Train Timetabling Problem Research Articles

First train timetable synchronization with interval trains in subway networks

As the subway system is beginning to operate interval trains in the early morning, it is necessary to discuss the first train timetable synchronization with interval trains. In this paper, with the consideration of both ordinary trains and interval trains, we study the first train timetable problem with minimum total transfer waiting time. The first train timetable model is formulated as a mixed integer programming problem and a genetic simulated annealing algorithm is developed to deal with the case study of Beijing subway network. The proposed model and the algorithm are effective in improving the performance of the first train timetable synchronization. After optimization, the total transfer waiting time and the number of long waiting time transfer directions for Beijing subway network have decreased significantly. The results indicate that operating interval trains is an effective way to solve the problem of long waiting time in early morning.

Train timetabling with the general learning environment and multi-agent deep reinforcement learning

This paper proposes a multi-agent deep reinforcement learning approach for the train timetabling problem of different railway systems. A general train timetabling learning environment is constructed to model the problem as a Markov decision process, in which the objectives and complex constraints of the problem can be distributed naturally and elegantly. Through subtle changes, the environment can be flexibly switched between the widely used double-track railway system and the more complex single-track railway system. To address the curse of dimensionality, a multi-agent actor–critic algorithm framework is proposed to decompose the large-size combinatorial decision space into multiple independent ones, which are parameterized by deep neural networks. The proposed approach was tested using a real-world instance and several test instances. Experimental results show that cooperative policies of the single-track train timetabling problem can be obtained by the proposed method within a reasonable computing time that outperforms several prevailing methods in terms of the optimality of solutions, and the proposed method can be easily generalized to the double-track train timetabling problem by changing the environment slightly.

Integrated line planning and train timetabling through price-based cross-resolution feedback mechanism

Railway line planning and train timetabling are two key planning steps that determine the operating cost and passenger service quality of a railway operator under the infrastructure capacity limitations. Traditionally, the line planning and train timetabling problems are solved sequentially at the strategic and tactical level, respectively. In this study, by introducing two types of binary decision variables, we first propose a unified integer linear programming (ILP) model for the integrated optimization of line planning and train timetabling. The line planning problem is modeled using ILP to satisfy passenger demand, whereas the cyclic train timetabling problem is formulated as a multi-commodity network flow model with a side track capacity constraint. The two types of binary decision variables are coupled by a cross-resolution consistency constraint, which ensures the conformity of the line planning and train timetabling decisions. Furthermore, a dual decomposition mechanism based on the Alternating Direction Method of Multipliers (ADMM) is developed to dualize the cross-resolution consistency and track capacity constraints, such that the original ILP model is decomposed into a line planning sub-problem and a set of train-specific sub-problems. After the linearization of the quadratic penalty terms in the ADMM, each sub-problem contains the Lagrangian relaxation price information based on the cross-resolution consistency constraint. Moreover, the primal and dual solutions are obtained by iteratively and efficiently solving the line planning sub-problem using a commercial solver, and each train-specific sub-problem through a tailored forward dynamic programming algorithm. Furthermore, a real-life case study is conducted based on the Beijing–Shanghai high-speed railway corridor to verify the efficiency and effectiveness of the proposed model and algorithm. The results of the numerical experiments demonstrate that the ADMM can achieve significantly smaller optimality gaps than Lagrangian relaxation, and the integrated optimization approach can improve the objective value by 5.78% on average compared with the sequential optimization approach.

