Article

A Stochastic Dynamic Programming Approach for Delay Management of a Single Train Line

Authors:
To read the full-text of this research, you can request a copy directly from the authors.

Abstract

Railway delay management considers the question of whether a train should wait for a delayed feeder train. Several works in the literature analyze these so-called wait-depart decisions. The underlying models range from rules of thumb to complete network optimizations. Almost none of them account for uncertainties regarding future delays. In this paper, we present a multi-stage stochastic dynamic programming (SDP) model to make wait-depart decisions in the presence of uncertain future delays. The SDP approach explicitly accounts for potential recourse actions at later stations in a look-ahead manner when making the decision in the current stage. The objective is to minimize the total delay experienced by passengers at their final station by recursively solving Bellman equations. We focus on a single train line but consider the effects on direct feeder and connecting trains. In an extensive numerical study, we compare the solution quality and computational effort of the SDP to other optimization approaches and simple heuristic decision rules that are frequently used in delay management. The SDP approach outperforms the other approaches in almost every scenario with regard to solution quality in reasonable time and seems to be a promising starting point for stochastic dynamic delay management with interesting future research opportunities.

No full-text available

Request Full-text Paper PDF

To read the full-text of this research,
you can request a copy directly from the authors.

... In the past, some of them were used for practice in Germany, e.g., RWT: a long-distance train is allowed to wait up to 3 min for a delayed long-distance train (Stelzer 2016). Dispatching rules, especially NW, RWT and AW, are, therefore, often used in numerical studies for comparison with optimization models that are harder to solve, taking advantage of the fast and easy computation; see, e.g., Dollevoet et al. (2012), , , Bauer and Schöbel (2014), Schön and König (2018). ...
... A stochastic dynamic program (SDP) incorporating delay distributions from statistical literature is developed in Schön and König (2018). Potential recourse actions for the decision process are determined on single train lines, considering effects on feeder and connecting trains. ...
... Stochastic (2014), Berger et al. (2011a, b), Biederbick and Suhl (2007), Ginkel and Schöbel (2007), Kliewer and Suhl (2011), König and Schön (2020), Lemnian et al. (2016), Rückert et al. (2017), Schachtebeck and Schöbel (2010), Schöbel (2001Schöbel ( , 2007Schöbel ( , 2009), Schön and König (2018) (2014), Dollevoet et al. (2012Dollevoet et al. ( , 2015 models are rather rare. In Berger et al. (2011b) a simulation platform using stochastic distribution functions is presented; Bender et al. (2013) briefly sketch a stochastic program and Schön and König (2018) model an SDP for a single train line. When considering the stochastic nature of delays, the question arises why not more stochastic approaches exist. ...
Article
Full-text available
Passengers traveling by train may need to change trains on their route. If the focal train of a passenger is late, the passenger might miss his connection and has to decide how to continue his trip. Delay management addresses the question whether the connecting train should wait (or not) for the delayed passengers. If the connecting train waits, delays would get transferred through the network. In literature, several works consider delays and their impact on railways and how to reschedule disturbed plans. We focus on works, aiming to minimize passenger inconvenience as it is done in delay management. In the last two decades, dozens of works considering the delay management problem have emerged, tackling the problem in different ways. In this paper, an overview on the existing literature is given, and a new classification is introduced. We provide a taxonomy scheme for railway problems at an operational level and show how the field of delay management fits to other parts of the planning process. Moreover, limitations of the delay management approaches are discussed and future research opportunities are suggested.
... In conjunction with simulations, the created models can be modified and analyzed to get conclusions, verify and validate the research [129]. [18,26,63,72,84,88,92,93,[97][98][99][100]102,103,113,114] Smart Decision Support Systems (SDSS) ...
