Periodic Event Scheduling Problem Research Articles

Periodic timetabling with integrated track choice for railway construction sites

We propose a mixed-integer linear programming model to generate and optimize periodic timetables with integrated track choice in the context of railway construction sites. When a section of a railway network becomes unavailable, the nearby areas are typically operated close to their capacity limits, and hence carefully modeling headways and allowing flexible routings becomes vital. We therefore discuss first how to integrate headway constraints into the Periodic Event Scheduling Problem (PESP) that do not only prevent overtaking, but also guarantee conflict-free timetables in general and particularly inside stations. Secondly, we introduce a turn-sensitive event-activity network, which is able to integrate routing alternatives for turnarounds at stations, e.g., turning at a platform vs. at a pocket track for metro-like systems. We propose several model formulations to include track choice, and finally evaluate them on six real construction site scenarios on the S-Bahn Berlin network.

Forward and Line-Based Cycle Bases for Periodic Timetabling

The optimization of periodic timetables is an indispensable planning task in public transport. Although the periodic event scheduling problem (PESP) provides an elegant mathematical formulation of the periodic timetabling problem that led to many insights for primal heuristics, it is notoriously hard to solve to optimality. One reason is that for the standard mixed-integer linear programming formulations, linear programming relaxations are weak, and the integer variables are of pure technical nature and in general do not correlate with the objective value. While the first problem has been addressed by developing several families of cutting planes, we focus on the second aspect. We discuss integral forward cycle bases as a concept to compute improved dual bounds for PESP instances. To this end, we develop the theory of forward cycle bases on general digraphs. Specifically for the application of timetabling, we devise a generic procedure to construct line-based event-activity networks and give a simple recipe for an integral forward cycle basis on such networks. Finally, we analyze the 16 railway instances of the benchmark library PESPlib, match them to the line-based structure, and use forward cycle bases to compute better dual bounds for 14 out of the 16 instances.

Passenger-centric periodic timetable adjustment problem for the utilization of regenerative energy

Optimizing the railway timetable to increase synchronous accelerating and braking processes can lead to an improvement in the usage of regenerative energy. However, such a synchronized timetable might result in little or unsuitable transfer connections for the passengers. This paper focuses on the optimization of railway periodic timetables, to increase usage of regenerative energy while ensuring passenger satisfaction. We work by extending the traditional Periodic Event Scheduling Problem (PESP) formulation, to address the problem of synchronization of acceleration and braking phases, (and re-used energy) and including passenger-related events (and their satisfaction). Three objectives are identified, in a resulting Mixed Integer Linear Programming (MILP) model: maximizing the overlapping times of accelerating and braking trains to achieve increased usage of regenerative energy, minimizing the total passengers’ generalized travel times (global passenger dissatisfaction), and minimizing the maximum increase in individual’s generalized travel time (local passenger dissatisfaction). A multi-step approach solves the trade-offs among three conflicting objectives. Results on a realistic case study show that the proposed approach can find optimized timetables, which compared to the currently-in-use timetable, can increase the usage of regenerative energy by over 1.5 times, save the average generalized travel time per passenger by 2 min, with only a minor increase on specific individual generalized travel time (up to 4 min). A detailed results analysis imply that to achieve a higher usage of regenerative energy, it is required to have a higher tolerance for the maximum increase in individual generalized travel time, while this is not necessary for the overall passenger generalized travel time, which can even be reduced when the maximum increase in individual generalized travel time becomes larger.

