The Ideal Train Timetabling Problem

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1 The Ideal Tran Tmetablng Problem Tomáš Robenek a,1, Shad Sharf Azadeh a, Janghang Chen b, Mchel Berlare a, Yousef Maknoon a a Transport and Moblty Laboratory (TRANSP-OR), School of Archtecture, Cvl and Envronmental Engneerng (ENAC), École Polytechnque Fédérale de Lausanne (EPFL), CH-1015 Lausanne, Swtzerland 1 E-mal: tomas.robenek@epfl.ch b Insttute of Sno-US Global Logstcs, Shangha Jaotong Unversty, P.R. Chna Abstract The am of ths paper s to analyze and to mprove the current plannng process of the passenger ralway servce. At frst, the state-of-the-art n research s presented. Gven the recent changes n legslature allowng compettors to enter the ralway ndustry n Europe, also known as lberalzaton of ralways, the current way of plannng does not reflect the stuaton anymore. The orgnal plannng s based on the accessblty/moblty concept provded by one carrer, whereas the compettve market conssts of several carrers that are drven by the proft. Moreover, the current practce does not defne the deal tmetables (the ntal most proftable tmetables) and thus t s assumed that the Tran Operatng Companes (TOCs) use ther hstorcal data (tran occupaton, tcket sales, etc.) n order to construct the deal tmetables. For the frst tme n ths feld, we tackle the problem of deal tmetables n ralway ndustry from passenger behavor pont of vew. We propose the Ideal Tran Tmetablng Problem (ITTP) to create a lst of tran tmetables for each TOC separately. The ITTP approach ncorporates the passenger demand n the plannng and ts am s to mnmze the passengers cost. The outcome of the ITTP s the deal tmetables (ncludng connectons between the trans), whch then serve as nputs for the tradtonal Tran Tmetablng Problem (TTP). Keywords Ideal Ralway Tmetablng, Cyclc - Noncyclc tmetable, Mxed Integer Lnear Programmng 1 Introducton The tme of domnance of one ral operatng company (usually the natonal carrer) over the markets n Europe s reachng to an end. The new EU regulaton (EU Drectve 91/440) allows open access to the ralway nfrastructure to companes other than those who own the nfrastructure, thus allowng the competton to exst n the market. Up to ths pont, the natonal carrers were subsdzed by local governments and ther purpose was to offer the accessblty and moblty to the publc (passengers). On the other hand, the goal of the prvate sector s to generate revenue,.e. to maxmze the captured demand. However, the passenger demand s subject to the human behavor that ncorporates several factors, to lst a few: senstvty to the tme of the departure related to the trp purpose 1

2 (weekday peak hours for work or school, weekends for lesure, etc.), comfort, percepton, etc. Moreover the passenger servce has to compete wth other transportaton modes (car, natonal ar routes, etc.) and thus faces even hgher pressure to create good qualty tmetables. STRATEGIC - several years TACTICAL - >= 1 year OPERATIONAL - < 1 year Actual Tmetables Tran Platformng Platform Assgnments Demand Lne Plannng Lnes Tran Tmetablng Actual Tmetables Rollng Stock Plannng Tran Assgnments Actual Tmetables Crew Plannng Crew Assgnments TOC Infrastructure Manager Fgure 1: Plannng overvew of ralway operaton If we have a closer look at the plannng horzon of the ralway passenger servce (as descrbed n Caprara et al. (2007) and vsualzed on Fgure 1), we can see that the ssue of deal departure tmes has been neglected n the past. The Tran Tmetablng Problem (TTP) does take as nput the deal tmetables (n ts non-cyclc verson), however the procedure of generatng such tmetables s mssng. Smlarly, n the cyclc verson of the TTP, the objectve functon that would consder passenger demand as the drver s undefned. We beleve that the