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Assessment of flexible timetables in real-time traffic management of a railway bottleneck Andrea D’Ariano a , Dario Pacciarelli b, * , Marco Pranzo c a Department of Transport and Planning, Delft University of Technology, The Netherlands b Dipartimento di Informatica e Automazione, Universita ` Roma Tre, Via della Vasca Navale, 79, 00146 Roma, Italy c Dipartimento di Ingegneria dell’Informazione, Universita ` di Siena, Italy Received 7 December 2006; received in revised form 30 July 2007; accepted 31 July 2007 Abstract A standard practice to improve punctuality of railway services is the addition of time reserves in the timetable to recover perturbations occurring in operations. However, time reserves reduce line capacity, and the amount of time reserves that can be inserted in congested areas is, therefore, limited. In this paper, we investigate the new concept of flexible timetable as an effective policy to improve punctuality without decreasing the capacity usage of the lines. The principle of a flexible timetable is to plan less in the timetable and to solve more inter-train conflicts during operations. The larger degree of free- dom left to real-time management offers better chance to recover disturbances. We illustrate a detailed model for conflict resolution, based on the alternative graph formulation, and analyze different algorithms for resolving conflicts, based on simple local rules or global optimization. We compare the solutions obtained for different levels of flexibility and buffer time inserted in the timetable. An extensive computational study, based on a bottleneck area of the Dutch railway network, confirms that flexibility is a promising concept to improve train punctuality and to increase the throughput of a railway network. Ó 2007 Elsevier Ltd. All rights reserved. Keywords: Flexible timetable; Conflict resolution; Real-time train scheduling 1. Introduction The management of railway traffic in congested areas is usually based on off-line timetable design. Building a timetable may require several months, during which many variants are discussed in depth and all possible conflicts among trains are solved. In real-time, trains are managed with strict adherence to the timetable. How- ever, when train runs are perturbed (causing primary delays), signaling and route conflicts may arise with respect to the scheduled train paths (thus propagating to other trains as secondary delays). For this reason, time reserves are usually inserted in the timetable in order to reduce the effects of minor perturbations (Carey 0968-090X/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.trc.2007.07.006 * Corresponding author. Tel.: +39 0655173238; fax: +39 065573030. E-mail addresses: [email protected] (A. D’Ariano), [email protected] (D. Pacciarelli), [email protected] (M. Pranzo). Available online at www.sciencedirect.com Transportation Research Part C 16 (2008) 232–245 www.elsevier.com/locate/trc

Assessment of flexible timetables in real-time traffic management of a railway bottleneck

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Page 1: Assessment of flexible timetables in real-time traffic management of a railway bottleneck

Available online at www.sciencedirect.com

Transportation Research Part C 16 (2008) 232–245

www.elsevier.com/locate/trc

Assessment of flexible timetables in real-time trafficmanagement of a railway bottleneck

Andrea D’Ariano a, Dario Pacciarelli b,*, Marco Pranzo c

a Department of Transport and Planning, Delft University of Technology, The Netherlandsb Dipartimento di Informatica e Automazione, Universita Roma Tre, Via della Vasca Navale, 79, 00146 Roma, Italy

c Dipartimento di Ingegneria dell’Informazione, Universita di Siena, Italy

Received 7 December 2006; received in revised form 30 July 2007; accepted 31 July 2007

Abstract

A standard practice to improve punctuality of railway services is the addition of time reserves in the timetable to recoverperturbations occurring in operations. However, time reserves reduce line capacity, and the amount of time reserves thatcan be inserted in congested areas is, therefore, limited. In this paper, we investigate the new concept of flexible timetable asan effective policy to improve punctuality without decreasing the capacity usage of the lines. The principle of a flexibletimetable is to plan less in the timetable and to solve more inter-train conflicts during operations. The larger degree of free-dom left to real-time management offers better chance to recover disturbances. We illustrate a detailed model for conflictresolution, based on the alternative graph formulation, and analyze different algorithms for resolving conflicts, based onsimple local rules or global optimization. We compare the solutions obtained for different levels of flexibility and buffertime inserted in the timetable. An extensive computational study, based on a bottleneck area of the Dutch railway network,confirms that flexibility is a promising concept to improve train punctuality and to increase the throughput of a railwaynetwork.� 2007 Elsevier Ltd. All rights reserved.

Keywords: Flexible timetable; Conflict resolution; Real-time train scheduling

1. Introduction

The management of railway traffic in congested areas is usually based on off-line timetable design. Buildinga timetable may require several months, during which many variants are discussed in depth and all possibleconflicts among trains are solved. In real-time, trains are managed with strict adherence to the timetable. How-ever, when train runs are perturbed (causing primary delays), signaling and route conflicts may arise withrespect to the scheduled train paths (thus propagating to other trains as secondary delays). For this reason,time reserves are usually inserted in the timetable in order to reduce the effects of minor perturbations (Carey

0968-090X/$ - see front matter � 2007 Elsevier Ltd. All rights reserved.

doi:10.1016/j.trc.2007.07.006

* Corresponding author. Tel.: +39 0655173238; fax: +39 065573030.E-mail addresses: [email protected] (A. D’Ariano), [email protected] (D. Pacciarelli), [email protected] (M. Pranzo).

