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informs ® Vol. 37, No. 2, March–April 2007, pp. 133–142 doi 10.1287/inte.1060.0246 issn 0092-2102 eissn 1526-551X 07 3702 0133 ©2007 INFORMS

The Schedules Track Maintenance to Improve Track Workers’ Safety

J. I. van Zante–de Fokkert Centre for Quantitative Methods, PO Box 414, 5600 AK , The Netherlands, [email protected] D. den Hertog University, PO Box 90153, 5000 LE Tilburg, The Netherlands, [email protected] F. J. van den Berg, J. H. M. Verhoeven ProRail bedrijfsplanning, PO Box 2038, 3500 GA , The Netherlands {[email protected], [email protected]}

After several fatal accidents, improving the safety of rail-track workers became a political priority in The Nether- lands. Maintaining the track was one of the most dangerous jobs in the country. ProRail divides its infrastructure into working zones to be taken out of service during maintenance. We developed a two-step method of con- structing a four-week maintenance schedule in which each working zone of the main lines is closed to trains at night exactly once. The alterations in trains’ departure and arrival times are within the restrictions imposed by train operators. Workers have accepted the resulting schedule, which provides them with a manageable maximum workload per night. Such a schedule is unique in Europe. It has been in operation since 2000 and has clearly proven its benefits. Key words: facilities–equipment planning: maintenance, replacement; government: services. History: This paper was refereed.

fter railroad track workers suffered several fatal support decision making with respect to railway Aaccidents, the Dutch government decided to pro- infrastructure (Harris and Ramsey 1994, Carey 1994, tect track workers in space and time while they were Hensher 1997, Higgins and Ferreira 1997, Higgins working on the tracks. It summarized this decision in 1998, Hooghiemstra et al. 1999, Odijk 1999, Cheung the statement “werkploeg is trein,” or a work crew et al. 1999). Higgins (1998) developed an optimiza- is equivalent to a train (in the sense that both re- tion model for determining the allocation of railway quire the same precautions to avoid collisions). This maintenance activities and crews that best minimizes means that if track workers are working in part of the disruption to and from scheduled trains. It mini- the corridor, that part should be closed to all traffic. mizes the time a given track segment’s service level is ProRail therefore divided its infrastructure into basic below a specified benchmark (by scheduling activities working zones and asked us to develop an efficient as early as possible). The method uses tabu search to maintenance schedule based on these working zones. find a solution to the problem. Cheung et al. (1999) The government delegated the management of the developed a method of scheduling enough preventive railways to ProRail (http://www.prorail.nl), which is maintenance work to avoid disrupting railway opera- responsible for the capacity, reliability, and safety of tions during the operational period (of about 19 hours rail infrastructure. We used operations research (OR) every day). It schedules the maintenance work during techniques to construct a four-week, cyclic, preven- the remaining time, which is less than five hours a tive maintenance schedule. Such a schedule is unique day. Moreover, it follows a set of rules and procedures in Europe. ProRail has used it since 2000, and it has in scheduling the maintenance work (for example, clearly proven its benefits. safety rules). Higgins (1998) assumed that work crews OR models and techniques have played an impor- can carry out maintenance activities during train ser- tant role in railway applications. Some OR models vice, which is not the case when “werkploeg is trein”

