The train platforming problem: The infrastructure management company perspective

If railway companies ask for station capacity numbers, their underlying question is in fact one about the platformability of extra trains. Train platformability depends not only on the infrastructure, buffer times, and the desired departure and arrival times of the trains, but also on route durations, which depend on train speeds and lengths, as well as on conflicts between routes at any given time. We consider all these factors in this paper. We assume a current train set and a future one, where the second is based on the expected traffic increase through the station considered. The platforming problem is about assigning a platform to each train, together with suitable in- and out-routes. Route choices lead to different route durations and imply different in-route-begin and out-route-end times. Our module platforms the maximum possible weighted sum of trains in the current and future train set. The resulting number of trains can be seen as the realistic capacity consumption of the schedule. Our goal function allows for current trains to be preferably allocated to their current platforms.Our module is able to deal with real stations and train sets in a few seconds and has been fully integrated by Infrabel, the Belgian Infrastructure Management Company, in their application called Ocapi, which is now used to platform existing and projected train sets and to determine the capacity consumption.     

Techniques for inserting additional trains into existing timetables

In this paper techniques for scheduling additional train services (SATS) are considered as is train scheduling involving general time window constraints, fixed operations, maintenance activities and periods of section unavailability. The SATS problem is important because additional services must often be given access to the railway and subsequently integrated into current timetables. The SATS problem therefore considers the competition for railway infrastructure between new services and existing services belonging to the same or different operators. The SATS problem is characterised as a hybrid job shop scheduling problem with time window constraints. To solve this problem constructive algorithm and meta-heuristic scheduling techniques that operate upon a disjunctive graph model of train operations are utilised. From numerical investigations the proposed framework and associated techniques are tested and shown to be effective.     

Integrated optimization on train scheduling and preventive maintenance time slots planning

We address the problem of simultaneously scheduling trains and planning preventive maintenance time slots (PMTSs) on a general railway network. Based on network cumulative flow variables, a novel integrated mixed-integer linear programming (MILP) model is proposed to simultaneously optimize train routes, orders and passing times at each station, as well as work-time of preventive maintenance tasks (PMTSs). In order to provide an easy decomposition mechanism, the limited capacity of complex tracks is modelled as side constraints and a PMTS is modelled as a virtual train. A Lagrangian relaxation solution framework is proposed, in which the difficult track capacity constraints are relaxed, to decompose the original complex integrated train scheduling and PMTSs planning problem into a sequence of single train-based sub-problems. For each sub-problem, a standard label correcting algorithm is employed for finding the time-dependent least cost path on a time-space network. The resulting dual solutions can be transformed to feasible solutions through priority rules. Numerical experiments are conducted on a small artificial network and a real-world network adapted from a Chinese railway network, to evaluate the effectiveness and computational efficiency of the integrated optimization model and the proposed Lagrangian relaxation solution framework. The benefits of simultaneously scheduling trains and planning PMTSs are demonstrated, compared with a commonly-used sequential scheduling method.     

A branch-and-price approach for solving the train unit scheduling problem

We propose a branch-and-price approach for solving the integer multicommodity flow model for the network-level train unit scheduling problem (TUSP). Given a train operator’s fixed timetable and a fleet of train units of different types, the TUSP aims at determining an assignment plan such that each train trip in the timetable is appropriately covered by a single or coupled train units. The TUSP is challenging due to its complex nature. Our branch-and-price approach includes a branching system with multiple branching rules for satisfying real-world requirements that are difficult to realize by linear constraints, such as unit type coupling compatibility relations and locations banned for coupling/decoupling. The approach also benefits from an adaptive node selection method, a column inheritance strategy and a feature of estimated upper bounds with node reservation functions. The branch-and-price solver designed for TUSP is capable of handling instances of up to about 500 train trips. Computational experiments were conducted based on real-world problem instances from First ScotRail. The results are satisfied by rail practitioners and are generally competitive or better than the manual ones.     

