CASPT 2015 Evaluatng the Qualty of Ralway Tmetables from Passenger Pont of Vew Tomáš Robenek Shad Sharf Azadeh Mchel Berlare Abstract In the ralway passenger servce plannng, the man focus s often on the feasblty of the solutons and/or the assocated costs of the Tran Operatng Company (TOC). The costs of TOCs are the drver for the noncyclc verson of the Tran Tmetablng Problem (TTP), whereas feasblty s the man concern of the cyclc verson of the same problem. Usually, the passengers for whom the servce s desgned are not taken nto consderaton, when creatng the tmetables. Ths could be one of the man reasons for whch the wllngness of passengers to use trans as ther mean of transport has reduced. In ths research, a choce based optmzaton approach s ntroduced that addresses ths ssue from passenger satsfacton pont of vew. We valdate our model usng a sem-real data of a major European ralway company. Keywords Ralway Passenger Servce Ideal Tmetable Cyclc vs. Non-Cyclc 1 Introducton In ths study, we gve attenton to the problematc of provdng the passenger servce n ralways. The offered product n ths case s the tmetable and the consumer s the passengers. 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 (weekday peak hours for work or school, weekends for lesure, etc.), comfort, percepton and others. Moreover the passenger servce has to compete wth other transportaton Transport and Moblty Laboratory (TRANSP-OR), School of Archtecture, Cvl and Envronmental Engneerng (ENAC), École Polytechnque Fédérale de Lausanne (EPFL), Staton 18, CH-1015 Lausanne, Swtzerland, E-mal: {tomas.robenek, shad.sharfazadeh, mchel.berlare}@epfl.ch
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 Fg. 1 Plannng overvew of ralway operaton The current TOC plannng horzon as descrbed by Caprara et al (2007) s vsualzed on Fgure 1. In the frst stage, the Lne Plannng Problem (LPP) decdes on whch lnes wll be operated and wth what frequences. The LPP s the only problem n the plannng horzon that actually takes nto account the demand n the form of an hourly Orgn Destnaton (OD) flows (Schöbel (2012)). The second stage n the plannng horzon, s the Tran Tmetablng problem (TTP), where two dfferent models exst: non-cyclc (Caprara et al (2002)) and cyclc (Peeters (2003)). The non-cyclc TTP takes as nput the deal tmetables and tres to resolve the track conflcts by mnmzng the tmetable shfts needed. However, the orgn of the deal tmetables s unclear as well as the punshment for the shfts. Smlarly, n the cyclc problem, the model takes as an nput the cycle and creates ether arbtrary feasble solutons or a feasble solutons based on user defned objectve functon. Typcally, the user defned functons are rather smplfed such as mnmzaton of travel tme, whch can not properly account for the passenger behavor. Both of the models can secure connectons between two trans, however wth no ncentve f the connecton s actually needed as none of the models takes passengers nto account. In the surveyed lterature, a model that ntegrates mode choce and cyclc verson of the TTP s presented (Cordone and Redaell (2011)). The objectve functon s maxmzaton of the demand captured by the ralway mode as opposed to other modes. The constrant that estmates the demand captured by the tmetable s usng a logt model, whose attrbutes form the total trp
length. The resultng formulaton s non-lnear and non-convex and s solved usng heurstcs. In our study, we propose to ntegrate the Route Choce Model (RCM) and the TTP formng a new plannng phase called the Ideal Tran Tmetablng Problem (ITTP). 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). In ths problem, we mmc the RCM by ntroducng a passenger cost related to a concerned tmetable. The objectve functon of ths problem s the passenger cost mnmzaton. The model wll allow tmetables of the TOC s tran lnes 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 (va passengers costs). 