No. 2002-43 ALLOCATION OF RAILWAY ROLLING STOCK FOR PASSENGER TRAINS By Erwin Abbink, Biana van den Berg, Leo Kroon and Mar Salomon April 2002 ISSN 0924-7815
Alloation of Railway Rolling Stok for Passenger Trains Erwin Abbink 1 Biana van den Berg 2 Leo Kroon 1,3 Mar Salomon 4 1 NS Reizigers, Department of Logistis, Utreht 2 Cap Gemini Ernst & Young, Amsterdam 3 Erasmus University Rotterdam, Rotterdam Shool of Management L.Kroon@fbk.eur.nl 4 Catholi University of Brabant, Department of Eonomis, Tilburg Abstrat For a ommerially operating railway ompany, providing a high level of servie for the passengers is of utmost importane. The latter requires a high puntuality of the trains and an adequate rolling stok apaity. Unfortunately, the latter is urrently (2002) one of the bottleneks in the servie provision by the main Duth railway operator NS Reizigers. Espeially during the morning rush hours, many passengers annot be transported aording to the usual servie standards due to a shortage of the rolling stok apaity. On the other hand, a more effetive alloation of the available rolling stok apaity seems to be feasible, sine there are also a few trains with some slak apaity. The effetiveness of the rolling stok apaity is determined mainly by the alloation of the train types and subtypes to the lines. Therefore, we desribe in this paper a model that an be used to find an optimal alloation of train types and subtypes to train series. This optimal alloation is more effetive than the manually planned one, whih is aomplished by minimizing the shortages of apaity during the rush hours. The model is implemented in the modeling language OPL Studio 3.1, solved by CPLEX 7.0, and tested on several senarios based on the 2001-2002 timetable of NS Reizigers. The results of the model were reeived positively, both by the planners and by the management in pratie, sine these results showed that a signifiant servie improvement over the manually planned alloation an be ahieved within a shorter throughput time of the involved part of the planning proess. - 1 -
1. Introdution In the Netherlands, the railway operator NS Reizigers transports nearly one million railway passengers every workday. At the busiest moment of the morning rush hours, whih is usually around eight o lok in the morning, about 250 trains are running at the same time. The total number of trains running per day is about 4800. The number of available arriages equals about 2600. Part of the lines (whih are alled train series in the remainder of this paper) of NS Reizigers in the Randstad, the metropolitan area in the western part of the Netherlands, is shown in Figure 1 below. Figure 1: The train series of NS Reizigers in the Randstad For a ommerially operating railway ompany, providing a high level of servie for the passengers is of utmost importane. The latter requires a high puntuality and an adequate rolling stok apaity. Unfortunately, the latter is urrently (2002) one of the bottleneks in the servie provision by NS Reizigers. Espeially during the morning rush hours, many passengers do not have a seat during parts of their journey due to a shortage of rolling stok apaity. This shortage is aused by a ombination of fators. First, the number of passengers has grown signifiantly during the last years. Seond, new units of rolling stok have been ordered, but these units have long delivery times. - 2 -
Given the urrent situation, it is important to look for the most effetive alloation of the available rolling stok apaity among the trains, espeially during the rush hours. In fat, a better alloation than the manually planned one seems to be feasible, sine at the same time there are also several trains with some slak apaity. Most researh on railway rolling stok deals with the routing and sheduling of loomotives and freight trains. For researh in these areas, see for example the overview by Cordeau et al. (1998). However, also the rolling stok irulation for passenger trains reeived some attention reently. Examples of researh in this area are provided by Ben- Khedher et al. (1998), Bruker et al. (1998), and Cordeau et al. (2001). The first paper deals with the sheduling of high speed train units of the Frenh railway operator SNCF. The other papers deal with the routing of loomotive hauled arriages. Shrijver (1993), Groot (1996), Blom (1998), and Van Montfort (1998) desribe researh fousing on the rolling stok irulation of Duth train units and loomotive hauled arriages. All of the mentioned papers mainly deal