TRAIN PLATFORMING PROBLEM Ľudmila JÁNOŠÍKOVÁ 1, Michal KREMPL 2

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1 GIS Ostrava Geonformatcs for Intellgent Transportaton Abstract TRAIN PLATFORMING PROBLEM Ľudmla JÁNOŠÍKOVÁ 1, Mchal KREMPL 2 1 Department of Transportaton Networks, Faculty of Management Scence and Informatcs, Unversty of Žlna, Unverztná 1, Žlna, Slovak Republc Ludmla.Janoskova@fr.unza.sk 2 Insttute of Transport, Faculty of Mechancal Engneerng, VŠB Techncal Unversty of Ostrava, 17. lstopu 15, Ostrava - Poruba, Czech Republc Mchal.Krempl.st@vsb.cz The tran platformng problem conssts n the allocaton of passenger trans to platforms n a ralway staton. One of the mportant problems a dspatcher has to solve, especally n a large ralway staton, s to decde, at whch platform track an approachng tran should arrve. There s a tool helpng hm n hs ob called the track occupancy plan. The plan specfes for each arrvng or departng tran the platform track along wth the tme slot durng whch the track wll be occuped by the tran. Ths paper deals wth a method for computer-ed desgn of the track occupancy plan. The problem s formulated as a b-crteron mxed nteger programmng problem. The frst obectve s to mnmse the devatons of the arrval and departure tmes proposed by the model from the tmes specfed by the tmetable. The second crteron mses the desrablty of the platform tracks to be assgned to the trans. The model s solved usng a lexcographc approach and the local branchng algorthm. The model was verfed by usng the real ta of Prague man staton. Results of the experments are ncluded. Keywords: tran platformng, schedulng, mxed nteger mathematcal programmng; multpleobectve programmng 1. INTRODUCTION The tran platformng problem s a subproblem of the generaton of a tmetable for a ralway company. The generaton of a tmetable s a herarchcal process. At the frst stage, a prelmnary tmetable for the whole network s proposed. In ths phase, a macroscopc vewpont at the ralway network s appled. Statons are consdered as black boxes. Capacty lmts of partcular statons and the movement of trans nsde the statons are not taken nto account. Then, at the second stage, a mcroscopc vewpont related to statons s appled. At every staton, the network tmetable s checked whether t s feasble wth respect to capacty, safety and tran operators preferences. Ths process results n a track occupancy plan whch specfes for each arrvng or departng tran the platform track along wth the tme slot durng whch the track wll be occuped by the tran. Cargo trans do not affect the plan snce they travel mostly n nght, when there are fewer passenger trans, they use dfferent tracks n the staton, and n case of conflctng movements they can wat at the entry sgnal. In the Czech and Slovak Republc, plannng tran movements through the staton s done by hand, usng planner s experence and a set of rules determned by a ralway company. The man goal of ths research s to desgn a more sophstcated approach whch would serve as a planner s decson supportng tool and result n a better track occupancy plan. Improvement n the plan qualty results n: 1. better management of tran operaton n the staton, namely: a) shorter tmes of routes occupaton by arrvng and departng trans, b) unform worklo of the nfrastructure elements, such as tracks, swtches, and platforms, whch les to a more robust plan resstant to random dsturbances; 2. hgher servce qualty perceved by passengers, namely: a) shorter dstances needed for changng trans,

2 GIS Ostrava Geonformatcs for Intellgent Transportaton b) more approprate platforms (platforms near to tcket sales ponts and to the staton entrance, platforms equpped by staton shops or caterng etc.), c) less probablty of changng the planned platform when the tran delays. 3. meetng tran operators requrements on arrval and departure tmes and platforms assgned to trans. Routng and schedulng trans at a staton has been studed by researchers n countres, where large, busy statons wth capacty constrants can be found. Bllonet (2003) dresses only the routng problem. The problem s modelled usng a graph theory and the nteger programmng formulaton of the resultng graph colourng problem s solved. However, the k colourng problem s not ndeed an optmsaton problem, t means any feasble soluton s acceptable and the problem formulaton does not reflect the soluton qualty, such as route lengths or platform preferences for ndvdual trans. Zwaneveld (1997) and Zwaneveld, Kroon, and van Hoesel (2001) formulate the problem of tran routng as a weghted node packng problem, usng bvalent programmng, whle the soluton algorthm apples the branch-and-cut method. A dsvantage of the above presented models s that the calculatons connected wth them are computatonally too complex and tme consumng. Another, practcally orented approach has gven up on applyng the nteger programmng methods, and