An integrated micro–macro approach to robust railway timetabling

With the increasing demand for railway transportation infrastructure managers need improved automatic timetabling tools that provide feasible timetables with enhanced performance in short computation times. This paper proposes a hierarchical framework for timetable design which combines a microscopic and a macroscopic model of the network. The framework performs an iterative adjustment of train running and minimum headway times until a feasible and stable timetable has been generated at the microscopic level. The macroscopic model optimizes a trade-off between minimal travel times and maximal robustness using an Integer Linear Programming formulation which includes a measure for delay recovery computed by an integrated delay propagation model in a Monte Carlo setting. The application to an area of the Dutch railway network shows the ability of the approach to automatically compute a feasible, stable and robust timetable. Practitioners can use this approach both for effective timetabling and post-evaluation of existing timetables.     

A three-level framework for performance-based railway timetabling

The performance of railway operations depends highly on the quality of the railway timetable. In particular for dense railway networks it can be challenging to obtain a stable robust conflict-free and energy-efficient timetable with acceptable infrastructure occupation and short journey times. This paper presents a performance-based railway timetabling framework integrating timetable construction and evaluation on three levels: microscopic, macroscopic, and a corridor fine-tuning level, where each performance indicator is optimized or evaluated at the appropriate level. A modular implementation of the three-level framework is presented and demonstrated on a case study on the Dutch railway network illustrating the feasibility of this approach to achieve the highest timetabling design level.     

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.     

A stochastic optimization model for transit network timetable design to mitigate the randomness of traveling time by adding slack time

Transit network timetabling aims at determining the departure time of each trip of all lines in order to facilitate passengers transferring either to or from a bus. In this paper, we consider a bus timetabling problem with stochastic travel times (BTP-STT). Slack time is added into timetable to mitigate the randomness in bus travel times. We then develop a stochastic integer programming model for the BTP-STT to minimize the total waiting time cost for three types of passengers (i.e., transferring passengers, boarding passengers and through passengers). The mathematical properties of the model are characterized. Due to its computational complexity, a genetic algorithm with local search (GALS) is designed to solve our proposed model (OPM). The numerical results based on a small bus network show that the timetable obtained from OPM reduces the total waiting time cost by an average of 9.5%, when it is tested in different scenarios. OPM is relatively effective if the ratio of the number of through passengers to the number of transferring passengers is not larger than a threshold (e.g., 10 in our case). In addition, we test different scale instances randomly generated in a practical setting to further verify the effectiveness of OPM and GALS. We also find that adding slack time into timetable greatly benefits transferring passengers by reducing the rate of transferring failure.     

A new approach for improving the performance of freight train timetabling of a single-track railway system

A new approach for improving the performance of freight train timetabling for single-track railways is proposed. Using the idea of a fixed-block signaling system, we develop a matrix representation to express the occupation of inter- and intra-station tracks by trains illustrating the train blocking time diagram in its entirety. Train departure times, dwell times, and unnecessary stopping are adjusted to reduce average train travel time and single train travel time. Conflicts between successive stations and within stations are identified and solved. A fuzzy logic system is further used to adjust the range of train departure times and checks are made to determine whether dwell times and time intervals can be adjusted for passenger and freight trains at congested stations to minimize train waiting times. By combining manual scheduling expertise with the fuzzy inference method, timetable efficiency is significantly improved and becomes more flexible.     