Train timetabling for a double-track urban rail transit line under dynamic passenger demand

Train timetable is of critical importance for an urban rail transit line, chiefly because it is the primary factor determining passenger perception of service quality. As it not only delivers efficient transit service to users but also significantly contributes to operator profitability, the train timetabling problem has been a widely studied subject in academic circles. Still, the existing models in the literature, for the most part, fail to sufficiently take into account train capacity, fleet size, and vehicle circulation. As a contribution to bridging this research gap, this study mainly focuses on the train timetabling problem in a congested urban rail corridor to adapt to dynamic behavior of passenger demand subject to operational and resource constraints. A nonlinear programming model is formulated to devise timetables with a view to minimizing the average waiting time per passenger. In the model, the congestion is represented by some passengers who may not be able to take the first incoming train due to limited train capacity. To evaluate the effectiveness of the proposed approach, a case study is performed on a metro line in Istanbul. According to study results, the optimized demand-oriented train timetable proved to be more advantageous when compared to its periodical counterpart prepared by traffic planners. Finally, a sensitivity analysis is conducted to attain the best trade-off between passenger satisfaction and operation cost.

A Scheduling Approach for the Combination Scheme and Train Timetable of a Heavy-Haul Railway

Heavy-haul railway transport is a critical mode of regional bulk cargo transport. It dramatically improves the freight transport capacity of railway lines by combining several unit trains into one combined train. In order to improve the efficiency of the heavy-haul transport system and reduce the transportation cost, a critical problem involves arranging the combination scheme in the combination station (CBS) and scheduling the train timetable along the trains’ journey. With this consideration, this paper establishes two integer programming models in stages involving the train service plan problem (TSPP) model and train timetabling problem (TTP) model. The TSPP model aims to obtain a train service plan according to the freight demands by minimizing the operation cost. Based on the train service plan, the TTP model is to simultaneously schedule the combination scheme and train timetable, considering the utilization optimal for the CBS. Then, an effective hybrid genetic algorithm (HGA) is designed to solve the model and obtain the combination scheme and train timetable. Finally, some experiments are implemented to illustrate the feasibility of the proposed approaches and demonstrate the effectiveness of the HGA.

A modelling approach to the train timetabling problem using adaptive headways for dynamic passenger demand

ABSTRACT Rail systems are preferred to find sustainable solutions to traffic problems in cities with high population density. To increase the use of public transport without reducing service quality, it is essential to construct a demand-based timetable. In this study, a mathematical model that meets the passenger demand in cases of over and under-saturation, considers structural-operational constraints, aims minimum energy consumption and machinist costs is presented. Passenger demands are forecasted using four popular techniques. In both over and under-saturated situations, it is possible to meet passenger demand by using adaptive headways and determining train specifications as coupled or decoupled. Moreover, a comfort coefficient is incorporated in the model to adjust the service quality of the trip. To demonstrate the performance of the proposed MILP model, Kayseri Ulasim A.S. timetabling problem is solved. The effect of different passenger demand levels, over and under-saturated cases, on headways and train types is analysed.

An integrated energy-efficient and transfer-accessible model for the last train timetabling problem

Subway is considered to be one of the most energy-intensive transportation modes for its high operating frequency. However, energy-efficient operations for the subway system are of great importance yet have not been paid much attention to. In this study, we first develop an integrated energy-efficient and transfer-accessible model to minimize the tractive energy consumption and maximize the number of last train connections, which could contribute to the development of high energy-efficient strategies and the construction of wide-accessibility timetables for the subway system. Four tractive modes, which are accelerating–braking (A–B) mode, accelerating–coasting–braking (A–Co–B) mode, accelerating–cruising–braking (A–Cr–B)​ mode, and the mixed mode, are proposed to facilitate the last train operations. A real-life case study of the Beijing subway network is solved by a tailored genetic algorithm. Results show that the A–B mode is the most energy-intensive with an energy consumption of 466.9 kWh, while the A–Co–B mode becomes the most energy-efficient (402.5 kWh). The A–Cr–B and the mixed modes consume 442.2 kWh and 412.8 kWh, respectively. The findings are of significant value for subway companies in addition to their academic merits.

