types of search behaviors of cycle-capacity iteration and binary search.

The formulations of lin-$MPTwC\left(\pi \right)$ are presented in Appendix B, while the readers are also referred to [1] for an exposition. In addition to performance evaluation, we also compare the two kinds of cross-operation treatments in detail and interpret their operational conditions to demonstrate the advantage of our approach. All methods are implemented using IBM ILOG CPLEX^® 20.1 in a Python 3.8 environment. The computing device is a laptop with AMD R7-5800H CPU and 16 GB RAM.

5.1. Input Data and Scheduling Parameters

The Guangzhou–Zhuhai Intercity Rail is a 115 km line consisting of 18 stations, with a 26 km branch separating at Xiaolan station, as shown in Figure 7. This is a rather typical line for demonstrating the application of our studied problem, in which the main line (shown in darker lines and dots) from Guangzhou South to Zhuhai also has prioritized cross-line traffic from Beijing, while the side line has relatively less traffic, and our numerical experiments are targeted at scheduling the periodic timetable of the main line with the presence of prioritized cross-line traffic. In the illustrated infrastructure layout, the black-colored dots represent the stations not allowing overtake and the orange-colored dots are those allowing overtake.

We solicited the historical timetable data for this line, from which we generated multiple typical operating scenarios corresponding to varied passenger demands. Since the solution performance of the large-input instance mainly relies on the lower-level solution of the periodic timetabling problem, we will mainly test the case of a 60 min nominal cycle length, which also fits the case of most scenarios applicable for our problem setting. The traffic conditions range from relaxed to busy of, with five to nine trains, including one first train and one cross-line service. For each case, the lower bound of the cycle length variable is estimated by computing the total time needed for separating the given number of trains—specifically, the lower bounds for the cases 60-1f1c4n, 60-1f1c6n, 120-1f2c11n, 120-1f2c13n, and 120-1f2c15n are 30, 35, 40, 45, and 50 min, respectively.

The overall line plans of these scenarios are provided in Figure 8 and

Table 2. The line plans for the operation scenarios.

Line Name	Line Concept
60-1f1c3n	$f_1^{†},f_2,f_3,f_4,f_9^{‡}$
60-1f1c4n	$f_1,f_2,f_3,f_4,f_5,f_9$
60-1f1c5n	$f_1,f_2,f_3,f_4,f_5,f_6,f_9$
60-1f1c6n	$f_1,f_2,f_3,f_4,f_5,f_6,f_7,f_9$
60-1f1c7n	$f_1,f_2,f_3,f_4,f_5,f_6,f_7,f_8,f_9$

† Represents the first train. ‡ Marks the cross-line operation.

. The first train operates at a fixed frequency of $f_1=1$, and a cross-line service $f_9$ is incorporated within the line plan. In 60 min-nominal-cycle-length periodic timetabling, we define the initial departure time of the cross-line service to be 30 min.

In terms of train operating parameters, when no overtake exists the minimal and maximal duration for dwell of any station is designated as 2 and 3 min, respectively. When an overtake exists, the minimal and maximal duration for dwell are assumed to be 5 and 10 min, respectively. This is because we allow a 2 min arrive–pass headway and a 3 min pass–depart headway in the case of overtakes, and a 3 min pass–pass headway and 5 min arrive–arrive headway are allowed. Additionally, we implement a 7% surplus for running times (in integer seconds) within station sections up to the optimization decision.

5.2. Solution Performance

We test the solution performances of the proposed solution methods along with the straight linearization of the original model and explore their solution behaviors. Then, we also briefly attend to the service balance issue of the same train types. For all computations, we limit the total solution time to within 3600 s.

5.2.1. Results

Table 3. Comparison of different methods.

Scenario	LNR-CPLEX			CC-NMR		BS-NMR		HB-NMR
	Time	r_Gap	Gap	Time	Gap	Time	Gap	Time	Gap
60-1f1c3n	91.19	0	0	2.71	4.63	2.27	0	2.63	0
60-1f1c4n	30.95	0	0	3.07	2.76	5.82	0	5.42	0
60-1f1c5n	542.05	0	0	1.08	9.84	21.5	0	19.01	0
60-1f1c6n	3600	6.7	0	1.3	9.01	63.27	0	68.48	0
60-1f1c7n	3600	12.28	3.64	1.3	8.33	2114.13	0	1246.94	0

Time: CPU time, in seconds. r_Gap (%): Relative gap in the branch-and-bound procedure. Gap (%): The gap between the current solution and the optimal solution.