133 van Zante–de Fokkert et al.: The Netherlands Schedules Track Maintenance to Improve Track Workers’ Safety 134 Interfaces 37(2), pp. 133–142, ©2007 INFORMS is applied. Cheung et al. (1999) assumed that each day With enforcement of protection in time, carrying out several hours were available for maintenance during corrective maintenance would cause an unacceptable which there were no trains, which is certainly not the disturbance for the trains. Under the new system, Pro- case in The Netherlands. Rail must combine all maintenance activities in a cer- tain working zone and carry them out during one The Safe Maintenance Project out-of-service period of about five and a half hours (net) every four weeks according to the maintenance In 2000, the Dutch railway network consisted of about schedule. 6,500 railway kilometers, more than 3,000 switches, We started the project in January 1999 and devel- and 400 yards. It handled about 5,000 trains per oped a four-week cyclic maintenance schedule for the day. In 2000, it transported over a million passengers main lines, where most disturbances occur (Figure 1). and 60,000 tons of freight per day. The network is Passenger traffic on the main lines is greatest dur- operated mainly by the Dutch railways (http://www. ing the daytime. If the working zones of these lines ns.nl) and is fully financed and owned by the Dutch were out of service in the daytime, then, for 90 per- government. cent of the zones, the disruption of railway traffic In 1995 in Mook, a train hit and killed three would exceed the restrictions on maximum delays for track workers carrying out maintenance activities. trains and on trains’ early departures and would even This accident caused a national investigation, which cause train cancellations. Therefore, ProRail cannot do showed that track workers had one of the most maintenance in these zones in the daytime. At night, dangerous jobs in The Netherlands. Their individ- ual risk (IR) was 3.4 deaths per 10,000 track work- ers per annum, three times higher than the IR in the building industry. The investigators concluded that to ensure safety, the railways should allow no trains on the parts of the rail infrastructure under mainte- nance. The track workers should be protected in space and time. ProRail decided to combine maintenance activities as much as possible and to change corrective main- tenance into preventive maintenance. In this way, it would minimize the number of times maintenance takes place and the total maintenance time, both of which would improve safety and reduce the incon- venience for train operators, passengers, and freight Utrecht transport. Kijfhoek ProRail divided the complete Dutch railway net- work into working zones according to an unambigu- ous method based on track functionality (den Hertog et al. 2005). A working zone is the part of the railway network taken out of service when track workers are working on it. We call this practice protection in space. After the working zones were in place, we devel- oped a maintenance schedule for the working zones, which we call protection in time. This schedule was based on a new concept for carrying out mainte- Figure 1: The network consists of the main railroads (black nondashed lines) and the peripheral railroads (dashed lines) of the Dutch railway net- nance. ProRail traditionally carried out maintenance work. The gray lines are the boundaries of The Netherlands including all where and when necessary (corrective maintenance). the isles. van Zante–de Fokkert et al.: The Netherlands Schedules Track Maintenance to Improve Track Workers’ Safety Interfaces 37(2), pp. 133–142, ©2007 INFORMS 135 however, passenger traffic is limited and the traffic wants additional zones free of traffic, potentially fur- consists mainly of freight trains. By grouping freight ther disrupting railway traffic. trains (one set of trains per hour in each direction), The track workers work for 10 contractors, each the railway can limit disruption. Therefore, ProRail maintaining a designated part of the Dutch railway performs maintenance in the zones at night. network. They must carry out the planned activities ProRail will schedule maintenance of the remain- within the time slots in the maintenance schedule. ing 10 percent of the main line zones, the peripheral Because the track workers’ capacity is limited at night railroads, and the yards in the daytime to balance the and during the weekend, the contractors want to limit track workers’ workloads between day and night. We the work scheduled at night and in particular on the will focus on the construction of the nightly main- weekend. Moreover, the workload at night has to be tenance schedule for the main lines. The train oper- balanced over the nights of the week as much as pos- ators (passenger and freight), ProRail Nieuwbouw sible. The number of switches to maintain determines (which lays and replaces tracks), and the track work- the workload. ers all impose restrictions on the maintenance sched- ule, some conflicting. Two-Step Solution Approach In the long run, train operators benefit from preven- We used a two-step solution method to draw up the tive maintenance because it reduces the risk of unfore- maintenance schedule. In the first step, we specify seen disruptions from track malfunctions. However, single-track grids (STGs), which are sets of working planned preventive maintenance activities can also zones that can be out of service simultaneously. In the disrupt service. Maintenance work at night, in partic- second step, we assign the STGs to nights. One reason ular, slows freight traffic, except for Saturday night, to assign STGs instead of working zones to nights is when there is almost no freight traffic. Trains can that train operators are used to the concept of STGs. be delayed, advanced, or even cancelled. We sought The order in which Steps 1 and 2 are carried out can to minimize the inconvenience and to restrict delays be sequential. However, reiteration between the steps and advances to some maximum time. For passenger is possible and can be necessary, for example, when trains, route changes are not permitted, but for freight the resulting workload per night is too high. trains, they are. Maintenance schedules should leave at least one of the (two or more) parallel tracks open. Step 1: SpecifyingSingle-Track Grids Train cancellations are acceptable only in extreme The main lines consist of more than 300 working cases. zones. Assuming that we can find one time slot of ProRail Nieuwbouw lays and replaces track, and five and a half hours per night, then 11 working zones their projects usually take several (consecutive) nights on average will be out of service simultaneously per in one or more working zones. Because these projects night in a four-week schedule. An STG is a set of are not considered as maintenance, they are not taken zones that can be out of service simultaneously (Fig- into account in the maintenance schedule. To carry ures 2, 3, and 4). The resulting disruptions to railway out their projects efficiently, ProRail Nieuwbouw often traffic are within the restrictions imposed by the train