with the operational problem of finding the most effiient shedule for a set of units of rolling stok, given a ertain alloation of the train types and subtypes to the train series. In ontrast with this, the urrent paper deals with the tatial problem of finding the most effetive alloation of the train types, subtypes, and units of rolling stok to the train series, suh that as many people as possible an be transported with a seat, espeially during the rush hours. The remaining part of this paper is strutured as follows. In Setion 2, we present some additional bakground information for providing a better understanding of the rolling stok alloation problem. In Setion 3, we give a rough desription of the planning proess related to the rolling stok irulation. The details of the alloation problem are desribed in Setion 4. Setion 5 desribes the assumptions underlying the model and the notation that we used to desribe the model. Setion 6 gives the details of the model. Setion 7 presents our omputational results based on several senarios involving the 2001-2002 timetable of NS Reizigers. The paper is finished in Setion 8 with onlusions and some subjets for further researh. 2. Bakground information Before we go into the details of the rolling stok alloation problem, we first desribe some bakground information pertaining to the rolling stok irulation of NS Reizigers. In the Netherlands, three different ategories of trains for passenger transportation an be distinguished: Interity trains, Inter-Regional trains, and Regional trains. The line system ontains the train series, eah of whih onneting an origin station and a destination station with a ertain frequeny. Eah train series belongs to one of the three train ategories. The timetable gives the arrival and departure times of the trains at the relevant stations. The timetable of NS Reizigers is yli with a yle length of one hour. - 3 -
Figure 2: Koploper train unit with 3 arriages Figure 3: Mat 64 train unit with 2 arriages NS Reizigers has a large variety of rolling stok available for passenger transportation. Some examples are given in Figures 2 and 3. The three main lasses to be distinguished are: power driven equipment, diesel driven equipment, and loomotive hauled arriages. Power driven and diesel driven equipment onsist of train units that an move individually without a loomotive. Eah train unit onsists of a ertain number of arriages that annot be split from eah other during the operations. Within eah lass, various train types an be distinguished. Power driven equipment omprises, among others, Koplopers for the Interity trains and Mat 64 for the Regional trains. The train types an be subdivided subsequently into different subtypes. Koplopers, for instane, an be subdivided into Koplopers with three arriages and Koplopers with four arriages. Similarly, Mat 64 an be subdivided into train units with two arriages and train units with four arriages. Train units of the same train type an be ombined into longer trains. Train units of different train types annot be ombined in one train. It should be noted that not the entire fleet of rolling stok is available for passenger transportation. Some train units are reserved for maintenane or bakup. Ledn 7 10 20 30 40 50 8 10 20 30 40 50 9 8865 8863 8827 8829 8831 8833 8818 8820 8822 8824 Ut 8856 8854 Figure 4: Part of the rolling stok irulation during the morning rush hours - 4 -
The way the train units are put together into longer trains within a rolling stok irulation is shown in Figure 4. This figure shows a time-spae diagram for part of the trains of the Regional train series 8800 between Leiden (Ledn) and Utreht (Ut). See also Figure 1 for the loations of the mentioned stations. Trains of the 8800 train series run twie per hour. The numbers at the top of the figure indiate the time axis. The dashed diagonal lines indiate the trains and the adjaent numbers are the train numbers. Eah line represents one train unit of the type Mat 64 with two arriages. Train 8820, for instane, is run with three train units. Upon arrival in Leiden, two of these train units return to Utreht on train 8833, and the third train unit remains in Leiden. This train unit is stored on the loal shunting yard, and is used only in the afternoon rush hours on train 8865, as is indiated at the end of the orresponding line. Something similar happens with one train unit of train 8822. Notie the Last-In-First-Out priniple that is used when storing train units on the shunting yard of