lack of the defnton of the deal tmetables and how to create them, s a major gap, caused by the lack of a competton n the prevous ralway market settngs. We assume that not takng the passengers wshes nto account, lead to the decrease of the ralway mode share n the transportaton market. And thus we propose to nsert an addtonal secton n the plannng horzon called the Ideal Tran Tmetablng Problem (ITTP). In the ITTP, we ntroduce a defnton of the deal tmetable as follows: the deal tmetable, conssts of tran schedules, such that the cost assocated wth travelng by tran, of all of the passengers s mnmzed. Such a tmetable would beneft both, passengers and the TOC n the respectve manner: t would ft passengers wshes, whch would lead to the ncrease of the demand and to ncrease the TOCs proft. The ITTP s usng the output of the LPP and serves as an nput to the tradtonal TTP and hence, t s placed between the two respectve problems (Fgure 2). The drver of ths problem s the passenger demand. The model wll allow tmetables of the lne to take the form of the non-cyclc or cyclc schedule. Moreover, we ntroduce a demand nduced connectons. The connectons between the trans are not pre-defned, but are subject to the demand. In the lterature the connectons are handled only n the cyclc verson of the TTP, 2

3 STRATEGIC - several years TACTICAL - >= 1 year OPERATIONAL - < 1 year Actual Tmetables Tran Platformng Platform Assgnments Demand Lne Plannng Lnes Ideal Tran Tmetablng Ideal Tmetables Tran Tmetablng Actual Tmetables Rollng Stock Plannng Tran Assgnments Actual Tmetables Crew Plannng Crew Assgnments TOC Infrastructure Manager Fgure 2: Modfed overvew of ralway operaton where the connectons are always nduced, wthout a proper reasonng. In ths manuscrpt, we ntroduce the current lterature on the topc (Secton 2) and before we formulate the Ideal Tran Tmetablng Problem mathematcally (Secton 4), we dscuss how to defne qualty of a tmetable from the passenger pont of vew,.e. how to form the objectve functon (Secton 3). We fnalze the paper by drawng some conclusons (Secton 5). 2 Lterature Revew The state-of-the-art lterature s mostly focused on the tradtonal plannng problems and consders the demand only n the ntal phase (.e. the LPP). Due to the extensveness of the lterature, we focus on revewng of the classcal TTP as the ITTP s goal s to provde better nformaton for the TTP usng the outcome of the LPP. The latest extensve lterature revew on LPP can be found n Schöbel (2012). The am of the TTP s to fnd a feasble (operatonal) tmetable for a whole ralway network,.e. there are no conflcts of the trans usng the tracks. In the non-cyclc verson, deal tmetables wth ther respectve profts or costs serve as the man nput. The TTP then shfts the departures for conflctng trans, such that the losses of the profts or ncrease of the costs are mnmzed. In the cyclc verson, the model searches for a frst feasble tmetable gven the sze of the cycle. The user can create hs/her own objectve functon, otherwse arbtrary soluton wll be selected. 2.1 Non-Cyclc TTP Most of the models, on the non-cyclc tmetablng, n the publshed lterature, formulate the problem ether as Mxed Integer Lnear Programmng (MILP) or Integer Lnear Programmng (ILP). The MILP model uses contnuous tme, whereas the ILP model dscretzes the tme. Due to the complexty of the problem, many heurstc approaches are consdered. Brannlund et al. (1998) use dscretzed tme and solve the problem wth lagrangan relaxaton of the track capacty constrants. The model s formulated as an ILP. Caprara et al. 3