Multi-objective re-synchronizing of bus timetable: Model,complexity and solution

This work is originally motived by the re-planning of a bus network timetable. The existing timetable with even headways for the network is generated using line by line timetabling approach without considering the interactions between lines. Decision-makers (i.e., schedulers) intend to synchronize vehicle timetable of lines at transfer nodes to facilitate passenger transfers while being concerned with the impacts of re-designed timetable on the regularity of existing timetable and the accustomed trip plans of passengers. Regarding this situation, we investigate a multi-objective re-synchronizing of bus timetable (MSBT) problem, which is characterized by headway-sensitive passenger demand, uneven headways, service regularity, flexible synchronization and involvement of existing bus timetable. A multi-objective optimization model for the MSBT is proposed to make a trade-off between the total number of passengers benefited by smooth transfers and the maximal deviation from the departure times of the existing timetable. By clarifying the mathematical properties and solution space of the model, we prove that the MSBT problem is NP-hard, and its Pareto-optimal front is non-convex. Therefore, we design a non-dominated sorting genetic (NSGA-II) based algorithm to solve this problem. Numerical experiments show that the designed algorithm, compared with enumeration method, can generate high-quality Pareto solutions within reasonable times. We also find that the timetable allowing larger flexibility of headways can obtain more and better Pareto-optimal solutions, which can provide decision-makers more choice.     

Automated real-time railway traffic control: an experimental analysis of reliability,resilience and robustness

Railway transportation provides sustainable, fast and safe transport. Its attractiveness is linked to a broad concept of service reliability: the capability to adhere to a timetable in the presence of delays perturbing traffic. To counter these phenomena, real-time rescheduling can be used, changing train orders and times, according to rules of thumb, or mathematical optimization models, minimizing delays or maximizing punctuality. In the literature, different indices of robustness, reliability and resilience are defined for railway traffic. We review and evaluate these indices applied to railway traffic control, comparing optimal rescheduling approaches such as Open Loop and Closed Loop control, to a typical First-Come-First-Served dispatching rule, and following the timetable (no-action). This experimental analysis clarifies the benefits of automated traffic control for infrastructure managers, railway operators and passengers. The timetable order, normally used in assessing a-priori reliability, systematically overestimates unreliability of operations that can be reduced by real-time control.     

An activity-based approach for scheduling multimodal transit services

This paper proposes a new activity-based transit assignment model for investigating the scheduling (or timetabling) problem of transit services in multi-modal transit networks. The proposed model can be used to generate the short-term and long-term timetables of multimodal transit lines for transit operations and service planning purposes. The interaction between transit timetables and passenger activity-travel scheduling behaviors is captured by the proposed model, as the activity and travel choices of transit passengers are considered explicitly in terms of departure time choice, activity/trip chain choices, activity duration choice, transit line and mode choices. A heuristic solution algorithm which combines the Hooke–Jeeves method and an iterative supply–demand equilibrium approach is developed to solve the proposed model. Two numerical examples are presented to illustrate the differences between the activity-based approach and the traditional trip-based method, together with comparison on the effects of optimal timetables with even and uneven headways. It is shown that the passenger travel scheduling pattern derived from the activity-based approach is significantly different from that obtained by the trip-based method, and that a demand-sensitive (with uneven headway) timetable is more efficient than an even-headway timetable.     

A probabilistic model for vehicle scheduling based on stochastic trip times

Vehicle scheduling plays a profound role in public transit planning. Traditional approaches for the Vehicle Scheduling Problem (VSP) are based on a set of predetermined trips in a given timetable. Each trip contains a departure point/time and an arrival point/time whilst the trip time (i.e. the time duration of a trip) is fixed. Based on fixed durations, the resulting schedule is hard to comply with in practice due to the variability of traffic and driving conditions. To enhance the robustness of the schedule to be compiled, the VSP based on stochastic trip times instead of fixed ones is studied. The trip times follow the probability distributions obtained from the data captured by Automatic Vehicle Locating (AVL) systems. A network flow model featuring the stochastic trips is devised to better represent this problem, meanwhile the compatibility of any pair of trips is redefined based on trip time distributions instead of fixed values as traditionally done. A novel probabilistic model of the VSP is proposed with the objectives of minimizing the total cost and maximizing the on-time performance. Experiments show that the probabilistic model may lead to more robust schedules without increasing fleet size.     

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.     