Leiden. This is beause the traks of the shunting yard an be approahed from one side only. 3. Planning proess The planning proess related to the rolling stok alloation and irulation starts after the line system and the timetable have been ompleted. This planning proess is initiated within the ommerial branh of NS Reizigers, where an overview of the preferred train types per train series is omposed. This overview takes into aount the train ategory of the train series, the tehnial possibilities of the train types and subtypes, their running time harateristis, the required apaity per train, and the passengers preferenes. The required apaity per train is based on the expeted numbers of passengers. These numbers are based on the ounting figures, whih are estimates by ondutors of the number of passengers per train. The statistial proedure, whih is used to transform the ounting figures of the ondutors into the required apaity per train, falls outside the sope of this paper. Apart from these ounting figures, an annual growth fator is taken into aount, and also a ertain omfort fator may be applied. The required apaity per train distinguishes between first lass and seond lass passengers. The trains that run in parallel at one partiular moment of the day make up a so-alled ross-setion. The eight o lok ross-setion indiates all trains that run at eight o lok in the morning, whih is usually the busiest moment of the day. The irulation time of a train series, inluding the return times at the endpoints, and its frequeny determine the number of ross-setion trains of the train series. For instane, the Regional train series 8800 that was shown in Figure 4 ontains four eight o lok ross-setion trains, namely the trains 8820, 8822, 8829, and 8831. This is aused by the fat that the irulation time between Utreht and Leiden and vie versa is about two hours and that there are two - 5 -
trains per hour in eah diretion. The eight o lok ross-setion trains of all train series together make up the omplete eight o lok ross-setion. As a first step in the planning proess of the rolling stok, an alloation of the rolling stok apaity is determined to the trains in the eight o lok ross-setion. Here the idea is that, if it is possible to determine an appropriate alloation of the rolling stok apaity during the morning rush hours, then this alloation will be appropriate during the other hours of the day as well. This is reasonable, sine in pratie the required apaity during the evening rush is usually less than during the morning rush: the evening rush lasts longer than the morning rush, and it has a lower peak. The alloation of the rolling stok apaity to the eight o lok ross-setion trains is an example of bottlenek planning, where the aim is to maximize the effetive rolling stok apaity. After the alloation of the rolling stok has been planned for the eight o lok rosssetion trains, the rolling stok irulation is also planned for the other trains during the day. The results of this proess are harts as shown in Figure 4 for eah train series, possibly inluding inter-onnetions between different train series. The planning of the rolling stok irulation for the other trains during the day is an effiieny problem, where the aim is to minimize the total number of arriage kilometers. After the rolling stok irulation has been planned throughout the day and for eah individual day of the week, it will also be balaned aross the days of the week: on eah shunting yard, the number of train units stored there during the night will be equaled to the number of train units that are required there on the next morning. This is aomplished by modifying the rolling stok irulation on the early morning trains or on the late evening trains, or by adding some so-alled deadheading trains. A final step in the planning proess is the organization of the maintenane of the rolling stok. 4. Problem definition The problem disussed in this paper is the alloation of the rolling stok apaity to the trains in the eight o lok ross-setion. To be more exat, the question is: how many units of eah train type and subtype should be deployed on eah train of the eight o lok ross-setion in order to maximize the effetive apaity for transporting passengers? When determining the alloation of equipment in a train series, the preferred equipment is to be taken into aount as muh as possible. It is, however, not always possible to deploy the most preferred equipment in a train series. As was mentioned earlier, NS Reizigers is faed with a shortage of rolling stok apaity. Hene, even if a partiular type of equipment is preferred, alternatives will often have to be hosen, sine not enough train units of the preferred type are available. - 6 -