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 Fg. 2 Modfed overvew of ralway operaton The structure of the manuscrpt s as follows: we ntroduce a defnton of a passenger cost (Secton 2), followed by a problem defnton and ts mathematcal formulaton (Secton 3). The model s tested on a Swss case study (Secton 4). The paper s fnalzed by drawng some conclusons and dscusson of possble extensons (Secton 5). 2 Passenger Cost 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 deal passenger tmetable: 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 Fg. 3 Scheduled Delay Functons As shown n Small (1982), the passengers are wllng to s hft ther arrval tme by 1 to 2 mnutes earler, f t wll save them 1 mnute of the n-vehcletme, 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 + 2.5 W T + 10 NT + SD [mn] (1)
WT 1 WT 2 Transfer 1 Transfer 2 VT 2 VT 1 VT 3 Orgn Destnaton Fg. 4 Example Network 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 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 + 2.5 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 27.81 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 passenger tmetable can be defned as follows: The deal passenger 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 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. 3 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 provde the deal tmetables,.e. to mnmze the passenger cost. 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 and ts segments g G l. Segment s a part of the lne between two statons, where the tran does not stop. Each lne has an assgned frequency expressed as the avalable trans v V l (lnes, segments and frequences are the output of the LPP). Based on the pool of lnes, the set of paths between every OD par p P can be generated. 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 and l < L p ), 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 (combnaton of ndces t) can use at most one path. If there s no path assgned to a gven passenger (due to the lmted capacty of the trans), t s assumed that the passenger would take the earlest possble shortest path outsde of the plannng horzon H. Wthn the path tself, 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 we know the exact tnerary of every passenger, we can measure the tran occupaton o l vg of every tran v of every lne l on each of ts segment g. Derved from the occupaton, number of tran unts u l v s assgned to each
tran. Ths value can be equal to zero, whch means that the tran s not runnng and the frequency of the lne can be reduced. 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 trans for the lne l (frequency) G l set of segments on lne l Input Parameters Followng s the lst of parameters used n the model: H end of the plannng horzon [mn] M suffcently large number (can take the value of H) m mnmum transfer tme [mn] c cycle [mn] π t deal arrval tme of a passenger t to hs destnaton [mn] r pl runnng tme between OD par on path p usng lne l [mn] h pl tme to arrve from the startng staton of the lne l to the orgn/transferrng pont of the OD par n the path p [mn] D t demand between OD par wth deal tme t [passengers] q value of the n vehcle tme [monetary unts per mnute] q w value of the watng tme n the relaton to the VOT [untless] f 1 coeffcent of beng late n the relaton to the VOT [untless] f 2 coeffcent of beng early n the relaton to the VOT [untless] a penalty for havng a tran transfer [mn] β capacty of a sngle tran unt [passengers] j maxmum length of the tran [tran unts] γ n-vehcle-tme of the shortest path between OD par [mn] η number of transfers n the shortest path for OD par [untless] C t penalty cost for not servng passenger t nsde of the plannng horzon H [monetary unts] 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 w tpl 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 value of the scheduled delay of a passenger wth deal tme t between OD par s tp the value of 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 (from ts frst staton) 