replaced them by the heurstcs, solvng the schedulng and routng problems at a tme (Carey and Carvlle, 2003). The algorthm ncorporates, or consders, the operatonal rules, costs, preferences and tre-offs, whch are appled by experts creatng plans manually. The shortcomng of ths approach s obvous: snce t s a heurstcs, the optmalty of the resultng plan s not guaranteed. Other way of research, e.g. (Bažant and Kavčka, 2009, Chakroborty and Vkram, 2008), has been drected at operatonal tran management. In real tme t s necessary to reflect the requrements of the operaton burdened wth rregulartes,.e. to re-schedule the arrvals and departures tmes, and/or re-route trans. In ths paper we propose a mxed nteger programmng (MIP), b-crtera model of the tran platformng problem. The problem can be solved by a lexcographc approach, where partcular crtera are ranked accordng to ther mportance. 2. PROBLEM FORMULATION The tran platformng problem conssts of the followng partal ssues. For each tran, a platform track must be specfed at whch the tran should arrve; the platform track assgnment determnes the route, on whch the tran approaches, arrval tme at the platform and departure tme from the platform need to be determned. The soluton should mnmse devatons from the planned arrval and departure tmes and mse the total preferences for platforms and routes. The nputs to the mathematcal programmng model are as follow: 1. track layout of the staton, whch s necessary for determnng feasble platform tracks for a tran and conflctng routes, 2. lst of trans, where the ta requred for each tran nclude: a) planned tme of ts arrval at the platform, b) planned tme of ts departure from the platform, c) lne on whch the tran arrves (n-lne) and departs (out-lne), d) lst of feasble platform tracks wth ther desrablty for the tran, e) category of the tran. All tme ta are gven n mnutes. Further on we present the formulaton of the MIP model. Frst we need to explan the symbols used:

3 GIS Ostrava Geonformatcs for Intellgent Transportaton Subscrpts whch n the mathematcal model represent obects,, tran k, k platform track Input parameters (constants) t planned arrval tme of tran at the platform t planned departure tme of tran t Cn I O c t mn t p k s k stanrd amount of tme passengers take to change trans (depends on partcular ralway staton) arrval lne track (n-lne) for tran departure lne track (out-lne) for tran category of tran ; c = 1 for regonal stoppng trans and ncreases wth the speed and dstance travelled by the tran. We have to dvde tran nto categores, because nternatonal fast express trans have obvously hgher mportance than the regonal ones. Delays of nternatonal trans can commt more traffc problems and extra costs than delays of regonal trans. mnmum dwell tme of a tran at the platform mum tme nterval, n whch two tran movements are tested for a conflct preference coeffcent; t reflects the desrablty of the assgnment of platform track k to tran number of swtches on the route of tran from the arrval lne track to platform track k and from platform track k to the departure lne track mn s number of swtches on the shortest tran route n the staton s number of swtches on the longest tran route n the staton a(l,k,l,k) coeffcent, whch has value true, f the route connectng lne l to platform track k conflcts wth the route connectng lne l to platform track k; f there exsts any route connectng lne l to track k and any route connectng lne l to track k such that these two routes do not conflct, then a(l,k,l,k) = false. If both trans use the same staton or lne tracks (.e. k = k or l = l), then a(l,k,l,k) = true. The exstence of route conflcts can be dentfed n vance from a detaled map of the track layout. We opted the concept of conflctng routes and conflct solvng from Carey and Carvlle (2003). If two trans are on conflctng routes we must ensure that there s at least a requred mnmum heway (tme nterval) between them, for safety and sgnallng reasons. For example, let h(,k,,k) be the mnmum heway requred between tran departng from track k and the next tran arrvng at track k. The superscrpts d and a denote departure and arrval, and the order of the superscrpts ndcates the order of the trans,.e., tran s followed by. Smlarly we have h(,k,,k), h(,k,,k) and h(,k,,k) for combnatons arrval arrval, arrval departure, departure departure. We need not ntroduce subscrpts to denote the n-lnes or outlnes used by trans snce for an arrvng tran the n-lne s alrey specfed by I, and for a departng tran the out-lne s specfed by O. The preference coeffcent p k may reflect: operator s preferences of platforms, the dstance of the track k to the connectng trans, the length of the route used by tran arrvng to or departng from platform track k. The smoother and shorter the route s, the less the possblty of a conflct wth other trans s, hence the probablty of delay propagaton decreases.