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and Kwiecinski, 1995). Time reserves can be divided into recovery time and buffer time. The former is obtainedby planning the travel time of the trains strictly larger than their minimum travel time, attained when travelingat maximum speed. The latter is a time reserve over the minimum separation time between consecutive trainpaths. Recovery times are introduced mainly to reduce the primary delays, while the latter are defined to limitthe propagation of secondary delays.

Larger time reserves allow increasing train punctuality, but reduce the capacity usage of the lines, i.e. themaximum number of trains which may be operated through a line per time period. In congested areas, theamount of time reserves that can be inserted in the timetable is limited. In fact, despite the big effort spentin the off-line development of timetables, in congested areas deviations from the timetable do frequently hap-pen, thus often requiring the restoration of a feasible schedule. As an example of this situation, Fig. 1 showstwo time-space diagrams reporting (a) the scheduled and (b) the actual behavior of the railway traffic observedin March 2003 in a congested part of the Dutch railway network, i.e. the line from the Riekerpolder junctionto the Schiphol and Hoofddorp stations. The vertical axis shows a 15 min time interval, from 07:55 to 08:10,while the horizontal axis reports the distance from Riekerpolder to Hoofddorp. The diagonal lines in the dia-gram represent train positions during a given time period. Diagram (a) is the same for all the days of theobserved period, while Diagram (b) shows the daily observed behavior of the trains.

Schaafsma (2001) proposed the new concept of Railway Dynamic Traffic Management for improving thetimetable robustness (i.e. the resilience to disturbance in operation), without decreasing the capacity of thelines. The basic idea is to keep traffic flowing in the bottleneck by avoiding unnecessary waiting time. Thisconcept has been developed within the Dutch railway companies in the last years (Middelkoop and Hemelrijk,2005; Schaafsma, 2005; Van den Top, 2005; Middelkoop and Loeve, 2006; Schaafsma and Bartholomeus,2007) resulting in a number of principles for the management of congested areas by planning less in theoff-line phase, and by postponing the resolution of possible conflicts among trains to the real-time traffic man-agement. These principles include flexibility of routing, arrival/departure times and sequencing. This informa-tion should be partially defined in the off-line timetable and then fixed in real-time, based on the status of thenetwork and on the current train positions. Schaafsma and Bartholomeus (2007) report on the first implemen-tation of some of these concepts in the Schiphol bottleneck, in the Dutch railway network. In their paper, theauthors limits the assessment to routing flexibility and to train resequencing using the First Come First Servedrule.

In this paper, we evaluate the benefits of using flexible arrival/departure times and advanced schedulingalgorithms for conflict resolution. In what follows, we refer to a rigid timetable to mean the traditionalone, in which all conflicts among trains are solved off-line and time reserves are used to absorb minor pertur-bations. We refer to a flexible timetable to denote a timetable in which less details are fixed in the plan andmore control decisions are left to the dispatcher. The concept of flexible timetable is described in Section 2.

Fig. 1. (a) Time-space diagram of the timetable, (b) time-space diagram of the observed trains behavior during March 2003 (Source:ProRail).

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In Section 3, we formally introduce the conflict resolution problem, its alternative graph formulation and theresolution algorithms. In Section 4, we report on the assessment of a flexible timetable under different conflictresolution algorithms and degrees of flexibility. We also evaluate the effects of flexibility in terms of through-put, showing that similar level of performance in terms of train delays can be attained by increasing the flex-ibility while reducing the buffer time, which corresponds to increase the throughput of the network.

2. Flexible timetable

The starting point of dynamic traffic management (Schaafsma and Bartholomeus, 2007) is to allow morefreedom to the real-time control, by relaxing some of the timetable specifications. According to this principle,a flexible timetable defines: (i) a set of feasible platform tracks for each train and for each station; (ii) timewindows of [minimum, maximum] arrival/departure times, a larger time window corresponding to havingmore flexibility; and (iii) a provisional (or even partial) order of trains at overtakes and junctions. The com-plete definition of these three elements is then postponed to real-time operations: (i) choice of the platform/passing track at each station; (ii) choice of the exact train arrival and departure times at each platform andrelevant timetable points of the network; and (iii) choice of the train orders at overtakes and junctions.

Since point (i) has been evaluated in practice in Schaafsma and Bartholomeus (2007), in this paper, wefocus on points (ii) and (iii), and investigate the effectiveness of flexible arrival/departure times as a strategyfor improving the timetable robustness under different resequencing policies. The idea is that a flexible time-table guarantees a better use of time reserves with respect to a rigid one, provided that advanced computerizedtraffic management systems are available to support human dispatchers in their tasks. In this case, a determin-istic plan of train operations is off-line developed for the less loaded traffic area, while this is defined in real-time for the congested area, based on the actual traffic situation in the railway network. A flexible timetablewould be similar to the one shown in Fig. 2, in which three trains A, B and C travel through a congested area.In this case, a deterministic plan of train operations is off-line developed for the outer low traffic area.