Article
Full-text available
Background: Industry 4.0 technologies have been widely used in the railway industry, focusing mainly on maintenance and control tasks necessary in the railway infrastructure. Given the great potential that these technologies offer, the scientific community has come to use them in varied ways to solve a wide range of problems such as train failures, train station security, rail system control and communication in hard-to-reach areas, among others. For this reason, this paper aims to answer the following research questions: what are the main issues in the railway transport industry, what are the technologic strategies that are currently being used to solve these issues and what are the technologies from industry 4.0 that are used in the railway transport industry to solve the aforementioned issues? Methods: This study adopts a systematic literature review approach. We searched the Science Direct and Web of Science database inception from January 2017 to November 2021. Studies published in conferences or journals written in English or Spanish were included for initial process evaluation. The initial included papers were analyzed by authors and selected based on whether they helped answer the proposed research questions or not. Results: Of the recovered 515 articles, 109 were eligible, from which we could identify three main application domains in the railway industry: monitoring, decision and planification techniques, and communication and security. Regarding industry 4.0 technologies, we identified 9 different technologies applied in reviewed studies: Artificial Intelligence (AI), Internet of Things (IoT), Cloud Computing, Big Data, Cybersecurity, Modelling and Simulation, Smart Decision Support Systems (SDSS), Computer Vision and Virtual Reality (VR). This study is, to our knowledge, one of the first to show how industry 4.0 technologies are currently being used to tackle railway industry problems and current application trends in the scientific community, which is highly useful for the development of future studies and more advanced solutions. Funding: Colombian national organizations Minciencias and the Mining-Energy Planning Unit.
... Pellegrini, Marlière, Pesenti, and Rodriguez (2015), Pellegrini, Pesenti, and Rodriguez (2019) formulated the train regulation problem into a mixed integer linear programming model and designed a heuristic algorithm to find the optimal train rescheduling plans. Schön and König (2018) presented a multi-stage stochastic dynamic programming model for the train regulation problem, in which uncertainties over future delays were considered. Based on the alternative graph method, Ariano, Corman, Pacciarelli, and Pranzo (2008) described the train regulation problem as a job shop scheduling problem. ...
Article
In high-frequency metro lines, train delays and substation peak power often occur, affecting safe and efficient train operation. In this paper, we propose real-time train regulation methods considering substation peak power reduction, in which runtimes and dwelltimes are adjusted to minimize the timetable and headway deviations and avoid multiple train accelerating. Firstly, we proposed two indirect indicators, i.e. overlapping time between accelerating phases and overlapping quantity between accelerating phases, which are minimized to suppress substation peak power in joint optimal train regulation models. The joint optimal train regulation models are based on the traditional real-time train regulation model considering the train traffic dynamics and control constraints. For the real-time requirement of train regulation, model predictive control (MPC) algorithms are designed to solve the formulated joint optimal control models, which generate the optimal train regulation strategies at each control cycle based on the real-time updated feedback system states. Finally, numerical examples based on one of the Guangzhou metro lines are implemented to verify the effectiveness and robustness of the proposed methods. The results show that the train regulation strategy with minimizing the overlapping quantity can not only suppress train delays and substation peak power, but also meet the real-time computation requirement.
... Altazin et al., 2020), stochastic dynamic programming (e.g. Schön and König, 2018) or by combining an optimisation model with passenger flow control strategies (e.g. Liu et al., 2020). ...
Article
Full-text available
Due to the multi-level nature of public transport networks, disruption impacts may spill-over beyond the primary effects occurring at the disrupted network level. During a public transport disruption, it is therefore important to quantify and control the disruption impacts for the total public transport network, instead of delimiting the analysis of their impacts to the public transport network level where this particular disruption occurs. We propose a modelling framework to quantify disruption impact propagation from the train network to the urban tram or bus network. This framework combines an optimisation-based train rescheduling model and a simulation-based dynamic public transport assignment model in an iterative procedure. The iterative process allows devising train schedules that take into account their impact on passenger flow re-distribution and related delays. Our study results in a framework which can improve public transport contingency plans on a strategic and tactical level in response to short- to medium-lasting public transport disruptions, by incorporating how the passenger impact of a train network disruption propagates to the urban network level. Furthermore, this framework allows for a more complete quantification of disruption costs, including their spilled-over impacts, retrospectively. We illustrate the successful implementation of our framework to a multi-level case study network in the Netherlands.