Seondly, the alloation is determined by the required apaity. Train units of different types usually have different apaities. Therefore, depending on the required apaities, one partiular type of equipment will be more suitable than another. Furthermore, it is a strit requirement that the length of eah train does not exeed the length of the shortest platform along the train s route. Hene, if a busy train series ontains a station with a short platform, then a train type with a large apaity per arriage (suh as double-dekers) will have to be hosen. Finally, there are restritions for the Regional train series regarding the running time harateristis. Espeially on a train series with stops at a relatively short distane of eah other, the alloated rolling stok should be able to aelerate and brake quikly. In most train series, the equipment is transferred to the next return train of the same train series at the terminal station. However, on a number of train series the equipment is transferred to another train series at the terminal station. If this is struturally the ase, then obviously the same train type must be alloated to both train series. On eah individual train, at most one train type an be alloated, sine train units of different train types annot be ombined into one train. Furthermore, on eah train series, it is desirable to have as few as possible train types and subtypes: the latter may lead to an inreased robustness of the railway system, beause the adjustments by traffi ontrol beome muh simpler. For eah train series, the maximum allowed numbers of train types and subtypes depend on the number of ross-setion trains in the train series. 5. Assumptions and notation In this setion we desribe the assumptions that were made for modeling the problem of alloating the rolling stok apaity to the train series. First, we assume the timetable, and the required apaities of eah train to be known. Based on these elements, for eah train series the relevant trains of the eight o lok ross-setion an be determined a priori. We also assume that for eah train series a list of allowed train types as well as all other relevant data elements have been provided. Next, we assume that we only have to alloate the available train units (power driven and diesel driven equipment) to the train series. In other words, the loomotive hauled arriages have been alloated to ertain train series already before running the model. This is a reasonable assumption, sine there are only a few train series that are appropriate for being servied in this way. The rolling stok alloation for these train series is assumed to remain the same as in the urrent situation in pratie. There are also a few train series on whih train units are ombined and split underway. For example, on the 700 train series, trains arriving in Zwolle from Groningen and Leeuwarden are ombined into one train bound for Amersfoort. In Amersfoort, this train is split again into one part bound for Amsterdam and one part bound for Shiphol. - 7 -
In the reverse diretion, the ombining and splitting proess takes plae in the reverse order. This proess leads to a omplex rolling stok irulation, whih falls outside the sope of our model. Furthermore, the urrently alloated type of rolling stok (Koplopers) is undoubtedly the most appropriate train type for these train series. Therefore, an alternative alloation of train types would be highly undesirable there. As a final part of this setion, we desribe the notation that we use to express the apaity alloation model. The train series (or lines) are denoted by l = 1,,L, and the ross-setion trains within train series l are represented by t = 1,,T l. Next, the different types and subtypes of train units are represented by τ = 1,, τ max, and by = 1,, τ, respetively. First and seond lass are denoted by =1,2, both for passengers and for apaities of train units. For eah train series l, the parameters M l and m l denote the maximum allowed number of train types and subtypes. The length of the shortest platform along the route of train series l is desribed by L l and the set of allowed train types for this train series is denoted by D l. For ross-setion train t of train series l, the expeted number of passengers in lass is given by P l,t, and the maximum number of shortages in lass is represented by U l,t,. The parameters λ andc, denote the length of eah train unit of subtype in meters, and the apaity for passengers in lass of eah train unit of subtype, respetively. The available number of train units of subtype and the train type to whih subtype belongs are denoted by N andτ. 6. Model In this setion, we desribe the model that we used for solving the apaity alloation problem. Here we only give a desription of the deision variables, the objetive funtion and the most relevant onstraints. In the Appendix of this paper, the remaining onstraints of the model an be found. 