Train scheduling for minimizing passenger waiting time with time-dependent demand and skip-stop patterns: Nonlinear integer programming models with linear constraints

This paper focuses on how to minimize the total passenger waiting time at stations by computing and adjusting train timetables for a rail corridor with given time-varying origin-to-destination passenger demand matrices. Given predetermined train skip-stop patterns, a unified quadratic integer programming model with linear constraints is developed to jointly synchronize effective passenger loading time windows and train arrival and departure times at each station. A set of quadratic and quasi-quadratic objective functions are proposed to precisely formulate the total waiting time under both minute-dependent demand and hour-dependent demand volumes from different origin–destination pairs. We construct mathematically rigorous and algorithmically tractable nonlinear mixed integer programming models for both real-time scheduling and medium-term planning applications. The proposed models are implemented using general purpose high-level optimization solvers, and the model effectiveness is further examined through numerical experiments of real-world rail train timetabling test cases.     

Passenger-demands-oriented train scheduling for an urban rail transit network

This paper considers the train scheduling problem for an urban rail transit network. We propose an event-driven model that involves three types of events, i.e., departure events, arrival events, and passenger arrival rates change events. The routing of the arriving passengers at transfer stations is also included in the train scheduling model. Moreover, the passenger transfer behavior (i.e., walking times and transfer times of passengers) is also taken into account in the model formulation. The resulting optimization problem is a real-valued nonlinear nonconvex problem. Nonlinear programming approaches (e.g., sequential quadratic programming) and evolutionary algorithms (e.g., genetic algorithms) can be used to solve this train scheduling problem. The effectiveness of the event-driven model is evaluated through a case study.     

Heuristic approaches for solving large-scale bus transit vehicle scheduling problem with route time constraints

This paper presents new models for multiple depot vehicle scheduling problem (MDVS) and multiple depot vehicle scheduling problem with route time constraints (MDVSRTC). The route time constraints are added to the MDVS problem to account for the real world operational restrictions such as fuel consumption. Compared to existing formulations, this formulation decreases the size of the problem by about 40% without eliminating any feasible solution. It also presents an exact and two heuristic solution procedures for solving the MDVSRTC problem. Although these methods can be used to solve medium size problems in reasonable time, real world applications in large cities require that the MDVSRTC problem size be reduced. Two techniques are proposed to decrease the size of the real world problems. For real-world application, the problem of bus transit vehicle scheduling at the mass transit administration (MTA) in Baltimore is studied. The final results of model implementation are compared to the MTA's schedules in January 1998. The comparison indicates that, the proposed model improves upon the MTA schedules in all respects. The improvements are 7.9% in the number of vehicles, 4.66% in the operational time and 5.77% in the total cost.     

Multiple-phase train trajectory optimization with signalling and operational constraints

The train trajectory optimization problem aims at finding the optimal speed profiles and control regimes for a safe, punctual, comfortable, and energy-efficient train operation. This paper studies the train trajectory optimization problem with consideration of general operational constraints as well as signalling constraints. Operational constraints refer to time and speed restrictions from the actual timetable, while signalling constraints refer to the influences of signal aspects and automatic train protection on train operation. A railway timetable provides each train with a train path envelope, which consists of a set of positions on the route with a specified target time and speed point or window. The train trajectory optimization problem is formulated as a multiple-phase optimal control model and solved by a pseudospectral method. This model is able to capture varying gradients and speed limits, as well as time and speed constraints from the train path envelope. Train trajectory calculation methods under delay and no-delay situations are discussed. When the train follows the planned timetable, the train trajectory calculation aims at minimizing energy consumption, whereas in the case of delays the train trajectory is re-calculated to track the possibly adjusted timetable with the aim of minimizing delays as well as energy consumption. Moreover, the train operation could be affected by yellow or red signals, which is taken into account in the train speed regulation. For this purpose, two optimization policies are developed with either limited or full information of the train ahead. A local signal response policy ensures that the train makes correct and quick responses to different signalling aspects, while a global green wave policy aims at avoiding yellow signals and thus proceed with all green signals. The method is applied in a case study of two successive trains running on a corridor with various delays showing the benefit of accurate predictive information of the leading train on energy consumption and train delay of the following train.     