Lage Horst- Kijfhoek Zwaluwe Gilze- Tilburg Boxtel Eindhoven Deurne Venlo

Figure 2: The corridor Kijfhoek–Venlo runs from Kijfhoek near the seaport to Venlo, which is near the border between The Netherlands and (Figure 1). The arrows show the directions of the railway traffic when both tracks of the corridor are in service. A corridor consists of several sections. A section connects two major yards; section Tilburg–Boxtel, for example, connects the Tilburg and Boxtel yards. van Zante–de Fokkert et al.: The Netherlands Schedules Track Maintenance to Improve Track Workers’ Safety 136 Interfaces 37(2), pp. 133–142, ©2007 INFORMS

Tilburg Boxtel Restrictions imposed 1 Working zones Timetable 1 1 by train operators

23 Step 1 Specification of single- Figure 3: The section connecting the Tilburg and Boxtel yards consists of track grids three working zones (1, 2, and 3). Working zone 1, for example, consists of all the tracks labeled with a 1. The small circles indicate the working- zone boundaries. Evaluation of timetables by train operators operators, that is, maximums on delayed departures and early departures. Timetable per Forbidden Zones in left and Allowed STG In principle, STGs should be based on work- STG nights per STG right track per STG combinations ing zones. However, the train operators traditionally based their work on sections (segments connecting Figure 5: In specifying STGs, we considered the working zones, the two major yards within a corridor). Therefore, we timetable, and the restrictions imposed by the train operators. define the STGs in terms of sections, which we later map onto the working zones. Hence, an STG defines Theoretically, a trade-off exists between workload a set of sections on a particular corridor (the railway balance and disturbance to the train traffic. However, infrastructure between two important yards). We can- ProRail decided that disturbance to the train traffic not take both tracks in these sections out of service at is the dominant criterion. So, the first goal is to find the same time because that would totally block traffic a schedule that minimizes disturbance to train traf- (Figure 4). fic. Because ProRail performs maintenance at night We will summarize the definitions: A corridor is when there are no passenger trains, it inconveniences the railway infrastructure between two very big and only the freight trains. Train operators require “almost important yards (Figure 2). A corridor consists of sev- no disturbance,” meaning that freight trains should eral sections. A section connects two major yards (Fig- depart no more than a certain number of minutes ear- ure 3). Each section contains one or more working lier and arrive no more than a certain number of min- zones. A working zone is a part of a section (Figure 3). utes later than the times in the original timetable. Such a working zone can be taken out of service The train operators carried out the first step with for maintenance because signalling techniques protect the help of a software system that supports the gen- such a working zone. An STG is a collection of sec- eration of the train timetables (Figure 5). They deter- tions for which we take either the left or the right mined the STGs in the following way. They assumed track out of service (Figure 4). that one track of a particular section, for example,

Lage Horst- Kijfhoek Dordrecht Zwaluwe Breda Gilze-Rijen Tilburg Boxtel Eindhoven Deurne Sevenum Venlo

Figure 4: An STG on the corridor Kijfhoek–Venlo, defined by the dotted lines, contains the sections Dordrecht– Lage Zwaluwe, Breda–Gilze-Rijen, Boxtel–Eindhoven, and Deurne–Horst-Sevenum (the dotted lines). On these sections, only the right track is in service. The arrows show the adjusted direction of the railway traffic from Venlo to Kijfhoek. The trains from Kijfhoek to Venlo, following their usual direction, are not disrupted by trains coming in the opposite direction because they can pass each other at sections where there are two tracks in service, for example, Kijfhoek–Dordrecht, Lage Zwaluwe–Breda, and Gilze-Rijen–Boxtel. When the left track of the STG is out of service, the situation is simply reversed. Both the left and the right tracks of an STG contain a set of zones. van Zante–de Fokkert et al.: The Netherlands Schedules Track Maintenance to Improve Track Workers’ Safety Interfaces 37(2), pp. 133–142, ©2007 INFORMS 137