4 GIS Ostrava Geonformatcs for Intellgent Transportaton In our model, coeffcent p k s set accordng to the followng formula: p k s 0.8 s s s k mn f track f track k k otherwse s the planned (or desred)track for tran slocatet the sameplatformas the plannedtrack Sets of obects K K() U set of all platform tracks set of feasble platform tracks for tran set of all arrvng, departng, and transt trans W() V V V V set of all connectng trans, whch has to wat for tran (, ) :, U,, t t t set of ordered pars of those trans that may arrve concurrently (, ) :, U,, t t t set of ordered pars of those trans that arrvng tran and departng tran may travel concurrently (, ) :, U,, t t t set of ordered pars of those trans that departng tran and arrvng tran may travel concurrently (, ) :, U,, t t t set of ordered pars of those trans that may depart concurrently Decson and auxlary varables of the model for U, k K : x k 1 0 f track k sassgnedto tran otherwse u v dfference between the planned and real arrval tme of tran at a platform, U dfference between the planned and real departure tme of tran from a platform, U The followng auxlary varables y are ntroduced for the couple of those trans and that may travel concurrently. They enable to express safety heways between conflctng trans. for, U, : y 1 0 f tran arrvesbeforetran arrves otherwse for, V for, V for, V : y : y : y f tran arrvesbeforetran departs otherwse f tran departsbeforetran arrves otherwse f tran departsbeforetran departs otherwse After these prelmnares, the mathematcal model can be wrtten as follows:

5 GIS Ostrava Geonformatcs for Intellgent Transportaton mnmse c u v (1) mse subect to U k x k U kk mn p (2) v t u t t U (3) Cn v t u t t U; W( ) (4) u, k,, k M 1 y M1 x M x t u t h 1 V, k K, k K : ai, k, I k k,, (5) u t u t h, k,, k My M1 x M1 x V, k K, k K : ai, k, I k k,, (6) Constrants (7) (12) are specfed for the other combnatons of arrval departure and have smlar meanng as (6). u t v t h, k,, k My M1 x M1 x u y k, U,, k K K (13) k k k, k,, k M1 y M1 x M x t v t h 1 1 x k kk ( ) k, U,, k K K (14), U,, I I, t t (15) 1 k U (16) u, v 0 U (17) x 0,1 U k k K (18) y 0,1, U, (19) 0,1 y V 0,1 y 0,1 y, (20), V (21), V (22) Model descrpton Obectve functon (1) mnmses the weghted devatons of the arrval and departure tmes proposed by the model from the tmes specfed by the tmetable. The weghts cause that long-dstance/hgh- speed trans wll respect planned tmes and regonal trans wll be postponed f necessary. The second crteron mses the desrablty of the platform tracks to be assgned to the trans. Constrant (3) ensures that a mnmum dwell tme needed for boardng and alghtng must be kept.