4 (2002, 2006), Fscher et al. (2008) and Cacchan et al. (2012) also use lagrangan relaxaton of the same constrants to solve the problem. In Cacchan et al. (2008), column generaton approach s tested. The approach tends to fnd better bounds than the lagrangan relaxaton. In Cacchan et al. (2010a), several ILP re-formulatons are tested and compared. In Cacchan et al. (2010b), the ILP formulaton s adjusted, n order to be able to schedule extra freght trans, whlst keepng the tmetables of the passengers trans fxed. In Cacchan et al. (2013), dynamc programmng, to solve the clque constrants, s used. In Carey and Lockwood (1995), a heurstc, that consders one tran at a tme and solves a MILP, based on the already scheduled trans, s ntroduced. Hggns et al. (1997) then show several more heurstcs to solve the MILP model. Olvera and Smth (2000) and Burdett and Kozan (2010), re-formulate the problem as a job-shop schedulng one. Erol (2009), Caprara (2010) and Harrod (2012), survey dfferent types of models for the TTP. None of the above formally defnes the deal tmetable. The models focus on the feasblty of the solutons,.e. the track occupaton constrants. Demand s omtted n the formulatons. 2.2 Cyclc TTP One of the frst papers, dealng wth cyclc tmetables s Serafn and Ukovch (1989). The paper brngs up the topc of cyclc schedulng based on the Perodc Event Schedulng Problem (PESP). The problem s solved wth an algorthm usng mplct enumeraton and network flow theory. In Nachtgall and Voget (1996) model for mnmzaton of the watng tmes n the ralway network, whlst keepng the cyclc tmetables (based on PESP), s solved usng branch and bound. The same model s solved usng genetc algorthms n Nachtgall (1996). The general PESP model s solved usng constrant generaton algorthm n Odjk (1996) and wth branch and bound n Lndner and Zmmermann (2000). In Kroon and Peeters (2003), varable trp tmes are consdered. Peeters (2003) then further elaborates on PESP and dscuss dfferent forms of the objectve functon. In Lebchen and Mohrng (2002), the PESP attrbutes are analyzed on the case study of Berln s underground and n Lebchen (2004) mplementaton of the symmetry n the PESP model s presented. Lndner and Zmmermann (2005) propose to use decomposton based branch and bound algorthm to solve the PESP. Kroon et al. (2007) and Shafa et al. (2012), deal wth robustness of cyclc tmetables. Lebchen and Mohrng (2004) propose to ntegrate network plannng, lne plannng and rollng stock schedulng nto the one perodc tmetablng model (based on PESP). Cam et al. (2007) and Kroon et al. (2014) ntroduce flexble PESP nstead of the fxed tmes of the events, tme wndows are provded. 3 Qualty of a Tmetable In order to fnd a good tmetable from the passenger pont of vew, we need to take nto account passenger behavor. Such a behavor can be modeled usng dscrete choce theory (Ben-Akva and Lerman (1985)). The base assumpton n dscrete choce theory s that the passengers maxmze ther utlty,.e. mnmze the cost assocated wth each alternatve and select the best one. We propose the followng costs assocated wth passengers deal tmetable: 4

5 n-vehcle-tme (VT) watng tme (WT) number of transfers (NT) scheduled delay (SD) The n-vehcle-tme s the (total) tme passengers spend on board of (each) tran. Ths tme allows the passengers to dstngush between the slow and the fast servces. The watng tme s the tme passengers spend watng between two consecutve trans n ther respectve transfer ponts. The cost percepton related to the watng tme s evaluated as double and a half of the n-vehcle-tme (see Wardman (2004)). The transfer(s) am at dstngushng between drect and nterchange servces. In lterature and practce, t s by addng extra travel (n-vehcle) tme to the overall journey. In our case, we have followed the example of Dutch Ralways (NS), where penalty of 10 mnutes per transfer s appled (see de Kezer et al. (2012)). Even though varety of studes show that number of nterchanges, dstance walked, weather, etc. play effect n the process, t