Fig. 3 describes how time windows are operatively used. We consider a train running on a line with threestations and a timetable with distributed time reserves. The train arrives at station 2 at time A with a largeearly arrival. In the rigid timetable, the time period from A to C is unused since the rigid arrival time is C.Thus, the train leaves the station at time E and arrives to station 3 at time G. With a flexible timetable, thearrival time at station 2 is the time window [B,C], and the departure time becomes the time interval [D,E].In this case, the train dwell time starts at time B, the train leaves the station at time D and reaches station3 at time F, if the headway with respect to the preceding train contains sufficient margin. This example showsthat the use of flexibility permits to preserve some of the previous time reserve (interval BC), although not all(interval AB is lost).

The introduction of time windows makes necessary to produce two different timetables. The operational

timetable, used by railway managers to control the railway traffic includes both the minimum and maximumarrival and departure times. The public timetable available to passengers includes the maximum arrival time

Fig. 2. An example of flexible timetable.

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Fig. 3. Operative use of the flexibility concept.

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and the minimum departure time only. In Fig. 3, [B,C] is the arrival time interval and C is the arrival timeavailable to passengers. Similarly, [D,E] is the departure time interval and D is the passenger departure time.Clearly, a drawback of the flexible timetable is that each travel time between two stations is increased by thesize of the time window, for example the travel time from station 2 to station 3 is [D,G] that is equal to thetravel time of the rigid timetable [E,G] plus the time window size [D,E]. The higher travel times should becompensated by an improved punctuality, due to the greater possibility of managing traffic in real-time,and making a better use of time reserves.

3. Conflict resolution problem

The real-time process of defining a feasible train schedule is called train dispatching or conflict resolution.The usual approach to conflict resolution is to identify and resolve potential conflicts one at a time. Decisionsupport systems are a prerequisite for online dispatching of train operations. Among the published results, wecite the papers of S�ahin (1999), Dorfman and Medanic (2004), Dessouky et al. (2006), and the survey papers ofCordeau et al. (1998), Oh et al. (2004), Tornquist (2005) and Ahuja et al. (2005).

S�ahin (1999) studies the real-time conflict resolution problem on a single-track railway. The problem is for-mulated as a job-shop scheduling problem with the objective of minimizing average secondary delays. Inter-train conflicts are detected and solved in the order they appear by using a look-ahead strategy. Dorfman andMedanic (2004) also address a conflict resolution problem on a single-track railway. They propose a discrete-event model for scheduling trains and preventing deadlocks. A greedy strategy, similar to human dispatcherbehavior, allows to quickly handle timetable perturbations and to obtain satisfactory sub-optimal schedules.Dessouky et al. (2006) propose a train dispatching system based on a branch and bound scheduling procedure.The resulting approach is able to find exact solutions for a single-track network with 14 train routes withintwo hours of computation time. The development of effective heuristics is addressed for real-time purposes.

In this section, we model the conflict resolution problem by means of the alternative graph formulation ofMascis and Pacciarelli (2002), and illustrate the formulation and the benefits of flexibility with a small exam-ple. Finally, we briefly describe the algorithms used to solve the problem.

3.1. Mathematical formulation of the problem

Railway networks, under the fixed block signaling system constraints, are organized into track segmentscalled block sections. Each block section can host at most one train at a time. A conflict occurs whenever

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two or more trains require the same block section at the same time. In the management of railway networks,the problem of finding a conflict-free schedule is faced by railway practitioners both in the timetable develop-ment and during operations.

The conflict resolution problem (CRP) is the real-time problem to find a conflict-free schedule compatiblewith the real-time status of the network and such that trains arrive and depart with the smallest possible delay.It should be noted that the definition of the conflict resolution problem in this study is slightly different fromthe one mostly used in practice. In fact, while the standard practice is to detect and solve conflicts one at a time,in our definition the aim of conflict resolution is to develop a new conflict-free plan, i.e. we consider the overallconflict resolution problem as a unique problem to be solved in the best way.

The combinatorial structure of CRP is similar to that of blocking job shop scheduling problem with no-swap allowed (Mascis et al., 2001, 2007). This implies that even answering to the question of whether a con-flict-free schedule compatible with the real-time train positions exists or not is an NP-complete problem, asproved by Mascis and Pacciarelli (2002). In the usual definition of the job-shop scheduling problem, a jobmust be processed by a prescribed sequence of machines, and each machine can process one job at a time.Blocking constraint represents the fact that a job, after the completion on a machine, cannot leave it untilthe subsequent machine becomes available for its processing. No-swap constraint imposes that a job can moveto the next machine only strictly after that the previous job left it, thus forbidding two jobs to exchange theirrespective machines. The processing of a job on a machine is called an operation. In terms of conflict resolutionproblem, jobs correspond to trains and machines to block sections. The processing time of an operation cor-responds to the running time of the associated train on the corresponding block section. Since a block sectioncannot host two trains simultaneously, whenever two trains require the same block section at the same time,there is a conflict. Solving the conflict corresponds to defining a processing order between incompatibleoperations.