... Y. Zinder et al. [27] developed a DP to schedule the two-way traffic between two stations connected by a single-track railway with a siding, but it considered only the simple scenario of two stations. C. Schön et al. [28] presented a multi-stage stochastic dynamic programming model to make wait-depart decisions in the presence of uncertain future delays for a single line railway and they proposed a DP to minimize the total passengers' delay at destination. T. Ghasempour et al. [29] presented an adaptive railway traffic controller for real-time operations based on approximate dynamic programming (ADP) to limit consecutive delays resulting from trains by sequencing them at critical locations and their case study only included five stations. ...
Article
Full-text available
A dynamic programming (DP) approach with adaptive state generation and conflicts resolution is developed to address the timetable-rescheduling problem (TRP) at relatively lower computation costs. A multi-stage decision-making model is first developed to represent the timetable-rescheduling procedure in high-speed railways. Then, an adaptive state generation method by reordering the trains at each station is proposed to dynamically create the possible states according to the states of previous stages, such that the infeasible states can be removed and the search space is reduced. Then, conflicts are resolved by retiming the arrival and/or departure times of trains. Furthermore, the state transfer equation is built and Bellman equation is developed to derive the solution to minimize the total delay time (TT). A series of simulation experiments and a real-world case study are used to evaluate the performance of the proposed method. The simulation experiments indicate that the proposed method is able to find the optimal timetable with appropriate overtaking at right stations and reduce the total delay by 62.7% and 41.5% with respect to the First-Come-First-Serve (FCFS) and First-Schedule-First-Serve (FSFS) strategy that are widely used in practice. Comparing to the intelligent scheduling method (e.g., Ant Colony Optimization and Particle Swarm Optimization), similar objective performance can be achieved at a much lower cost of computation time, which make the proposed method more applicable to the TRP in daily operation of high-speed railway.
... Each of the algorithms studied assumes full information about the future railway states; an extension to an online problem with information available as time passes by would be a more realistic setting and allow for competitive analysis of decision-making strategies. Moreover, uncertainty is currently modeled in the decision of the players but not in the future railway states (based on stochastic prediction; see Corman and Kecman 2018) and/or stochastic optimization (see Schön and König 2019). This latter possibility would enable introducing risk-averse or risk-taking behavior in some of the players. ...
Article
Full-text available
In the last decade, optimization models for railway traffic rescheduling mostly focused on incorporating an increasing detail of the infrastructure, with the goal of proving feasibility and quality from the point of view of the managers of the infrastructure (tracks and stations). Different approaches that manage only the passenger flows instead focus more explicitly on the quality of service perceived by the passengers. This paper investigates microscopic railway traffic optimization models and algorithms, merging these two streams of research. In particular, we analyze the characterization of an equilibrium point between the reordering choices of train dispatchers in railway traffic optimization and the route choice of passengers in the available services of the railway transport network. We describe how passenger choice at stations along the route intertwines deeply with the problem of rescheduling trains over tracks and station resources in a very complicated setting that might not exhibit equilibrium points in general. Delaying trains and/or dropping passenger connections and/or giving particular route advice to passengers might influence the behavior of traffic controllers and passengers, determining a trade-off between the delays of trains, weighted by the passenger load, and the travel time of passengers. We study this problem with a game theoretical approach, focusing on the solutions corresponding to Nash equilibria of a game involving passengers and infrastructure managers. The proposed game theoretical approach is able to easily consider information and interdependence of the actions of multiple stakeholders. Computational results based on a real-world Dutch railway network quantify the trade-off between the minimization of train delays and passenger travel times and the performance, stability, and convergence of the equilibrium point given different algorithms and information available. The final aim of this work is to study the impact of effective implementations of railway traffic management and dissemination of information to passengers and operators.