s rather dffcult to ncorporate n optmzaton models. Thus usng the appled value (by NS) wll brng ths research closer to the ndustry. The scheduled delay s ndcatng the tme of the day passengers want to travel,.e. followng the assumpton that the demand s tme dependent. For example: most of the people have to be at ther workplace at 8 a.m. Snce t s mpossble to provde servce that would secure deal arrval tme to the destnaton for everyone, scheduled delay functons are appled (Fgure 3). Scheduled Delay f_2 Ideal Tme f_1 Tme Fgure 3: Scheduled Delay Functons As shown n Small (1982), the passengers are wllng to shft ther arrval tme by 1 to 2 mnutes earler, f t wll save them 1 mnute of the n-vehcle-tme, smlarly they would shft ther arrval by 1/3 to 1 mnute later for the same n-vehcle-tme savng. If we would consder the boundary case, the lateness (f 1 = 1) s perceved equal to the n-vehcle-tme and earlness (f 2 = 0.5) has half of the value (as seen on Fgure 3). To estmate the perceved cost (qualty) of the selected tnerary n a gven tmetable for a sngle passenger, we sum up all the characterstcs: C = V T W T + 10 NT + SD [mn] (1) For a better understandng, consder the followng example usng network on Fgure 4: passenger s tnerary conssts of takng 3 consecutve trans n order to go from hs orgn to 5

6 WT 1 WT 2 Transfer 1 Transfer 2 VT 2 VT 1 VT 3 Orgn Destnaton Fgure 4: Example Network hs destnaton, he has to change tran twce. If he arrves to hs destnaton earler than hs deal tme, hs SD wll be: ( ) deal tme arrval tme SD e = argmax, 0 (2) 2 We use argmax functon as one tran lne has several trans per day scheduled and the passenger selects the one closest to hs desred travelng tme. On the other hand, f he arrves later than hs deal tme, then hs SD wll be: SD l = argmax (0, arrval tme deal tme) (3) The overall scheduled delay s then formed: SD = argmn (SD e, SD l ) (4) Hs overall perceved cost wll be the followng: C = V T W T + 10 NT + SD [mn] (5) trans transfers The resultng value s n mnutes, however t s often desrable to estmate the cost n monetary values for prcng purposes. In such a case, natonal surveys estmatng respectve naton s value of tme (VOT) exst. The VOT s gven n naton s currency per hour, for nstance n Swtzerland the VOT for commuters usng publc transport s swss francs per hour (Axhausen et al. (2008)). To make the cost n monetary unts, smply multply the whole Equaton 1 by the VOT/60. The am of our research s not to calbrate the weghts n Equaton 1, but to provde better tmetables n terms of the departure tmes. The weghts serve as an nput for our problem and thus can be changed at any tme. Addng everythng up, the deal tmetable from the passenger pont of vew can be defned as follows: The deal tmetable conssts of tran departure tmes that passengers global costs are mnmzed,.e. the most convenent path to go from an orgn to a destnaton traded-off by a tmely arrval to the destnaton for every passenger. Smlar concept, mprovng qualty of tmetables has been done n Vansteenwegen and Oudheusden (2006, 2007). Ther approach has been focused on relable connectons for 6

7 transferrng passengers, whereas n our framework we focus on the overall satsfacton of every passenger. Other concept smlar to ours has been used n the delay management, namely n Kana et al. (2011) and Sato et al. (2013). However ther defnton of dssatsfacton of passengers omts the scheduled delay. 