A tailored branch-and-price approach for a joint tramp ship routing and bunkering problem

This paper deals with a practical tramp ship routing problem while taking into account different bunker prices at different ports, which is called the joint tramp ship routing and bunkering (JSRB) problem. Given a set of cargoes to be transported and a set of ports with different bunker prices, the proposed problem determines how to route ships to carry the cargoes and the amount of bunker to purchase at each port, in order to maximize the total profit. After building an integer linear programming model for the JSRB problem, we propose a tailored branch-and-price (B&P) solution approach. The B&P approach incorporates an efficient method for obtaining the optimal bunkering policy and a novel dominance rule for detecting inefficient routing options. The B&P approach is tested with randomly generated large-scale instances derived from real-world planning problems. All of the instances can be solved efficiently. Moreover, the proposed approach for the JSRB problem outperforms the conventional sequential planning approach and can incorporate the prediction of future cargo demand to avoid making myopic decisions.     

A heuristic for the train pathing and timetabling problem

In a railroad system, train pathing is concerned with the assignment of trains to links and tracks, and train timetabling allocates time slots to trains. These important tasks were traditionally done manually, but there is an increasing move toward automated software based on mathematical models and algorithms. Most published models in the literature either focus on train timetabling only, or are too complicated to solve when facing large instances. In this paper, we present an optimization heuristic that includes both train pathing and train timetabling, and has the ability to solve real-sized instances. This heuristic allows the operation time of trains to depend on the assigned track, and also lets the minimum headway between the trains to depend on the trains’ relative status. It generates an initial solution with a simple rule, and then uses a four-step process to derive the solution iteratively. Each iteration starts by altering the order the trains travel between stations, then it assigns the services to the tracks in the stations with a binary integer program, determines the order they pass through the stations with a linear program, and uses another linear program to produce a timetable. After these four steps, the heuristic accepts or rejects the new solution according to a Threshold Accepting rule. By decomposing the original complex problem into four parts, and by attacking each part with simpler neighborhood-search processes or mathematical programs, the heuristic is able to solve realistic instances. When tested with two real-world examples, one from a 159.3 km, 29-station railroad that offers 44 daily services, and another from a 345 km, eight-station high-speed rail with 128 services, the heuristic obtained timetables that are at least as good as real schedules.     

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.     

Optimizing train stopping patterns and schedules for high-speed passenger rail corridors

High-speed railway (HSR) systems have been developing rapidly in China and various other countries throughout the past decade; as a result, the question of how to efficiently operate such large-scale systems is posing a new challenge to the railway industry. A high-quality train timetable should take full advantage of the system’s capacity to meet transportation demands. This paper presents a mathematical model for optimizing a train timetable for an HSR system. We propose an innovative methodology using a column-generation-based heuristic algorithm to simultaneously account for both passenger service demands and train scheduling. First, we transform a mathematical model into a simple linear programming problem using a Lagrangian relaxation method. Second, we search for the optimal solution by updating the restricted master problem (RMP) and the sub-problems in an iterative process using the column-generation-based algorithm. Finally, we consider the Beijing–Shanghai HSR line as a real-world application of the methodology; the results show that the optimization model and algorithm can improve the defined profit function by approximately 30% and increase the line capacity by approximately 27%. This methodology has the potential to improve the service level and capacity of HSR lines with no additional high-cost capital investment (e.g., the addition of new tracks, bridges and tunnels on the mainline and/or at stations).     

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.     

Real-time energy consumption minimization in railway networks

A new timetable must be calculated in real-time when train operations are perturbed. Although energy consumption is becoming a central issue both from the environmental and economic perspective, it is usually neglected in the timetable recalculation. In this paper, we formalize the real-time Energy Consumption Minimization Problem (rtECMP). It finds in real-time the driving regime combination for each train that minimizes energy consumption, respecting given routing and precedences between trains. In the possible driving regime combinations, train routes are split in subsections for which one of the regimes resulting from the Pontryagin’s Maximum Principle is to be chosen. We model the trade-off between minimizing energy consumption and total delay by considering as objective function their weighted sum. We propose an algorithm to solve the rtECMP, based on the solution of a mixed-integer linear programming model. We test this algorithm on the Pierrefitte-Gonesse control area, which is a critical area in France with dense mixed traffic. The results show that the problem is tractable and an optimal solution of the model tackled can often be found in real-time for most instances.     