6 GIS Ostrava Geonformatcs for Intellgent Transportaton Constrant (4) states that connectng tran wth real departure least u Cn t t. v t has to wat n staton to tme at Constrants (5) (12) ensure that a mnmum heway wll be kept between conflctng trans. More precsely, constrant (5) states that f trans and have planned arrval tmes wthn t and tran arrves at platform track k before tran arrves at track k,.e. x 1, x 1, y 1, (23) k k and trans are on conflctng routes (.e. a(i,k,i,k) s true), then tran s allowed to arrve at least h(,k,,k) mnutes later than tran. If at least one of the condtons (23) s not met (e.g. tran s not assgned to track k), then constrant (5) becomes rrelevant as the rght-hand sde s negatve (M s a sutably pcked hgh postve number). If tran s followed by tran ( y 0 ), then s allowed to arrve at least h(,k,,k) mnutes later than, whch s ensured by constrant (6). Constrants (7) (12) have a smlar meanng for the other combnatons of arrval departure. Constrants (13) (14) ensure that a tran wll not be dspatched to an occuped track. If tran s followed by tran ( y 0 ) and both trans arrve at the same track k, then s allowed to arrve at least h(,k,,k) mnutes after tran leaves track k, whch s expressed by constrant (13). Constrant (14) holds for the reverse order of trans,. Constrant (15) states that same n-lne. y s 1 f tran s followed by tran at the arrval and both trans travel on the Constrant (16) ensures that each tran s always dspatched to exactly one platform track. The remanng oblgatory constrants (17) (22) specfy the defnton domans of the varables. Ths multple-crtera optmsaton problem was solved usng the lexcographc approach, where the obectve functons are ranked accordng to ther mportance. In the problem at hand, the frst obectve functon (.e. to meet the tmetable) s more mportant that the second one (.e. to respect track preferences). Ths orderng reflects how decsons are currently me n practce. The soluton technque conssts of two steps. In the frst step the problem (1), (3) (22) s solved gvng the best value of the weghted sum of devatons Then the constrant u v best c f1 U (24) best f 1. s ded and the model (2) (22), (24) s solved. Because both MIP problems are hard and the optmal solutons cannot be found wthn a reasonable tme lmt, we decded to mplement the local branchng heurstc (Fschett, Lod, and Salvagnn, 2009) usng the general optmsaton software Xpress. 3. CASE STUDY The model was verfed by usng the real ta of Prague man staton and the tmetable vald for the years 2004/2005. Prague man staton s a large staton that at the gven tme h 7 platforms, 17 platform tracks and 8 arrval or departure lne tracks. Accordng to the tmetable 2004/2005, the staton dealt wth 288 regular passenger trans per a weeky. We could use any tmetable for valton, however we used the tmetable vald for 2004/2005 because we knew that t was done wth some mstakes. We wanted to demonstrate that our model s vald, can detect every possble conflct n the tmetable and suggest ts soluton. Snce the model wth 288 trans contans varables and constrants, t s not possble to solve t to optmalty n a reasonable tme. That s why the decomposton of the problem must be done. The plannng perod (a y) was dvded nto shorter tme perods. They were chosen n such a way so that the mornng

7 GIS Ostrava Geonformatcs for Intellgent Transportaton and evenng peak hours were taken as a whole and the rest of the y was dvded nto shorter perods wth approxmately the same number of trans. The resultng tme ntervals can be seen n Table 1. For every tme nterval, the mathematcal programmng model was solved usng the lexcographc approach descrbed n the prevous secton. In case that the exact algorthm (branch and bound method) not fnsh n a predetermned computatonal tme (30 mnutes) then the local branchng heurstc was appled. The computatonal experments were performed for a shorten change tran tme whch s 4 mnutes n Prague man statons, as well as for the normal change tme (8 mnutes). The results for the shortened tme are reported n Table 1 and for the