4 Mathematcal Formulaton In ths secton, we present a mxed nteger programmng formulaton for the Ideal Tran Tmetablng Problem. The am of ths problem s to defne and to provde the deal tmetables as nput for the tradtonal TTP. It s not well sad n the TTP, what deal means. It s only brefly mentoned, that supposedly, those are the tmetables, that brng the most proft to the TOCs (ths assumpton s n lne wth the compettve market). Generally speakng, the more of the demand captured, the hgher the proft. Thus the ITTP s goal s to desgn TOC s tmetables, such that the captured passenger demand s maxmzed (objectve, but not the form of the objectve functon). The nput of the ITTP s the demand that takes the form of the amount of passengers that want to travel between OD par I and that want to arrve to ther destnaton at ther deal tme t T. Apart of that, there s a pool of lnes l L along wth the lnes frequences expressed as the avalable tran unts v V l (both results of the LPP) and the set of paths between every OD par p P. The path s called an ordered sequence of lnes to get from an orgn to a destnaton ncludng detals such as the runnng tme from the orgn of the lne to the orgn of the OD par h pl (where l = 1), the runnng tme from an orgn of the OD par to a transferrng pont between two lnes r pl (where l = 1), the runnng tme from the orgn of the lne to the transferrng pont n the path h pl (where l > 1), the runnng tme from one transferrng pont to another r pl (where l > 1 and l < L p ) and the runnng tme from the last transferrng pont to a destnaton of the OD par r pl (where l = L p ). Note that the ndex p s always present as dfferent lnes usng the same track mght have dfferent runnng tmes. Part of the ITTP s the routng of the passengers through the ralway network. Usng a decson varable x tp, we secure that each passenger (t) can use exactly one path. Smlarly, wthn the path, passenger can use exactly one tran on every lne n the path (decson varable y tplv ). These decson varables, among others, allow us to backtrace the exact tnerary of every passenger. The tmetable s understood as a set of departures for every tran on every lne (values of d l v). The tmetable can take form of a non-cyclc or a cyclc verson (dependng f the cyclcty constrants are actve, see below). Snce the nput demand s determnstc, the objectve functon would be to mnmze the total travel tme of every passenger. However, as dscussed n the prevous secton, passengers are not smply mnmzng the total travel tme, but the overall cost of the journey by tran. Thus the objectve functon takes form of the Equaton 5 weghted by the demand and the value of tme. It could be objected that snce the demand s determnstc, the problem does not reflect the realty. However,the goal of the ITTP s to take the determnstc demand and desgn such tmetables that would ft the estmated demand and avod loss of the estmated number of passengers. In order to ncrease the forecasted determnstc demand, external factors lke 7

8 servce on board, destnatons served, renewed vehcle park, etc. would have to be changed. We can formulate the ITTP as follows: Sets Followng s the lst of sets used n the model: I set of orgn-destnaton pars T set of deal tmes for OD par P set of possble paths between OD par L set of operated lnes L p set of lnes n the path p V l set of avalable vehcles on lne l The lnes and the set of avalable vehcles per lne V l s an output from the Lne Plannng Problem based on the selected frequences wthn the problem. Input Parameters Followng s the lst of parameters used n the model: M suffcently large number (for daly plannng n mnutes, the value can be 1440) m mnmum transfer tme c cycle r pl runnng tme between OD par on path p usng lne l h pl tme to arrve from the startng staton of the lne l to the frst pont (that nvolves ths one) n the path p of the OD par D t demand between OD wth deal tme t q w value of the watng tme (=2.5) q t value of the n vehcle tme (VOT) f 1 coeffcent of beng early (SD e = 0.5) f 2 coeffcent of beng late (SD l = 1) a penalty for havng a tran transfer (=10 mn) Decson Varables Followng s the lst of decson varables used n the