Breda–Gilze-Rijen (Figure 2), would be out of ser- at the Centre for Quantitative Methods (CQM) that vice during five and a half hours at night and they minimized the (maximum) delay for a predefined modified the train timetable to prevent trains from STG on a particular corridor. They also extended this travelling on this track. Next, they searched for other model to deal with predefined STGs on two or more sections for which they can take one track out of ser- connected corridors. Stuurman (1996) developed a vice without violating the restrictions on departure similar model. However, she assumed the STG to be a and arrival times. It is striking that when one track decision variable, that is to be defined by the model, of a certain section is taken out of service, there are while maximizing the number of single-track sections. often other sections for which one track can be taken All these models use an enormous amount of infras- out of service with almost no added delays. All train tructure data and hence require efficient interfaces operators evaluated the resulting deviations from the with existing information systems. When we devel- original, unhindered timetable. If the differences in oped our maintenance schedule, we did not have time departure times and arrival times violated the restric- to develop such interfaces. However, when we have tions, then they adjusted the defined STGs. to create a new maintenance schedule, we can use We assumed that each zone was contained in one these models to create efficient STGs. STG. Clearly, we could not schedule STGs on the same corridor for maintenance in the same night because Step 2: Assigning of Single-Track Grids to Nights the combined disturbances would violate the restric- In Step 2, we assigned the STGs to nights. We took tions. Even STGs on different corridors sometimes into account the allowed STG combinations (from cannot be combined, for example, when different cor- Step 1) and the contractor’s workload per night. To ridors have sections in common. We assumed that the avoid completely blocking corridors, we assigned the STGs were symmetric, which means that the timetable left and right tracks of an STG to different nights allows either the left or the right track to be out of (Figure 6). service. We did this to avoid having to construct both Implementing Step 2 manually is hardly possible. left- and right-track timetables. We developed a mixed-integer-programming model The data required for Step 1 are: (Appendix) to carry it out and implemented the —the working zones within each section, model as a software tool. —the original, unadjusted timetables, and The data required for Step 2 are: —the train operators’ restrictions on departure and —The nights to which the STG may not be assigned, arrival times. —The zones contained in the left and the right The results of Step 1 are: tracks for each STG, —A timetable for each STG, specifying either the —The pairs of STGs that can be scheduled for the right or left track out of service at night for five and same night, and a half hours; —The workload and contractor for each zone. —A list of forbidden nights for each STG, which An example of an STG with forbidden nights is means that the STG may not be assigned to these for- the extra STG that should be planned for a weekend bidden nights; night. —The working zones (left and right) for each STG, The results of Step 2 are: plus an extra STG containing zones not in the defined —The assignments of all STGs’ left and right tracks STGs; and to nights, —Whether the STGs in each pair of STGs can be —The nightly contractor workload, which depends assigned to the same night or not. on the number of switches and kilometers to be main- The extra STG contains zones that are hard to main- tained according to the schedule, and tain without too much nuisance. Maintenance of this —A schedule for each night. extra STG is planned for a weekend night during which there is almost no train traffic. Results We did not computerize the generation of STGs. Using our optimization tool, we can evaluate sev- Kingsale (2000) and Pelsers (2001) developed a model eral maintenance schedules, examining the trade-off van Zante–de Fokkert et al.: The Netherlands Schedules Track Maintenance to Improve Track Workers’ Safety 138 Interfaces 37(2), pp. 133–142, ©2007 INFORMS

Forbidden Zones in left and Allowed STG Contractor, switches, and nights per STG right track per STG combinations kilometers per zone

Step 2 Assign left and right tracks of STGs to nights

Evaluate contractors’ Evaluate train operators’ workload per night possibilities for each STG combination