6.1 Deision variables The most important deision variables of the model are the variables Nl,t, and S l,t,. The variable Nl,t, denotes the number of train units of subtype that are alloated to rosssetion train t of train series l. The variable S l, t, represents the shortages in lass on ross-setion train t of train series l. The latter is explained in detail in Setion 6.2. Other deision variables in the model are the binary alloation variables A l,t, τ, A l,t,, A l, τ, and A l,. For example, the variables Al,t, τ have the following meaning: A l,t,τ = (1 / 0) if train type τ (is / is not) alloated to ross-setion train t of train series l. The deision variables A l,t,, A l, τ, and Al, have a similar meaning. These binary variables are defined only for allowed ombinations of indies. For example, if train type - 8 -
τ does not belong to the set of allowed train types of train series l, then the variables Al,t,τ and Al, τ are not defined, and the same holds for the variables Al,t, and Al, if τ = τ. 6.2 Objetive funtion The objetive funtion of the apaity alloation model basially fouses on minimizing the weighted total number of shortages on the trains. Here the shortages in lass of train t are equal to the expeted number of passengers in lass of train t that do not have a seat during (part of) their train. This objetive funtion an be expressed as in (1): l t min (1) w l, t, Sl, t, Subjet to S l, t, Pl, t, C, Nl, t, for all l, t, (2) S 0 for all l, t, (3) l, t, And all other onstraints (desribed in Setion 6.3 and in the Appendix) The other onstraints to be satisfied are desribed in Setion 6.3 and in the Appendix. Constraints (2) speify that the shortages on a ertain train are not less than the differene between the required and the alloated apaity. Constraints (3) are required in order to exlude the possibility that shortages on a ertain train are ompensated by an exess of apaity on another train. Sine Constraints (2) and (3) are the only onstraints involving the variables S l,t,, any solution in whih none of the Constraints (2) and (3) is satisfied with equality for a ertain variable S l,t, an be improved. From the foregoing, it follows that S l,t, indeed equals the shortages in lass on train t of train series l. The parameters w l,t, are used to give a ertain weight to the shortages on the trains, e.g. based on the lengths of the trains. In order to get an aurate representation of the total number of kilometers traveled by passengers without a seat, it may be neessary to split eah train into a number of onseutive trips. In fat, shortages usually do not our on the whole length of the route of a train, but only on ertain parts of it. Note that the weights w l,t, may be hosen in suh a way that the objetive does not ount the absolute number of shortages on the trains, but the relative ones (that is: the shortages as a perentage of the required apaity). Indeed, by hoosing w l,t, = 1/ P l,t,, and denoting the relative shortages in lass on train t of train series l by R l,t,, we obtain - 9 -
l t min (1 ) R l, t, C, Nl, t, Rl, t, 1 - for all l, t, (2 ) P l, t, R 0 for all l, t, (3 ) l, t, Note that this is the definition of the inverse utilization rather than the utilization as defined usually. However, in an optimization model, this linear inverse utilization an be handled more easily. A disadvantage of the alternative objetive funtion may be the fat that now a ertain perentage of shortages on a busy train is as bad as a ertain perentage of shortages on a quiet train. Nevertheless, these relative shortages are used in pratie as an alternative riterion to evaluate feasible apaity alloations. 6.3 Constraints In this setion we desribe the most relevant onstraints of the model. A desription of the other onstraints of the model an be found in the Appendix. The latter onstraints mainly handle the seletion of orret ombinations of alloation variables. The most relevant onstraints of the model are the following: N for all l, t (4) l,t, 1 A l,t,τ = 1 for all l, t (5) τ λ N l, S l t, U l, t,, t Ll for all l, t (6), for all l, t, (7) l τ t Nl, t, N for all (8) A, M for all l (9) l τ l Al, m l for all l (10) First, Constraints (4) speify that at least one train unit should be alloated to eah rosssetion train. Seond, Constraints (5) state that on eah train exatly one train type is to - 10 -