model: C t the total cost of a passenger wth deal tme t between OD par w t the total watng tme of a passenger wth deal tme t between OD par w tp the total watng tme of a passenger wth deal tme t between OD par usng path p the watng tme of a passenger wth deal tme t between OD par on the lne l that s part of the path p,.e. the watng tme n the transferrng pont, when transferrng to lne l x tp 1 f passenger wth deal tme t between OD par chooses path p; 0 otherwse s t the scheduled delay of a passenger wth deal tme t between OD par s tp the scheduled delay of a passenger wth deal tme t between OD par travelng on the path p d l v the departure tme of a tran v on the lne l 8

9 y tplv 1 f a passenger wth deal tme t between OD par on the path p takes the tran v on the lne l; 0 otherwse zv l dummy varable to help modelng the cyclcty correspondng to a tran v on the lne l Routng Model The ITTP model can be decomposed nto 2 parts: routng and prcng. The routng takes care of the feasblty of the soluton, whereas prcng takes care of the cost attrbutes. At frst, we present the routng of the passengers the Routng Model (RM): mn D t C t (6) I t T x tp = 1, I, t T, (7) p P y tplv = 1, I, t T, p P, l L p, (8) v V ( l d l v d l ) v 1 = c z l v, l L, v V : v > 1, (9) C t 0, I, t T, (10) d l v 0, l L, v V l, (11) x tp (0, 1), I, t T, p P, (12) y tplv (0, 1), I, t T, p P, l L p, v V l, (13) z l v N, l L, v V l. (14) The objectve functon (6) s mnmzng the passengers costs. Constrants (7) secure that every passenger s usng exactly one path to get from hs/her orgn to hs/her destnaton. Smlarly constrants (8) make sure that every passenger takes exactly one tran on each of the lnes n hs/her path. Constrants (9) model the cyclcty usng nteger dvson. When solvng the non-cyclc verson of the problem, these constrants have to be removed. Constrants (10) (14) set the domans of decson varables. Prcng Constrants To make the ITTP complete, we need to expand the Routng Model wth the prcng constrants. We wll add the prcng constrants n blocks of attrbutes that create the cost of a passenger. s tp f 2 ( s tp f 1 t s t s tp M (1 x tp ), (( ) ) I, t T, p P, (15) d L v + h L + r p L t ( ) M 1 y tp L v, I, t T, p P, v V L, (16) ( )) d L v + h L + r p L ( ) M 1 y tp L v, I, t T, p P, v V L, (17) s t 0, I, t T, (18) 9

10 s tp 0, I, t T, p P. (19) The frst block of constrants takes care of the scheduled delay (SD). In our model we have 2 types of scheduled delay: SD for every path (constrants (19)) and SD that s lnked to the path, whch wll be the fnal selected path of a gven passenger(s) wth a gven deal tme (constrants (18)). As descrbed n the Secton 3, the constrants (16) model the earlness of the passengers (Equaton 2) and constrants (17) model the lateness (Equaton 3). Snce both of the above constrants are actve at the same tme, the Equaton 4 s mplctly taken care of (greater equal sgn). Constrants (15) make sure that only one SD s selected not necessarly the lowest one as t depends on the cost of the whole path,.e. the path wth the smallest overall cost wll be selected for the gven OD par wth a gven deal tme. w t w tp M (1 x tp ), I, t T, p P, (20) w tp = l L p \1, I, t T, p P, (21) (( ) )) d l v + h pl (d l v + hpl + r pl + m ) ( ) I, t T, p P, l L p : M (1 y tpl v M 1 y tplv, l > 1, l = l 1, v V l, v V l, (( ) )) (22) d l v + h pl (d l v + hpl + r pl + m ) ( ) I, t T, p P, l L p : +M (1 y tpl v + M 1 y tplv, l > 1, l = l 1, v V l, v V l, (23) w t 0, I, t T, (24) w tp 0, I, t T, p P, (25) 0, I, t T, p P, l L p. (26) The second block of constrants s modelng the watng tme (WD). There are 3 types of watng tme: the fnal selected watng tme n the best path (constrants (24)), the total watng tme of every path (constrants (25)) and the watng tme at every