Real time management of a metro rail terminus

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Abstract

This paper addresses a scheduling problem arising in the real time management of a metro rail terminus. It mainly consists in routing incoming trains through the station and scheduling their departures with the objective of optimizing punctuality and regularity of train service. The purpose of this work is to develop an automated train traffic control system, able to directly implement most traffic control actions, without the authorization of the local area manager. The scheduling problem is modeled as a bicriteria job shop scheduling problem with additional constraints. The two objective functions, in lexicographical order, are the minimization of tardiness/earliness and the headway optimization. The problem is solved in two steps. At first a heuristic builds a feasible solution by considering the first objective function. Then the regularity is optimized without deteriorating the first objective function. Computational results show that the system is able to manage the terminus very efficiently.

Introduction

Railway traffic optimization is experiencing an increasing interest both among researchers and practitioners. Solving problems of practical interest in this field requires using detailed models, able to represent real and different railway traffic situations, and developing efficient algorithms to be used as decision support system in traffic control operation. Railway scheduling problems have been studied by using different techniques, including linear programming, integer or non-linear programming, graph theory and dynamic programming. Among the published results, we cite the papers by Carey, 1994, Carey and Lockwood, 1995, Higgins et al., 1997, Cai et al., 1998, Adenso-Díaz et al., 1999, Şahin, 1999, Dorfman and Medanic, 2004 and the survey paper of Cordeau et al. (1998). Most of published results focus on scheduling trains at off-line planning level and deal with simplified models, in which stations have often unlimited capacity. As observed by Carey and Carville (2003), busy stations may be the most complex part of the network to schedule. This is the case in particular when dealing with metropolitan rail networks, in which scheduling decisions along the lines are typically simple, whereas traffic control at terminal stations is made more difficult because of the smaller area available and heavy traffic conditions.

This paper deals with the real time management of a metro rail terminus. The management and control of rail operations is usually based on off-line generated timetables for every train, and consists in operating in real time with strict adherence to these timetables. For a metro rail terminus, it mainly consists in routing incoming trains through the station and scheduling their departures according to the off-line timetable. However, when incoming trains are heavily delayed it is necessary to reschedule their departure times in order to provide service continuity and punctuality as much as possible. To a large extent, this task is carried out by human operators all over the world. A local area manager is in charge of setting routes and scheduling train departures with the objective of pursuing punctuality and regularity of the train service as much as possible. Computer support, when available, consists in most cases of a control panel describing the current situation of the network. On the other hand, there are several attempts to develop computerized decision support systems allowing a more efficient and easier management process.

There is a limited amount of published works on routing and scheduling trains at busy stations. Carey (1994) proposes a mixed integer programming (MIP) formulation for the problem, Carey and Carville (2003) provide heuristic algorithms that sequentially considers one train at a time and defines for it an arrival time, a departure time and a platform. If there is a conflict in the schedule, arrival or departure times for some train are increased until there is no longer a conflict. Carey and Crawford (2007) extend the single-station scheduling algorithms to find and resolve conflicts on lines between stations, and carry out an extensive computational study for a corridor with several stations.

Zwaneveld et al. (2001) consider the routing of trains through railway stations, given the detailed layout of the stations and a tentative timetable. They formulate the problem as a node-packing problem, and develop a branch and cut algorithm to determine a feasible path for as many trains as possible. Two additional objectives are considered in lexicographical order, namely the minimization of the number of shunting movements, and the maximization of train preferences for certain platforms or routes. Kroon et al. (1997) study the computational complexity of several variants of this routing problem.

In this paper we report on the implementation of scheduling algorithms for a real time Train Management System (TMS), able to route and schedule train movements through a metro line terminus. We report in particular on the results of a research project on the management of rail traffic at an underground metro rail terminus, in Italy. The scheduling algorithm developed within the project produces a plan of movements for all trains circulating in the terminus, with the objective of optimizing punctuality and regularity of the train service. According to railway practitioners preferences, punctuality is considered more important than regularity. In fact, respecting the off-line timetable would imply automatically respecting the regularity of train service, while the converse is not true. For this reason, in this paper the two objective functions are considered in lexicographical order.