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 o l vg tran occupaton of a tran v of the lne l on a segment g u l v number of tran unts of a tran v on the lne l αv l 1 f a tran v on the lne l s beng operated; 0 otherwse Routng Model The ITTP model can be decomposed nto 2 parts: routng and cost estmaton. The routng takes care of the feasblty of the soluton, whereas cost estmaton takes care of the passenger cost attrbutes. At frst, we present the Routng Model (RM): o l vg = I mn D t C t (6) I t T x tp 1, I, t T, (7) p P = x tp, I, t T, p P, l L p, (8) y tplv v V l ( d l v d l ) v 1 = c z l v, l L, v V l : v > 1, (9) t T p P y tplv D t, l L, v V l, g G l, (10) u l v β o l vg, l L, v V l, g G l, (11) α l v j u l v, l L, v V l, (12) C t 0, I, t T, (13) d l v 0, l L, v V l, (14) x tp (0, 1), I, t T, p P, (15)
y tplv (0, 1), I, t T, p P, l L p, v V l, (16) o l vg 0, l L, v V l, g G l, (17) u l v (0, j), l L, v V l, (18) α l v (0, 1), l L, v V l, (19) z l v N, l L, v V l. (20) The objectve functon (6) ams at mnmzng the passenger cost. Constrants (7) secure that every passenger s usng at most 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, f ths path s beng used. 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) keep track of a tran occupaton. Constrants (11) verfy that the tran capacty s not exceeded on every stretch/segment of the lne. Constrants (12) assgn tran drvers,.e. f a tran v on the lne l s beng operated or not. Constrants (13) (20) set the domans of decson varables. Cost Estmatng Constrants To make the ITTP complete, we need to expand the Routng Model wth the cost estmatng constrants. We wll add the cost related constrants n blocks of attrbutes that create the cost of a passenger. s tp f 1 s t s tp (( d L v + h L ( ( s tp f 2 π t d L v M (1 x tp ), ) ) I, t T, p P, (21) + r p L π t ( ) M 1 y tp L v, I, t T, p P, v V L, )) (22) + r p L + h L ( M 1 y tp L v ), I, t T, p P, v V L, (23) s t 0, I, t T, (24) s tp 0, I, t T, p P. (25) 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 (25)) 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 (24)). As descrbed n the Secton 2, the constrants (22) model the earlness of the passengers (Equaton 2) and constrants (23) model the lateness (Equaton 3). Constrants (21) make sure that only one SD s selected (Equaton 4)
not necessarly the lowest one as t depends on the cost of the whole tnerary (constrants (33)),.e. the path wth the smallest overall cost wll be selected for the gven OD par wth a gven deal tme. These constrants also allow us to avod the non-lnearty n the estmaton of the fnal passenger cost (constrants (33)). w tpl w tpl w t w tp M (1 x tp ), I, t T, p P, w tp = l L p \1 (( ) d l v + h pl (d l v + hpl + r pl ) ( M (1 y tpl v M (( ) d l v + h pl (d l v + hpl + r pl ) ( +M (1 y tpl v + M w tpl, I, t T, p P, )) + m 1 y tplv )) + m 1 y tplv (26) (27) I, t T, p P, ), l L p : l > 1, l = l 1, v V l, v V l, (28) I, t T, p P, ), l L p : l > 1, l = l 1, v V l, v V l, (29) w t 0, I, t T, (30) w tp 0, I, t T, p P, (31) w tpl 0, I, t T, p P, l L p. (32) 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 (30)), the total watng tme of every path (constrants (31)) and the watng tme at every transferrng pont n every path (constrants (32)). The constrants (28) and (29) 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 (27) add up all the watng tmes n one path to estmate the total watng tme n a gven path. Constrants (26) make sure that only one WT s selected (smlarly as constrants (21) for SD). C t q q w w t + q a p P +q p P r pl l L p x tp ( L p 1) x tp + q s t, I, t T, (33)
