The Ideal Train Timetabling Problem

Transcription

1 The Ideal Tran Tmetablng Problem Tomáš Robenek Janghang Chen Mchel Berlare Transport and Moblty Laboratory, EPFL May 2014

2 The Ideal Tran Tmetablng Problem May 2014 Transport and Moblty Laboratory, EPFL The Ideal Tran Tmetablng Problem Tomáš Robenek, Janghang Chen, Mchel Berlare Transport and Moblty Laboratory École Polytechnque Fédérale de Lausanne Staton 18 CH-1015 Lausanne phone: fax: {tomas.robenek, janghang.chen, May 2014 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. However, gven the recent changes n legslature allowng compettors n the ralway ndustry, the current way of plannng s not suffcent 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 and thus t s assumed that they evolve ncrementally, based on a hstorcal data (tran occupaton, tcket sales, etc.). And thus, we ntroduce a defnton of an deal tmetable that s expressed usng the passenger cost. In order to create the tmetables tself, we propose to nsert the Ideal Tran Tmetablng Problem (ITTP) that s solved for each Tran Operatng Company (TOC) separately, nto the plannng process. The ITTP approach ncorporates the passenger demand n the plannng and ts am s to mnmze the passenger cost(s). The outcome of the ITTP s the deal tmetables (ncludng connectons between the trans and weghted by the demand), whch then serve as an nput for the tradtonal Tran Tmetablng Problem (TTP). The TTP takes nto account wshes of each TOC (the deal tmetables) and creates global feasble tmetable for the gven ralway network, whle mnmzng the changes of the TOCs wshes. The ITTP s n lne wth the new market structure and t can produce both: non-cyclc and cyclc

3 The Ideal Tran Tmetablng Problem May 2014 tmetables. The model s tested on the data provded by the Israel Ralways (IR). The nstance conssts of a full demand OD Matrx of an average workng day n Israel durng The results are compared to the current tmetable of IR. Due to the large complexty of the model, t s solved usng the Column Generaton methodology. Keywords Ralway Optmzaton, Tmetablng, Demand, Ideal Tmetable, Passenger Utlty

4 The Ideal Tran Tmetablng Problem May Introducton The tme of domnance of one ral operatng company (usually the natonal carrer) over the markets n Europe s reachng to an end. Wth the new EU regulaton, the track management and tran operatng companes (TOC) have to be separated subjects. Thus allowng competton (prvate sector) to enter 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. In transportaton sector, revenue s generated by transportng passengers or goods. The prospectve passengers and goods are consdered to be the demand to capture, hence TOCs goal s to maxmze the captured demand. In case of goods, the demand s more flexble and ts market s more or less already open for the prvate sector, unlke for passengers. The man drver of goods demand s cost and n some cases also the trp tme. On the other hand, the passenger demand s also senstve to the tme of the departure related to the trp purpose (weekday peak hours for work or school, weekends for lesure, etc.) and others (comfort, percepton, etc.). Wth the change of the market, the TOCs have to adapt to a new busness model. In ths paper, we descrbe the current way of plannng of the passenger ralway servce, and dscuss, how the demand s taken nto account n the current plannng process (Secton 2). After the analyss, we ntroduce current lterature on the topc (Secton 3) and elaborate on a new problem to be nserted n the plannng process, n a way that the new objectve s properly taken nto account (Secton 4). Due to the complexty of the problem, we decompose the model and solve t usng column generaton methodology (Secton 5). At the end of the paper, descrpton of the case study s shown (Secton 6) and we fnalze the paper wth conclusons and future work (Secton 7). 2 Ralway Plannng In ths secton, we present the current state-of-the-art of the research n the plannng of passenger ralway servce. Snce plannng a ralway operaton s a complex task, due to the large soluton space, t s parttoned nto several problems that are solved sequentally (Caprara et al. (2007)). These problems and the sequence n whch they are solved, can be seen on Fgure 1. The logcal frst step, n the plannng of passenger ralway servce, should be the ralway network 1