One aim of the project is to move a step further in the direction of automating the train traffic control process, by enabling the TMS to implement most traffic control actions, without the authorization of the local area manager. To this aim, detailed optimization models are necessary, in order to guarantee that a solution, which is feasible for the optimization model, is always also physically feasible. Our optimization model incorporates a detailed description of the network topology, including railway signal aspects and safety rules. Differing from other authors, we explicitly include blocking constraints in the optimization model, which requires that a train, having reached the end of a track segment, cannot enter the subsequent segment if the latter is occupied by another train, thus preventing other trains from entering the former segment. In fact, omitting such constraints might cause deadlock situations in the terminus. To achieve this, we model the problem by using the alternative graph formulation of Mascis and Pacciarelli (2002), which allows modeling job shop scheduling problems with blocking constraints, and we adapt it to deal with specific constraints arising in the rail terminus.

The solution procedure described in this paper, consists in developing a plan of movements by first considering the punctuality function only. In a second step, the plan is improved by optimizing the regularity of train service, without affecting the punctuality objective function. More precisely, the overall decision problem is approached by dividing it into a routing/sequencing problem and a scheduling problem.

The routing/sequencing problem consists in assigning a path to each train in the station and in solving conflicts among trains by sequencing their movements with the objective of optimizing train punctuality. We model this problem by means of an alternative graph, and solve it with a fast heuristic, able to find a feasible solution within the strict time limits necessary for real time purposes. Our scheduling policy considers one train at a time, sequentially, and it is similar to the sequential heuristic of Carey and Carville (2003). A postprocessing phase allows to improve upon the first solution. In fact, the approach of scheduling one train at a time is analogous to the manual method of local area managers. As observed by Carey and Carville (2003), it performs well in practice, and makes the scheduling choices more understandable and acceptable to human experts.

Hence, the scheduling problem consists in defining an exact timing for train movements, without changing the routing and the sequencing, with the objective of improving the regularity of the service in terms of headway, and without degrading train punctuality. We show that this second problem can be solved at optimality with a polynomial time algorithm.

The paper is organized as follows. In Section 2, the routing/sequencing problem is introduced using the typical layout of a metro rail terminus. In Section 3, the alternative graph formulation of the problem is described. Sections 4 Routing and sequencing phase, 5 Scheduling phase deal with the routing/sequencing algorithm and the scheduling algorithm, respectively. In Section 6, our computational experience is reported. Some conclusions follow in Section 7. For the sake of clarity, Table 1 summarizes the notation we use throughout the paper.

Section snippets

Problem description routing/sequencing

The problem consists in defining a schedule in real time for all the trains circulating in the terminus, plus a given number of incoming trains. In particular, for each train it is necessary to define a routing in the terminus, and a departure time from the terminus in a given time horizon. The routing problem consists in assigning a path to each train from its current position, or from the station entrance, to the terminus exit point.

Fig. 1 describes a typical example of an Italian metro rail

Sequencing models

The sequencing problem described in this paper has been modeled as a blocking job shop problem by using the alternative graph model of Mascis and Pacciarelli (2002), a generalization of the disjunctive graph of Roy and Sussman (1964). With this model a set of jobs (trains) compete for the usage of a set of resources (routes). A train path through the station corresponds to an ordered list of resources to traverse, which can be modeled with a sequence of operations. Let R be the set of problem

Routing and sequencing phase

We next describe the routing algorithm. The main problem to face when assigning paths to incoming trains is that a train entering the station from platforms B1 or B2 may affect the departure of earlier trains, due to the incompatibility between routes. The basic idea of the routing algorithm is to build a first solution in a greedy fashion, by assigning to each train a sequence of routes and blocks, in such a way that the train can reach the exit with the smallest possible earliness/tardiness,

Scheduling phase

Given the precedence graph resulting from routing/sequencing phase, in the scheduling phase the departure times of the trains are adjusted in order to improve the regularity of the train service. The purpose of this phase is to optimize the headway without affecting the punctuality and without changing the routing and the sequencing determined in the previous phase. Hence, the scheduling problem consists in determining new departure times h1,h2,,h|N| for the trains such that the train

Computational results

This section deals with the performance of the routing/sequencing algorithm with respect to the punctuality objective function, and with the performance of the scheduling algorithm with respect to the regularity objective function. We report in particular on our experience with a practical case study, concerning the real time management of rail traffic at an Italian metro rail terminus (see Fig. 1). All the computational experiments are carried out by using a detailed simulator of the terminus,

Conclusion and future research

This paper addresses the real time scheduling of train service in a Metro rail terminus. Since the problem is bi-objective in nature, the train schedules are generated in two steps. The first step addresses the problem of routing and sequencing trains through the station with the objective of optimizing the punctuality, which it is solved with a fast heuristic. The second step addresses the problem of scheduling train departures with the objective of optimizing the regularity of train service,

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