... Corman et al. (2017) integrate delay management and train scheduling, and develop a model and fast heuristics to minimize the time passengers spend in the system. Schön and König (2018) introduce a stochastic version of DMP on a single line where delays are affected by uncertainty, and propose a dynamic programming approach to minimize the total passengers' delay at destination. ...
Article
The delay management problem arises in public transportation networks, often characterized by the necessity of connections between different vehicles. The attractiveness of public transportation networks is strongly related to the reliability of connections, which can be missed when delays or other unpredictable events occur. Given a single initial delay at one node of the network, the delay management problem is to determine which vehicles have to wait for the delayed ones, with the aim of minimizing the dissatisfaction of the passengers. In this paper, we present strengthened mixed integer linear programming formulations and new families of valid inequalities. The implementation of branch-and-cut methods and tests on a benchmark of instances taken from real networks show the potential of the proposed formulations and cuts.
Article
Accurate train arrival delay predictions can provide timely information for passengers and train dispatchers. Previous work mainly focused on predicting the delay of a single train, which is not enough to assist dispatchers, because making more comprehensive decisions considering more trains needs more future delay information of a group of trains. Therefore, this paper proposes a Bayesian optimization-based multi-output deep learning model, which includes a fully connected neural network (FCNN) and two long–short-term memory (LSTM) components, to predict the arrival delays of multiple trains simultaneously. The proposed model is calibrated and validated with the train operation data from the Wuhan–Guangzhou (W-G) high-speed railway. The test results show that the mean absolute error and the root mean square error of the proposed model is 1.379 and 2.021 min, over the four subsequent trains. Moreover, the proposed model outperforms the standard train delay prediction benchmark model.
Article
Full-text available
The increased need for transportation worldwide has led to intense competition among several transportation types. Thus, considering factors affecting the choice of transportation means of passengers is essential. In many countries, railways have been losing market share in both freight and passenger transport, especially against highways. Railway transport systems must regain their declining share for the sake of the economy and sustainability. For this reason, many studies have been conducted to eliminate delays in high-speed trains, the speed of which is the most important criterion for preference. This study determines the reasons for train delays in a bid to make the high-speed train project successful in Turkey and for trains to serve better. Furthermore, regression analysis and the Pythagorean fuzzy analytic hierarchy process (AHP) analysis were performed according to the most effective criteria. The most effective criteria were determined as maintenance, repair, and closure due to renewal. Additionally, various suggestions regarding the effect of the obtained causes on train delays were put forward.
Article
As a tour agency plans a tour schedule with rail transport, it is necessary to evaluate the capacity of the rail transport network to determine the number of passengers in the tour group that can be served. This study develops a stochastic rail transport network (SRTN) according to the train timetable and tour schedule, such that each available train’s loading capacity (that is, the number of cars) may be fully and partially reserved for other requests. Under the consideration of a tour group including q passengers, the network reliability accordingly means the probability that the q-passenger group can successfully depart from a source station for a sink station via the SRTN. In addition, the passengers might not punctually change trains at a transfer station owing to train arrival delay. Because the travel agency cannot control the situation of train arrival delay while planning a tour schedule, the uncertainty of train arrival delay results in an acceptable delay probability that the passengers traveling by non-direct trains can successfully change trains and this must be considered for the network reliability evaluation. An algorithm integrating the minimal paths and the modified Recursive Sum of Disjoint Product are integrated to solve the addressed problem. A real case of a rail transport network in Taiwan is adopted to demonstrate the applicability of the proposed approach. The travel agency can view the network reliability as a reference indicator to evaluate the number of passengers in the group that can be served.