Constraint Answer Set Programming: Integrational and Translational (or SMT-based) Approaches

AbstractConstraint answer set programming or CASP, for short, is a hybrid approach in automated reasoning putting together the advances of distinct research areas such as answer set programming, constraint processing, and satisfiability modulo theories. CASP demonstrates promising results, including the development of a multitude of solvers: acsolver, clingcon, ezcsp, idp, inca, dingo, mingo, aspmt2smt, clingo[l,dl], and ezsmt. It opens new horizons for declarative programming applications such as solving complex train scheduling problems. Systems designed to find solutions to constraint answer set programs can be grouped according to their construction into, what we call, integrational or translational approaches. The focus of this paper is an overview of the key ingredients of the design of constraint answer set solvers drawing distinctions and parallels between integrational and translational approaches. The paper also provides a glimpse at the kind of programs its users develop by utilizing a CASP encoding of Traveling Salesman problem for illustration. In addition, we place the CASP technology on the map among its automated reasoning peers as well as discuss future possibilities for the development of CASP.

Train timetabling in rail transit network under uncertain and dynamic demand using Advanced and Adaptive NSGA-II

It is critical to design an adaptable and stable train timetable for long-term use in rail transit network that not only meets the dynamicity of passenger demand in different hours within one day, but also meets the uncertainty of passenger demand in different days. In this study, a scenario-based train timetabling framework is constructed to classify the possibilities of passenger demand in multiple days into a set of scenarios based on profile and volume of passenger demand. On this basis, multi-scenario demand input method (MM) is introduced to deal with the uncertainty of passenger demand, which is different from one-scenario method (OM) and average-scenario method (AM). A MM-based mixed-integer linear programming model is formulated for the bi-objective train timetabling problem under uncertain and dynamic demand at acyclic network level, in which multi-scenario small-granularity passenger demand follows actual distribution processed from historical data. The two objectives are to minimize train service cost and penalized passenger waiting time from perspectives of enterprises and passengers. Advanced and Adaptive NSGA II (AANSGA-II) is proposed to cope with the high-complexity bi-objective problem, which applies advanced population sorting based on neighborhood distance, adaptive genetic operation based on scoring mechanism and improved population initialization based on boundary individuals. The model and algorithm are testified by a small-scale numerical experiment on a virtual line and a large-scale real-world instance in Shenyang Metro network. As a result, MM-based train timetables are generally better than AM-based and OM-based train timetables in reducing generalized cost and raising robustness. Besides, AANSGA-II is more applicable than NSGA-II and CPLEX in shortening computation time at the same time of improving computation result.

An Integer Linear Programming model for integrated train stop planning and timetabling with time-dependent passenger demand

We consider the problem of jointly optimizing train stop plan and timetable for a set of trains on a railway corridor, with focus on time-dependent passenger demand: these are very relevant elements in railway operations, and closely related to the service quality provided to the passengers. The train stop planning problem requires to determine the stopping pattern for each train, while the train timetabling problem calls for finding departure and arrival times of each train at each visited station. The solution of both problems determines the train services available to the passengers, with the corresponding stops, the departure and arrival times, and the total travel time. To define train services that are convenient for the passengers, it is important to satisfy the passenger demand: in particular, we take into consideration the desired departure time intervals of passengers from their origin station towards their destination station. Our goal is to determine a solution to the integrated problem that includes these three aspects (stop plan, timetable, time-dependent passenger demand). The integrated problem is solved at the tactical level, and includes constraints imposing that the demand is satisfied in the desired time intervals, and the train capacity is respected. The solution of the problem defines the train services, by determining the passenger flow on trains and the time intervals in which passengers have to be served.We formulate an Integer Linear Programming (ILP) model for this integrated problem, and solve it with the commercial optimization solver CPLEX on a real-world instance of the Wuhan–Guangzhou high-speed railway line in China, by considering different time interval lengths in which the demand must be satisfied. In addition, we compare the obtained results with those determined by solving a model that neglects the time-dependency of the passenger demand. The computational results show that the train stop plan and timetable are deeply influenced by the desired departure time intervals of the passengers, and that neglecting time-dependency results in more than 30% of the passengers departing in a different time interval than the desired one.