5 The Ideal Tran Tmetablng Problem May 2014 STRATEGIC - several years TACTICAL - >= 1 year OPERATIONAL - < 1 year Actual Tmetables Tran Platformng Platform Assgnment s Demand Lne Plannng Lnes Tran Tmetablng Actual Tmetables Rollng Stock Plannng Tran Assgnment s Actual Tmetables Crew Plannng Crew Assgnment s TOC IM Fgure 1: Plannng overvew of ralway operaton desgn. However, snce most of the ralway nfrastructure has been already bult (startng n the early 19th century) and only small parts of the network are beng buld nowadays, t s often omtted from the plannng process as such. Moreover, the decson, what new parts to add n the network, s often poltcal and handled by the local authortes. The plannng horzon then starts wth the lne desgn followed by the tmetable desgn, rollng stock and crew schedulng and tran platformng (Caprara et al. (2007)). In the Lne Plannng Problem (LPP), the man nput s demand per Orgn/Destnaton (n the form of the OD Matrx) and the global ralway nfrastructure (network). Based on ths nput, selecton of the most sutable lnes from the pool of pre-processed potental lnes, connectng these orgns and destnatons, s undertaken. Apart from that, expected frequences and capactes of the lnes are selected as well. Ths problem s handled by the TOC and t s usually solved every few years (the nfrastructure and the OD matrx do not change fast). For more nformaton about the LPP, please refer to the latest paper surveyng the problem - Schöbel (2012). The Tran Tmetablng Problem (TTP) exsts n two settngs: non-cyclc (Caprara et al. (2002)) and cyclc (Peeters (2003)). The dfference between the two s, that trans wth the cyclc tmetables leave the statons at the begnnng of every cycle,.e. f the cycle s one hour, the trans then leave the staton every hour n the same mnute. The two optons also dffer n ther respectve nputs: the pre-created deal tmetables (consstng of the desred tmes and the tme wndows, that gve degree of freedom to the problem, n order to fnd a feasble soluton) for the non-cyclc verson; frequences and fxed departures for some of the trans n the cyclc 2

6 The Ideal Tran Tmetablng Problem May 2014 verson. To the best of our knowledge, how to create the deal tmetables (non-cyclc tmetablng), s nowhere to be found n the publshed lterature. Even a defnton, of what such an deal tmetable would be, s non exstent. In case of the cyclc tmetablng, the model only searches for a feasble soluton, wthout consderng what could be the best start of the cycle. We assume, that the common practce n the ndustry s to use the "hstorcal" tmetables and modfy them n every new plannng of tmetables, usng the gven TOCs tran occupaton data. 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 (a defnton of an deal tmetable, to the best of our knowledge, does not exst, even though the deal tmetables are used n the non-cyclc TTP) 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 (Secton 4). 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. STRATEGIC - several years TACTICAL - >= 1 year OPERATIONAL - < 1 year Actual Tmetables Tran Platformng Platform Assgnment s Demand Lne Plannng Lnes Ideal Tran Tmetablng Ideal Tmetables Tran Tmetablng Actual Tmetables Rollng Stock Plannng Tran Assgnment s Actual Tmetables Crew Plannng Crew Assgnment s TOC IM Fgure 2: Modfed overvew of ralway operaton 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 3