transferrng pont n every path (constrants (26)). The constrants (22) and (23) are complementary constrants that model the watng tme n the transferrng ponts n every path. In other words, these two constrants fnd the two best connected trans n the two tran lnes n the passengers path. Constrants (21) add up all the watng tmes n one path to estmate the total watng tme n a gven path. Constrants (20) make sure that only one WT s selected (smlarly as constrants (15) for SD). C t = q v q w w t + q v a +q v p P p P r pl l L p x tp ( L p 1) x tp + q v s t, I, t T. (27) At last, constrants (27) combne all the attrbutes together as n Equaton 5 multpled by the VOT. The complete ITTP model can be seen n Appendx A. 10

11 5 Conclusons and Future Work In ths research, we survey the lterature on the current plannng horzon for the ralway passenger servce and we dentfy a gap n the plannng horzon demand based (deal) tmetables. We then defne a new way, how to measure the qualty of a tmetable from the passenger pont of vew and ntroduce a defnton of such an deal tmetable. We present a formulaton of a mxed nteger lnear problem that can desgn such tmetables. The new Ideal Tran Tmetablng Problem fts nto the current plannng horzon of ralway passenger servce and s n lne wth the new market structure and the current trend of puttng passengers back nto consderaton, when plannng a ralway servce. The novel approach not only desgns tmetables that ft the best the demand, but also creates by tself connecton between two trans, when needed. Moreover, the output conssts of the routng of the passengers and thus the tran occupaton can be extracted and be used effcently, when plannng the rollng stock assgnment (.e. the Rollng Stock Plannng Problem). The ITTP can create both non-cyclc and cyclc tmetables. In the future work, we wll focus on effcent solvng of the problem. The new defnton of a qualty of a tmetable (the passenger pont of vew) creates a lot of opportuntes for future research: effcent handlng of the TOC s fleet, better delay management, robust tran tmetablng passenger-wse, etc. References Axhausen, K. W., Hess, S., Köng, A., Abay, G., Bates, J. J. and Berlare, M. (2008). Income and dstance elastctes of values of travel tme savngs: New swss results, Transport Polcy 15(3): Ben-Akva, M. and Lerman, S. (1985). Dscrete Choce Analyss, The MIT Press, Cambrdge Massachusetts. Brannlund, U., Lndberg, P. O., Nou, A. and Nlsson, J. E. (1998). Ralway tmetablng usng lagrangan relaxaton, Transportaton Scence 32(4): Burdett, R. and Kozan, E. (2010). A sequencng approach for creatng new tran tmetables, OR Spectrum 32(1): Cacchan, V., Caprara, A. and Fschett, M. (2012). A lagrangan heurstc for robustness, wth an applcaton to tran tmetablng, Transportaton Scence 46(1): Cacchan, V., Caprara, A. and Toth, P. (2008). tmetablng on a corrdor, 4OR 6(2): A column generaton approach to tran Cacchan, V., Caprara, A. and Toth, P. (2010a). Non-cyclc tran tmetablng and comparablty graphs, Operatons Research Letters 38(3): Cacchan, V., Caprara, A. and Toth, P. (2010b). Schedulng extra freght trans on ralway networks, Transportaton Research Part B: Methodologcal 44(2): Cacchan, V., Caprara, A. and Toth, P. (2013). Fndng clques of maxmum weght on a generalzaton of permutaton graphs, Optmzaton Letters 7(2):