Passenger Perspectives in Railway Timetabling: A Literature Review

When looking at railway planning, a discrepancy exists between planners who focus on the train operations and publish fixed railway schedules, and passengers who look not only at the schedules but also at the entirety of their trip, from access to waiting to on-board travel and egress. Looking into this discrepancy is essential, as assessing railway performances by merely measuring train punctuality would provide an unfair picture of the level of service experienced by passengers. Firstly, passengers’ delays are often significantly larger than the train delays responsible for the passengers to be late. Secondly, trains’ punctuality is often strictly related to too tight schedules that in turn might translate into knock-on delays for longer dwelling times at stations, trip delays for increased risk of missing transfer connections, and uncertain assessment of the level of service experienced, especially with fluctuating passenger demand. A key aspect is the robustness of railway timetables. Empirical evidence indicates that passengers give more importance to travel time certainty than travel time reductions, as passengers associate an inherent disutility with travel time uncertainty. This disutility may be broadly interpreted as an anxiety cost for the need for having contingency plans in case of disruptions, and may be looked at as the motivator for the need for delay-robust railway timetables. Interestingly, passenger-oriented optimisation studies considering robustness in railway planning typically limit their emphasis on passengers to the consideration of transfer maintenance. Clearly, passengers’ travel behaviour is far more complex and multi-faceted and thus several other aspects should be considered, as becoming more and more evident from passenger surveys. The current literature review starts by looking at the parameters that railway optimisation/planning studies are focused on and the key performance indicators that impact railway planning. The attention then turns to the parameters influencing passengers’ perceptions and travel experiences. Finally, the review proposes guidelines on how to reduce the gap between the operators’ railway planning and performance measurement on the one hand and the passengers’ perception of the railway performance on the other hand. Thereby, the conclusions create a foundation for a more passenger-oriented railway timetabling ensuring that passengers are provided with the best service possible with the resources available.     

Train routing and timetabling problem for heterogeneous train traffic with switchable scheduling rules

This paper proposes a mathematical model for the train routing and timetabling problem that allows a train to occasionally switch to the opposite track when it is not occupied, which we define it as switchable scheduling rule. The layouts of stations are taken into account in the proposed mathematical model to avoid head-on and rear-end collisions in stations. In this paper, train timetable could be scheduled by three different scheduling rules, i.e., no switchable scheduling rule (No-SSR) which allows trains switching track neither at stations and segments, incomplete switchable scheduling rule (In-SSR) which allows trains switching track at stations but not at segments, and complete switchable scheduling rule (Co-SSR) which allows trains switching track both at stations and segments. Numerical experiments are carried out on a small-scale railway corridor and a large-scale railway corridor based on Beijing–Shanghai high-speed railway (HSR) corridor respectively. The results of case studies indicate that Co-SSR outperforms the other two scheduling rules. It is also found that the proposed model can improve train operational efficiency.     

Scheduling heterogeneous train traffic on double tracks with efficient dispatching rules

To further improve the utilization rate of railway tracks and reduce train delays, this paper focuses on developing a high-efficiency train routing and timetabling approach for double-track railway corridors in condition that trains are allowable to travel on reverse direction tracks. We first design an improved switchable policy which is rooted in the approaches by Mu and Dessouky (2013), with the analysis of possible delays caused by different path choices. Then, three novel integrated train routing and timetabling approaches are proposed on the basis of a discrete event model and different dispatching rules, including no switchable policy (No-SP), Mu and Dessouky (2013)’s switchable policy (Original-SP) and improved switchable policy (Improved-SP). To demonstrate the performance of the proposed approaches, the heterogeneous trains on Beijing–Shanghai high speed railway are scheduled by aforementioned approaches. The case studies indicate that in comparison to No-SP and Original-SP approaches, respectively, the Improved-SP approach can reduce the total delay of trains up to 44.44% and 73.53% within a short computational time. Moreover, all of the performance criteria of the Improved-SP approach are usually better than those of other two approaches.     

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.     