7 The Ideal Tran Tmetablng Problem May 2014 or cyclc schedule. Moreover, we ntroduce a demand nduced connectons. The connectons between the trans are not pre-set, but are subject to the demand. In the lterature the connectons are handled only n the cyclc verson of the TTP, where the connectons are always nduced, wthout a proper reasonng. Returnng to the tradtonal TTP: the model s modfyng the tmetables for all scheduled trans, such that safety regulatons n the ralway network are mantaned. The problem s mnmzng the shfts by takng nto account each tmetable s cost or proft. As the ITTP wll provde, not only the deal tmetables, but also ther costs, t s then automatcally ready to be ntegrated nto the current plannng process. The desgn of the TTP suggests, that t serves manly to the Infrastructure Manager (IM) to secure the safety and the feasblty of the network, whlst maxmzng the wshes of TOCs. Ths problem s solved for every new tmetable,.e. typcally every year (n Europe usually n December, EU drectve from 2004 oblges all TOCs n Europe to do so on the same day; n some countres (France, Great Brtan) twce a year). The TTP n general s not able to solve all conflcts, specfcally wthn the tran statons, where a mcroscopc approach s needed. To handle these conflcts, the Tran Platformng Problem (TPP) s solved (Caprara et al. (2007)). The TPP takes as nput the desgned actual tmetables for tran statons and creates the routngs through the statons. Ths problem s consdered operatonal (routngs can be changed throughout the operaton of the actual tmetable) and t s handled by the IM. The Rollng Stock Plannng Problem (RSPP) s takng care of a fleet of a TOC,.e. what s the tran composton (numbers of 1st and 2nd class coaches) to be able to satsfy the demand and publshed tmetable wthout exceedng the avalable rollng stock (Caprara et al. (2007)). Ths problem s as well operatonal and n the jursdcton of the TOC. The last step n the plannng horzon s the Crew Plannng Problem, whch assgns crew to the scheduled trans, subject to unon rules and other wokng restrctons. The goal s to mnmze the sze of the crew needed for a global daly operaton of the servce. Ths problem s operatonal and handled by the TOC. For more detaled descrpton of the complete ralway plannng horzon, please refer to Caprara et al. (2007) or to Husman et al. (2005). In ths secton, we have dentfed several drawbacks of the current state-of-the-art plannng process. The cause (of these drawbacks) s the recent market change n the ralway ndustry,.e. movng from the accessblty/moblty concept provded by one carrer to the compettve 4

8 The Ideal Tran Tmetablng Problem May 2014 market consstng of several carrers, that are drven by the proft. The buldng stone of a compettve market s the demand. And thus n ths paper, we propose a new plannng phase (ITTP) based on the demand to create the most attractve tmetables for the passengers (both cyclc and non-cyclc) ncludng the connectons between the trans n the network (also based on the demand). 3 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). In order to be able to nsert the ITTP n the plannng horzon, we have surveyed the LPP and the TTP (both non-cyclc and cyclc versons). Apart from that, we have also looked nto the lterature on demand nteracton (outsde of the tradtonal plannng scope) n the passenger ralway servce (namely: revenue management, dynamc prcng, dscrete choce models, etc.). 3.1 Lne Plannng Problem The lterature s bascally dvded nto two parttons by passenger based and cost based objectve functon. One of the frst models, that maxmzes the drect travelers, can be found n Busseck et al. (1997b). The model s maxmzng the amount of drect passengers and uses one bnary decson varable (1 f the lne s selected to be n the soluton; 0 otherwse). In order to solve the model effcently, vald nequaltes are ntroduced. Ths model s also used by Hooghemstra et al. (1999). The phd thess Busseck (1997) extends the methodologes of solvng the drect passengers objectve and moreover evaluates the mnmzaton of operatonal costs (and ts technques) as defned by Claessens. In Claessens and van Djk (1995) and Claessens et al. (1998) dfferent approach s presented: nstead of maxmzng drect travelers, the mnmzaton of costs s the objectve. Addtonal decson varables on frequency and length of the trans (n terms of the number of carrages) are used. Unfortunately ths leads to a non-lnear model and thus lnear reformulaton s developed nstead (one decson varable representng the combnaton of the above). Another lnearzatons and addtonal technques to solve ths model are shown n Zwaneveld (1997), Goossens et al. (2001) and Busseck et al. (2004). Goossens et al. (2006) also works wth the cost optmal model and extends the approach by ntroducng multple lne types (Goossens (2004) further extends the method). Both of the above types of the model (passenger and cost based) are then presented agan n Busseck et al. (1997a). 5

9 The Ideal Tran Tmetablng Problem May 2014 In Barber et al. (2008), another type of the model maxmzng passenger coverage s presented. The man dfference, comparng to the others, s that the lnes are constructed from scratch, nstead of usng the set of preprocessed lnes. Dfferent knd of model, mnmzng the passengers travel tme s presented n Pfetsch and Borndörfer (2005), Schöbel and Scholl (2006) and Borndörfer et al. (2007, 2008). Lastly, the LPP s ntegrated wth TTP n Kasp and Ravv (2013). The model s mnmzng the total tme passengers spend n the network. As a soluton method cross-entropy metaheurstc s used. The latest paper surveyng the publshed lterature on LPP s Schöbel (2012). 3.2 Tran Tmetablng Problem Non-Cyclc Most of the models, on the non-cyclc tmetablng, n the publshed lterature, formulate the problem ether as MILP or 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. (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 job-shop schedulng. Erol (2009), Caprara (2010) and Harrod (2012), survey dfferent types of models for the TTP. 6