We next formulate CRP with an alternative graph. An alternative graph consists of a triple G ¼ ðN ; F ; AÞ,where N is a set of nodes, F is a set of fixed arcs and A is a set of pairs of alternative arcs. A dummy node 0,called start node, precedes all other nodes, and a dummy node n, called finish node, follows all other nodes.

The passing of train Ti through a particular block section Bj, i.e., an operation, is associated with a node ij

of the alternative graph. The route of train Ti is, therefore, a chain of nodes. An arc (ij,hk) represents a pre-cedence relation between two nodes ij and hk. The starting time Thk of operation hk is constrained to beThk P Tij + pijhk, where pijhk is the weight of arc (ij,hk). Train routes are represented with fixed arcs (ij, ik),corresponding to the precedence constraints between consecutive operations. The arc weight pijik indicatesthe running time of the train Ti through the block section Bj. Since a block section cannot host two trainsat the same time, whenever two jobs require the same block section, there is a potential conflict. In this case,a processing order must be defined between the incompatible operations, and we model this by introducing inthe graph a suitable pair of alternative arcs (see Fig. 4). Each alternative arc models a possible precedence rela-tion between two operations. This arc is weighted with the setup time, which is the time between the entranceof the head of a train in a block section and the exit of its tail (the last axle) from the previous one, plus addi-tional time margins to release the occupied route (see, e.g. Nie and Hansen, 2005).

A selection S is a set of arcs obtained from A choosing at most one arc from each pair. In a selection S, GðSÞindicates the graph (N,F [ S). We denote the value of a longest path from node ij to node hk in GðSÞ bylS(ij,hk). A feasible solution is obtained by solving each possible conflict among trains. In terms of alternativegraph formulation, this solution corresponds to the selection of an arc for each alternative pair in such a way

Fig. 4. The alternative graph formulation for two trains at a junction.

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that the resulting graph has no cycles. In fact, a cycle represents an operation preceding itself, which is infea-sible (i.e. it represents a deadlock situation).

The scheduling problem CRP is to assign a starting time Tij to each node ij 2 N, i.e. to plan the exact time inwhich each train will enter a block section, such that all fixed precedence relations, and exactly one for eachpair of the alternative precedence relations, are satisfied. A feasible assignment of starting times to nodes isTij = lS(0, ij), for ij 2 N.

The objective of our scheduling problem is to minimize the starting time of the dummy finish nodeTn = lS(0, n). We next explain this objective function in terms of the CRP. With a flexible timetable, for eachtrain Ti stopping at platform Sj, a maximum arrival time aij, a minimum dwell time sij and a minimum depar-ture time dij are associated. Ti is not allowed to depart from Sj before dij and is considered late if arriving afteraij. We define the total delay of Ti at Sj as the difference between its actual arrival time Tij and aij. We divide thetotal delay into two parts as follows. In a pre-processing phase, we first compute the earliest possible arrivaltime sij of Ti at Sj, computed when the train travels at maximum speed and by disregarding the presenceof other trains. If sij > aij, then the quantity sij � aij is an unavoidable delay that cannot be recovered byreal-time rescheduling of train operations. We call max{0, sij � aij} the primary delay of Ti. The quantitymax{0, Tij � max{sij,aij}} is called secondary delay, which is due to the resolution of conflicts with the othertrains in the network. We do not take into account the primary delays in the scheduling phase, and definea modified due date for Ti at Sj as max{sij,aij}. In the alternative graph, the reduction of the secondary delayscorresponds to the minimization of the maximum lateness when using these modified due dates.

The alternative graph formulation of the arrival/departure times, as shown in Fig. 5, requires the introduc-tion of two nodes for each train stopping at a platform Sj. Node ij represents the event that train Ti starts thedwell time at platform Sj, while node ik represents its departure, i.e. its entrance in the following block sectionBk. A fixed arc (ij, ik), weighted with the minimum dwell time sij, takes into account the minimum stoppingtime of Ti at Sj. We represent the minimum departure time as an arc from the dummy start node 0 to nodeik, weighted with the minimum departure time dij. The maximum secondary delay can be computed as the dif-ference between Tij and the modified due date max{sij,aij}. We model it with an arc from node ij to the dummyfinish node n, weighted � max{sij,aij}, that represents the contribution to our objective function. With thismodel minimizing lS(0,n) corresponds to minimizing the maximum secondary delay.

We define the quantity d = mini,j{dij = aij + sij � dij} as the timetable flexibility. In a rigid timetable, d = 0holds. In our computational experiments, we study the effects of flexibility starting from a rigid timetable. Weconstruct a flexible timetable from a rigid one by increasing the flexibility d, from 0 on. To ensure fairness inthe comparison, we set the maximum arrival times aij equal to the arrival times of the rigid timetable, sinceincreasing these values would immediately result in reduced train delays without changing their actual arrivaltimes. Thus, increasing the flexibility d corresponds to setting the minimum departure times dij equal to therigid departure times Dij minus d (see Fig. 5). In other words, in our experiments, the effect of a larger flex-ibility d simply corresponds to the possibility for a train to leave a station earlier than scheduled in the rigidtimetable.