be alloated. This is important sine train units of different train types annot be ombined into one single train. Constraints (6) desribe that on eah train the total length of the alloated train units should not exeed the length of the shortest platform along the route. Constraints (7) guarantee that the shortages on eah ross-setion train respet the upper limit U l,t,. Next, Constraints (8) desribe that for eah subtype the number of alloated train units should not exeed the available number of units. Aording to Constraints (9), the number of train types alloated to eah train series should not exeed the upper bound M l. Similarly, Constraints (10) speify that the number of subtypes alloated to eah train series should not exeed the upper bound m l. 7. Computational results In order to test the model, we implemented it in OPL Studio 3.1 using CPLEX 7.0. The hardware that we applied was a Pentium III proessor with 1 GHz and 256 Mb RAM. We studied the alloation of the available rolling stok to 50 Regional train series of the 2001-2002 timetable of NS Reizigers. This is almost the omplete set of Regional train series of NS Reizigers. The number of ross-setion trains in this data set equals 188, whih gives a model with about 2800 deision variables and 6600 onstraints. Most of the input data for the model, suh as the expeted numbers of passengers per train, are the same as in the manual solution, so that the rolling stok alloations obtained by the model an be ompared with the manually planned one. The weights w l,t, for the shortages were set to 1 for seond lass shortages and to 2 for first lass shortages, but we also experimented with the relative shortages as desribed in Setion 6.2. 7.1 Manual solution The harateristis of the manual solution are summarized in Tables 1 and 2. In Table 1, the olumns # Short. 1 and # Short. 2 denote the number of trains with shortages in the first and seond lass, respetively. The olumns Short. 1 and Short. 2 denote the total number of shortages in the first and seond lass, respetively. Table 1: Manual solution, Shortages # Short. 1 # Short. 2 Short. 1 Short. 2 Total Weighted 32 75 375 4869 5244 5619 Table 2: Manual solution, Utilization Min. util. 1 Min. util. 2 Avg. util. 1 Avg. util. 2 Max. util. 1 Max. util. 2 0,15 0,46 0,98 0,95 1,50 1,61-11 -
Table 2 evaluates the manual solution in terms of the utilization of the apaity. Here the utilization on a ertain train is defined as the ratio of the required apaity and the alloated apaity. Note that this is different from the relative shortages as defined in Setion 6.2. The average utilization equals the unweighted average over the trains. 7.2 Senarios In our omputational analysis, we studied four senarios, whih differ from eah other mainly by the maximum numbers of train types and subtypes that are allowed per train series. Within these senarios, we also varied the available rolling stok apaities. We used the urrent operational rolling stok apaity, but we also assumed an infinite apaity to be available. The latter is interesting in order to get an idea of the influene of the shortage of rolling stok apaity. Table 3 summarizes the studied senarios. Table 3: Different senarios Senario 1 Senario 2 Senario 3 Senario 4 Max. # of types 1 1 1 1 or 2 Max. # of subtypes 1 2 1 or 2 1, 2 or 3 Rolling stok apaity Current or Infinite Current or Infinite Current or Infinite Current or Infinite In Senario 3, the number of allowed subtypes per train series depends on the number of ross setion trains per train series. For train series with up to four trains, only one train type is allowed. For train series with more than four trains, two different train types are allowed. The applied rules in Senarios 1 and 3 are more restritive than in the manual solution. In Senario 4 we set the maximum numbers of train types and subtypes per train series equal to the numbers that are used in the manual solution. All senarios were solved using the objetive funtion that minimizes the absolute number of shortages. Only Senario 4 was also solved using the objetive funtion that minimizes the relative shortages, as was explained in Setion 6.2. 7.3 Results Tables 4 and 5 show the results that were obtained by solving the alloation model to optimality using the urrent rolling stok apaities. Computation times are not displayed, sine all omputation times were less than 1 minute. When we started this projet with another version of CPLEX and with less sophistiated hardware, the omputation times ran into hours or even days to get aeptable solutions, but nowadays we find optimal solutions usually in a ouple of seonds. - 12 -