10 The Ideal Tran Tmetablng Problem May Cyclc 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 va proposed algorthm. 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 and n Nachtgall (1996) usng genetc algorthms. Another algorthm, based on constrant generaton, to solve the PESP formulaton s presented n Odjk (1996). In Lndner and Zmmermann (2000), branch and bound algorthm s also appled to solve the PESP. In Kroon and Peeters (2003), varable trp tmes are consdered. Peeters (2003) further elaborates on PESP and n Lebchen (2004) mplementaton of the symmetry n the PESP model s dscussed. In Lebchen and Mohrng (2002), the PESP attrbutes are analyzed on the case study of Berln s underground. 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. 3.3 Demand Related Apart of the classcal problems (shown on Fgure 1), other addtonal technques lke revenue management, dynamc prcng or dscrete choce models can be used to affect the demand. Especally the revenue management, whch has been proven effectve n the arlne ndustry. In Ben-Khedher et al. (1998), the decson tool RalCap, used by the french natonal carrer SNCF, s descrbed. The man responsblty of the tool s to adjust the tran capacty (by addng new unt to the tran, drop empty extra unts or open them for reservatons on double-unt trans, open an optonal tran to reservatons and assgn t an tnerary-compatble fleet type) based on the current reservatons, ODs and forecasted demand. The tool s maxmzng the expected ncremental proft subject to operatonal constrants (manly avalablty of the rollng stock and ts routng through the network). In Cherc et al. (2004), model to maxmze the demand captured by tran s presented. The 7

11 The Ideal Tran Tmetablng Problem May 2014 resultng tmetable s cyclc (comng from constrants). The model ntegrates modal choce logt wth 3 alternatves - bus, tran and car (utlty conssts of travel tme, monetary cost, walkng tme, average watng tme and comfort). Coeffcents are estmated by a revealed preference. Snce the MILP s non-convex 2 methods are tested: branch and bound and heurstc approach. Both are tested on a regonal network n Italy, wth real schedules as nput. It s shown that wth the current schedule, only 4 % of the populaton can choose tran. In Lythgoe and Wardman (2002), analyss of the demand for travel from and to arport and a formulaton of a dscrete choce model are ntroduced. Whelan and Johnson (2004) are showng a dscrete choce model to decrease the overcrowdng on the trans by addng a specal tcket costs, when at the same tme not reducng the total amount of passengers transported,.e. smoothng the demand along the tme horzon. In Cordone and Redaell (2011), ntegraton of the modal choce model and classcal cyclc TTP s presented. In L et al. (2006), smulaton framework based on the dynamc prcng s dscussed. In Crever et al. (2012) the am s to cover the demand wth maxmzng the proft usng dfferent prcng strateges. The model s usng preset booked schedules, whch are then utlzed on an operatonal level. Abe et al. (2007) descrbes the revenue management (RM) n the ralway ndustry wth case studes of RM around the world. Comparson wth the RM n arlne ndustry s elaborated as well. In Bharll and Rangaraj (2008), revenue management for Indan Ralways s descrbed. Armstrong and Messner (2010) shows the overvew of RM n the ralway ndustry (both freght and passenger). Wang et al. (2012) descrbes a MILP model for RM. 4 Ideal Tran Tmetablng Problem The am of ths problem s to defne and to provde the deal tmetables as an 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 8