Economical Speed for Optimizing the Travel Time and Energy Consumption in Train Scheduling using a Fuzzy Multi-Objective Model

Speed is one of the most influential variables in both energy consumption and train scheduling problems. Increasing speed guarantees punctuality, thereby improving railroad capacity and railway stakeholders’ satisfaction and revenues. However, a rise in speed leads to more energy consumption, costs, and thus, more pollutant emissions. Therefore, determining an economic speed, which requires a trade-off between the user’s expectations and the capabilities of the railway system in providing tractive forces to overcome the running resistance due to rail route and moving conditions, is a critical challenge in railway studies. This paper proposes a new fuzzy multi-objective model, which, by integrating micro and macro levels and determining the economical speed for trains in block sections, can optimize train travel time and energy consumption. Implementing the proposed model in a real case with different scenarios for train scheduling reveals that this model can enhance the total travel time by 19% without changing the energy consumption ratio. The proposed model has little need for input from experts’ opinions to determine the rates and parameters.

Energy-efficient timetabling and rolling stock circulation planning based on automatic train operation levels for metro lines

We investigate the energy-efficient train timetabling and rolling stock circulation planning problem for metro lines based on flexible selections of train operation levels, which involve running levels (various running times and speed profiles) and dwell levels (various dwell times at stations). The travel demand of passengers is involved in the energy-efficient train timetabling problem via service patterns, i.e., different operation headways are used in peak and off-peak hours to satisfy the passenger demand. A mixed integer nonlinear programming (MINLP) formulation is presented to optimize train timetables and circulation plans of rolling stocks simultaneously with the aim of minimizing the headway deviations relating to service patterns, the headway variations among neighboring train services, the needed depot movements for rolling stocks, and the overall energy consumption. To increase the regularity (i.e., the consistency of operation level selections of train services) of train timetables and to improve the computational efficiency, two extensions of the energy-efficient timetabling formulation are proposed, i.e. the energy-efficient train timetabling models with pre-fixed selections and optimal identical (or consistent) selections of the operation levels. The proposed MINLP formulations are transformed into mixed integer linear programming formulations, which could be effectively solved, e.g., by Cplex and Gurobi solvers. To investigate the effectiveness of the proposed models, we perform computational experiments based on the real-world data of the Beijing Yizhuang line. The numerical results illustrate that the energy consumption of the daily train timetable obtained by the proposed energy-efficient train timetabling model with flexible selection of operation levels is reduced by 9.90% when compared with that of the train timetables without operation level selections. Moreover, the energy savings of the energy-efficient train timetables with optimal identical selection and pre-fixed selection of operation levels are 7.67% and 5.75%, respectively. However, the computation time of the energy-efficient train timetabling models with operation level selections is significantly longer than that without operation level selections.

Scheduling local and express trains in suburban rail transit lines: Mixed–integer nonlinear programming and adaptive genetic algorithm

We investigate the train timetabling problem in suburban rail transit lines by considering (1) the traditional stopping mode (TSM), in which all trains stop at each station, and (2) the express/local stopping mode (ELM), in which express trains can skip certain low–demand stations. We first propose two mixed–integer linear programming models for the train timetabling problem under the TSM with and without capacity constraints. Next, we develop two mixed–integer nonlinear programming models under the ELM with and without “overtaking”; thus, a total of four optimization models are proposed. The objective is to minimize the passenger travel time (PTT). Owing to the NP–hardness of the studied problem, we propose an adaptive genetic algorithm (A–GA) that can efficiently solve the four proposed models. The A–GA is customized to solve the train timetabling problem with train capacity, overtaking, and other operational constraints, reducing the PTT. To evaluate the performance of the proposed algorithm, we conduct numerical experiments on 60 randomly generated realistic instances and a real–world case study based on Shanghai Metro Line 16. The computational results for the realistic instances indicate that our A–GA can obtain near–optimal solutions with significantly less computation time than an established commercial solver. The computational results from the real-world case study quantify the benefits of considering the combination of the ELM and overtaking strategies in train timetabling. Furthermore, we perform a sensitivity analysis on key parameters of our mathematical formulations. The results provide insights to railway managers on how to set key parameters when applying the proposed formulations and solution methodology in practice.