Article
Delay management for railways is concerned with the question of whether a train should wait for a delayed feeder train or depart on time. The answer should not only depend on the length of the delay but also consider other factors, such as capacity restrictions. We present an optimization model for delay management in railway networks that accounts for capacity constraints on the number of passengers that a train can effectively carry. While limited capacities of tracks and stations have been considered in delay management models, passenger train capacity has been neglected in the literature so far, implicitly assuming an infinite train capacity. However, even in open systems where no seat reservation is required and passengers may stand during the journey if all seats are occupied, physical space is naturally limited, and the number of standing seats is constrained for passenger safety reasons. We present a mixed-integer nonlinear programming formulation for the delay management problem with passenger rerouting and capacities of trains. Our model allows the rerouting of passengers missing their connection due to delays or capacity constraints. We linearize the model in exact and approximate ways and experimentally compare the different approaches with the solution of a reference model from the literature that neglects capacity constraints. The results demonstrate that there is a significant impact of considering train capacity restrictions in decisions to manage delays.
Article
Full-text available
Transparency in transport processes is becoming increasingly important for transport companies to improve internal processes and to be able to compete for customers. One important element to increase transparency is reliable, up-to-date and accurate arrival time prediction, commonly referred to as estimated time of arrival (ETA). ETAs are not easy to determine, especially for intermodal freight transports, in which freight is transported in an intermodal container, using multiple modes of transportation. This computational study describes the structure of an ETA prediction model for intermodal freight transport networks (IFTN), in which schedule-based and non-schedule-based transports are combined, based on machine learning (ML). For each leg of the intermodal freight transport, an individual ML prediction model is developed and trained using the corresponding historical transport data and external data. The research presented in this study shows that the ML approach produces reliable ETA predictions for intermodal freight transport. These predictions comprise processing times at logistics nodes such as inland terminals and transport times on road and rail. Consequently, the outcome of this research allows decision makers to proactively communicate disruption effects to actors along the intermodal transportation chain. These actors can then initiate measures to counteract potential critical delays at subsequent stages of transport. This approach leads to increased process efficiency for all actors in the realization of complex transport operations and thus has a positive effect on the resilience and profitability of IFTNs.
Article
Rescheduling trains in dense railway systems to cope in real time with limited disturbances is a challenging problem with multiple conflicting objectives and various types of decisions. Based on the French railway system in the Paris region, this paper proposes an approach combining multi-objective optimization, to select rescheduling decisions, and macroscopic simulation, to compute the objectives associated to these decisions. Possible decisions include canceling or short-turning trains and skipping or adding stops. Three main objectives are optimized to propose multiple solutions to the decision makers: The recovery time, the quality of service for passengers and the number of decisions. Two greedy heuristics are presented whose results on actual data are compared with a full enumeration method. The multi-objective feature of the approach is also analyzed. The implementation and successful validation in real life of a decision-support tool, that is now implemented, is discussed.
Article
Purpose Environmental issues have become an important concern in modern supply chain management. The structure of closed-loop supply chain (CLSC) networks, which considers both forward and reverse logistics, can greatly improve the utilization of materials and enhance the performance of the supply chain in coping with environmental impacts and cost control. Design/methodology/approach A biobjective mixed-integer programming model is developed to achieve the balance between environmental impact control and operational cost reduction. Various factors regarding the capacity level and the environmental level of facilities are incorporated in this study. The scenario-based method and the Epsilon method are employed to solve the stochastic programming model under uncertain demand. Findings The proposed stochastic mixed-integer programming (MIP) model is an effective way of formulating and solving the CLSC network design problem. The reliability and precision of the Epsilon method are verified based on the numerical experiments. Conversion efficiency calculation can achieve the trade-off between cost control and CO 2 emissions. Managers should pay more attention to activities about facility operation. These nodes might be the main factors of costs and environmental impacts in the CLSC network. Both costs and CO 2 emissions are influenced by return rate especially costs. Managers should be discreet in coping with cost control for CO 2 emissions barely affected by return rate. It is advisable to convert the double target into a single target by the idea of “Efficiency of CO 2 Emissions Control Reduction.” It can provide managers with a way to double-target conversion. Originality/value We proposed a biobjective optimization problem in the CLSC network considering environmental impact control and operational cost reduction. The scenario-based method and the Epsilon method are employed to solve the mixed-integer programming model under uncertain demand.