12 The Ideal Tran Tmetablng Problem May 2014 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 t p, 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 t plv ). 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 d l v. The tmetable can take form of a non-cyclc or a cyclc verson (dependng f the cyclcty constrants are actve, see Secton 4.1). Snce the nput demand s statc, the ntutve objectve functon would be to mnmze the total travel tme of every passenger. However, the nformaton about the deal arrval tme t s present and hence to maxmze the demand, we have to combne the total travel tme and the tmelness of the arrval to the destnaton. To express the tmelness, we borrow the concept of the scheduled delay from the traffc flow theory (see Arnott et al. (1990)). The logc behnd t, s as follows: f a passenger arrves to hs/her destnaton on hs/her deal tme, then hs/her scheduled delay (s t p ) s equal to zero, otherwse lnear delay functons are appled. There are two cases: Beng early s t p = (t t ) f 2 Beng late s t p = (t t) f 1 where t s the tme of the arrval nto the destnaton. We assume that the scheduled delay s perceved dfferently, when beng late and early. The calbraton of the rato between beng late and early s a subject for further analyss (n our example n Fgure 3, we use f 1 = 2 and f 2 = 1). The scheduled delay s dfferent for every OD par wth dfferent deal tmes t usng dfferent paths p. Snce t s much more attractve to express the objectve functon n monetary unts, for further estmaton of the proft (not a subject of ths paper) and full ntegraton wth the TTP (the objectve s to mnmze the tmetable shfts subject to the proft), we multply the value of the 9

13 The Ideal Tran Tmetablng Problem May 2014 Scheduled Delay f_2 Ideal Tme f_1 Tme Fgure 3: Scheduled Delay Concept scheduled delay (n tme unts) by the value of tme q 2 (monetary unts per tme unt). The same goes for the total travel tme, where t s splt nto n vehcle tme (tme unts) multpled by the q 2 and watng tme (tme unts) multpled by the value of watng tme q 1 (monetary unts per tme unts). Accordng to Wardman (2004) and Axhausen et al. (2008), the tme spent watng s perceved dfferently (two tmes more) than the tme spent n vehcle. Lastly, usng a drect tran, nstead of a several trans wth nterchanges, s more attractve to the passengers (see Axhausen et al. (2008)). To take care of ths attrbute, we ntroduce a mnmum transfer tme m. Snce t s as well a watng tme, t s multpled by the same value of q 1. As we allow unlmted number of transfers, t s then multpled by the sze of the path mnus one ( L p 1), as the transfers happen n-between two lnes. In the end, we can combne all of the above attrbutes nto a one cost C t : C t = q 1 w t + q 1 m p P x t p ( L p 1) + q 2 p P r pl l L p x t p + q 2 p P s t p x t p (1) In the fnal form of the objectve functon, the above cost s weghted by the demand D t. As the problem s no longer concerned by the accessblty and moblty of the passengers, we don t need to take nto account the maxmum travel tme between an orgn and a destnaton. Based on the above assumptons, we can formulate the deal tmetable: The deal tmetable conssts of such tran departures that the passengers global 10

14 The Ideal Tran Tmetablng Problem May 2014 costs are mnmzed,.e. the fastest most convenent path to get from the orgn to the destnaton traded-off by a tmely arrval to the destnaton for every passenger. 4.1 Mathematcal Formulaton In ths secton, we present a mxed nteger programmng formulaton for the deal tran tmetablng problem. Input Parameters Followng s the lst of parameters used n the model: I set of orgn-destnaton pars t T set of deal tmes for OD par l L set of operated lnes v V l set of avalable vehcles on lne l p P set of possble paths between OD par l L p set of lnes n the path p 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 orgn of the par D t demand between OD wth deal tme t m mnmum transfer tme c cycle q 1 value of the watng tme q 2 value of the n vehcle tme 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. Decson Varables Followng s the lst of decson varables used n the model: C t the total cost of the passengers wth deal tme t between OD par w t the total watng tme of the passengers wth deal tme t between OD par w t p the total watng tme of the passengers wth deal tme t between OD par usng path p 11