An analysis of the parameterized complexity of periodic timetabling

Public transportation networks are typically operated with a periodic timetable. The periodic event scheduling problem (PESP) is the standard mathematical modeling tool for periodic timetabling. PESP is a computationally very challenging problem: For example, solving the instances of the benchmarking library PESPlib to optimality seems out of reach. Since PESP can be solved in linear time on trees, and the treewidth is a rather small graph parameter in the networks of the PESPlib, it is a natural question to ask whether there are polynomial-time algorithms for input networks of bounded treewidth, or even better, fixed-parameter tractable algorithms. We show that deciding the feasibility of a PESP instance is NP-hard even when the treewidth is 2, the branchwidth is 2, or the carvingwidth is 3. Analogous results hold for the optimization of reduced PESP instances, where the feasibility problem is trivial. Moreover, we show W[1]-hardness of the general feasibility problem with respect to treewidth, which means that we can most likely only accomplish pseudo-polynomial-time algorithms on input networks with bounded tree- or branchwidth. We present two such algorithms based on dynamic programming. We further analyze the parameterized complexity of PESP with bounded cyclomatic number, diameter, or vertex cover number. For event-activity networks with a special—but standard—structure, we give explicit and sharp bounds on the branchwidth in terms of the maximum degree and the carvingwidth of an underlying line network. Finally, we investigate several parameters on the smallest instance of the benchmarking library PESPlib.

An iterative heuristic for passenger-centric train timetabling with integrated adaption times

In this paper we present a method to construct a periodic timetable from a tactical planning perspective. We aim at constructing a timetable that is feasible with respect to infrastructure constraints and minimizes total perceived passenger travel time. In addition to in-train and transfer times, our notion of perceived passenger time includes the adaption time (waiting time at the origin station). Adaption time minimization allows us to avoid strict frequency regularity constraints and, at the same time, to ensure regular connections between passengers’ origins and destinations. The combination of adaption time minimization and infrastructure constraints satisfaction makes the problem very challenging.The described periodic timetabling problem can be modelled as an extension of a Periodic Event Scheduling Problem (PESP) formulation, but requires huge computing times if it is directly solved by a general-purpose solver for instances of realistic size. In this paper, we propose a heuristic approach consisting of two phases that are executed iteratively. First, we solve a mixed-integer linear program to determine an ideal timetable that minimizes the total perceived passenger travel time but neglects infrastructure constraints. Then, a Lagrangian-based heuristic makes the timetable feasible with respect to infrastructure constraints by modifying train departure and arrival times as little as possible. The obtained feasible timetable is then evaluated to compute the resulting total perceived passenger travel time, and a feedback is sent to the Lagrangian-based heuristic so as to possibly improve the obtained timetable from the passenger perspective, while still respecting infrastructure constraints. We illustrate the proposed iterative heuristic approach on real-life instances of Netherlands Railways and compare it to a benchmark approach, showing that it finds a feasible timetable very close to the ideal one.

Timetable merging for the Periodic Event Scheduling Problem

We propose a new mixed integer programming based heuristic for computing new benchmark primal solutions for instances of the PESPlib. The PESPlib is a collection of instances for the Periodic Event Scheduling Problem (PESP), comprising periodic timetabling problems inspired by real-world railway timetabling settings, and attracting several international research teams during the last years. We describe two strategies to merge a set of good periodic timetables. These make use of the instance structure and minimum weight cycle bases, finally leading to restricted mixed integer programming formulations with tighter variable bounds. Implementing this timetable merging approach in a concurrent solver, we improve the objective values of the best known solutions for the smallest and largest PESPlib instances by 1.7 and 4.3 percent, respectively.

Finding robust periodic timetables by integrating delay management

This paper defines and solves a mathematical model for finding robust periodic timetables by proposing an extension of the Periodic Event Scheduling Problem (PESP). In order to model delayed and non-nominal travel times already in the timetabling step, the aim of this paper is to integrate delay management into the periodic timetabling problem and investigating the resulting problem (RPT). After revisiting both (PESP) and delay management individually, we introduce a periodic delay management model – an auxiliary model capable of evaluating periodic timetables with respect to delay resistance. Having introduced periodic delay management, we define the robust periodic timetabling problem (RPT). Due to the high complexity of the robust periodic timetabling problem we propose two different simplifications of the problem and introduce solution algorithms for both of them. These solution algorithms are tested against timetables found by standard procedures for periodic timetabling with respect to their delay-resistance. The computational results show that our algorithms yield timetables which can cope better with occurring delays, even on large-scale datasets and with low computational effort.