12 Cam, G., Fuchsberger, M., Laumanns, M. and Schupbach, K. (2007). Perodc ralway tmetablng wth event flexblty. Caprara, A. (2010). Almost 20 years of combnatoral optmzaton for ralway plannng: from lagrangan relaxaton to column generaton, n T. Erlebach and M. Lübbecke (eds), Proceedngs of the 10th Workshop on Algorthmc Approaches for Transportaton Modellng, Optmzaton, and Systems, Vol. 14 of OpenAccess Seres n Informatcs (OASIcs), Schloss Dagstuhl Lebnz-Zentrum fuer Informatk, Dagstuhl, Germany, pp Caprara, A., Fschett, M. and Toth, P. (2002). Modelng and solvng the tran tmetablng problem, Operatons Research 50(5): Caprara, A., Kroon, L. G., Monac, M., Peeters, M. and Toth, P. (2007). Passenger ralway optmzaton, n C. Barnhart and G. Laporte (eds), Handbooks n Operatons Research and Management Scence, Vol. 14, Elsever, chapter 3, pp Caprara, A., Monac, M., Toth, P. and Guda, P. L. (2006). A lagrangan heurstc algorthm for a real-world tran tmetablng problem, Dscrete Appled Mathematcs 154(5): IV ALIO/EURO Workshop on Appled Combnatoral Optmzaton. Carey, M. and Lockwood, D. (1995). A model, algorthms and strategy for tran pathng, The Journal of the Operatonal Research Socety 46(8): pp de Kezer, B., Geurs, K. and Haarsman, G. (2012). Interchanges n tmetable desgn of ralways: A closer look at customer resstance to nterchange between trans., n AET (ed.), Proceedngs of the European Transport Conference, Glasgow, 8-10 October 2012 (onlne), AET. Erol, B. (2009). Models for the Tran Tmetablng Problem, Master s thess, TU Berln. Fscher, F., Helmberg, C., Jan sen, J. and Krosttz, B. (2008). Towards solvng very large scale tran tmetablng problems by lagrangan relaxaton, n M. Fschett and P. Wdmayer (eds), ATMOS th Workshop on Algorthmc Approaches for Transportaton Modelng, Optmzaton, and Systems, Schloss Dagstuhl - Lebnz-Zentrum fuer Informatk, Germany, Dagstuhl, Germany. Harrod, S. S. (2012). A tutoral on fundamental model structures for ralway tmetable optmzaton, Surveys n Operatons Research and Management Scence 17(2): Hggns, A., Kozan, E. and Ferrera, L. (1997). Heurstc technques for sngle lne tran schedulng, Journal of Heurstcs 3(1): Kana, S., Shna, K., Harada, S. and Tom, N. (2011). An optmal delay management algorthm from passengers vewponts consderng the whole ralway network, Journal of Ral Transport Plannng & Management 1(1): Robust Modellng of Capacty, Delays and Reschedulng n Regonal Networks. Kroon, L. G., Dekker, R. and Vromans, M. J. C. M. (2007). Cyclc ralway tmetablng: a stochastc optmzaton approach, Proceedngs of the 4th nternatonal Dagstuhl, ATMOS conference on Algorthmc approaches for transportaton modelng, optmzaton, and systems, ATMOS 04, Sprnger-Verlag, Berln, Hedelberg, pp

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14 Small, K. A. (1982). The schedulng of consumer actvtes: Work trps, The Amercan Economc Revew 72(3): pp Vansteenwegen, P. and Oudheusden, D. V. (2006). Developng ralway tmetables whch guarantee a better servce, European Journal of Operatonal Research 173(1): Vansteenwegen, P. and Oudheusden, D. V. (2007). Decreasng the passenger watng tme for an ntercty ral network, Transportaton Research Part B: Methodologcal 41(4): Wardman, M. (2004). Publc transport values of tme, Transport Polcy 11(4): A ITTP Model mn D t C t I t T C t = q v q w w t + q v a x tp ( L p 1) s tp f 2 +q v p P ( s tp f 1 t p P r pl l L p x tp + q v s t, I, t T, x tp = 1, I, t T, p P y tplv = 1, I, t T, p P, l L p, v V ( l d l v d l ) v 1 = c z l v, l L, v V : v > 1, M (1 x tp ), I, t T, p P, s t s tp (( ) ) d L v + h L + r p L t ( ) M 1 y tp L v, I, t T, p P, v V L, ( )) d L v + h L + r p L ( ) M 1 y tp L v, I, t T, p P, v V L, w t w tp M (1 x tp ), I, t T, p P, w tp =, I, t T, p P, l L p \1 (( ) )) d l v + h pl (d l v + hpl + r pl + m ) ( ) I, t T, p P, l L p : M (1 y tpl v M 1 y tplv, (( ) )) l > 1, l = l 1, v V l, v V l, d l v + h pl (d l v + hpl + r pl + m ) ( ) I, t T, p P, l L p : +M (1 y tpl v + M 1 y tplv, l > 1, l = l 1, v V l, v V l, 14

15 C t 0, I, t T, d l v 0, l L, v V l, x tp (0, 1), I, t T, p P, y tplv (0, 1), I, t T, p P, l L p, v V l, z l v N, l L, v V l, s t 0, I, t T, s tp 0, I, t T, p P, w t 0, I, t T, w tp 0, I, t T, p P, 0, I, t T, p P, l L p. 15