C t 1 x tp C t, I, t T, (34) p P C t = q q w (m + c + η ) + q a η + q γ + ( H + c + γ + a η π t ) f1 q, I, t T. (35) At last, constrants (33) combne all the attrbutes together as n Equaton 5 multpled by the VOT. If a passenger t can not be served wthn the plannng horzon, the constrants (34) become actve and penalze the passenger wth a cost assocated to hs shortest path realzed wth the frst possble path outsde of the plannng horzon (n the next cycle) constrants (35). 4 Case Study In order to test the ITTP model, we have selected the network of S-trans n canton Vaud, Swtzerland as our case study. The reduced network s represented on Fgure 5 (as of tmetable 2014). We consder only the man statons n the network (n total 13 statons). A smple algorthm n Java has been coded, n order to fnd all the possble paths between every OD par. The algorthm allowed maxmum of 3 consecutve lnes to get from an orgn to a destnaton. The travelng tmes have been extracted from the Swss Federal Ralways (SBB) webste. The mnmum transfer tme between two trans has been set to 4 mnutes. Vallorbe 13 Yverdon-Les-Bans 12 Payerne 1 S1 S2 S3 S4 S11 S21 S31 Cossonay 11 2 Palézeux 3 Pudoux-Chexbres Lausanne Renens 8 7 Montreux 9 Morges Vevey 6 5 10 Allaman Vlleneuve 4 Fg. 5 Network of S-trans n canton Vaud, Swtzerland
Lne ID From To Departures S1 S2 S3 S4 S11 S21 S31 1 Yverdon-les-Bans Vlleneuve 6:19 7:19 8:19 2 Vlleneuve Yverdon-les-Bans 5:24 6:24 7:24 8:24 3 Vallorbe Palézeux 5:43 6:43 7:43 8:43 4 Palézeux Vallorbe 6:08 7:08 8:08 5 Allaman Vlleneuve 6:08 7:08 8:08 6 Vlleneuve Allaman 6:53 7:53 8:53 7 Allaman Palézeux 5:41 6:41 7:41 8:41 8 Palézeux Allaman 6:35 7:35 8:35 9 Yverdon-les-Bans Lausanne 5:26* 6:34 7:34 8:34 10 Lausanne Yverdon-les-Bans 5:55 6:55 7:55 8:55 11 Payerne Lausanne 5:39 6:39 7:38* 8:39 12 Lausanne Payerne 5:24 6:24 7:24 8:24 13 Vevey Pudoux-Chexbres 6:09 7:09 8:09 14 Pudoux-Chexbres Vevey 6:31* 7:36 8:36 Table 4 Lst of S-tran lnes n canton Vaud, Swtzerland In Table 4, you can fnd the lst of all S-tran lnes of the canton Vaud n the tmetable of 2014. There are 7 lnes that run n both drectons. Each combnaton of a lne and ts drecton has ts unque ID number. Column from marks the orgn staton of the lne as well as column to marks ts destnaton. The columns departures show the currently operated tmetable (.e. departures from the orgn of the lne) n the mornng peak hour (5 a.m. to 9 a.m.), whch s the tme horzon used n our study. Trans that dd not follow the cycle (marked wth a star *) were set to a cycle value, n order to not volate the cyclcty constrants (the tmetables n Swtzerland are cyclc wth a cycle of one hour). The SBB s operatng the Stadler Flrt tran unts on the lnes S1, S2, S3 and S4. In our case study, we have homogenzed the fleet and thus use ths type of a tran also for the rest of the lnes. The capacty of ths unt s 160 seats and 220 standng people. The maxmum amount of tran unts per tran s 2 (as SBB never uses more unts). The amount of tran unts per tran remans the same along the lne, but t mght change at the end statons (we don t go nto further detals as ths s the task of the Rollng Stock Problem). The demand and ts dstrbuton has been estmated based on the SBB report and observaton (more detals n Appendx A). In total there are 10 077 passengers n the network for the current stuaton. The coeffcents of the passenger cost are as descrbed n Secton 2. 4.1 Results In all of the experments, we have run 3 types of the ITTP model: current, cyclc and non-cyclc. The current model reflects the currently operated SBB tmetable as n Table 4 (the decson varables d have been set to the values n the table). Subsequently, the cyclc model does not have the departure tmes as a hard constrant and thus the CPLEX can look for better values than those