An Iterative Heuristic for Passenger-Centric Train Timetabling with Integrated Adaption Times

In this paper we present a method to construct a periodic timetable from a tactical planning perspective. We aim at constructing a timetable that is feasible with respect to infrastructure constraints and minimizes average perceived passenger travel time. In addition to in-train and transfer times, our notion of perceived passenger time includes the adaption time (waiting time at the origin station). Adaption time minimization allows us to avoid strict frequency regularity constraints and, at the same time, to ensure regular connections between passengers’ origins and destinations. The combination of adaption time minimization and infrastructure constraints satisfaction makes the problem very challenging. The described periodic timetabling problem can be modelled as an extension of a Periodic Event Scheduling Problem (PESP) formulation, but requires huge computing times if it is directly solved by a general-purpose solver for instances of realistic size. In this paper, we propose a heuristic approach consisting of two phases that are executed iteratively. First, we solve a mixed-integer linear program to determine an ideal timetable that mini- mizes the average perceived passenger travel time but neglects infrastructure constraints. Then, a Lagrangian-based heuristic makes the timetable feasible with respect to infra- structure constraints by modifying train departure and arrival times as little as possible. The obtained feasible timetable is then evaluated to compute the resulting average per- ceived passenger travel time, and a feedback is sent to the Lagrangian-based heuristic so as to possibly improve the obtained timetable from the passenger perspective, while still respecting infrastructure constraints. We illustrate the proposed iterative heuristic approach on real-life instances of Netherlands Railways and compare it to a benchmark approach, showing that it finds a feasible timetable very close to the ideal one.

A concurrent approach to the periodic event scheduling problem

We introduce a concurrent solver for the periodic event scheduling problem (PESP). It combines mixed integer programming techniques, the modulo network simplex method, satisfiability approaches, and a new heuristic based on maximum cuts. Running these components in parallel speeds up the overall solution process. This enables us to significantly improve the current upper and lower bounds for all benchmark instances of the library PESPlib.

Solving cyclic train timetabling problem through model reformulation: Extended time-space network construct and Alternating Direction Method of Multipliers methods

The cyclic train timetabling problem aims to synchronize limited operational resources toward a master periodic schedule of transport services. By introducing an extended time-space network construct, this paper proposes a new type of integer programming model reformulation for the cyclic train timetabling problem on a double-track railway corridor at the macroscopic level. This reformulation method also holds the promises to be applied in a broader set of routing and scheduling problems with periodic activity requirements. We also hope that this space-time network extension technique, as a special version of variable splitting methods in the dual decomposition literature, could potentially bridge the modeling gaps between cyclic and non-cyclic timetables. Specifically, the existing mathematical programming model for the periodic event scheduling problem (PESP) is transformed into a multi-commodity network flow model with two coupled schedule networks and side track capacity constraints. In addition, two dual decomposition methods including Lagrangian relaxation and Alternating Direction Method of Multipliers (ADMM), are adopted to dualize the side track capacity constraints. For each train-specific sub-problem in an iterative primal and dual optimization framework, we develop an enhanced version of forward dynamic programming to find the time-dependent least cost master schedule across the time-space network over multiple periods. ADMM-motivated heuristic methods with adjusted penalty parameters are also developed to obtain good upper bound solutions. Based on real-world instances from the Beijing-Shanghai high-speed railway corridor, we compare the numerical performance between the proposed reformulation and the PESP model that involves the standard optimization solver.