3.2. Illustrative example

In this subsection, we illustrate the alternative graph formulation of CRP for a small example with threetrains and the railway network depicted on the top of Fig. 6. In the network, there are seven block sectionsand a station. The timetable is shown in Fig. 7. At time t = 0, there are three trains in the network. Train A is a

Fig. 5. The alternative graph formulation of a scheduled stop.

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Fig. 6. A railway network and the alternative graph formulation.

Fig. 7. Optimal solutions of CRP for three values of d.

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passenger train, going from block section 1 to block section 7, and stopping at station S. Train B is a passengertrain going from block section 2 to block section 7 with a stop at the station. Finally, train C is a freight traingoing from block section 4 to block section 7.

The alternative graph is reported at the bottom of Fig. 6. Each node of the alternative graph represents theevent that a train enters a block section and is denoted by the pair (train, block section). For example, A3 andB3 are different nodes referring to block section 3. Each alternative pair is associated with the usage of a com-mon block section. Trains A and B share block sections 2, 3 and 7. Train C shares only block section 7 withtrains A and B. For each couple of trains sharing a block section there is a pair of alternative arcs representingthe decision to be taken on the precedence relation between the trains. Arcs (0, A2), (0,B3) and (0, C5) repre-sent the time needed for each train to reach the end of the current block section, which is the off-line entrancein the next block section plus the entry delay qA, qB, and qC, respectively. Arcs (0, A7) and (0, B7) represent theminimum departure time from the station. Finally, arcs (AS, n), (Aout, n), (BS, n), (Bout, n) and (Cout, n)

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represent the modified due dates of the trains at the station and at the exit point of the network (end of section7). The weight of all horizontal arcs corresponds to the running time of the trains on the block sections. Forthe sake of simplicity, all the setup times (alternative arcs weights) are set equal to zero. Note that the initialposition of train B implies that train A is not allowed to precede train B on block sections 2, 3 and 7, and we,therefore, have the selected alternative arcs (B3,A2), (B7,A3) and (Bout,A7).

We next describe the timetable of Fig. 7. The values Tij, for j = 1, . . . ,out, are associated with the time trainTi should enter block section Bj or Ti should start the dwell time at the stop. In columns 2, 3 and 4, the time-table for the three trains is reported. These are the off-line scheduled times at which each train should entereach block section. The off-line exit order is ACB. In columns 5–13, three different optimal solutions ofCRP are reported for three different values of d. In the three cases, train A enters the network with a delayqA = 47, train B is on time (qB = 0) and train C enters the network with a delay qC = 17. The flexibility variesfrom d = 0, i.e. the rigid timetable, to d = 10 and d = 25. The optimal output sequence is BCA for d = 0 andd = 10, while this is BAC for d = 25. The primary delay, for each train and each relevant point, is computed asthe shortest path in the graph of Fig. 6 disregarding the presence of other trains (without considering the alter-native arcs). Precisely, sAS = sAout = 34, sBS = sBout = 0 and sCout = 17. The maximum [average] secondarydelay in the three cases is 41, 31 and 18 [is 19, 13 and 10] for d = 0, d = 10 and d = 25, respectively.

The main flexibility effect is to allow train B (on time) to depart earlier from the station and to enable trainA, that is late, to arrive earlier at the stop, thus also leaving the station with a smaller delay. For increasingflexibility values, train A is able to recover a large part of its delay, thus making more convenient to restore theoff-line order between train A and C. This example shows that flexibility is effective only if there are earlytrains hindering subsequent late trains. In such cases, larger flexibility enables early trains to anticipate theirmovements, thus enabling subsequent late trains to reduce their respective delays. This fact shows that a nec-essary condition for flexible timetables to be useful in practice is that sufficient recovery time is included in thetimetable. If the timetable is designed in such a way that no train can reach a station earlier than scheduled inthe rigid timetable, then allowing larger flexibility cannot produce significant effects in practice.

3.3. Conflict resolution algorithms

In this section, four algorithms for CRP are described, based on the formulation in Section 3. We considerthree greedy heuristics and a branch and bound algorithm.

The first heuristic is the First Come First Served (FCFS) dispatching rule, usually adopted in railway prac-tice, which is simply to give precedence to the train arriving first at a block section.

The second heuristic, called First Leave First Served (FLFS), is as follows. When two trains require thesame block section, we first compute the time needed to each train to enter and traverse the block section.Then, precedence is given to the train that is able to leave the block section first.