15 The Ideal Tran Tmetablng Problem May 2014 w t pl the watng tme of the passengers wth deal tme t between OD par on the lne l that s part of the path p x t p 1 f the passengers wth deal tme t between OD par choose path p; 0 otherwse s t p scheduled delay of the passengers wth deal tme t between OD par travelng on the path p dv l the departure tme of a tran v on the lne l y t plv 1 f the passengers wth deal tme t between OD par on the path p take the tran v on the lne l; 0 otherwse z l v frequency wthn cyclcty Model s t p The mathematcal formulaton then looks as follows: C t = q 1 w t + q 1 m w t pl w t pl f 2 (( d L v +q 2 p P r pl l L p w t mn p P I x t p + q 2 w t p (( ) ( dv l + h pl d l M (1 y t pl v (( ) ( dv l + h pl d l +M (1 y t pl v + h L t T D t C t (2) x t p ( L p 1) p P s t p x t p (3) p P x t p = 1, I, t T, (4) y t plv = 1, I, t T, p P, l L p, v V l M (1 ) x t p, I, t T, p P, (6) w t p = v + hpl l L p \1 (5) w t pl, I, t T, p P, (7) + r pl + m )) ), I, t T, p P, l L p : ) M ( 1 y t plv v + hpl l > 1, l = l 1, v V l, v V l, + r pl + m )) ), I, t T, p P, l L p : ) + M ( 1 y t plv l > 1, l = l 1, v V l, v V l, ) + r ) p L t M (1 ) y t p L v, I, t T, p P, v V L, (10) (8) (9) 12

16 The Ideal Tran Tmetablng Problem May 2014 s t p f 1 (t ( d L v + h L + +r p L )) ( ) M 1 y t p L v, I, t T, p P, v V L, (11) d l v d l v 1 = c zl v, l L, v V : v > 1, (12) w t 0, I, t T, (13) w t p 0, I, t T, p P, (14) w t pl 0, I, t T, p P, l L p, (15) x t p (0, 1), I, t T, p P, (16) s t p 0, I, t T, p P, (17) d l v 0, l L, v V l, (18) y t plv (0, 1), I, t T, p P, l L p, v V l, (19) z l v N, l L, v V l. (20) The objectve functon (2) s mnmzng the passengers costs. Constrant (3) calculates the cost of the soluton. Constrants (4) secure that every passenger s usng exactly one path to get from hs/her orgn to hs/her destnaton. Smlarly constrants (5) make sure that every passenger takes exactly one tran on each of the lnes n hs/her path. Constrants (6) select the best path n terms of the watng tme. Constrants (7) add up all watng tmes along the gven path. Constrants (8) and (9) set the proper watng tme n the transferrng statons. Constrants (10) and (11) are complementary constrants (one at a tme s actve) that calculate the scheduled delay n passengers destnatons. Lastly, constrants (12) are handlng the cyclcty of the created tmetables (f removed, the created tmetables would take non-cyclc form). Constrants (13)-(20) set the domans of decson varables. 5 Soluton Approach The mxed nteger programmng formulaton of the ITTP has a large soluton space, whch makes the problem dffcult to solve. Moreover, the presence of the bg M constrants lead to a weak lower bound. Thus n the followng secton, we decompose the mxed nteger model and formulate t as a set parttonng problem. 13

17 The Ideal Tran Tmetablng Problem May Set Parttonng Model Let Ω be the set of feasble assgnments of a demand between all OD pars wth all deal tmes. Note that a feasble assgnment represents the assgnment of a sngle demand between a gven OD par wth a gven deal tme to a gven path. Input Parameters The followng nput parameters are used n the set parttonng model: a Ω set of all possble assgnments I set of orgn-destnaton pars t T set of all tme steps t T set of tmes that there s a demand between OD par l L set of operated lnes c cycle C a cost of the assgnment a 1 f OD par at tme t s assgned n assgnment a, B t a = 0 otherwse. 1 f the assgnment a s usng lne l at tme t, Ea lt = 0 otherwse. Decson Varables Followng s the lst of decson varables used n the model: λ a x t l 1 f assgnment a s a part of the soluton, 0 otherwse. 1 f there s a tran scheduled on lne l at tme t, 0 otherwse. Model mn a Ω a Ω C a λ a a Ω B t a λ a = 1, I, t T, (22) Ea lt λ a x t l, l L, t T, (23) (21) 14