Optimal synchronization and coordination of actual passenger-rail timetables

This study provides a novel solution for the synchronized and coordinated railway scheduling optimization (SCSO) problem by the determination of the departure times of a public transit network. Railway timetable optimization is dealt with maximizing the number of synchronized meetings to allow for smooth transfers at interchanges. The developed model uses binary variables to record the number of synchronized meetings considering the importance of transfer stations and rail lines without the need to apply the modeling of passenger assignments. The model allows for a permissible and flexible transfer waiting time for making a connection between rails instead of the commonly used and assumed values. The solution of the mixed-integer programing problem of larger-sized railway networks is based on a synchronized and coordinated scheduling optimization genetic algorithm (SCSO-GA) with a local search strategy (LSS). This solution method is proved to be more efficient and accurate than the CPLEX solver. In addition it is proven to be a periodic event-scheduling problem (PESP) solver. The model is tested computationally on the Beijing urban rail transit network. The results demonstrate the advantage of the novel approach over other methods.

Periodic railway timetabling with sequential decomposition in the PESP model

With continuously increasing capacity utilization of railway networks as well as growing requirements on service quality and reliability, railway timetabling is becoming increasingly difficult. Although most timetables are still constructed manually in practice, the demand for advanced automatic timetabling techniques is evident. Long computation times, however, are a major impediment for the use of optimization-based timetabling tools within today's planning process. Focusing on the construction of periodic timetables via the periodic event scheduling problem (PESP), the paper introduces a new decomposition technique to speed up automatic timetabling. The approach is based on solving a sequence of smaller subproblems and can be parameterized to reach a suitable compromise between the two extremes of either simultaneous or sequential planning. Computational results on large timetabling instances for Switzerland's railway network show very promising results. In particular, finding feasible as well as near optimal timetables can be considerably accelerated compared to solving the PESP using the standard MILP formulation.

Designing alternative railway timetables under infrastructure maintenance possessions

Increasing supply in railway networks comes at the cost of an increased need for infrastructure maintenance. This also means adjusting the timetable due to long maintenance or constructions’ possessions. In this article, we introduce the Train Timetable Adjustment Problem (TTAP), which for given station and open-track possessions, finds an alternative timetable that minimizes the deviation from the original timetable. We propose a mixed integer linear programming (MILP) model for solving TTAP, and apply retiming, reordering, short-turning and cancellation to generate alternative timetables. The model represents an extended periodic event scheduling problem (PESP) formulation and introduces new constraints for cancelling and retiming train lines, while short-turning is being applied in a preprocessing step. In order to solve larger and more complex instances, we use a row generation approach to add station capacity constraints. The model solves real-life instances with multiple possessions for a large area of the Dutch railway network in reasonable time, and could be up-scaled to the complete Dutch network. Additionally, it may be applicable for disruption management after some modifications. Operators and infrastructure managers could use it to automatically generate optimal alternative timetables on the macroscopic level in case of maintenance or construction works and thus, coordinate traffic for the complete network.

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 state-of-the-art realization of cyclic railway timetable computation

We describe the periodic event scheduling problem (PESP) based on periodic event networks and extend it by symmetry constraints. The modeling power of the PESP is discussed for automatic calculation of feasible periodic railway timetables. Including the described extensions, complete modeling of integrated regular-interval timetables is possible. Encoding the PESP to propositional logic enables the usage of efficient SAT solvers for solving the PESP. However, optimizing timetables by linear programming is possible, too. As almost all real-world timetable problems are heavily overconstrained, methods for automatic resolving of conflicts are described. Since there is still a lack of efficient conflict resolving algorithms for large-scale intermeshed railway networks, we introduce several strategies for efficiently resolving conflicts and intelligently decomposing timetable problems and discuss the trade-off between computation time and reduction of the solution space. These strategies allow quick adaption to small changes as well. The described methods are implemented in the software system TAKT. We apply the enhanced TAKT to a timetabling study and present some key figures and results.

Improving the modulo simplex algorithm for large-scale periodic timetabling

The periodic event scheduling problem (PESP), in which events have to be scheduled repeatedly over a given period, is a complex and well-known discrete problem with numerous real-world applications. The most prominent of them is to find periodic timetables in public transport. Although even finding a feasible solution to the PESP is NP-hard, recent achievements demonstrate the applicability and practicability of the periodic event scheduling model. In this paper we propose different approaches to improve the modulo network simplex algorithm (Nachtigall and Opitz, 2008 [17]), which is a powerful heuristic for the PESP problem, by exploiting improved search methods in the modulo simplex tableau and larger classes of cuts to escape from the many local optima. Numerical experiments on large-scale railway instances show that our algorithms not only perform better than the original method, but even outperform a state-of-the-art commercial MIP solver.