The third heuristic is the greedy algorithm AMCC described by Mascis and Pacciarelli (2002). This is toforbid one alternative arc at a time, which would introduce the largest delay. Precisely, AMCC enlarges aselection S, initially empty, at each step by choosing a pair of unselected alternative arcs ((ik,uj), (uk, ij))(see Fig. 4) such that the quantity lS(0, ik) + pikuj + lS(uj,n) is the maximum among all the unselected alterna-tive arcs. Selecting arc (ik,uj) would increase lS(0, n) more than any other unselected alternative arc. For thisreason, AMCC forbids this arc and selects its alternative, i.e. arc (uk, ij) is added to S.

The branch and bound algorithm is described in D’Ariano et al. (2007), we, therefore, only summarize itsmain components. The initial solution is the best solution found by the heuristics described above.

The lower bound used to evaluate partial selections is an adaptation of the single machine Jackson’s Pre-emptive Schedule (JPS) of Carlier (1982) to the railway scheduling problem. This lower bound is computed foreach block section.

Implication rules are a key tool to speed up branch and bound algorithms, since they reduce the number ofbranches needed to prove the optimality of a solution. The idea is to prove that, given a partial selection S andan unselected pair of alternative arcs ((ik,uj), (uk, ij)) 2 A, no improvement of the current best solution is pos-sible if arc (uk, ij) is selected. Thus, arc (ik,uj) can be added to S. We identified two kinds of implication rules:dynamic and static. Static rules are computed off-line on the basis of the topology of the railway network.Dynamic rules are computed during the execution of the solution procedure, based on the current selection

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S and on the values lS(0, ik) and lS(ik,n), for all ik 2 N. The rationale of these rules is that, given an upperbound UB to the optimum, if a lower bound to any extension of S [ {(uk, ij)} is larger or equal than UB, then(ik,uj) can be added to S. Dynamic implication rules have been successfully used in solving the job-shop prob-lem, both with infinite capacity buffers (Carlier and Pinson, 1989, 1994) and with blocking constraints (Mascisand Pacciarelli, 2002). When dealing with a large number of machines, as in CRP, the dynamic rules of Mascisand Pacciarelli (2002) may require a significant computational effort when using the complete matrix of lon-gest paths lS(ij,hk). We therefore do not include them in our branch and bound. We only use the rulesdescribed by Carlier and Pinson (1994), which provide an efficient algorithm for finding a maximal set of arcsthat can be immediately selected. We also use the static implication rules introduced by D’Ariano et al. (2007)for the resolution of train scheduling problems.

The branching scheme of our algorithm is binary and consists of selecting an alternative arc pair on thebasis of the AMCC rule. The search strategy is a mixed strategy in which we alternate some repetitions ofthe depth-first visit to a selection of the most promising node, i.e., the open node of the branch and boundtree with smallest lower bound.

4. Computational experience

In this section, we assess flexible timetable in combination with different conflict resolution algorithms. Wereport on our computational experience with a large set of experiments on randomly generated perturbations.The influence of timetable flexibility has been tested on practical size instances derived from the dispatchingarea of Schiphol (Fig. 8). This is a bottleneck area of the Dutch railway network that includes the under-ground station of Schiphol, beneath the international airport of Amsterdam.

The Schiphol railway network is composed of 86 block sections and includes 16 platforms and two trafficdirections. Trains enter/leave the network from/to ten access points: the high speed line HSL (block sections20 and 59), the station of Nieuw Vennep (Nvp) (block sections 33 and 70), the yard of Hoofddorp station(HfdY) (block sections 18 and 71), and the two stations Amsterdam Lelylaan (Asdl) (block sections 46 and50) and Amsterdam Zuid WTC (Asdz) (block sections 77 and 45). Between Schiphol and Amsterdam, thereis the Riekerpolder junction (block sections 26, 27, 28, 29, 85 and 75 in Fig. 8). Specifically, the Hoofddorp

Fig. 8. The Schiphol dispatching area.

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station (Hfd) has two platforms, which are dedicated to freight trains, while the Schiphol station (Shl) has sixplatforms, which are dedicated to passenger and freight trains. The two traffic directions are largely indepen-dent except around Amsterdam Lelylaan station (block sections 38, 40, 42, 44, 46, 47, 48, 49 and 50) and at theborder of Hoofddorp yard (block section 4).

In a first set of experiments, we considered the typical rolling stock characteristics and a provisional Dutchtimetable for year 2007 (Hemelrijk et al., 2003). This is a rigid timetable with a limited amount of time reservesto recover delays, due to the high number of trains that is not far from the network capacity saturation. Thistimetable is hourly, cyclical and contains 27 trains per direction, for a total of 54 trains. It is worthwhileobserving that the actual number of trains per hour scheduled at Schiphol during year 2007 is 20 trains perdirection, but we chose this more challenging provisional timetable to assess the effects of flexibility undermore severe traffic conditions.

Starting from the rigid timetable for 2007, we construct a flexible timetable by replacing the scheduled arri-val/departure times with maximum arrival/minimum departure times. To ensure fairness when comparing theresults for different level of flexibility, we set the maximum arrival times constant and equal to the rigid arrivaltimes in all experiments. In fact, increasing these values would immediately result in reduced train delays with-out changing their actual arrival times.