Table 4: Current apaities, Shortages Sen. # Short. 1 # Short. 2 Short. 1 Short. 2 Total Weighted 1 35 92 283 5512 5795 6079 2 27 69 237 3770 4007 4245 3 37 87 304 4826 5130 5435 4 26 69 239 3657 3896 4136 Table 5: Current apaities, Utilization Sen. Min. util. 1 Min. util. 2 Avg. util. 1 Avg. util. 2 Max. util. 1 Max. util. 2 1 0,14 0,48 0,76 1,01 1,41 1,49 2 0,14 0,48 0,73 0,96 1,27 1,49 3 0,14 0,41 0,76 1,00 1,39 1,49 4 0,14 0,41 0,72 0,95 1,39 1,43 Furthermore, if a higher number of different types and subtypes are allowed per train series, then this obviously has a positive influene on the shortages. On the other hand, a higher number of different types and subtypes per train series will have a negative effet on the robustness of the alloation, as was explained earlier. Note that the evaluation of the optimal solution for Senario 3 is rather similar to the evaluation of the manual solution. However, the optimal solution for Senario 3 may be more robust in pratie than the manual solution, due to the appliation of striter rules. The optimal solutions for Senarios 2 and 4 are better than the manual solution. Here the total number of shortages is more than 20% lower than in the manual solution. Also the numbers of trains with shortages are lower than in the manual solution. Tables 6 and 7 show the results that were obtained by applying the apaity alloation model in the ase of infinite rolling stok apaities. Obviously, there are still shortages in this ase. These are due to the restritions that are posed by the lengths of the platforms in the stations, whih prohibit the deployment of longer trains. Table 6: Infinite apaities, Shortages Sen. # Short. 1 # Short. 2 Short. 1 Short. 2 Total Weighted 1 15 19 196 1779 1975 2173 2 11 20 154 1797 1951 2106 3 12 20 160 1797 1957 2118 4 11 20 154 1797 1951 2106-13 -
Table 7: Infinite apaities, Utilization Sen. Min. util. 1 Min. util. 2 Avg. util. 1 Avg. util. 2 Max. util. 1 Max. util. 2 1 0,14 0,46 0,64 0,81 1,41 1,43 2 0,14 0,41 0,62 0,81 1,41 1,43 3 0,14 0,48 0,63 0,81 1,41 1,43 4 0,14 0,41 0,63 0,80 1,41 1,43 Sine these shortages are not aused by a lak of rolling stok apaity, they may be alled unavoidable: these shortages an only be avoided by running more trains, by extending the infrastruture, or by relaxing some of the other onstraints that are to be satisfied. However, suh options annot be put into pratie on the short run. It follows that the appliation of the rolling stok alloation model gives a redution of the avoidable shortages by about 35% (e.g. 1-(4007-1951)/(5244-1951) for Senario 2). Tables 8 and 9 show the results that were obtained by applying the alloation model to Senario 4 with the urrent rolling stok apaities. However, here we studied the influene of the objetive of minimizing the relative shortages (as desribed in Setion 6.2) instead of the objetive of minimizing the absolute shortages. Table 8: Senario 4, Current apaities, Shortages Obj. # Short. 1 # Short. 2 Short. 1 Short. 2 Total Weighted Util. 26 69 191 4325 4516 4516 Short 26 69 239 3657 3896 4136 Table 9: Senario 4, Current apaities, Utilization Obj. Min. util. 1 Min. util. 2 Avg. util. 1 Avg. util. 2 Max. util. 1 Max. util. 2 Util. 0,14 0,48 0,71 0,94 1,39 1,49 Short. 0,14 0,41 0,72 0,95 1,39 1,43 These tables show that the alternative objetive leads to a solution where the numbers of trains with shortages are the same, but where the total number of shortages is higher. Of ourse, the latter was to be expeted, sine the alternative objetive does not expliitly fous on minimizing the absolute shortages. The solutions obtained by both objetive funtions turn out to be very similar. Nevertheless, it seems to be more natural to fous on minimizing the absolute shortages than on minimizing the relative ones. 8. Conlusions and final remarks The availability of rolling stok is urrently (2002) one of the bottleneks in the servie provision by the Duth railway operator NS Reizigers. Espeially during the morning - 14 -