18 The Ideal Tran Tmetablng Problem May 2014 x t l n l, l L, t T mn(t+c,t) t =t x t l 1, l L, t T, (25) λ a {0, 1}, a Ω, (26) x t l {0, 1}, l L, t T. (27) (24) The objectve functon (21) s mnmzng the costs of passengers. Constrants (22) ensure that there s exactly one assgnment for every passenger n the optmal soluton. Constrants (23) lnk the scheduled trans wth used assgnments. Constrants (24) dctate that the amount of scheduled trans on every lne does not exceed the amount of avalable physcal trans. Constrants (25) are cyclcty constrants. Constrants (26) and (27) set the domans of decson varables. In ths new formulaton, we have reduced the amount of decson varables and got rd of the bg M constrants. However, the soluton space s now even bgger and thus n order to avod the "exploson" of the soluton space (tme), we propose to solve the lnear programmng relaxaton of the above problem usng column generaton, as descrbed n the next secton. 5.2 Column Generaton In the lnear programmng (LP) relaxaton of the set parttonng problem the domans of λ a and x t l are extended to [0, 1]. Despte the large number of varables t s possble to solve the LP relaxaton usng a column generaton algorthm. In a column generaton algorthm we mantan a restrcted master problem (RMP) that only consders a small subset Ω 1 Ω of all the possble varables. New varables are added to Ω 1 untl we can decde that no varable n Ω \ Ω 1 can mprove the soluton that results from only usng the varables n Ω 1. In the frst teraton of column generaton, the RMP s solved usng the set Ω 1 consstng of passenger assgnments n the ntal feasble soluton provded by the CPLEX after solvng the orgnal formulaton usng the current publshed tmetables of a TOC. Thereafter, n each successve teraton of the column generaton process, the followng dual varables are passed to the subproblem for dentfyng feasble assgnments wth negatve reduced cost: α t dual varables for constrant 22 β t l dual varables for constrant 23 We do not need to consder the dual varables correspondng to constrants (24) and (25) snce these constrants do not nvolve the λ a varables and therefore the assocated dual varables do 15

19 The Ideal Tran Tmetablng Problem May 2014 Algorthm 1: Branch and Prce Data: data fle, Ω, f nshed - boolean, duals - float Result: Ω 1 Ω, soluton 1 begn 2 Ω 1 ntalsoluton 3 duals 4 soluton 5 repeat 6 duals solvemaster(ω 1 ) 7 f nshed true 8 for N do 9 temp solvesubproblem(, t, duals) 10 f reducedcost(temp) < 0 then 11 Ω 1 temp 12 f nshed f alse 13 untl f nshed 14 soluton solvemaster(ω 1 ) 15 f soluton Z then 16 ub solvemaster(ω 1, ntegral) 17 f soluton = ub then 18 break 19 soluton branch&bound(soluton) 20 prnt soluton not mpact the reduced cost of the λ a varables. Based on the dual varables from RMP, the subproblem generates new columns to enter the actve pool of columns Ω 1 by calculatng the most negatve reduced cost column for each vessel separately n each teraton of the column generaton process. When there are no columns wth negatve reduced cost for any subproblem to enter Ω 1, the column generaton termnates. The column generaton n pseudocode can be seen n Lnes (1) (13) n Algorthm 1. For mathematcal justfcaton of column generaton, please refer to Barnhart et al. (1998), Desaulners et al. (2005) and Fellet (2010). 16

20 The Ideal Tran Tmetablng Problem May Sub-Problem In each teraton of column generaton, we solve a sub-problem for every OD par, every deal tme t T and every path p P. In each subproblem, the objectve s to dentfy the feasble assgnment for that partcular OD par wth deal tme wth the most negatve reduced cost to be added to the current pool of actve columns Ω 1 n the restrcted master problem. Note that the ndex and t s removed from all decson varables, snce t s solved separately for each OD par wth deal tme t. Input Parameters Followng s the lst of parameters used n the model: t deal travel tme for OD par p path of the sub-problem t T set of all tme steps l L the sequence of lnes used to get from the orgn to the destnaton r l runnng tme of lne l h l runnng tme to get from the startng staton of the lne l to the frst staton on the same lne ncluded n the current path m the mnmum transfer tme q 1 the value of tme spent watng q 2 the value of tme spent n vehcle Decson Varables Followng s the lst of decson varables used n the model: beta t l 1 f lne l s used at tme t, 0 otherwse. w the total watng tme of the passengers w l the watng tme of the passengers when transferrng to lne l s scheduled delay of the passengers C the cost of the passengers Model mn C α + β t l betat l (28) l L t T 17