Assignment of left and right Number of switches and kilometers to Schedule per night tracks of STGs to nights be maintained per night per contractor

Figure 6: In Step 2, we assigned STGs to nights. between the number of nights in the schedule and the used CPLEX 6.5 to solve it. For 16 nights, the model contractor’s workload (Figures 7a and 7b). The train contained about 2,000 (integer) variables and 2,000 operators evaluate the schedules for their effect on constraints. Solving one scenario took about 20 min- railway traffic. The contractors judge the distribution utes on a Pentium 266 MHz, 64 MB computer. of their workload over the nights. We extended the model to evaluate different sce- Depending on the scenario considered, the model’s narios, for example, prioritizing nights by weighting objective and constraints can change, requiring changes the workload differently over the nights. We can also in parameters (appendix). We modeled this mixed- minimize the overall maximum workload instead of integer-programming problem in AIMMS (1999) and the sum of the individual maximum workloads (or

Week 1 Week 2 Week 3 Week 4 14

12

10

8

6

4

2

0 MTWT FS SMTWTF SSSSSS MTWTF MTWTF

Figure 7a: In this example of workload distribution (number of switches) for a contractor in a four-week main- tenance schedule with 16 utilized nights, the maximum workload is 13 switches in the nights from Sunday to Monday (third week) and from Sunday to Monday (fourth week). This example is based on artificial data because the real numbers are confidential. van Zante–de Fokkert et al.: The Netherlands Schedules Track Maintenance to Improve Track Workers’ Safety Interfaces 37(2), pp. 133–142, ©2007 INFORMS 139

Week 1 Week 2 Week 3 Week 4 25

20

15

10

5

0 MTWT F S SMTWT F S SMTWT FSMTWTFSS S

Figure 7b: In this example of workload distribution (number of switches) for a contractor in a four-week main- tenance schedule with 10 utilized nights, the maximum workload is 20 switches in the nights from Friday to Saturday (first week) and from Friday to Saturday (third week). This example is based on artificial data because the real numbers are confidential. we can minimize a combination of the two). We can for track building and renewal projects, which were maximize or minimize the number of nights (overall sufficient, and we can “copy” the timetables for all or per contractor) on which we schedule no activities. maintenance nights every four weeks. After we opti- We can base the workload on the number of switches, mized this scenario, the workload for the contractors on the number of kilometers, or on a combination. was still acceptable. Moreover, the contractors had a We can measure absolute workloads or relative work- schedule that was in line with the labor agreements: loads. weeks 1 and 3, night work; and weeks 2 and 4, day We compared the different scenarios with the help work. of diagrams showing the workload distribution of the With the approval of all parties involved, we contractors over the maintenance nights. The systems divided the 28 nights into three sets: six “red” nights produce diagrams of the number of switches and of to be used only for planned maintenance, 18 “green” the number of kilometers to be maintained per night nights to be used for work other than maintenance— and per contractor (Figures 7a and 7b). laying and replacing track—and four “yellow” nights We distributed the chosen maintenance schedule to be used for both kinds of work. In addition to the widely among maintenance contractors, train oper- working zones scheduled for maintenance, additional ators, and ProRail Nieuwbouw personnel. With the zones may be out of service (within the restrictions lists of working zones scheduled to be out of ser- imposed by the train operators) for other purposes. vice each night, we distributed maps showing the track sections planned for maintenance each night ConcludingRemarks (Figure 8). The maintenance project we worked on concerned The first schedule agreed upon consisted of 16 several parties with different, sometimes conflicting, nights for maintenance (out of 28). However, it turned interests. With the help of OR techniques, we put the out that the remaining 12 nights did not allow enough first maintenance schedule for railway infrastructure time for laying and replacing track. We had two into operation in 1999. Nowhere in Europe had any options: to take some additional working zones out infrastructure implemented such a concept. of service during the 16 maintenance nights for that The maintenance schedule for the main lines has work or to reduce the number of maintenance nights proven its benefits. First, the new maintenance sched- to 10. To avoid a lot of work developing timeta- ule made it possible for ProRail to implement the bles, we chose the second option. This left 18 nights policy “werkploeg is trein,” improving trackworkers’ van Zante–de Fokkert et al.: The Netherlands Schedules Track Maintenance to Improve Track Workers’ Safety 140 Interfaces 37(2), pp. 133–142, ©2007 INFORMS

