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.     

Distributed model predictive control for railway traffic management

Every day small delays occur in almost all railway networks. Such small delays are often called “disturbances” in literature. In order to deal with disturbances dispatchers reschedule and reroute trains, or break connections. We call this the railway management problem. In this paper we describe how the railway management problem can be solved using centralized model predictive control (MPC) and we propose several distributed model predictive control (DMPC) methods to solve the railway management problem for entire (national) railway networks. Furthermore, we propose an optimization method to determine a good partitioning of the network in an arbitrary number of sub-networks that is used for the DMPC methods. The DMPC methods are extensively tested in a case study using a model of the Dutch railway network and the trains of the Nederlandse Spoorwegen. From the case study it is clear that the DMPC methods can solve the railway traffic management problem, with the same reduction in delays, much faster than the centralized MPC method.     

Optimizing the demand captured by a railway system with a regular timetable

The railway systems in various European countries adopt regular timetables, in which the trains arrive and depart at constant intervals. In fact, their simple structure provides several advantages both to the passengers and to the management of the service. The design of such timetables has recently received a certain attention in the literature, but the standard model aims to optimize the service for a fixed demand. We relax this unrealistic assumption, taking into account the reciprocal influence between the quality of the timetable and the amount of transport demand captured by the railway. This results into a mixed-integer non linear model with a non-convex continuous relaxation. We solve it by a branch-and-bound algorithm based on a piecewise-linear overestimate of the objective function and a heuristic algorithm which iteratively applies the standard fixed-demand model and a demand-estimation model, feeding each one with data based on the solution obtained from the other one at the previous iteration. The computational results presented concern both random instances and a real-world regional network located in Northwestern Italy.     

Optimizing the freight train connection service network of a large-scale rail system

This paper presents a formulation and solution for the train connection services (TCSs) problem in a large-scale rail network in order to determine the optimal freight train services, the frequency of services, and the distribution of classification workload among yards. TCS problem is modeled as a bi-level programming problem. The upper-level is intended to find an optimal train connection service, and the lower-level is used for assigning each shipment to a sequence of train services and determining the frequency of services.Our model solves the TCS problem of the China railway system, which is one of the largest railway systems in the world. The system consists of 5544 stations, and over 520,000 shipments using this system for a year period. A subnetwork is defined with 127 yards having some minimum level of reclassification resources and 14,440 demands obtained by aggregating 520,000 shipments to the subnetwork. We apply a simulated annealing algorithm to the data for optimal computation after pre-processing and get an excellent result. Comparing our optimal solution with the existing plan result, there are improvements of about 20.8% in the total cost.     

A bi‐level programming approach — Optimal transit fare under line capacity constraints

The fare of a transit line is one of the important decision variables for transit network design. It has been advocated as an efficient means of coordinating the transit passenger flows and of alleviating congestion in the transit network. This paper shows how transit fare can be optimized so as to balance the passenger flow on the transit network and to reduce the overload delays of passengers at transit stops. A bi‐level programming method is developed to optimize the transit fare under line capacity constraints. The upper‐level problem seeks to minimize the total network travel time, while the lower‐level problem is a stochastic user equilibrium transit assignment model with line capacity constraints. A heuristic solution algorithm based on sensitivity analysis is proposed. Numerical example is used to illustrate the application of the proposed model and solution algorithm.     

Solving train formation problem using simulated annealing algorithm in a simplex framework

The train formation plan (TFP) determines the train services and their frequencies and assigns the demands. The TFP models are often formulated as a capacitated service network design problem, and the optimal solution is normally difficult to find. In this paper, a hybrid algorithm of the Simplex method and simulated annealing is proposed for the TFP problem. The basic idea of the proposed algorithm is to use a simulated annealing algorithm to explore the solution space, where the revised Simplex method evaluates, selects, and implements the moves. In the proposed algorithm, the neighborhood structure is based on the pivoting rules of the Simplex method that provides an efficient method to reach the neighbors of the current solution. A state‐of‐the‐art method is applied for parameters tuning by using the design of experiments approach. To evaluate the proposed model and the solution method, 25 test problems have been simulated and solved. The results show the efficiency and the effectiveness of the proposed approach. The proposed approach is implemented to develop the TFP in the Iranian railway as a case study. It is possible to save significant time and cost through solving the TFP problem by using the proposed algorithm and developing the efficient TFP plan in the railway networks. Copyright © 2012 John Wiley & Sons, Ltd.     

Designing personal rapid transit (PRT) networks

This paper presents a heuristic method for designing a PRT network. Because the PRT system operating characteristics and performance measures differ widely from those of conventional transit technologies, an algorithm for the PRT network design problem (NDP) is derived by using concepts from some current NDP algorithms. We minimize the sum of passenger travel time cost, construction cost, vehicle cost and operational costs, subject to an available budget of guideway, a maximum number of vehicles and given link capacities. Starting with a well-connected initial network, the algorithm eliminates and adds links iteratively as it searches for a near-optimal solution. If this solution satisfies the budget constraint, it is considered to be acceptable. Otherwise, additional links are deleted until a feasible near-optimal solution is obtained. The link elimination phase of the algorithm only considers half of the links at a time which greatly decreases computing time. None of the links in an acceptable solution will be overloaded.     