A stochastic model for the integrated optimization on metro timetable and speed profile with uncertain train mass

The integrated timetable and speed profile optimization model has recently attracted more attention because of its good achievements on energy conservation in metro systems. However, most previous studies often ignore the spatial and temporal uncertainties of train mass, and the variabilities of tractive force, braking force and basic running resistance on energy consumption in order to simplify the model formulation and solution algorithm. In this paper, we develop an integrated metro timetable and speed profile optimization model to minimize the total tractive energy consumption, where these real-world operating conditions are explicitly considered in the model formulation and solution algorithm. Firstly, we formulate a two-phase stochastic programming model to determine the timetable and speed profile. Given the speed profile, the first phase determines the timetable by scheduling the arrival and departure times for each station, and the second phase determines the speed profile for each inter-station with the scheduled arrival and departure times. Secondly, we design a simulation-based genetic algorithm procedure incorporated with the optimal train control algorithm to find the optimal solution. Finally, we present a simple example and a real-world example based on the operation data from the Beijing Metro Yizhuang Line in Beijing, China. The results of the real-world example show that, during peak hours, off-peak hours and night hours, the total tractive energy consumptions can be reduced by: (1) 10.66%, 9.94% and 9.13% in comparison with the current timetable and speed profile; and (2) 3.35%, 3.12% and 3.04% in comparison with the deterministic model.     

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.     

A hierarchical line planning approach for a large-scale high speed rail network: The China case

Planning a set of train lines in a large-scale high speed rail (HSR) network is typically influenced by issues of longer travel distance, high transport demand, track capacity constraints, and a non-periodic timetable. In this paper, we describe an integrated hierarchical approach to determine line plans by defining the stations and trains according to two classes. Based on a bi-level programming model, heuristics are developed for two consecutive stages corresponding to each classification. The approach determines day-period based train line frequencies as well as a combination of various stopping patterns for a mix of fast trunk line services between major stations and a variety of slower body lines that offer service to intermediate stations, so as to satisfy the predicted passenger transport demand. Efficiencies of the line plans described herein concern passenger travel times, train capacity occupancy, and the number of transfers. Moreover, our heuristics allow for combining many additional conflicting demand–supply factors to design a line plan with predominantly cost-oriented and/or customer-oriented objectives. A range of scenarios are developed to generate three line plans for a real-world example of the HSR network in China using a decision support system. The performance of potential train schedules is evaluated to further examine the feasibility of the obtained line plans through graphical timetables.     

Improving recovery-to-optimality robustness through efficiency-balanced design of timetable structure

To improve the service quality of the railway system (e.g., punctuality and travel times) and to enhance the robust timetabling methods further, this paper proposes an integrated two-stage approach to consider the recovery-to-optimality robustness into the optimized timetable design without predefined structure information (defined as flexible structure) such as initial departure times, overtaking stations, train order and buffer time. The first-stage timetabling model performs an iterative adjustment of all departure and arrival times to generate an optimal timetable with balanced efficiency and recovery-to-optimality robustness. The second-stage dispatching model evaluates the recovery-to-optimality robustness by simulating how each timetable generated from the first-stage could recover under a set of restricted scenarios of disturbances using the proposed dispatching algorithm. The concept of recovery-to-optimality is examined carefully for each timetable by selecting a set of optimally refined dispatching schedules with minimum recovery cost under each scenario of disturbance. The robustness evaluation process enables an updating of the timetable by using the generated dispatching schedules. Case studies were conducted in a railway corridor as a special case of a simple railway network to verify the effectiveness of the proposed approach. The results show that the proposed approach can effectively attain a good trade-off between the timetable efficiency and obtainable robustness for practical applications.     

Development of hybrid optimization of train schedules model for N-track rail corridors

From a capacity perspective, efficient utilization of a railway corridor has two main objectives; avoidance of schedule conflicts, and finding a proper balance between capacity utilization and level of service (LOS). There are several timetable tools and commercial rail simulation packages available to assist in reaching these objectives, but few of them offer both automatic train conflict resolution and automatic timetable management features for the different types of corridor configurations. This research presents a new rescheduling model to address some of the current limitations. The multi-objective linear programming (LP) model is called “Hybrid Optimization of Train Schedules” (HOTS), and it works together with commercial rail simulation tools to improve capacity utilization or LOS metrics. The HOTS model uses both conflict resolution and timetable compression techniques and is applicable to single-, double-, and multiple-track corridors (N-track networks), using both directional and bi-directional operations. This paper presents the approach, formulation and data requirements for the HOTS model. Single and multi-track case studies test and demonstrate the model’s train conflict resolution and timetable compression capabilities, and the model’s results are validated by using RailSys simulation package. The HOTS model performs well in each tested scenario, providing comparable results (either improved or similar) to the commercial packages.     