We study the network by simulating different real-time traffic conditions, and by rescheduling train oper-ations for different kinds of initial perturbations. We model an initial perturbation by assigning an input delayto some trains, i.e. by letting some trains enter late into the network. In total, we generate 60 perturbationschemes. Each instance is generated by varying the number of delayed trains from 7 to 27 and choosingthe delayed trains randomly within the first half hour of the cyclical timetable. For each delayed train, we gen-erate a random value of the input delay on the basis of the Gaussian or the uniform distribution in a range[0,max], for max by varying from 200 to 1800 s. In our computational experience, we do not introduce prior-ities among trains and do not consider other perturbations, such as temporary unavailability of someresources (trains, stations, block sections, etc.). For each instance, we compute four CRP solutions by usingthe three initial heuristics and the branch and bound algorithm described in Section 3.3, truncated after 120 sof computation. The algorithms are implemented in C++ language and executed on a laptop equipped with a1.6 GHz processor.

Our experiments are organized in two subsections. In Section 4.1, we evaluate the solutions produced bydifferent scheduling algorithms in terms of maximum and average secondary delays for varying the level offlexibility. In Section 4.2, we analyze the benefit of flexibility with four timetables with different amount ofbuffer time and throughput. In this second set of experiment we used only our branch and bound algorithm.

4.1. Analysis of different CRP algorithms

In the first set of experiments, we evaluate the effectiveness of four CRP algorithms for varying the level offlexibility. Given a value of the flexibility d, we set the minimum departure times equal to the rigid departuretimes minus d. Five timetables are generated for d = {0,30,60,90,120}, where d = 0 corresponds to the rigidtimetable. For each of the 60 perturbation schemes, we generate an instance of CRP for d = {0, 30,60,90,120},thus yielding a total of 300 CRP instances.

Fig. 9 shows the maximum and average secondary delays obtained by the four algorithms for different val-ues of the flexibility d. Each value reported in the figure is the average over the 60 perturbation schemes. Forsome instances, one or more of the three initial algorithms may fail to find a feasible solution. In order to com-pare the different results, we consider a failure penalty of 10 min for the maximum secondary delay and 1 min-ute for the average secondary delay.

As expected, all the four algorithms take advantage of an increasing flexibility, even if the three initial heu-ristics exhibit less consistent behaviors. The limited branch and bound clearly outperforms the other algo-rithms. This algorithm finds proven optimal solutions in 297 cases out of 300 with the time limit of 120 s.The average number of branches is 116 while and the average computation time is 1.94 s. In the three openinstances, the limited branch and bound algorithm is able to improve the solutions found by the three initialheuristics. For all the heuristics, the deviation from the lower bound is over 85%, while this deviation decreases

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Fig. 9. Maximum and average secondary delays for the four algorithms.

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to 26% when using the limited branch and bound algorithm. Moreover, the best solution is always foundwithin the first 30 s of computation.

The First Come First Served (FCFS) and the First Leave First Served (FLFS) rules, being the most ‘‘local’’rules, give the worst results. Both maximum and average secondary delays are more than double with respectto the values found by the limited branch and bound. FCFS is unable to find a feasible solution 10 times out of300, FLFS 14 times out of 300, AMCC 23 times out of 300 instances. However, when AMCC finds a feasiblesolution, this algorithm is very close to the optimal solution, and almost always better then the solutions foundby FCFS and FLFS. The computation time of the three heuristics is significantly shorter than for the limitedbranch and bound algorithm. FCFS only needs 0.01 s on average to compute a solution, FLFS requires 0.02 sand AMCC 0.5 s, which make these heuristics potentially useful if very strict time limits are imposed for com-puting a solution.

Fig. 10 shows the maximum and average secondary delays obtained by the limited branch and bound, asobserved at the stops in Hoofddorp platforms, in Schiphol platforms and at the exit points of the dispatchingarea. It can be observed that Hoofddorp station is less critical than Schiphol station. Besides, some trainsgather an additional delay from Schiphol to the exit of the dispatching area.

Increasing the flexibility from 0 to 120 s causes a decrease of the delay at all stations/exit points of morethan 30% both for the maximum and average secondary delays. However, the flexibility contribution is incon-stant. When increasing the flexibility from 0 to 30 s, the maximum and average secondary delays decrease ofaround 10%, while when increasing the flexibility from 90 to 120 s, the effect is a decrease of around 5% forboth values.

This set of experiments shows that the use of advanced optimization algorithms for conflict resolution isimportant to obtain significant benefits in terms of maximum and average secondary delays. In fact, all algo-rithms are able to exploit the flexibility in order to reduce delays, but the branch and bound algorithm makesthe best use of flexibility when compared to the heuristic solution.

4.2. Analysis of different timetables

In this subsection, we study the effects of flexibility in four timetables, differing each other for the level ofbuffer time. In this second set of experiments we use the same set of 60 initial perturbations of the previoussubsection, and study only the CRP solutions provided by the limited branch and bound algorithm. We refer

Fig. 10. Maximum and average secondary delays at the two intermediate stations and exit points.