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).     

A bilevel flow model for hazmat transportation network design

In this work we consider the following hazmat transportation network design problem. A given set of hazmat shipments has to be shipped over a road transportation network in order to transport a given amount of hazardous materials from specific origin points to specific destination points, and we assume there are regional and local government authorities that want to regulate the hazmat transportations by imposing restrictions on the amount of hazmat traffic over the network links. In particular, the regional authority aims to minimize the total transport risk induced over the entire region in which the transportation network is embedded, while local authorities want the risk over their local jurisdictions to be the lowest possible, forcing the regional authority to assure also risk equity. We provide a linear bilevel programming formulation for this hazmat transportation network design problem that takes into account both total risk minimization and risk equity. We transform the bilevel model into a single-level mixed integer linear program by replacing the second level (follower) problem by its KKT conditions and by linearizing the complementary constraints, and then we solve the MIP problem with a commercial optimization solver. The optimal solution may not be stable, and we provide an approach for testing its stability and for evaluating the range of its solution values when it is not stable. Moreover, since the bilevel model is difficult to be solved optimally and its optimal solution may not be stable, we provide a heuristic algorithm for the bilevel model able to always find a stable solution. The proposed bilevel model and heuristic algorithm are experimented on real scenarios of an Italian regional network.     

Integrating robust timetabling in line plan optimization for railway systems

We propose a heuristic algorithm to build a railway line plan from scratch that minimizes passenger travel time and operator cost and for which a feasible and robust timetable exists. A line planning module and a timetabling module work iteratively and interactively. The line planning module creates an initial line plan. The timetabling module evaluates the line plan and identifies a critical line based on minimum buffer times between train pairs. The line planning module proposes a new line plan in which the time length of the critical line is modified in order to provide more flexibility in the schedule. This flexibility is used during timetabling to improve the robustness of the railway system. The algorithm is validated on the DSB S-tog network of Copenhagen, which is a high frequency railway system, where overtakings are not allowed. This network has a rather simple structure, but is constrained by limited shunt capacity. While the operator and passenger cost remain close to those of the initially and (for these costs) optimally built line plan, the timetable corresponding to the finally developed robust line plan significantly improves the minimum buffer time, and thus the robustness, in eight out of ten studied cases.     

Transit network design for small-medium size cities

This paper proposes a novel heuristic to solve the network design problem for public transport in small-medium size cities. Such cities can be defined as those with a diameter of a few kilometers with up to a few hundred thousand residents. These urban centers present a specific spatial configuration affecting the land use and mobility system. Transportation demand is widespread in origin and concentrated in a small number of attraction points close to each other. This particular structure of demand (‘many-to-few’) suggests the need for specific methodologies for the design of a transit system at a network level. In this paper, such design methodologies are defined in terms of models and solution procedures and tested on a selected case study. The solution methods show promising results. The key variables of the model are the routes and their frequencies. The constraints of the problem affect the overall demand to be served, the quality of the proposed service (transfer, load factors) and the definition of routes.     

Traffic signal timing problems with environmental and equity considerations

Traffic signal timings in a road network can not only affect total user travel time and total amount of traffic emissions in the network but also create an inequity problem in terms of the change in travel costs of users traveling between different locations. This paper proposes a multi‐objective bi‐level programming model for design of sustainable and equitable traffic signal timings for a congested signal‐controlled road network. The upper level of the proposed model is a multi‐objective programming problem with an equity constraint that maximizes the reserve capacity of the network and minimizes the total amount of traffic emissions. The lower level is a deterministic network user equilibrium problem that considers the vehicle delays at signalized intersections of the network. To solve the proposed model, an approach for normalizing incommensurable objective functions is presented, and a heuristic solution algorithm that combines a penalty function approach and a simulated annealing method is developed. Two numerical examples are presented to show the effects of reserve capacity improvement and green time proportion on network flow distribution and transportation system performance and the importance of incorporating environmental and equity objectives in the traffic signal timing problems. Copyright © 2013 John Wiley & Sons, Ltd.     

Modeling and solving discrete network design problem with stochastic user equilibrium

In this paper, we address the discrete network design problem, which determines the addition of new roads to existing transportation network to optimize the transportation system performance. Road users are assumed to follow the traffic assignment principle of stochastic user equilibrium. A mixed‐integer nonlinear nonconvex problem is developed to model this discrete network design problem with stochastic user equilibrium. The original problem is relaxed into a convex mixed‐integer nonlinear program, whose solution provides a lower bound of the original problem. The relaxed problem is then embedded into two proposed global optimization solution algorithms to obtain the global optimal solution of the problem. Copyright © 2016 John Wiley & Sons, Ltd.     