Flexible Connections in PESP Models for Cyclic Passenger Railway Timetabling

In this paper we describe how rolling stock and passenger connections in a cyclic railway timetable can be modeled in a flexible way within the model for the Periodic Event-Scheduling Problem (PESP). Usually, PESP models assume that the constraints for rolling stock or passenger connections specify in detail which trains should connect with each other. However, the flexibility described in this paper allows the model to choose which trains should connect with each other in a rolling stock or passenger connection. We express the connection times in a minimum matching of arriving and departing trains in terms of certain process time variables. We also describe an abstract framework demonstrating that, under certain conditions, the extra flexibility can be modeled purely in terms of PESP constraints. The concept of flexible rolling stock and passenger connections is illustrated by an example based on three intercity lines of Netherlands Railways.

Solving a periodic single-track train timetabling problem by an efficient hybrid algorithm

Train timetabling with minimum delays is the most important operating problem in any railway industry. This problem is considered to be one of the most interesting research topics in railway optimization problems. This paper deals with scheduling different types of trains in a single railway track. The primary focus of this paper is on the periodic aspects of produced timetables and the proposed modeling is based on the periodic event scheduling problem (PESP). To solve large-scale problems, a hybrid meta-heuristic algorithm based on simulated annealing (SA) and particle swarm optimization (PSO) is proposed and validated using some numerical examples and an Iranian case study that covers the railway line between two cities of Isfahan and Tehran.

The periodicity and robustness in a single-track train scheduling problem

The most important operating problem in any railway industry is to produce robust train timetables with minimum delays. The train scheduling problem is defined as an application of job shop scheduling which is considered to be one of the most interesting research topics. This paper deals with scheduling different types of trains in a single railway track. The authors have focused on the robust and periodic aspects of produced timetables. This paper is also concerned with some applicable constraints, such as the acceleration and deceleration times, station capacity and headway constraints. The periodic timetable for railways is modeled based on the periodic event scheduling problem (PESP). Furthermore, a fuzzy approach is used to reach a tradeoff among the total train delays, the robustness of schedules, and the time interval between departures of trains from the same origins. To solve large-scale problems, a meta-heuristic algorithm based on simulated annealing (SA) is utilized and validated using some numerical examples on a periodic robust train scheduling problem. Finally, a robustness measure is defined in order to assure the effectiveness of the proposed SA to find robust solutions.

Periodic railway timetabling with event flexibility

AbstractThis article addresses the problem of generating conflict‐free periodic train timetables for large railway networks. We follow a two‐level approach, where a simplified track topology is used to obtain a macrolevel schedule, and the detailed topology is considered locally on the microlevel. To enlarge the solution space in the interface of the two levels, we propose an extension of the well‐known periodic event scheduling problem (PESP) such that it allows to generate flexible time slots for the departure and arrival times instead of exact times. This flexible periodic event scheduling problem (FPESP) formulation increases the chance to obtain feasible solutions (exact train routings) subsequently on the microlevel, in particular for stations with dense peak traffic. Total trip time and the time slot sizes are used as multiple objectives and weighted and/or constrained to allocate the flexibility where it is most useful. Tests on medium‐size instances of the Swiss Federal Railways' 2007 service intention demonstrate the advantage of the FPESP model, while it only moderately increases its solution time in most cases. A trade‐off analysis of the two conflicting objectives is presented and discussed for different scenarios. Finally, we present the Flexbox model, which generalizes the single event flexibilities of the FPESP to arbitrary subsets of events. These Flexboxes can make use of natural dependencies in the timetable and can considerably increase the total flexibility of the timetable. © 2010 Wiley Periodicals, Inc. NETWORKS, 2011