The multi-objective railway timetable rescheduling problem

Unexpected disruptions occur for many reasons in railway networks and cause delays, cancelations, and, eventually, passenger inconvenience. This research focuses on the railway timetable rescheduling problem from a macroscopic point of view in case of large disruptions. The originality of our approach is to integrate three objectives to generate a disposition timetable: the passenger satisfaction, the operational costs and the deviation from the undisrupted timetable. We formulate the problem as an Integer Linear Program that optimizes the first objective and includes $\epsilon$-constraints for the two other ones. By solving the problem for different values of $\epsilon$, the three-dimensional Pareto frontier can be explored to understand the trade-offs among the three objectives. The model includes measures such as canceling, delaying or rerouting the trains of the undisrupted timetable, as well as scheduling emergency trains. Furthermore, passenger flows are adapted dynamically to the new timetable. Computational experiments are performed on a realistic case study based on a heavily used part of the Dutch railway network. The model is able to find optimal solutions in reasonable computational times. The results provide evidence that adopting a demand-oriented approach for the management of disruptions not only is possible, but may lead to significant improvement in passenger satisfaction, associated with a low operational cost of the disposition timetable.     

Computing 4D near-optimal trajectories for dynamic air traffic flow management with column generation and branch-and-price

Abstract

The current air traffic system faces recurrent saturation problems. Numerous studies are dedicated to this issue, including the present research on a new dynamic regulation filter holding frequent trajectory optimisations in a real-time sliding horizon loop process. We consider a trajectory optimisation problem arising in this context, where a feasible four-dimensional (4D) trajectory is to be built and assigned to each regulated flight to suppress sector overloads while minimising the cost of the chosen policy. We model this problem with a mixed integer linear programme and solve it with a branch-and-price approach. The pricing sub-problem looks for feasible trajectories in a dynamic three-dimensional (3D) network and is solved with a specific algorithm based on shortest path labelling algorithms and on dynamic programming. Each algorithm is tested on real-world data corresponding to a complete traffic day in the European air traffic system; experimental results, including computing times measurement, validate the solution process.     

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.     

User perspectives in public transport timetable optimisation

The present paper deals with timetable optimisation from the perspective of minimising the waiting time experienced by passengers when transferring either to or from a bus. Due to its inherent complexity, this bi-level minimisation problem is extremely difficult to solve mathematically, since timetable optimisation is a non-linear non-convex mixed integer problem, with passenger flows defined by the route choice model, whereas the route choice model is a non-linear non-continuous mapping of the timetable. Therefore, a heuristic solution approach is developed in this paper, based on the idea of varying and optimising the offset of the bus lines. Varying the offset for a bus line impacts the waiting time passengers experience at any transfer stop on the bus line.In the bi-level timetable optimisation problem, the lower level is a transit assignment calculation yielding passengers’ route choice. This is used as weight when minimising waiting time by applying a Tabu Search algorithm to adapt the offset values for bus lines. The updated timetable then serves as input in the following transit assignment calculation. The process continues until convergence.The heuristic solution approach was applied on the large-scale public transport network in Denmark. The timetable optimisation approach yielded a yearly reduction in weighted waiting time equivalent to approximately 45 million Danish kroner (9 million USD).     

A delay propagation algorithm for large-scale railway traffic networks

In scheduled railway traffic networks a single delayed train may cause a domino effect of secondary delays over the entire network, which is a main concern to planners and dispatchers. This paper presents a model and an algorithm to compute the propagation of initial delays over a periodic railway timetable. The railway system is modelled as a linear system in max-plus algebra including zero-order dynamics corresponding to delay propagation within a timetable period. A timed event graph representation is exploited in an effective graph algorithm that computes the propagation of train delays using a bucket implementation to store the propagated delays. The behaviour of the delay propagation and the convergence of the algorithm is analysed depending on timetable properties such as realisability and stability. Different types of delays and delay behaviour are discussed, including primary and secondary delays, structural delays, periodic delay regimes, and delay explosion. A decomposition method based on linearity is introduced to deal with structural and initial delays separately. The algorithm can be applied to large-scale scheduled railway traffic networks in real-time applications such as interactive timetable stability analysis and decision support systems to assist train dispatchers.     