Data
Full-text available
Railway traffic is operated according to a detailed schedule, specifying for each train its path through the network plus arrival and departure times at its scheduled stops. During daily operations, disturbances perturb the plan and dispatchers take action in order to keep operations feasible and to limit delay propagation. This article presents a thorough assessment of the possible application of an optimization-based framework for the evaluation of different timetables and proactive railway traffic management over a large network, considering stochastic disturbances. Two types of timetables are evaluated in detail: “ regular” and “ shuttle” timetables. The former is the regular plan of operations for normal traffic conditions, while the latter plan is designed to be robust against widespread disturbances, such as adverse weather, track blockage, and other operational failures. A test case is presented on a large Dutch railway network with heavy traffic, for which we compute by microsimulation detailed train movements at the level of block signals and at a precision of seconds. When comparing the timetables, a trade-off is found between the minimization of train delays, due to potential conflicts and due to delayed rolling stock and crew duties, and the minimization of passenger travel time between given origins and destinations.
Article
Optimization models for railway traffic rescheduling tackle the problem of determining, in real-time, control actions to reducing the effect of disturbances in railway systems. In this field, mainly two research streams can be identified. On the one hand, train scheduling models are designed to include all conditions relevant to feasible and efficient operation of rail services, from the viewpoint of operations managers. On the other hand, delay management models focus on the impact of rescheduling decisions on the quality of service perceived by the passengers. Models in the first stream are mainly microscopic, while models in the second stream are mainly macroscopic.This paper aims at merging these two streams of research by developing microscopic passenger-centric models, solution algorithms and lower bounds. Several fast heuristic methods are proposed, based on alternative decompositions of the model. A lower bound is proposed, consisting of the resolution of a set of min-cost flow problems with activation constraints. Computational experiments, based on multiple test cases of the real-world Dutch railway network, show that good quality solutions and lower bounds can be found within a limited computation time.
Article
In order to improve the robustness of a railway system in station areas, this paper introduces an iterative approach to successively optimize the train routing through station areas and to enhance this solution by applying some changes to the timetable in a tabu search environment. We present our vision on robustness and describe how this vision can be used in practice. By introducing the spread of the trains in the objective function for the route choice and timetabling module, we improve the robustness of a railway system. Using a discrete event simulation model, the performance of our algorithms is evaluated based on a case study for the Brussels’ area. The computational results indicate an average improvement in robustness of 6.2% together with a decrease in delay propagation of about 25%. Furthermore, the effect of some measures like changing the train offer to further increase the robustness is evaluated and compared.
Data
This paper deals with the development of decision support systems for traffic management of large and busy railway networks in case of severe disturbances. Railway operators typically structure the control of complicated networks into the coordinated control of several local dispatching areas. A dispatcher takes rescheduling decisions on the trains running on its local area while a coordinator addresses global issues that may arise between areas. While several advanced train dispatching models and algorithms have been proposed to support the dispatchers' task, the coordination problem did not receive much attention in the literature on train scheduling. This paper presents new heuristic algorithms for both local dispatching and coordination and compares centralized and distributed procedures to support the task of dispatchers and coordinators. We adopt dispatching procedures driven by optimization algorithms and based on local or global information and decisions. Computational experiments on a Dutch railway network, actually controlled by ten dispatchers, assess the performance of the centralized and distributed procedures. Various traffic disturbances, including entrance delays and blocked tracks, are analyzed on various time horizons of traffic prediction. Results show that the new heuristics clearly improve the global performance of the network with respect to the state of the art.