21 The Ideal Tran Tmetablng Problem May 2014 C = q 1 w + q 1 m ( L 1) + q 2 r l + q 2 s (29) l L beta t l = 1, l L, (30) t T w = w l, (31) w l ( t beta t l + h ) ( l t beta t l 1 + h l 1 + r l 1 + m ), l L : l > 1, t, t T : l L\1 t t + h l 1 + r l 1 (32) w l ( t beta t l + h ) ( l t beta t l 1 + h l 1 + r l 1 + m ), l L : l > 1, t, t T : t t + h l 1 + r l 1 (33) s f 2 (t beta t L + h L ) t, t T, (34) s f 1 t ( t beta t L + h L ), t T, (35) h 0, (36) beta t l {0, 1}, l L, t T. (37) The objectve functon (28) s lookng for a column wth the most negatve reduced cost. Constrant (29) calculates the cost of the soluton. Constrants (30) make sure that passengers take exactly one tran on each lne n the path. Constrants (31) calculate the total watng tme. Constrants (32) and (33) set the proper watng tme n the transferrng statons. Constrants (34) and (35) are complementary constrants (one at a tme s actve) that calculate the scheduled delay n passengers destnatons. Constrants (36) and (37) set the domans of decson varables. 6 Case Study As our case study, we use the data from the Israel Ralways (IR), kndly provded by Mor Kasp and Tal Ravv, who have cleaned the data and used them n ther study (Kasp and Ravv (2013)). The data consst of an hourly demand from 6 a.m. to 1 a.m. for every orgn destnaton par n the IR network (Fgure 4). The demand has been extracted from the tcket sells and tran counts n the year 2008 and t s a representatve sample of an average workng day (over the whole year of 2008) n Israel. We further modfy the data for our specfc use. The demand s smoothed nto mnutes usng the posson process. Moreover, snce the scheduled delay s related to the arrval nto the destnaton, 18

22 The Ideal Tran Tmetablng Problem May 2014 Fgure 4: Israel Ralways Network we shft the demand s deal tme from the orgn to the destnaton, by addng up the shortest travel tme for the specfc OD (assumng that the passengers mnmze ther cost). For the values of tme, we are usng the swss values as estmated n Axhausen et al. (2008), due to the unavalablty of the srael values. The IR network conssts of 48 passenger statons, whch adds up to 2256 OD pars. As our 19

23 The Ideal Tran Tmetablng Problem May 2014 benchmark, we are usng the current IR tmetables (year 2014), from whch we have extracted 36 undrectonal lnes (n our study we do not take nto account dfferent stoppng patterns wthn the same lnes) that are beng operated by 389 trans (t s an odd number, as we do not take nto account trans scheduled between 1 a.m. and 6 a.m., as no demand data exst for ths perod of tme). The data processng s stll undergong, however we have not seen a path between an OD par that would requre more then one nterchange. The sze of the problem justfes the use of column generaton. 7 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 ntroduce a defnton of such an deal tmetable and formulate a mxed nteger lnear problem that can desgn such tmetables. Snce the proposed formulaton s complex (large amount of decson varables, bg M constrants), we propose to decompose the problem and solve t usng column generaton methodology. At the current stage, the mplementaton s undergong and hence no results are beng provded. We plan to use a demand data obtaned from the Israel Ralways as mentoned n the above secton. The current tmetables of IR are cyclc, thus we wll be able to measure the cost savngs between our cyclc tmetables and the ones of IR. Moreover, usng our non-cyclc verson of the model, we wll be also able to measure the actual cost of the cyclc tmetables. 8 References Abe, I., M. I. of Technology. Technology and P. Program (2007) Revenue Management n the Ralway Industry n Japan and Portugal: A Stakeholder Approach, Massachusetts Insttute of Technology, Engneerng Systems Dvson, Technology and Polcy Program. Armstrong, A. and J. Messner (2010) Ralway revenue management: Overvew and models (operatons research), Workng Papers, MRG/0019, Department of Management Scence, Lancaster Unversty. Arnott, R., A. de Palma and R. Lndsey (1990) Economcs of a bottleneck, Journal of Urban Economcs, 27 (1) , ISSN

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