120 3 Passenger Coverage [%] 100 80 Passenger Cost [MCHF] 2 1 60 0 10 20 30 40 50 Number of Passengers (a) Passenger coverage as a functon of the demand for the current model 0 0 10 20 30 40 50 Number of Passengers [thousands] (b) Passenger cost as a functon of the demand for the current model at ɛ 100*% Dffernce n Pax Cost [kchf] 200 150 100 50 non-cyclc cyclc 0 0 10 20 30 40 50 Number of Passengers [thousands] (c) The dfference n passenger cost of the cyclc and non-cyclc model as compared to the current model at ɛ 100*% Fg. 6 Results of the SBB. The non-cyclc model dffers from the cyclc one by removng the cyclcty constrants. In order to speed up CPLEX, we would frst solve the current verson and gve ts soluton as a warm start for the cyclc model and solve t. Further along, we would gve the soluton of the cyclc model as a warm start to the non-cyclc model. Moreover, we have run the models for several levels of passenger densty, startng from the real-lke volume up untl the pont where the passenger coverage decreases to a level of 70%. The passenger coverage as a functon of the demand for the current model (the coverage s more or less the same for the other two models) can be found on Fgure 6(a). As t can be seen, the congeston starts at the amount of cca. 27 000 passengers and that the coverage goes down almost lnearly. The total passenger cost growth can be observed on Fgure 6(b) (we plot only the current model as the other two models yeld smlar values). The passenger cost grows rather exponentally and ts functon can be splt nto two lnear parts: non-congested (gradual slope) and congested (steep slope). Ths mght be useful for practtoners as t would allow them to predct the
passenger cost. Subsequently, we plot the relatve dfference of cyclc and noncyclc tmetables as opposed to the current tmetable on Fgure 6(c). In general, the cyclc model tends to fnd slghtly better tmetables than the current model (n the congested cases the beneft even dramatcally ncreases). The non-cyclc tmetable, on the other hand, s more flexble and copes the functon of the total passenger cost (Fgure 6(b)) and acheves more sgnfcant savngs. Ths s due to the fact that the trans do not have to follow the cyclc frequency and thus are more densely scheduled, for nstance n the most congested case, the average headway between two consecutve trans on a same lne s 22.6 mnutes, wth mnmum value of 1 mnute and maxmum value of 238 mnutes. 5 Conclusons and Future Work In ths research, we defne a new way, how to measure the qualty of a tmetable from the passenger pont of vew and ntroduce a defnton of an deal tmetable. We then present a formulaton of a mxed nteger lnear problem that can desgn the deal 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 passengers, but that also creates by tself connectons 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. We test the model on a sem-real data of the S-tran network of Canton Vaud n Swtzerland. Our model was able to fnd a better tmetable compared to the current SBB tmetable, where the acheved savngs, whlst keepng the tmetables cyclc, were around 3 000 CHF and around 7 000 CHF, n the case of the non-cyclc tmetable. Furthermore, we have focused on explotng the passenger congeston. Our study shows that two lnear functons, for congested and uncongested network, can be constructed and thus the passenger cost can be predcted. Most nterestngly, we show that the mprovements of the noncyclc tmetables as compared to the cyclc tmetables, are flexble (they copy the total passenger cost functons) due to the fact that these tmetables allow hgher tran densty. The average tran headway of the most congested case was reduced from the cycle (60 mnutes) down to 22 mnutes. Moreover the non-cyclc model was able to acheve around 160 000 CHF of savngs even though the network s dense. These savngs are expected to be even hgher for less dense networks. Due to ths fact, we would propose to combne the ITTP wth the Lne Plannng Problem n the future. In the future work, we wll focus on effcent solvng of the problem and extenson of the plannng horzon,.e. to be able to solve the problem for a whole day. Ths would allow us to explore, f the non-cyclc tmetables could