Metaheuristics for efficient aircraft scheduling and re-routing at busy terminal control areas

Intelligent decision support systems for the real-time management of landing and take-off operations can be very effective in helping air traffic controllers to limit airport congestion at busy terminal control areas. The key optimization problem to be solved regards the assignment of airport resources to take-off and landing aircraft and the aircraft sequencing on them. The problem can be formulated as a mixed integer linear program. However, since this problem is strongly NP-hard, heuristic algorithms are typically adopted in practice to compute good quality solutions in a short computation time. This paper presents a number of algorithmic improvements implemented in the AGLIBRARY solver (a state-of-the-art optimization solver to deal with complex routing and scheduling problems) in order to improve the possibility of finding good quality solutions quickly. The proposed framework starts from a good initial solution for the aircraft scheduling problem with fixed routes (given the resources to be traversed by each aircraft), computed via a truncated branch-and-bound algorithm. A metaheuristic is then applied to improve the solution by re-routing some aircraft in the terminal control area. New metaheuristics, based on variable neighbourhood search, tabu search and hybrid schemes, are introduced. Computational experiments are performed on an Italian terminal control area under various types of disturbances, including multiple aircraft delays and a temporarily disrupted runway. The metaheuristics achieve solutions of remarkable quality, within a small computation time, compared with a commercial solver and with the previous versions of AGLIBRARY.     

Timetable optimization models and methods for minimizing passenger waiting time at public transit terminals

This paper focuses on developing mathematical optimization models for the train timetabling problem with respect to dynamic travel demand and capacity constraints. The train scheduling models presented in this paper aim to minimize passenger waiting times at public transit terminals. Linear and non-linear formulations of the problem are presented. The non-linear formulation is then improved through introducing service frequency variables. Heuristic rules are suggested and embedded in the improved non-linear formulation to reduce the computational time effort needed to find the upper bound. The effectiveness of the proposed train timetabling models is illustrated through the application to an underground urban rail line in the city of Tehran. The results demonstrate the effectiveness of the proposed demand-oriented train timetabling models, in terms of decreasing passenger waiting times. Compared to the baseline and regular timetables, total waiting time is reduced by 6.36% and 10.55% respectively, through the proposed mathematical optimization models.     

Ferry service network design: optimal fleet size,routing, and scheduling

The study formulated a ferry network design problem by considering the optimal fleet size, routing, and scheduling for both direct and multi-stop services. The objective function combines both the operator and passengers’ performance measures. Mathematically, the model is formulated as a mixed integer multiple origin–destination network flow problem with ferry capacity constraints. To solve this problem of practical size, this study developed a heuristic algorithm that exploits the polynomial-time performance of shortest path algorithms. Two scenarios of ferry services in Hong Kong were solved to demonstrate the performance of the heuristic algorithm. The results showed that the heuristic produced solutions that were within 1.3% from the CPLEX optimal solutions. The computational time is within tens of seconds even for problem size that is beyond the capability of CPLEX.     

Train scheduling simulation that minimises operational conflicts due to service constraints

This paper presents an attempt made to facilitate re‐scheduling of trains to minimize operational delays and accommodate uniform headways for off peak sub urban services subject to resource constraints such as locomotive availability, poor track conditions and stations without siding facilities. The paper describes the computer simulation model designed to optimize train schedules on single‐track rail lines. Using this simulation program it is possible to plan and optimize timetables for railway networks with train runs within short time periods for both single track and double track conditions. The paper describes the capabilities of presenting the results of the simulation runs. These include the time‐distance graph, the network with train movements, dialog boxes with information about selected trains. The programme is capable of changing the starting point, departure time, train destinations and adding or deleting a stop etc. from the user interface. Four objects of array variables are used in the simulation process to keep train and station data. Two object arrays are used for the train movements in up and down directions. The stations' data are stored in the other two object arrays. One of these arrays of stations contains all the stations of the line while the other one contains only the stations with siding facilities. A case study that covers a 61 km long single‐track line with 14 stations is presented to highlight the model capabilities.     