Train timetabling with stop-skipping, passenger flow, and platform choice considerations

In conventional railway planning processes, stop-skipping decisions are often made at the line planning stage, which is executed prior to train timetabling and platform assignment. However, stop-skipping can shorten passenger journey time and also save on train operating costs. Hence, integrating train timetabling, stop-skipping, and platform choice decisions can help generate train timetables with improved passenger convenience and higher train operating efficiency. Integrating these decisions is a challenging task, as these decisions affect passenger train transfer behavior, which in turn affects the entire passenger flow. This study is a first attempt at integrating these decisions while simultaneously taking into account the passenger flow. We consider a train timetabling problem on a single, one-way track with stop-skipping, platform choice, and passenger flow considerations, and we formulate it as a constrained minimum-cost multi-commodity network flow problem on a time–space network. We analyze the problem’s complexity and develop a Lagrangian relaxation heuristic to solve the problem. We conduct a computational study with randomly generated data that captures the characteristics of the Beijing–Shanghai high-speed railway line. The computational results report the effectiveness of our Lagrangian relaxation heuristic and how the railway’s service capacity and passenger traffic intensity affect the solution.

First train timetabling and bus service bridging in intermodal bus-and-train transit networks

Subway system is the main mode of transportation for city dwellers and is a quite significant backbone to a city's operations. One of the challenges of subway network operation is the scheduling of the first trains each morning and its impact on transfers. To deal with this challenge, some cities (e.g. Beijing) use bus ‘bridging’ services, temporarily substituting segments of the subway network. The present paper optimally identifies when to start each train and bus bridging service in an intermodal transit network. Starting from a mixed integer nonlinear programming model for the first train timetabling problem, we linearize and reformulate the model using the auxiliary binary variables. Following that, the bus bridging model is developed to cooperate with the first train operation for reducing long transfer waiting times. After realizing the low computational efficiency of solving the integrated model, a tailored algorithm is designed to optimally solve the first train timetabling and bus service bridging problems. The exact models and algorithms are applied to the Beijing subway network to test their effectiveness and computational efficiency. Numerical results show that our approaches decrease the total passenger waiting time by 53.4% by a combined effect of adjusting the first train departure times and operating 27 bridging buses on 7 routes.

A column-generation-based matheuristic for periodic and symmetric train timetabling with integrated passenger routing

In this study, the periodic train timetabling problem is formulated using a time-space graph formulation that exploits the properties of a symmetric timetable. Three solution methods are proposed and compared where solutions are built by what we define as a dive-and-cut-and-price procedure. An LP relaxed version of the problem with a subset of constraints is solved using column generation where each column corresponds to the train paths of a line. Violated constraints are added by separation and a heuristic process is applied to help to find integer solutions. The passenger travel time is computed based on a solution timetable and Benders’ optimality cuts are generated allowing the method to integrate the routing of the passengers. We propose two large neighborhood search methods where the solution is iteratively destroyed and repaired into a new one and one random iterative method. The problem is tested on the morning rush hour period of the Regional and InterCity train network of Zealand, Denmark. The solution approaches show robust performance in a variety of scenarios, being able to find good quality solutions in terms of travel time and path length relatively fast. The inclusion of the proposed Benders’ cuts provide stronger relaxations to the problem. In addition, the graph formulation covers different real-life constraints and has the potential to easily be extended to accommodate more constraints.