perform better off-peak hours and n the context of the whole day. 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 or ntegraton wth other phases of the plannng horzon. References Axhausen KW, Hess S, Köng A, Abay G, Bates JJ, Berlare M (2008) Income and dstance elastctes of values of travel tme savngs: New swss results. Transport Polcy 15(3):173 185, DOI http://dx.do.org/10.1016/j.tranpol.2008.02.001 Ben-Akva M, Lerman S (1985) Dscrete Choce Analyss. The MIT Press, Cambrdge Massachusetts Caprara A, Fschett M, Toth P (2002) Modelng and solvng the tran tmetablng problem. Operatons Research 50(5):851 861, DOI http://dx.do.org/10.1287/opre.50.5.851.362 Caprara A, Kroon LG, Monac M, Peeters M, Toth P (2007) Passenger ralway optmzaton. In: Barnhart C, Laporte G (eds) Handbooks n Operatons Research and Management Scence, vol 14, Elsever, chap 3, pp 129 187 Cordone R, Redaell F (2011) Optmzng the demand captured by a ralway system wth a regular tmetable. Transportaton Research Part B: Methodologcal 45(2):430 446, DOI http://dx.do.org/10.1016/j.trb.2010.09.001, URL http://www.scencedrect.com/scence/artcle/p/s0191261510001049 Kana S, Shna K, Harada S, Tom N (2011) An optmal delay management algorthm from passengers vewponts consderng the whole ralway network. Journal of Ral Transport Plannng & Management 1(1):25 37, DOI http://dx.do.org/10.1016/j.jrtpm.2011.09.003, robust Modellng of Capacty, Delays and Reschedulng n Regonal Networks de Kezer B, Geurs K, Haarsman G (2012) Interchanges n tmetable desgn of ralways: A closer look at customer resstance to nterchange between trans. In: AET (ed) Proceedngs of the European Transport Conference, Glasgow, 8-10 October 2012 (onlne), AET Peeters L (2003) Cyclc Ralway Tmetable Optmzaton. ERIM Ph.D. seres Research n Management, Erasmus Research nst. of Management (ERIM) Sato K, Tamura K, Tom N (2013) A mp-based tmetable reschedulng formulaton and algorthm mnmzng further nconvenence to passengers. Journal of Ral Transport Plannng & Management 3(3):38 53, DOI http://dx.do.org/10.1016/j.jrtpm.2013.10.007, robust Reschedulng and Capacty Use Schöbel A (2012) Lne plannng n publc transportaton: models and methods. OR Spectrum 34:491 510, DOI 10.1007/s00291-011-0251-6 Small KA (1982) The schedulng of consumer actvtes: Work trps. The Amercan Economc Revew 72(3):pp. 467 479
Swss Federal Ralways (2013) SBB: Facts and Fgures 2013. Tech. rep. Vansteenwegen P, Oudheusden DV (2006) Developng ralway tmetables whch guarantee a better servce. European Journal of Operatonal Research 173(1):337 350, DOI http://dx.do.org/10.1016/j.ejor.2004.12.013 Vansteenwegen P, Oudheusden DV (2007) Decreasng the passenger watng tme for an ntercty ral network. Transportaton Research Part B: Methodologcal 41(4):478 492, DOI http://dx.do.org/10.1016/j.trb.2006.06.006 Wardman M (2004) Publc transport values of tme. Transport Polcy 11(4):363 377, DOI http://dx.do.org/10.1016/j.tranpol.2004.05.001 A Demand Generaton The total amount of passengers n the network has been estmated n the followng manner: the populaton of Swtzerland s 8 211 700 habtants and the populaton of Canton Vaud s 755 369 habtants, whch leads to a rough rato of 1:10. Applyng ths rato to a reported amount of passenger journeys per day by SBB (n total one mllon for the whole SBB network), we arrve to a demand volume of 100 000 passenger journeys per day n canton Vaud. However not all of these journeys are beng realzed usng S-trans. Snce almost all of the trans n Canton Vaud have to pass through ts captal cty Lausanne, we can derve the rato, between the S-trans and other class trans passng through Lausanne, of 40:60 percent, whch leaves us wth a 40 000 passenger journeys per day usng S-trans n Canton Vaud. Furthermore, the SBB report provdes hourly dstrbuton of passengers on a regonal servces from Monday to Frday. Accordng to ths report 25 percent of the journeys are beng realzed n the mornng peak hour, whch gves us cca. 10 000 passenger journeys n the mornng peak hour for the S-tran network of Canton Vaud. 42 208 475 275 5 a.m. 6 a.m. 7 a.m. 8 a.m. 9 a.m. Fg. 7 The hourly dstrbuton of the passenger groups In order to ease the sze of the generated lp fle(s), the passengers have been splt nto 1 000 passenger groups (ndces t) of varyng szes. These groups have been dvded