Scheduling freight trains traveling on complex networks

In the US, freight railways are one of the major means to transport goods from ports to inland destinations. According to the Association of American Railroad’s study, rail companies move more than 40% of the nation’s total freight. Given the fact that the freight railway industry is already running without much excess capacity, better planning and scheduling tools are needed to effectively manage the scarce resources, in order to cope with the rapidly increasing demand for railway transportation. This research develops optimization-based approaches for scheduling of freight trains. Two mathematical formulations of the scheduling problem are first introduced. One assumes the path of each train, which is the track segments each train uses, is given and the other one relaxes this assumption. Several heuristics based on mixtures of the two formulations are proposed. The proposed algorithms are able to outperform two existing heuristics, namely a simple look-ahead greedy heuristic and a global neighborhood search algorithm, in terms of railway total train delay. For large networks, two algorithms based on the idea of decomposition are developed and are shown to significantly outperform two existing algorithms.     

Ant colony optimization for the real-time train routing selection problem

This paper deals with the real-time problem of scheduling and routing trains in a railway network. In the related literature, this problem is usually solved starting from a subset of routing alternatives and computing the near-optimal solution of the simplified routing problem. We study how to select the best subset of routing alternatives for each train among all possible alternatives. The real-time train routing selection problem is formulated as an integer linear programming formulation and solved via an algorithm inspired by the ant colonies’ behavior. The real-time railway traffic management problem takes as input the best subset of routing alternatives and is solved as a mixed-integer linear program. The proposed methodology is tested on two practical case studies of the French railway infrastructure: the Lille terminal station area and the Rouen line. The computational experiments are based on several practical disturbed scenarios. Our methodology allows the improvement of the state of the art in terms of the minimization of train consecutive delays. The improvement is around 22% for the Rouen instances and around 56% for the Lille instances.     

Reinforcement learning approach for train rescheduling on a single-track railway

Optimal rail network infrastructure and rolling stock utilization can be achieved with use of different scheduling tools by extensive planning a long time before actual operations. The initial train timetable takes into account possible smaller disturbances, which can be compensated within the schedule. Bigger disruptions, such as accidents, rolling stock breakdown, prolonged passenger boarding, and changed speed limit cause delays that require train rescheduling. In this paper, we introduce a train rescheduling method based on reinforcement learning, and more specifically, Q-learning. We present here the Q-learning principles for train rescheduling, which consist of a learning agent and its actions, environment and its states, as well as rewards. The use of the proposed approach is first illustrated on a simple rescheduling problem comprising a single-lane track with three trains. The evaluation of the approach is performed on extensive set of experiments carried out on a real-world railway network in Slovenia. The empirical results show that Q-learning lead to rescheduling solutions that are at least equivalent and often superior to those of several basic rescheduling methods that do not rely on learning agents. The solutions are learned within reasonable computational time, a crucial factor for real-time applications.     

Integrating capacity analysis with high-speed railway timetabling: A minimum cycle time calculation model with flexible overtaking constraints and intelligent enumeration

Compared with most optimization methods for capacity evaluation, integrating capacity analysis with timetabling can reveal the types of train line plans and operating rules that have a positive influence on improving capacity utilization as well as yielding more accurate analyses. For most capacity analyses and cyclic timetabling methods, the cycle time is a constant (e.g., one or two hours). In this paper, we propose a minimum cycle time calculation (MCTC) model based on the periodic event scheduling problem (PESP) for a given train line plan, which is promising for macroscopic train timetabling and capacity analysis. In accordance with train operating rules, a non-collision constraint and a series of flexible overtaking constraints (FOCs) are constructed based on variations of the original binary variables in the PESP. Because of the complexity of the PESP, an iterative approximation (IA) method for integration with the CPLEX solver is proposed. Finally, two hypothetical cases are considered to analyze railway capacity, and several influencing factors are studied, including train regularity, train speed, line plan specifications (train stops), overtaking and train heterogeneity. The MCTC model and IA method are used to test a real-world case involving the timetable of the Beijing–Shanghai high-speed railway in China.     