reports the solution performances of the various methods proposed in this study, including straight linearization of $MPTwC\left(\pi \right)$ (LNR-CPLEX), cycle-capacity iterative enumeration (CC-NMR), binary search-guided iterative enumeration (BS-NMR), and hybrid enumeration (HB-NMR). The key performance metrics of the solution are indicated by Time, r_Gap, and Gap. Time stands for the CPU time for solving the given problem based on these algorithms, in seconds. r_Gap (%) represents the relative gap due to the branch-and-bound method in the CPLEX solution procedure. Gap (%) denotes the gap between the resulting solution and the optimal objective value. The robustness of the resulting timetables is dictated by the minimization of cycle length, which indicates the room for buffer time distribution, so cycle length assumes the most importance—the smaller the better (within the total solution time). Given the same minimal cycle length, the algorithm consuming less CPU time is considered better. It is noteworthy that the value of Gapindicates the absolute distance between the current objective value and the minimal cycle length, meaning they are equivalent in terms of demonstrating the solution performance. The minimal cycle lengths in our case studies are 2190, 2350, 2650, 3103, and 3596 s.

The first conclusion that can be drawn from the resulting table is that solving the quadratic periodic timetabling problem via pure linearization is rather difficult, as a small instance of five and six trains over 18 stations can take up to 91.19 and 30.95 s, respectively. The time required for case 60-1f1c3n is triple that for 60-1f1c4n, likely due to the larger search space introduced by the difference between the nominal cycle length and the target minimal cycle length. The optimal minimal cycle for 60-1f1c3n is 2050 s, which is 300 s greater than that for 60-1f1c4n. After that, the solution difficulty is increased considerably by the extra auxiliary defined to linearize the original quadratic constraints. This is demonstrated by the solution time of 542.05 s for the case of 60-1f1c5n and those that follow. This shows the computational trade-off between the difficulty introduced by the cross-line operation constraints and that introduced by linearizing the quadratic cyclicity constraints.

As shown in the column CC-NMR, the cycle-capacity iterative approach can generally not solve the problem to optimality, and our results suggest a rather “random” gap between 2.76% and 9.84%. On the other hand, it has a rather fast computational time. The reasons for this are twofold. First of all, the cycle-capacity approach only conducts limited iterations, which does not require much time at the higher level. As the resulting consumed capacity has a visible optimality gap, the lower-level branch-and-bound solution is not significantly challenged. As per our observation in the individual solution process, given the different cycle length inputs, it can be much harder to solve the same instance or to identify its feasibility. This is also shown by the sharp increase in solution time from 63.27 to 2114.13 s in the cases of 60-1f1c6n and 60-1f1c7n. With that said, however, the cycle-capacity iteration can greatly improve the solution process, as shown in the cases of 60-1f1c5n and 60-1f1c7n, whereas in the cases of 60-1f1c3n, 60-1f1c4n, and 60-1f1c6n, the cycle-capacity iteration has little to no contribution in terms of shortening the overall solution time. Recall that these implicit enumeration approaches work at a rather high level, where the lower solution is also subject to the difficulty of periodic timetabling with cross-line operations (which is much harder to solve than traditional periodic train timetabling; see later subsections). Due to this fact, the cycle-capacity approach may help reduce the solution time because it enables the whole solution procedure to avoid cases that are unnecessary to examine or difficult to solve. This results in a close-to-minor (11.6%) or considerable (41%) solution time drop in both the cases of 60-1f1c5n and 60-1f1c7n, highlighting the value of the investment in cycle-capacity iterations in order to achieve these time savings.

Returning to Figure 6, the numerical experiments indeed show the non-coincidence between the curves of cycle length and consumed capacity at the minimal cycle length. From our experimental experience, this is perhaps due to the fact that running time and scheduled waiting times and variable running times within station sections are scheduled to enable the minimization of working cycle lengths. At an initial stage where the cycle length is not stringent, train paths can be rather well aligned without having to be separated by just the nominal headway times, and there is space in the resulting timetable that can be squeezed to give a compressed timetable whose consumed capacity can be successfully used as the next cycle length. This is perhaps the case of 60-1f1c3n. With the decrease in cycle length, train separations are increasingly strict and variable running times are used in different sections in this process, and thus the resulting timetable is no longer possible, as shown in the cases of 60-1f1c5n, 60-1f1c6n, and 60-1f1c7n. Adhering to this idea, heuristics can be generated by first fixing the time needed within station sections and for separating overtakings. Next, a sensitivity analysis is performed on variable running times and overtake separations to further reduce the cycle length (or consumed capacity in the jargon of capacity analysis).

5.2.2. Solution Behaviors

To show the solution behaviors of the proposed approaches, we take the example of case 60-1f1c5n. As cycle-capacity iteration mainly serves the purpose of perturbing the solution with the next feasible cycle length, we mainly demonstrate BS-NMR and HB-NMR. The solution processes of these approaches are illustrated in Figure 9.