Zwolle

Amsterdam

Utrecht

Kijfhoek

Boxtel

Venlo

Figure 8: In this example map, thick lines indicate the track sections scheduled to be out of service in the Monday night of each first week, and the short lines crossing the track lines indicate yards. safety. Between 1999 and 2004, the individual risk Finally, the timetable developers’ workload decreased dropped from 3.4 to 2.0 deaths per 10,000 track work- because the plan for the six (red) nights is fixed. ers per annum. The fatalities since 1999 were not We expect that ProRail will soon put into opera- caused by accidents as in Mook because trains are tion a full schedule for the remaining working zones, no longer running on the tracks workers work on. the peripheral railroads, and the yards. Work on these Moreover, the maintenance schedule enabled a tran- zones takes place during the daytime as much as pos- sition from corrective to preventive maintenance. sible. We schedule zones that can be maintained only The four-week, cyclic, preventive maintenance sched- at night at the top of the main line schedule, filling ule has reduced the number of unplanned failures. the nights with low workloads first. van Zante–de Fokkert et al.: The Netherlands Schedules Track Maintenance to Improve Track Workers’ Safety Interfaces 37(2), pp. 133–142, ©2007 INFORMS 141

Passenger traffic is heavy during the daytime. Be- Variables cause the railway network is highly connected in wn binary variable that indicates whether night n The Netherlands, the smallest disturbance can cause is used in the schedule wn = 1 , n ∈ N , or not enormous inconvenience. For the peripheral railroads (wn = 0 . and the yards, the parties involved must cooperate to xstn binary variable that indicates whether track side specify which zones can be out of service and when. t of STG s is assigned to night n xstn = 1 , s ∈ S, The use of optimization models to generate STGs t ∈ T , n ∈ N , or not (xstn = 0 . in the first step of the solution approach deserves fur- yc maximum number of switches to be maintained ther research. Kingsale (2000) and Pelsers (2001) have per night by contractor c over all nights, c ∈ C. z maximum number of kilometers to be main- researched such an approach. Further studies should c tained per night by contractor c over all nights, also consider section-based STGs. c ∈ C.

Model Appendix   (P) Min yc + 1 − zc + M wn The Mixed-Integer-Programming Model c∈C n∈N We developed the mixed-integer-programming model s.t. xstn = 1 ∀ s ∈ S t ∈ T (1) to construct the maintenance schedule. n∈N xstn ≤ Rsn ∀ s ∈ S n ∈ N (2) t∈T Sets   x + x ≤ 1 C contractors. s1tn s2tn t∈T t∈T S single-track grids (STGs). ∀ s1s2 ∈ P n∈ N (3) T = left right track sides.   x · Q ≤ y ∀ c ∈ Cn∈ N (4) N nights. stn stc c s∈S t∈T P ⊂ S × S set of permitted STG combination, i.e.,   xstn · Vstc ≤ zc ∀ c ∈ C n∈ N (5) s1s2 ∈ P, s1 = s2, when STGs s1 and s2 can be s∈S t∈T combined during one night. max yc ≤ Qc ∀ c ∈ C (6) 1   · x ≤ w ∀ n ∈ N (7) Parameters 2S stn n s∈S t∈T weight factor that indicates whether the work- xstn ∈ 0 1 ∀ s ∈ S t ∈ Tn∈ N (8) load is measured by the number of switches w ∈ 0 1 ∀ n ∈ N (9) = 1 or by the number of kilometers = 0 n or by a weighted combination of the two 0 < The objective function consists of two parts. First, <1 . we minimize the number of nights with planned maintenance in the schedule. After that, we minimize Rsn binary parameter that indicates whether STG the sum of the maximum scheduled workload of the s can be assigned to night n Rsn = 1 or not (R = 0 , s ∈ S, n ∈ N . contractors. Constraints (1) and (2) ensure that the sn left and the right tracks of all STGs are assigned to Q number of switches to be maintained on track stc exactly one allowed night. Moreover, they should be side t of STG s by contractor c, s ∈ S, t ∈ T , assigned to different nights. Constraints (3) ensure c ∈ C. that only combinable STGs are scheduled in the same V number of kilometers to be maintained on stc night. Constraints (4) and (5) determine the maxi- track side t of STG s by contractor c, s ∈ S, t ∈ T , mum number of switches and the maximum number c ∈ C. of kilometers to be maintained per night by a par- Qmax maximum number of switches that can be c ticular contractor. The number of switches that one maintained per night by contractor c, c ∈ C. contractor can maintain simultaneously in one night M a big number. is limited (constraints (6)). Constraints (7) determine van Zante–de Fokkert et al.: The Netherlands Schedules Track Maintenance to Improve Track Workers’ Safety 142 Interfaces 37(2), pp. 133–142, ©2007 INFORMS