A hybrid optimization-simulation approach for robust weekly aircraft routing and retiming

We address the robust weekly aircraft routing and retiming problem, which requires determining weekly schedules for a heterogeneous fleet that maximizes the aircraft on-time performance, minimizes the total delay, and minimizes the number of delayed passengers. The fleet is required to serve a set of flights having known departure time windows while satisfying maintenance constraints. All flights are subject to random delays that may propagate through the network. We propose to solve this problem using a hybrid optimization-simulation approach based on a novel mixed-integer nonlinear programming model for the robust weekly aircraft maintenance routing problem. For this model, we provide an equivalent mixed-integer linear programming formulation that can be solved using a commercial solver. Furthermore, we describe a Monte-Carlo-based procedure for sequentially adjusting the flight departure times. We perform an extensive computational study using instances obtained from a major international airline, having up to 3387 flights and 164 aircraft, which demonstrates the efficacy of the proposed approach. Using the simulation software SimAir to assess the robustness of the solutions produced by our approach in comparison with that for the original solutions implemented by the airline, we found that on-time performance was improved by 9.8–16.0%, cumulative delay was reduced by 25.4–33.1%, and the number of delayed passengers was reduced by 8.2–51.6%.     

Non-discriminatory train dispatching in a rail transport market with multiple competing and collaborative train operating companies

Train dispatching is vital for the punctuality of train services, which is critical for a train operating company (TOC) to maintain its competitiveness. Due to the introduction of competition in the railway transport market, the issue of discrimination is attracting more and more attention. This paper focuses on delivering non-discriminatory train dispatching solutions while multiple TOCs are competing in a rail transport market, and investigating impacting factors of the inequity of train dispatching solutions. A mixed integer linear programming (MILP) model is first proposed, in which the inequity of competitors (i.e., trains and TOCs) is formalized by a set of constraints. In order to provide a more flexible framework, a model is further reformulated where the inequity of competitors is formalized as the maximum individual deviation of competitors’ delay cost from average delay cost in the objective function. Complex infrastructure capacity constraints are considered and modelled through a big M-based approach. The proposed models are solved by a standard MILP solver. A set of comprehensive experiments is conducted on a real-world dataset adapted from the Dutch railway network to test the efficiency, effectiveness, and applicability of the proposed models, as well as determine the trade-off between train delays and delay equity.     

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.     

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.     

Collaborative truck scheduling and appointments for trucking companies and container terminals

Appointment systems for truck arrivals at container terminals have been applied in many ports to reduce truck congestion. This study suggests a new appointment process by which trucking companies and terminals collaboratively determine truck operation schedules and truck arrival appointments. This study formulates a mathematical model involving a sub-problem for each trucking company to determine the optimal dispatching schedules for trucks and the other sub-problem for the terminal to estimate the expected truck system time in each time interval. An iterative collaboration process is proposed based on a decomposed mathematical formulation. Numerical experiments are conducted to investigate the performance of the decision process and the robustness of the process in practical operation conditions.     

Optimizing train operational plan in an urban rail corridor based on the maximum headway function

The train operational plan (TOP) plays a crucial role in the efficient and effective operation of an urban rail system. We optimize the train operational plan in a special network layout, an urban rail corridor with one terminal yard, by decomposing it into two sub-problems, i.e., the train departure profile optimization and the rolling stock circulation optimization. The first sub-problem synthetically optimizes frequency setting, timetabling and the rolling stock circulation at the terminal without a yard. The maximum headway function is generated to ensure the service of the train operational plan without considering travel demand, then we present a model to minimize the number of train trips, and design a heuristic algorithm to maximize the train headway. On the basis of a given timetable, the rolling stock circulation optimization only involves the terminal with a yard. We propose a model to minimize the number of trains and yard–station runs, and an algorithm to find the optimal assignment of train-trip pair connections is designed. The computational complexities of the two algorithms are both linear. Finally, a real case study shows that the train operational plan developed by our approach enables a better match of train headway and travel demand, and reduces the operational cost while satisfying the requirement of the level of service.