In the presented figures, the lower bounds of the binary search-guided solution process are illustrated as gray curves with triangles, the upper bounds are depicted as black curves with squares, the gaps are depicted as dashed red lines with disks, and the consumed capacities are presented as orange curves with diamonds. In Figure 9a, due to the logarithmic speed of binary search, the solution gap quickly drops to about 10%, performing five iterations in approximately 5 s. In the meantime, the HB-NMR used 12 cycle-capacity iterations to generate a solution with around an 8% gap with the optimal solution. The solutions of these two approaches are now identical in binary search mode. However, the BS-NMR takes a long time to generate new solutions and verify optimality while HB-NMR makes little progress in each relatively shorter iteration (compared to its BS-NMR counterpart). In general, the binary search exploration displays satisfactory performance in terms of avoiding unnecessary data points. However, the verification of optimality by solving a single-valued problem becomes rather slow, especially at small gap values.

5.2.3. Service Balance

To ensure the event spread (regularity) of services of the same type within a cycle, the following constraints are usually defined:

$$T/f_{\ell }-{\Lambda }_{\ell }\le {\pi }_j-{\pi }_i+T·{\rho }_{ij}\le T/f_{\ell }+{\Lambda }_{\ell },\forall ij\in {\mathcal{A}}_{reg,\ell },\ell \in \mathcal{L},$$

(21)

where $f_{\ell }$ and ${\Lambda }_{\ell }$ represent the frequency of services from the same type of service $\ell \in \mathcal{L}$ and the deviation of such an arc, respectively. As we are considering the minimal-cycle-length timetabling problem, this requirement is not very realistic as it must be further “stretched” to a periodic timetable of nominal cycle length. Due to their widespread application, they are given a brief consideration here. For each of the proposed cases, we exchange the last local service with $f_1$ and test the compatibility of the service balance with our problem setting of cross-line operations (while briefly overlooking the imperfect stop pattern). The results are reported in

Table 4. Results when implementing service balance.

Scenario	Cycle Length	CPU Time
60-1f1c3n	2050	2.1
60-1f1c4n	2350	3.56
60-1f1c5n	2735	11.89
60-1f1c6n	3000	49.98
60-1f1c7n	3325	472.49

.

In Table 4, cycle length indicates the computed minimal cycle length for operating the given scenario, in seconds. Time represents the CPU time needed to solve the problem, in seconds. Notably, the solutions of the first four scenarios implementing service balance requirements are moderately accelerated with respect to all methods, except for the heuristic cycle-capacity iteration. In the heavy-traffic case, the solution time is even significantly reduced. The possible cause of this reduction in solution time may include the more homogeneous stop pattern, as we substituted one heterogeneous service with an existing one from the current line plan. In this way, despite the expansion of the constraint system in the form of the service balance requirements, the other train separation constraints become easier to tackle. Additionally, the even distribution of services can either increase or decrease the minimal cycle length, with the goal of journey time minimization. This underscores the nonuniformity between the total journey time, cycle length, and service balance.

5.3. Cross-Line Operations

A brief idea is conveyed in Section 2, explaining the potential shortcomings in such a problem context when searching for a minimal-cycle periodic timetable. To explain this more concretely, we conduct a case study to review this. The original instance of $MPTwC(\pi ,T)$ allows the flexible cross-line operation times to be restored using Equation (13). Then, the alternate treatment of fixing the cross-line operations can be performed by dropping constraints (13) as well as the prime sign in Equation (12), which implements the following constraints:

$${\pi }_i={\pi }_i^{\prime },\forall i\in {\mathcal{E}}_s,s\in S_{cross}.$$

Therefore, we refer to this operating scenario as $MPTwC\left(\pi \right)$ with partially fixed periodic times, or f$-MPTwC\left(\pi \right)$. The instance and input settings are the same as those for $MPTwC\left(\pi \right)$. The case study scenario is the 60 min-cycle periodic timetable with eight trains, namely, 60-1f1c6n.

The computational results are illustrated in Figure 3. In the presented figure, the train paths are plotted in different colors in a two-dimensional plane, consisting of a horizontal axis representing time and a vertical axis representing stations. We follow the convention in plotting the periodic train paths, where services crossing a cycle are then resumed at the start of a new cycle. The stations are copied to show the distinction of arrival and departure, where a stop at a station is demonstrated by a tilted dashed line segment, and we let the arrival and departure equalize at the origin station Guangzhou South and destination station Zhuhai.