Article
The task of delay management is to decide whether connecting trains should wait for delayed feeder trains or depart on time in order to minimize the passengers’ delay. To estimate the effect of the wait-depart decisions on the travel times, most delay management models assume that passengers’ routes are predefined. However, in practice, passengers can adapt their routes to the wait-depart decisions and arising changes in the timetable. For this reason, in this paper we assume that passengers’ demand is given in form of pairs of origins and destinations (OD-pairs) and take wait-depart decisions and decisions on passengers’ routes simultaneously. This approach, called delay management with re-routing, was introduced in Dollevoet et al. (Transp. Sci. 46(1):74–89, 2012) and we build our research upon the results obtained there. We show that the delay management problem with re-routing is strongly NP-hard even if there is only one OD-pair. Furthermore, we prove that even if there are only two OD-pairs, the problem cannot be approximated with constant approximation ratio unless P=NP. However, for the case of only one OD-pair we propose a polynomial-time algorithm. We show that our algorithm finds an optimal solution if there is no reasonably short route from origin to destination which requires a passenger to enter the same train twice. Otherwise, the solution found by the algorithm is a 2-approximation of an optimal solution and the estimated travel time is a lower bound on the objective value.
Article
Railway conflict detection and resolution is the daily task faced by dispatchers and consists of adjusting train schedules whenever disturbances make the timetable infeasible. The main objective pursued by dispatchers in this task is the minimization of train delays, while train operating companies are also interested in other indicators of passenger dissatisfaction. The two objectives are conflicting whenever train delay reduction requires cancellation of some connected services, causing extra waiting times to transferring passengers. In fact, the infrastructure company and the train operating companies discuss on which connection to keep or drop in order to reach a compromise solution.This paper considers the bi-objective problem of minimizing train delays and missed connections in order to provide a set of feasible non-dominated schedules to support this decisional process. We use a detailed alternative graph model to ensure schedule feasibility and develop two heuristic algorithms to compute the Pareto front of non-dominated schedules. Our computational study, based on a complex and densely occupied Dutch railway network, shows that good coordination of connected train services is important to achieve real-time efficiency of railway services since the management of connections may heavily affect train punctuality. The two algorithms approximate accurately the Pareto front in a limited computation time.
Article
In this paper we survey the main studies dealing with the Train Timetabling Problem in its nominal and robust versions. Roughly speaking, the nominal version of the problem amounts of determining ”good” timetables for a set of trains (on a railway network or on a single one-way line), satisfying the so-called track capacity constraints, with the aim of optimizing an objective function that can have different meanings according to the requests of the railway company (e.g. one can be asked to schedule the trains according to the timetables preferred by the Train Operators or to maximize the passenger satisfaction). Two are the main variants of the nominal problem: one is to consider a cyclic (or periodic) schedule of the trains that is repeated every given time period (for example every hour), and the other one is to consider a more congested network where only a non-cyclic schedule can be performed. In the recent years, many works have been dedicated to the robust version of the problem. In this case, the aim is to determine robust timetables for the trains, i.e. to find a schedule that avoids, in case of disruptions in the railway network, delay propagation as much as possible. We present an overview of the main works on Train Timetabling, underlining the differences between models and methods that have been developed to tackle the nominal and the robust versions of the problem.
Article
The paper studies a train scheduling problem faced by railway infrastructure managers during real-time traffic control. When train operations are perturbed, a new conflict-free timetable of feasible arrival and departure times needs to be re-computed, such that the deviation from the original one is minimized. The problem can be viewed as a huge job shop scheduling problem with no-store constraints. We make use of a careful estimation of time separation among trains, and model the scheduling problem with an alternative graph formulation. We develop a branch and bound algorithm which includes implication rules enabling to speed up the computation. An experimental study, based on a bottleneck area of the Dutch rail network, shows that a truncated version of the algorithm provides proven optimal or near optimal solutions within short time limits.