rush hours, many passengers annot be transported aording to the usual servie standards. In this paper we therefore desribed a model that an be used to find an optimal alloation of the available train types and subtypes to the train series, thereby aiming at minimizing the shortages during the morning rush hours. The model was implemented in the modeling language OPL Studio and solved by CPLEX 7.0. The first experienes with the model were reeived quite positively. In Senarios 2 and 4, both the total shortages and the numbers of trains with shortages obtained by the model were signifiantly lower than in the manual solution. Both the planners and the management of the Logistis department of NS Reizigers were satisfied by the obtained alloations, not only beause of the improved quality of the obtained alloations, but also beause of the redution in the throughput time of the planning proess that will be enabled by the appliation of the model. Also the possibility to analyze several alloation senarios instead of just one is onsidered as quite useful. We intend to extend the model in the near future in order to make it even more useful for pratial appliation. In partiular, we will fous on train series involving ombining and splitting of train units, and we will fous on ombining the rolling stok alloation problem for the morning rush hours and the rolling stok alloation problem for the afternoon rush hours into one single model. A further extension of the model may also involve the inorporation of the loomotive hauled arriages. Currently, we are also working on the development of models that an be used to determine appropriate rolling stok irulations throughout the day and aross the days of the week, given the rolling stok distribution obtained by the apaity alloation model of the urrent paper. The latter researh is a follow-up of the researh arried out by Shrijver (1993), Groot (1996), Blom (1998), and Van Montfort (1998). Appendix In this Appendix we desribe the logial onstraints that have to be added to the model in order to obtain solutions that will be feasible in pratie. These onstraints were omitted in the main text intentionally to keep the desription there as simple as possible. A N M A for all l, t, (11) l, t, l, t, l, t, l, t, A A for all l, t, (12) l, t, l, t,τ A A for all l, t (13) l, l,τ A A for all l, t, τ (14) l, t,τ l,τ A A for all l, t, (15) l, t, l, - 15 -
Here Constraints (11) desribe that a positive number of train units N l,t, of subtype an be alloated to ross-setion train t of train series l if and only if A l,t, = 1. Here the parameter M l,t, is a fixed positive upper bound on the number of train units of subtype on train t of train series l. Constraints (12) guarantee that a ertain subtype an be alloated to train t of train series l only if the orresponding train type τ has been alloated to that train as well. Similarly, Constraints (13) guarantee that subtype an be alloated to train series l only if the orresponding train type τ has been alloated to that train series as well. Next, onstraints (14) speify that train type τ an be alloated to train t of train series l only if this type τ has been alloated to train series l. Constraints (15) have a similar interpretation for the subtypes. Aknowledgements The third author was partly sponsored by the Human Potential Program of the European Union under ontrat no. HPRN-CT-1999-00104 (AMORE). Referenes Ben-Khedher, N., J. Kintanar, C. Queille, and W. Strainling. Shedule optimization at SNCF: from oneption to day of departure. Interfaes, 28: 6-23, 1998. Blom, M. Minimum irulation of railway stok on a network, an integer programming approah. Master s thesis, University of Amsterdam, 1998. Bruker, P., J. Hurink, and T. Rolfes. Routing of railway arriages. Osnabrüker Shriften zur Mathematik, Reihe P, Heft 205, 1998. Cordeau, J.-F., P. Toth and D. Vigo. A survey of optimization models for train routing and sheduling. Transportation Siene, 32: 380-404, 1998. Cordeau, J.-F., F. Soumis, and J. Desrosiers. Simultaneous assignment of loomotives and ars to passenger trains. Operations Researh, 49: 531-548, 2001. Groot, R. Minimum irulation of railway stok, an integer programming algorithm. Master s thesis, University of Amsterdam, 1996. Montfort, J. van. Optimizing railway arriage irulation with integer linear programming. Master s thesis, University of Amsterdam, 1997. Shrijver, A. Minimum irulation of railway rolling stok. CWI Quarterly, 6: 205-217, 1993. - 16 -