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Fig. 11. Maximum and average total delays for the four timetables.

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to the original rigid timetable as the timetable B. Removing all the buffer time from timetable B, we obtain aless robust timetable A in which the 27 trains per direction are compressed in approximately 3200 s, whichcorrespond to a throughput of 31 trains per hour per direction. This is also the maximum number of trainsthat can circulate in the network within a hour when all trains run at their scheduled speed. However, time-table A still contains a limited amount of time reserve as recovery time. The largest recovery time is 240 s for atrain which requires a minimum of 790 s to traverse the network, plus 240 s of cumulated minimum dwell timeat the two intermediate stations. We also developed two new timetables, called C and D, obtained from A bydoubling and triplicating the amount of buffer time of timetable B. This corresponds to a throughput of 25and 23 trains per hour per direction in timetable C and D, respectively. In the four timetables, the speed profile(and the recovery time) of each train is kept as similar as possible to the original timetable, so that the amountof primary delay associated to a certain entry delay is approximately the same when varying the timetable andthe level of flexibility. Specifically, the maximum primary delay is 544, 534, 533, and 530 s for timetable A, B, C

and D, respectively, on average over the 60 instances and four timetables, while the average primary delay isapproximately 67 s for timetables A, B and C, and 66 s for timetable D.

To assess the timetable robustness, for each perturbation scheme we computed a solution with the limitedbranch and bound algorithm, for varying the level of flexibility from 0 to 180 s. The results are shown inFig. 11 where the maximum and average total delay are reported for each timetable (A, . . . ,D) and each valueof flexibility (d 2 {0, 30,60,90,120,150,180}). Each value in the plot is the average result over the 60 pertur-bation schemes. As expected, the maximum and average total delays in the case with larger buffer time aresignificantly smaller. However, increasing the flexibility from 0 to 30 s is always more effective than from150 to 180 s.

From Fig. 11, it is interesting to compare the effects of different levels of buffer time and flexibility againstthe total delay (maximum or average) and the throughput. In fact, timetables with different levels of buffertimes and flexibility exhibits similar maximum or average delays (see, e.g. timetable A with d = 90 and B withd = 0, or B with d = 120 and D with d = 30). Since different buffer time corresponds to different throughput,we can evaluate the flexibility in terms of equivalent buffer time and increased line capacity. Specifically, sim-ilar values of maximum delay are obtained by adding four trains per hour and increasing the flexibility of 90 s,while similar values of average delay are obtained by adding two trains per hour and increasing the flexibilityof 90 s. Clearly, no general conclusion can be drawn from this limited set of experiments, but the results clearlyshow that timetable flexibility combined with advanced real-time train scheduling algorithms are promisingtools that merit further investigation to increase the actual capacity of the lines and to improve the trainpunctuality.

5. Conclusions

In this paper, we discussed the concept of flexible timetables in the management of a complex railway net-work. We provided a detailed model of the conflict resolution problem, which can be used to develop fast andeffective solution algorithms. Three greedy heuristics and a branch and bound algorithm for conflict resolutionhave been developed and tested on a bottleneck area of the Dutch railway network.

From our computational experience, several conclusions can be drawn.

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(1) Flexible timetables are preferable to rigid ones, since flexibility offers more freedom to solve conflicts andincreases punctuality without decreasing the throughput.

(2) When comparing timetables with different values of buffer time and flexibility and similar behavior interms of maximum or average delay, flexible timetables exhibit a larger throughput, thus making a betteruse of existing capacity with respect to rigid timetables.

(3) The use of advanced optimization algorithms for conflict resolution improves the benefits of flexibletimetables in terms of delay minimization.

As for future research directions, a number of issues remain that need further development. In fact, up tonow the three principles of dynamic traffic management have been evaluated separately and on a single casestudy. Hence, there is a need to test the three principles simultaneously on different networks and timetables toassess the potential and limitations of these principles more in general. It should also be interesting to includeadditional constraints in the model, such as inter-train connections, constraints to the maximum allowed sec-ondary delay, and others. Adding new constraints to the alternative graph model is not difficult, but the effectsof such constrains still have to be evaluated in terms of quality of heuristic solutions, computational effort ofthe branch and bound algorithm, effects of flexibility and so on. A further, new research direction shouldaddress the process of designing flexible timetables: What is the best amount of flexibility required in practice?How to distribute time margins in a flexible timetable? Which is the best trade-off between the amount of buf-fer time, recovery time and flexibility? Where should the buffer time be located? New concepts are necessary toanswer such questions, but deeper analysis of these points will be useful from theoretical as well as from prac-tical points of view.

Acknowledgement

This work was partially supported by ProRail (The Netherlands), and by the European Commission, Grantnumber IST-2001-34705, project ‘‘COMBINE2- enhanced COntrol center for fixed and Moving Block sIgNal-ling systEm 2’’. The authors wish to thank the anonymous referees for their helpful comments.

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