normal change tme n Table 2. Table 1. Results of experments for the decomposed plannng perod and shortened change tme Tme nterval trans varables constrants Value of 1st obectve functon Value of 2nd obectve functon Delay on arrval Delay on depart. trans allocated to dfferent platform [-] [-] [-] [-] [-] [-] [mn] [mn] [-] 0:00-5: :00-8: :00-10: :00-12: :00-15: :00-18: :00-24: Table 2. Results of experments for the decomposed plannng perod and normal change tme Tme nterval trans varables constrants Value of 1st obectve functon Value of 2nd obectve functon Delay on arrval Delay on depart. trans allocated to dfferent platform [-] [-] [-] [-] [-] [-] [mn] [mn] [-] 0:00-5: :00-8: :00-10: :00-12: :00-15: :00-18: :00-24: The results of computatonal experments show that the tmetable was not correct wth regard to safety requrements. There were some trans travellng on conflctng routes concurrently. That s why ther desred arrvng or departng tmes could not be kept. Moreover, n some cases the orgnal tmetable not respect desred tme passengers need to change trans. However the model respects such connectons. The best soluton proposed by the model wth the shortened change tme delays 3 trans at arrval by 5 mnutes and

8 GIS Ostrava Geonformatcs for Intellgent Transportaton 12 trans at departure by 17 mnutes n total, and dspatches 31 (11 %) trans to platform tracks dfferent from the planned ones. Departures of 7 trans are postponed by 1 mnute and 5 trans by 2 mnutes. For the normal change tme, 3 trans are delayed at arrval by 5 mnutes and 16 trans are delayed at departure by 27 mnutes n total (8 trans by 1 mnute, 6 trans by 2 mnutes, 1 tran by 3 mnutes and 1 tran by 4 mnutes). 32 trans are dspatched to platform tracks dfferent from the planned ones. Other experments were performed to nvestgate: how decomposton of plannng perod nfluence the computatonal tme and the qualty of obtaned soluton wthn 30 mnutes lmt for computng, how tran delays nfluence the track occupancy plan, effcency of the branch and bound and local branchng methods. 4. CONCLUSIONS In the paper, a mxed nteger programmng model for the tran platformng problem at a passenger ralway staton s descrbed. The model proposes a track occupancy plan that respects safety constrants for tran movements and relatons between connectng trans, mnmses devatons of the arrval and departure tmes from the tmetable and mses the desrablty of the platform tracks to be assgned to the trans. The model could serve as a planner s decson supportng tool. Acknowledgements Ths research was supported by the Scentfc Grant Agency of the Mnstry of Educaton of the Slovak Republc and the Slovak Acemy of Scences under proect VEGA 1/0296/12 Publc servce systems wth far access to servce and by the Slovak Research and Development Agency under proect APVV Desgnng of Far Servce Systems on Transportaton Networks. REFERENCES Bažant, M. and Kavčka, A. (2009) Artfcal neural network as a support of platform track assgnment wthn smulaton models reflectng passenger ralway statons. Proceedngs of the Insttuton of Mechancal Engneers, rt F: Journal of Ral and Rapd Transt 223 (5), Bllonet, A. (2003) Usng nteger programmng to solve the tran-platformng problem. Transportaton Scence 37 (2), Carey, M. and Carvlle, S. (2003) Schedulng and platformng trans at busy complex statons. Transportaton Research rt A 37, Chakroborty, P. and Vkram, D. (2008) Optmum assgnment of trans to platforms under partal schedule complance. Transportaton Research rt B: Methodologcal 42 (2), FICO TM Xpress Optmzaton Sute [onlne]. Avalable from: [Accessed 10 October 2011]. Fschett, M., Lod, A., and Salvagnn, D. (2009) Just MIP t!. In: Manezzo, V., Stützle, T., Voß, S. (eds.), Matheurstcs. Sprnger, New York, Zwaneveld, P.J. (1997) Ralway Plannng - routng of trans and allocaton of passenger lnes. Ph.D. thess, Erasmus Unversty Rotterm, Rotterm, the Netherlands. Zwaneveld, P.J., Kroon, L.G., and van Hoesel, S.P.M. (2001) Routng trans through a ralway staton based on a node packng model. European Journal of Operatonal Research 128,