Optimal scheduling of a taxi fleet with mixed electric and gasoline vehicles to service advance reservations

This study addresses the problem of scheduling a fleet of taxis that are appointed to solely service customers with advance reservations. In contrast to previous studies that have dealt with the planning and operations of a taxi fleet with only electric vehicles (EVs), we consider that most taxi companies may have to operate with fleets comprised of both gasoline vehicles (GVs) and plug-in EVs during the transition from GV to (complete) EV taxi fleets. This paper presents an innovative multi-layer taxi-flow time-space network which effectively describes the movements of the taxis in the dimensions of space and time. An optimization model is then developed based on the time-space network to determine an optimal schedule for the taxi fleet. The objective is to minimize the total operating cost of the fleet, with a set of operating constraints for the EVs and GVs included in the model. Given that the model is formulated as an integer multi-commodity network flow problem, which is characterized as NP-hard, we propose two simple but effective decomposition-based heuristics to efficiently solve the problem with practical sizes. Test instances generated based on the data provided by a Taiwan taxi company are solved to evaluate the solution algorithms. The results show that the gaps between the objective values of the heuristic solutions and those of the optimal solutions are less than 3%, and the heuristics require much less time to obtain the good quality solutions. As a result, it is shown that the model, coupled with the algorithms, can be an effective planning tool to assist the company in routing and scheduling its fleet to service reservation customers.     

Nonlinear programming methods based on closed-form expressions for optimal train control

This paper proposes a novel approach to solve the complex optimal train control problems that so far cannot be perfectly tackled by the existing methods, including the optimal control of a fleet of interacting trains, and the optimal train control involving scheduling. By dividing the track into subsections with constant speed limit and constant gradient, and assuming the train’s running resistance to be a quadratic function of speed, two different methods are proposed to solve the problems of interest. The first method assumes an operation sequence of maximum traction – speedholding – coasting – maximum braking on each subsection of the track. To maintain the mathematical tractability, the maximum tractive and maximum braking functions are restricted to be decreasing and piecewise-quadratic, based on which the terminal speed, travel distance and energy consumption of each operation can be calculated in a closed-form, given the initial speed and time duration of that operation. With these closed-form expressions, the optimal train control problem is formulated and solved as a nonlinear programming problem. To allow more flexible forms of maximum tractive and maximum braking forces, the second method applies a constant force on each subsection. Performance of these two methods is compared through a case study of the classic single-train control on a single journey. The proposed methods are further utilised to formulate more complex optimal train control problems, including scheduling a subway line while taking train control into account, and simultaneously optimising the control of a leader-follower train pair under fixed- and moving-block signalling systems.     

A constraint generation algorithm for the construction of periodic railway timetables

This paper addresses the problem of constructing periodic timetables for train operations. We use a mathematical model consisting of periodic time window constraints by means of which arrival and departure times can be related pairwise on a clock, rather than on a linear time axis. Constructing a timetable, then, means solving a set of such constraints. This problem is known to be hard, i.e. it is NP-complete. We describe a new algorithm to solve the problem based on constraint generation and work out a real-life example. It appears that, for problem instances of modest, yet non-trivial, size, the algorithm performs very well, which opens a way to thorough performance analysis of railway systems by studying a large number of possible future timetables.     

Train commuters’ scheduling preferences: Evidence from a large-scale peak avoidance experiment

We study the trip scheduling preferences of train commuters in a real-life setting. The underlying data have been collected during large-scale peak avoidance experiment conducted in the Netherlands, in which participants could earn monetary rewards for traveling outside peak hours. The experiment included ca. 1000 participants and lasted for multiple months. Holders of an annual train pass were invited to join the experiment, and a customized smartphone app was used to measure the travel behavior of the participants. We find that compared to the pre-measurement, the relative share of peak trips decreased by 22% during the reward period, and by 10% during the post-measurement. By combining multiple complementary data sources, we are able to specify and estimate (MNL and panel latent class) departure time choice models. These yield plausible estimates for the monetary values that participants attach to reducing travel time, schedule delays, the number of transfers, crowdedness, and unreliability.