whether or not STGs are planned in a particular night Higgins, A. 1998. Scheduling of railway track maintenance activi- in the schedule. When at least one of the STGs is ties. J. Oper. Res. Soc. 49(10) 1026–1033. assigned to night n, then the binary variable w is Higgins, A., L. Ferreira. 1997. Modelling the number and location n of sidings on a single line railway. Comput. Oper. Res. 24(3) forced to be one. Moreover, the left side of this equa- 209–220. tion is at most one, because we divide by the total Hooghiemstra, J. S., L. G. Kroon, M. A. Odijk, M. Salomon, P. J. number of STG-track sides. Finally, constraints (8) and Zwaneveld. 1999. Decision support systems support the search for win-win solutions in railway network design. Interfaces (9) ensure that the decision variables for the assign- 29(2) 15–32. ment and the used nights are binary. There is no need Kingsale, S. J. P. 2000. Het enkelspoorraster probleem (in Dutch). to explicitly add the requirements that y and z have Master’s thesis, Hogeschool Holland, Diemen, The Nether- c c lands. to be integer because these requirements are automat- Odijk, M. A. 1999. Sensitivity analysis of a railway station track ically fulfilled by the model. layout with respect to a given timetable. Eur. J. Oper. Res. 112(3) 517–530. Acknowledgments Pelsers, M. L. M. 2001. Het automatisch genereren van enkelspoor- We thank the two anonymous referees for their many valu- rasters voor een railinfra-onderhoudsrooster (in Dutch). Mas- able comments on earlier versions of this paper. ter’s thesis, , Tilburg, The Netherlands. Stuurman, M. 1996. Gelijktijdige buitendienststellingen (in Dutch). Master’s thesis, Faculty of Economics and Econometrics, Uni- References versity of Amsterdam, Amsterdam, The Netherlands. AIMMS. 1999. AIMMS, The User’s Guide. Paragon Decision Tech- nology B. V., Haarlem, The Netherlands. Carey, M. 1994. Extending a train pathing model from one-way to Drs. B. J. Klerk, chairman of the board, ProRail two-way track. Transportation Res. B 28(5) 395–400. bedrijfsplanning, Postbus 2038, 3500 GA Utrecht, The Cheung, B. S. N., K. P. Chow, L. C. K. Hui, A. M. K. Yong. 1999. Railway track possession assignment using constraint satisfac- Netherlands, writes: “This is a verification letter to let tion. Engrg. Appl. Artificial Intelligence 12(5) 599–611. you know that the techniques and models described den Hertog, D., J. I. van Zante–de Fokkert, S. Sjamaar, R. Beusmans. in the paper ‘The Netherlands Schedules Track Main- 2005. Optimal working zone division for safe track main- tenance to Improve Track Workers’ Safety,’ written by tenance in The Netherlands. Accident Anal. Prevention 37(5) 890–893. J. I. van Zante–de Fokkert, D. den Hertog, F. J. van den Harris, N. G., J. B. H. Ramsey. 1994. Assessing the effects of railway Berg, and J. H. M. Verhoeven, were and are very valu- infrastructure failure. J. Oper. Res. Soc. 45(6) 635–640. able for our organisation. The resulting schedule has Hensher, D. A. 1997. A practical approach to identifying the mar- been operative in The Netherlands since 2000 and has ket potential for high speed rail: A case study in the Sydney- Canberra corridor. Transportation Res. A 31(6) 431–446. proven to be very beneficial for our organisation.”