Passenger oriented railway disruption management by adapting timetables and rolling stock schedules

In passenger railway operations, unforeseen events require railway operators to adjust their timetable and their resource schedules. The passengers will also adapt their routes to their destinations. When determining the new timetable and rolling stock schedule, the railway operator has to take passenger behavior into account. The operator should increase the capacity of trains for which the operator expects more demand than on a regular day. Furthermore, the operator could increase the frequency of the trains that serve stations with an additional demand.This paper describes a real-time disruption management approach which integrates the rescheduling of the rolling stock and the timetable by taking the changed passenger demand into account. The timetable decisions are limited to additional stops of trains at stations at which they normally would not call. Several variants of the approach are suggested, with the difference in how to determine which additional stops should be executed.Real-time rescheduling requires fast solutions. Therefore a heuristic approach is used. We demonstrate the performance of the several variants of our algorithm on realistic instances of Netherlands Railways, the major railway operator in the Netherlands.     

Energy-efficient metro train rescheduling with uncertain time-variant passenger demands: An approximate dynamic programming approach

In a heavily congested metro line, unexpected disturbances often occur to cause the delay of the traveling passengers, infeasibility of the current timetable and reduction of the operational efficiency. Due to the uncertain and dynamic characteristics of passenger demands, the commonly used method to recover from disturbances in practice is to change the timetable and rolling stock manually based on the experiences and professional judgements. In this paper, we develop a stochastic programming model for metro train rescheduling problem in order to jointly reduce the time delay of affected passengers, their total traveling time and operational costs of trains. To capture the complexity of passenger traveling characteristics, the arriving ratio of passengers at each station is modeled as a non-homogeneous poisson distribution, in which the intensity function is treated as time-varying origin-to-destination passenger demand matrices. By considering the number of on-board passengers, the total energy usage is modeled as the difference between the tractive energy consumption and the regenerative energy. Then, we design an approximate dynamic programming based algorithm to solve the proposed model, which can obtain a high-quality solution in a short time. Finally, numerical examples with real-world data sets are implemented to verify the effectiveness and robustness of the proposed approaches.     

An integrated metro operation optimization to minimize energy consumption

Energy efficient techniques are receiving increasing attention because of rising energy prices and environmental concerns. Railways, along with other transport modes, are facing increasing pressure to provide more intelligent and efficient power management strategies.This paper presents an integrated optimization method for metro operation to minimize whole day substation energy consumption by calculating the most appropriate train trajectory (driving speed profile) and timetable configuration. A train trajectory optimization algorithm and timetable optimization algorithm are developed specifically for the study. The train operation performance is affected by a number of different systems that are closely interlinked. Therefore, an integrated optimization process is introduced to obtain the optimal results accurately and efficiently.The results show that, by using the optimal train trajectory and timetable, the substation energy consumption and load can be significantly reduced, thereby improving the system performance and stability. This also has the effect of reducing substation investment costs for new metros.     

A bi-level model for single-line rail timetable design with consideration of demand and capacity

This paper proposes a bi-level model to solve the timetable design problem for an urban rail line. The upper level model aims at determining the headways between trains to minimize total passenger cost, which includes not only the usual perceived travel time cost, but also penalties during travel. With the headways given by the upper level model, passengers’ arrival times at their origin stops are determined by the lower level model, in which the cost-minimizing behavior of each passenger is taken into account. To make the model more realistic, explicit capacity constraints of individual trains are considered. With these constraints, passengers cannot board a full train, but wait in queues for the next coming train. A two-stage genetic algorithm incorporating the method of successive averages is introduced to solve the bi-level model. Two hypothetical examples and a real world case are employed to evaluate the effectiveness of the proposed bi-level model and algorithm. Results show that the bi-level model performs well in reducing total passenger cost, especially in reducing waiting time cost and penalties. And the section loading-rates of trains in the optimized timetable are more balanced than the even-headway timetable. The sensitivity analyses show that passenger’s desired arrival time interval at destination and crowding penalty factor have a high influence on the optimal solution. And with the dispersing of passengers' desired arrival time intervals or the increase of crowding penalty factor, the section loading-rates of trains become more balanced.