A rolling horizon algorithm for auto-carrier transportation

This paper introduces a rolling horizon algorithm to plan the delivery of vehicles to automotive dealers by a heterogeneous fleet of auto-carriers. The problem consists in scheduling the deliveries over a multiple-day planning horizon during which requests for transportation arrive dynamically. In addition, the routing of the auto-carriers must take into account constraints related to the loading of the vehicles on the carriers. The objective is to minimize the sum of traveled distances, fixed costs for auto-carrier operation, service costs, and penalties for late deliveries. The problem is solved by a heuristic that first selects the vehicles to be delivered in the next few days and then optimizes the deliveries by an iterated local search procedure. A branch-and-bound search is used to check the feasibility of the loading. To handle the dynamic nature of the problem, the complete algorithm is applied repeatedly in a rolling horizon framework. Computational results on data from a major European logistics service provider show that the heuristic is fast and yields significant improvements compared to the sequential solution of independent daily problems.     

A parallel algorithm for train rescheduling

One of the crucial factors in achieving a high punctuality in railway traffic systems, is the ability to effectively reschedule the trains when disturbances occur. The railway traffic rescheduling problem is a complex task to solve both from a practical and a computational perspective. Problems of practically relevant sizes have typically a very large search space, making them time-consuming to solve even for state-of-the-art optimization solvers. Though competitive algorithmic approaches are a widespread topic of research, not much research has been done to explore the opportunities and challenges in parallelizing them. This paper presents a parallel algorithm to efficiently solve the real-time railway rescheduling problem on a multi-core parallel architecture. We devised (1) an effective way to represent the solution space as a binary tree and (2) a novel sequential heuristic algorithm based on a depth-first search (DFS) strategy that quickly traverses the tree. Based on that, we designed a parallel algorithm for a multi-core architecture, which proved to be 10.5 times faster than the sequential algorithm even when run on a single processing core. When executed on a parallel machine with 8 cores, the speed further increased by a factor of 4.68 and every disturbance scenario in the considered case study was solved within 6 s. We conclude that for the problem under consideration, though a sequential DFS approach is fast in several disturbance scenarios, it is notably slower in many other disturbance scenarios. The parallel DFS approach that combines a DFS with simultaneous breadth-wise tree exploration, while being much faster on an average, is also consistently fast across all scenarios.     

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.     

A discrete-convex programming approach to the simultaneous optimization of land use and transportation

Three design problems are discussed in this article. First, it is shown that the network design problem with congestion reduces to an all-or nothing traffic assignment problem under some assumptions on the congestion function and the investment cost function. Second, the land use design problem is formulated as an extension of the Koopmans-Beckmann problem and a heuristic is proposed to solve this problem. Third, it is shown that the seemingly more complex problem of designing jointly a land-use plan and a transportation network reduces to a pure land-use design problem. All that is needed to solve the joint optimization problem is a shortest path algorithm and a heuristic to solve the land use design problem. Computational experience is reported for each algorithm.     

A tabu search algorithm to solve the integrated planning of container on an inter-terminal network connected with a hinterland rail network

Transport demand for containers has been increasing for decades, which places pressure on road transport. As a result, rail transport is stimulated to provide better intermodal freight transport services. This paper investigates mathematical models for the planning of container movements in a port area, integrating the inter-terminal transport of containers (ITT, within the port area) with the rail freight formation and transport process (towards the hinterland). An integer linear programming model is used to formulate the container transport across operations at container terminals, the network interconnecting them, railway yards and the railway networks towards the hinterland. A tabu search algorithm is proposed to solve the problem. The practical applicability of the algorithm is tested in a realistic infrastructure case and different demand scenarios. Our results show the degree by which internal (ITT) and external (hinterland) transport processes interact, and the potential for improvement of overall operations when the integrated optimization proposed is used. Instead, if the planning of containers in the ITT system is optimized as a stand-alone problem, the railway terminals may suffer from longer delay times or additional train cancellations. When planning the transport of 4060 TEU containers within one day, the benefits of the ITT planning without considering railway operations account for 17% ITT cost reduction but 93% railway operational cost growth, while the benefits of integrating ITT and railway account for a reduction of 20% in ITT cost and 44% in railway operational costs.     

Solution methods for the taxi pooling problem

In Taiwan, taxi pooling is currently performed by some taxi companies using a trial-and-error experience-based method, which is neither effective nor efficient. There is, however, little in the literature on effective models and solution methods for solving the taxi pooling problem. Thus, in this study we employ network flow techniques and a mathematical programming method to develop a taxi pooling solution method. This method is composed of three models. First, a fleet routing/scheduling model is constructed to produce fleet/passenger routes and schedules. A solution algorithm, based on Lagrangian relaxation, a sub-gradient method and a heuristic to find the upper bound of the solution, is proposed to solve the fleet routing/scheduling model. Then, two single taxi-passenger matching models are constructed with the goals of decreasing number of passenger transfers and matching all passengers and taxis. These two taxi-passenger matching models are directly solved using a mathematical programming solver. For comparison with the solution method, we also develop another heuristic by modifying a heuristic recently proposed for solving a one-to-many taxi pooling problem. The performance of the solution method and the additional heuristic are evaluated by carrying out a case study using real data and suitable assumptions. The test results show that these two solution methods could be useful in practice.     

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.