A priority-based ADMM approach for flexible train scheduling problems

In this study, we focus on how to optimize multi-type train scheduling problems in a practically available and computationally efficient manner. When solving the complicated train timetabling problem with overtaking operations for a congested high-speed rail corridor, a simplified approach is to predesignate a thin time interval from its origin station for each train. This setting is difficult and laborious in practice. What is more serious is that an imperfect setting may cause the expected solutions to be unreachable in many instances. To fundamentally get out of this thorny trouble, this study proposes a flexible scheduling framework that extends narrow time limitations to one-hour periods, which allows the trains to depart from any timestamp within the relaxed period. Further, a group of incompatibility constraints with a more compact form is formulated to mirror the coupling relation between train headway and regularity requirements. However, the relaxed time window increases the difficulty of obtaining the optimal or near-optimal solution under the standard Lagrangian-based decomposition framework. Therefore, an improved alternating direction method of multipliers (ADMM) approach is introduced for effectively eliminating the computational barriers that are due to its increased complexity. Specifically, a priority-based search sequence is employed by applying the associated dual costs during the solution of train-level path subproblems individually. Finally, a set of numerical experiments is conducted to demonstrate the effectiveness and availability of the proposed approach.

Train timetabling with dynamic and random passenger demand: A stochastic optimization method

Considering the dynamics and randomness of passenger demand, this paper investigates a train timetabling problem in the stochastic environment for an urban rail transit system. With the scenario-based representation of passenger distribution, an integer nonlinear programming (INLP) model is first formulated to simultaneously optimize the total number of train services, headway settings and speed profile selection decision during the planning time horizon, in which the expected total service cost is treated as the objective function. Through an analysis of the features of the nonlinear constraints, a reformulation method is proposed to develop an equivalent integer linear programming (ILP) model that can be easily solved by commercial software. Moreover, a variable neighborhood search algorithm is developed to find the approximate optimal solutions for large-scale problems within the tolerable computing time. Finally, two sets of numerical experiments, with the operation environments of a simple urban rail transit line and Fuzhou Metro Line 1, are implemented to verify the solution quality and effectiveness of the proposed methods.

Modeling and solving the last-shift period train scheduling problem in subway networks

Train arrivals and departures should be scheduled over a certain period when talking about train timetabling problems. For the midnight train operations, passengers significantly concern about the network transfer issue. Currently, some existing studies address the last train timetabling problem by only optimizing the timetable for the last train on a single subway line whereas this study takes into consideration the complete last-shift period. We first put forward a last-shift train scheduling model aiming to minimize the transfer waiting time and maximize the network connectivity. Two genetic-based algorithms, an integer-coded genetic algorithm (ICGA) and a binary-coded genetic algorithm (BCGA) are developed. The relevance and applicability of the algorithms have been demonstrated by several testing networks and real-world implementation. The ICGA and the branch-and-bound approaches show high efficiency in obtaining the optimal solutions for a small network, while the BCGA approach that bases on an integer-programming model shows low efficiency in addressing problems of sparse solution spaces. However, the branch-and-bound approach has limited ability in solving medium-sized networks. On the contrary, the ICGA generates satisfactory results in solution quality and computational efficiency when applied to large-sized networks.

Efficient dispatching system of railway vehicles based on internet of things technology

Abstract With the rapid development of national economy and logistics industry, more and more railway transportation is used. Therefore, the design and development of the railway vehicle dispatching information system and the optimization of the utilization rate of vehicle logistics space are of great practical significance for the railway vehicle enterprises to improve their economic benefits. Based on the Internet of things technology, this paper designs a logistics vehicle scheduling system which includes the functions of vehicle enterprise information management, scheduling and transportation management. The system includes basic information management, goods transportation management, acceptance review, scheduling planning, transportation tracking, report management and other functional modules. Users can realize department management, employee management, goods collection and transportation, and efficient vehicle scheduling and arrangement in railway vehicle enterprises through the system. Through collecting the train information and establishing the database, the train scheduling problem is solved on the database platform. The experimental results show that the optimal solution is 350.36 and the algorithm is stable. Finally, according to the simulation results, the effectiveness of the railway vehicle scheduling system based on the cultural relic networking technology is verified.