In Figure 3a, the minimal cycle length is 3255 s, and that of the second case is 3103 s. The latter requires more than 2 min to accommodate all the train paths, which shows one of the advantages of adopting recoverable cross-line operation times in a minimal-cycle-length timetable. However, using the second treatment method, the distribution of running-time margins and the buffer times between the trains scheduled between the first train C7601 and the cross-line service D941 have much better adaptability. The departure time of train C941 is 1550 s and should be restored to 1800 s in the nominal cycle length periodic timetable, which incurs a gap of 250 s, and the gap between the minimal and nominal cycle length is 497 s. Restoring cross-line train D941 means that the trains depart after it has only 497 − 250 = 247 s of space for stability maximization. Recall that our problem goal is to minimize the total cycle length for periodic train timetabling, which in turns tells us that this gap is the maximum possible for train D941, given a fixed nominal cycle length (3600 s in our case). Therefore, our treatment of the cross-line services actually finds the maximal possible space of adaptation for the trains scheduled between the first and the cross-line service, which in the meantime reduces the space for trains departing after the cross-line service. From the perspective of this particular case, the results seem rather favorable for all trains scheduled between the first train C7601 and the cross-line train D941 and those occupying positions later than D941. This might not be true for other scenarios, which suggests a need to balance the reserved space for robustness adjustment in minimal-cycle-length periodic timetables.

6. Conclusions and Future Work

6.1. Discussion and Conclusions

To tackle the periodic train timetable optimization problem in the case of cross-line operations in complex railway networks, we develop a mixed-integer nonlinear programming model that accommodates flexible running times, dwell times, train sequences, and over-running stations. Recoverable operation times of the cross-line operations are proposed using the original operation times in the form of a set of constraints that define the allowed regions for the cross-line operations. Three solution approaches are proposed to solve the problem, which mainly pivot around the quadratic constraints for expressing the periodicity of variable cycle lengths. The first solution builds the connection between the cycle length and consumed capacity of periodic timetables, where the consumed capacity of a periodic timetable must be a feasible cycle length for some different periodic timetable. The second solution idea relies on the one-dimensionality of the variable of cycle length, for which efficient binary search techniques can be utilized. The third solution is a combination of both. Using timetable data from the Guangzhou–Zhuhai Intercity Rail in China, we implement the solution algorithms for the optimization model to assess its practical effectiveness and compare its performances with linearization technique.

The computational experiments demonstrate that our proposed formulations and algorithms can indeed address the problem of periodic timetabling when prioritized cross-line trains are to be operated on a local railway line. The comparative advantage lies in that our approach does not have to apply fixed operational times for the prespecified cross-line services. In particular, when operators intend to strengthen the robustness of a periodic timetable, fixed cross-line services present considerable difficulty in adjusting the train paths, while our recoverable approach does not hinder such practices.

The results demonstrate that cycle-capacity iteration generally cannot solve the problem to optimality, but it can be combined with the binary search technique to increase solution quality in larger-scale problems. In comparison with the straight linearization approach, exploring a one-dimensional functional space is much cheaper than introducing linearization variables and inequalities in the linear number of quadratic constraints and is thus more efficient. Furthermore, our solution approach works at the high level and allows for multi-commodity flow reformulations and high-performance computing such as ADMM [2].

There is a major difference between fixed and recoverable cross-line operational times in modeling the problem. In the former approach, the resulting timetable cannot be adjusted for increasing its robustness. As in Figure 3a, the trains departing at Guangzhou South from 0 to 30 min are boxed by trains C7601 and D941 and no room is left for adjustment. As illustrated in Figure 3b, the trains departing after 30 min cannot be adjusted either because it would hurt the feasibility of the planned train paths. In contrast the recoverable approach allows for 250 s and 247 s for enhancing the robustness of trains before and after cross-line service D941. Therefore, the proposed recoverable approach has more operational flexibility than applying fixed times.

6.2. Future Research Directions

Certain limitations warrant consideration in our current study. The proposed model contains recoverable operational constraints for cross-line services in general, which may hinder the feasibility of scheduling periodic trains given the requirements on train paths, which have not been clarified in our study due to the limited scope of this paper. To address these limitations, future research can be conducted in the following directions:

• A key direction is to investigate the effective allocation of remaining times within the nominal cycle to achieve an optimal distribution of running-time slacks and buffer times in the periodic timetable. With the pre-existence of cross-line operations and first services, future studies could focus on developing strategies to distribute buffer times and running-time margins more effectively, enhancing timetable resilience against disruptions.
• The sensitivity of cross-line services’ operational times and operational parameters to the minimal cycle length of the periodic timetable was underexplored. Future research could analyze how variations in pre-scheduled operational parameters impact the minimal cycle length, particularly in networked operations, to improve the robustness and applicability of our approach in other railway systems.
• Building on our approach to cross-line operations, increasing the number and types of prioritized cross-line services can limit the possibility of operating local periodic train lines. This introduces the question of the applicability of local periodic timetabling in the presence of prioritized cross-line services in complex network settings.