nto hourly rates (Fgure 7) accordng to the SBB report (Swss Federal Ralways (2013)) and smoothed nto mnutes usng non-homogenous Posson process. Snce we use concept of an deal arrval tme to the destnaton, the generated arrval tme at the orgn has been shfted, by addng up the shortest path travel tme between the OD par, to the destnaton of the passengers. In order to generate real-lke OD flows (ndex ), we consder the followng probabltes: p(d = 7) = 0.5 probablty of a destnaton beng Lausanne p(d = 8) = 0.2 probablty of a destnaton beng Renens p(d = other) = 0.3 probablty of a destnaton beng other than Lausanne or Renens p(o = any) = 1/12 probablty of an orgn beng any staton (except the already selected destnaton)
Snce Lausanne s the bggest cty n the Canton wth all the lnes, except the lne 13 and 14, passng through t, t has the largest probablty of beng a destnaton (many people also use Lausanne as a transfer pont to hgher class trans). The cty wth the second hghest probablty s Renens, because t s the closest staton to one of the bggest unverstes n Swtzerland and from the network dagram (Fgure 5), we can see the most of the lnes stop there, whch suggests hgh demand. The rest of the statons have equal probablty of beng a destnaton (0.3/11), whch s rather small as n the mornng peak hour people travel towards ther work/school n bg ctes. On the other hand, the probablty of beng an orgn s unformly dstrbuted and dependent on ts destnaton (orgn can not be the same as a destnaton). The fnal probablty p(o = o, D = d) for every OD par can be seen n Table 6. In order to reach the total demand, the average sze of a group should be ρ = the total demand dvded by the number of groups. In the current scenaro ρ = 10 000/1 000 = 10. In our study, we use 3 dfferent classes of groups: small, medum and large. The sze of the small group s drawn from the unform dstrbuton U(1, 0.6ρ) and appled to ODs wth a probablty p(o = o, D = d) [0, 1.5) %. The sze of the medum group follows U(0.6ρ+1, ρ) and s appled to ODs wth a probablty p(o = o, D = d) [1.5, 3) %. The largest group sze follows a dstrbuton U(ρ+1, 2ρ) and s appled to a probablty p(o = o, D = d) [3, 4.5) %.
1 2 3 4 5 6 7 8 9 10 11 12 13 1 0 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.041666667 0.016666667 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 2 0.002272727 0 0.002272727 0.002272727 0.002272727 0.002272727 0.041666667 0.016666667 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 3 0.002272727 0.002272727 0 0.002272727 0.002272727 0.002272727 0.041666667 0.016666667 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 4 0.002272727 0.002272727 0.002272727 0 0.002272727 0.002272727 0.041666667 0.016666667 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 5 0.002272727 0.002272727 0.002272727 0.002272727 0 0.002272727 0.041666667 0.016666667 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 6 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0 0.041666667 0.016666667 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 7 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0 0.016666667 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 8 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.041666667 0 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 9 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.041666667 0.016666667 0 0.002272727 0.002272727 0.002272727 0.002272727 10 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.041666667 0.016666667 0.002272727 0 0.002272727 0.002272727 0.002272727 11 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.041666667 0.016666667 0.002272727 0.002272727 0 0.002272727 0.002272727 12 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.041666667 0.016666667 0.002272727 0.002272727 0.002272727 0 0.002272727 13 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.002272727 0.041666667 0.016666667 0.002272727 0.002272727 0.002272727 0.002272727 0 Table 6 Orgn Destnaton dstrbutons