0 0 0 Inegraed Train Timeabling and Rolling Sock Scheduling Model Based on Time-ependen emand for Urban Rail Transi Yixiang Yue (Corresponding auhor) School of Traffic and Transporaion, Beijing iaoong Universiy, No. Shang Yuan Cun, Hai ian isric, Beijing, China. 000 Phone: 0; Fax: -0- Email: yixiangyue@yahoo.com unao Han School of Traffic and Transporaion, Beijing iaoong Universiy, No. Shang Yuan Cun, Hai ian isric, Beijing, China. 000 Phone: 00; Email: 0@bju.edu.cn Shifeng Wang China Railway Elecrificaion Survey esign & Research Insiue Co. Ld., No. iangdu Road, He ong isric, Tianjin, China. 000 Phone: 0 Email: wangshifeng@jedi.com.cn Xiang Liu eparmen of Civil and Environmenal Engineering Rugers, The Sae Universiy of New ersey CoRE 0, Frelinghuysen Road, Piscaaway, N 0-0 Email: xiang.liu@rugers.edu Word coun:,words ex + ( figures + ables) x 0 words (each) =, words Submission ae: Augus s, 0 Paper submied for peer review a he h Transporaion Research Board Annual Meeing.
0 0 ABSTRACT The congesion problem of urban ransporaion is becoming increasingly criical for many meropolises. The Urban Rail Transi (URT) sysem has araced subsanial aenion due o is safey, high speed, high capaciy and susainabiliy. Wih a focus on providing a holisic modelling framework for rain scheduling problems, his paper proposes a novel opimizaion mehodology ha inegraes boh rain imeabling and rolling sock scheduling based on imedependen passenger flow demands. We paricularly consider he rade-off beween waiing imes of passengers and operaion coss of he URT sysem. Using rain pahs and rolling sock indicaors as decision variables, his problem is formulaed as a bi-level programming model. The simulaed-annealing (SA) based heurisic algorihm is employed o solve he proposed model and generae approximae opimal soluions. Numerical examples are developed o demonsrae he performance of he proposed approaches. The calculaion resuls and comparisons indicae ha, even for he large-scale Beijing rail ransi operaions, he SA-based algorihm can efficienly produce he approximae opimal scheduling sraegies wihin accepable compuaional limis, demonsraing he pracical value of our proposed approaches. Keywords: Urban rail ransi, Passenger flow, Train imeable, Rolling sock schedule, Bi-level programming, Simulaed annealing algorihm
0 0 0 0 INTROUCTION Wih he emergence of a more social economy, many rail ransi neworks have recenly been pu ino operaion or are under consrucion in many meropolises o saisfy large passenger ravel demands. Therefore, he efficien operaion of rail ransi lines has become an imporan issue for all URT sysems. The planning process for public ransporaion usually consiss of several consecuive phases. The process begins wih nework design, which is ypically followed by line planning, imeabling, and vehicle and crew scheduling (). Obaining a high qualiy railway operaion plan will ake several ieraions. Alhough here are many available models and algorihms o address each sep, he enire muli-sep process is sill very challenging and compuaionally cumbersome. Therefore, formulaing and solving a large-scale inegraed railway planning problem has long been a pursui for academics and praciioners. In compiling he rain diagram for urban rail sysem, i is imporan o consider he inerval ime of rains, he circulaion and he running roue of rains (). Min e al. (0) consider he rain-conflic resoluion problem for which he opimal conflic-free imeable and use a columngeneraion mehod o compue i (). Corman e al. (0) consider a bi-objecive problem of minimizing rain delays and missed connecions o provide a se of feasible non-dominaed schedules (). Kecman e al. (0) propose underlying algorihms auomaically idenify roue conflics wih conflicing rains, deermine accurae arrival and deparure imes/delays a saions (). These papers focus on arranging rains depending on conflics and use some useful algorihms o solve i. Line planning is he firs imporan sraegic elemen in he railway operaion planning process. Goossens e al. (00, 00) consider a model formulaion of he line-planning problem where oal operaing coss are o be minimized and he model is solved wih a branch-and-cu approach () (). Kaspi and Raviv (0) formulae an inegraed line planning and imeabling model wih he objecive of minimizing boh user inconvenience and operaional coss (). These researches lay he foundaion for our sudy of imeable and rolling sock. eveloping opimizaion models for consrucing periodic ransi imeables and synchronized schedules is anoher research direcion in he field. Carey and Crawford (00) design a series of heurisics for finding and resolving rains conflics so o saisfy various operaional consrains and objecives (). Zhou and Zhong (00, 00) formulae rain scheduling models which consider segmen and saion headway capaciies as limied resources, and developed algorihms o minimize boh passenger waiing imes and oal rain ravel imes (0) (). Goverde (00) describe a railway imeable sabiliy measure by using a max-plus sysem heory and analyzed rain delay propagaion processes (). Focusing on reducing passenger waiing ime a sops and ransfers, Liebchen (00) adaps a periodic even-scheduling approach and a well-esablished graph model o opimize he Berlin subway imeable (). Wong e al. (00) concenrae on he synchronizaion beween he differen lines of an URT nework o minimize passengers ransfer imes (). There are also a number of sudies relaed o rolling sock scheduling. Schlake e al. (0) conduc an analysis of he effec of lean producion mehods on main-line railway operaions (). Nielsen e al. (0) deal wih real-ime disrupion managemen of railway rolling sock (). Thorlacius e al. (0) propose an inegraed rolling sock planning model ha simulaneously akes ino accoun all pracical requiremens for rolling sock planning (). A number of recen sudies have pu more aenion on developing inegraed opimizaion models for imeable and passenger flow or rolling sock schedule. Niu and Zhou (0) focus on
0 0 opimizing a passenger rain imeable in a heavily congesed urban rail corridor (). Anoher paper of hem (0) focuses on he demand-oriened passenger rain-imeable opimizaion for a rail corridor under ime-dependen demand and skip-sop paerns (). Yang e al. (0) propose a new collaboraive opimizaion mehod for boh rain sop planning and rain scheduling problems on he acic level (0). Yue e al. (0) propose an innovaive mehodology using a column-generaion-based heurisic algorihm o simulaneously accoun for boh passenger service demands and rain scheduling (). Alhough a few researchers have aemped o explore inegraed railway planning, simulaneously accouning for line planning, imeabling and rolling sock scheduling is rarely seen in he lieraure. Line plan, imeable and rolling sock usage of inegraed opimizaion is he high-prioriy problem ha mus be solved on mos busy URT lines. In his paper, we propose a new mehodology o simulaneously accoun for oal passengers waiing imes, rain imeabling cos and rolling sock usage for URT lines. The framework of our proposed mehodology is illusraed in Figure. In our model, he inpus are URT line daa and secion-specific passenger flow. The decision variables correspond o he rain pah and rolling sock rajecory. In boh he model and algorihm, he firs sep is o opimize rain frequency while guaraneeing passenger flow consrains. Following his, he model simulaneously schedules rain imeables and he rolling sock usage while guaraneeing consrains relaed o passenger flow, rain flow and rolling sock size. The model oupu is a near-opimal rain imeable rolling sock scheduling. Inpu : Urban rail ransi nework (saions, secions, racks) Nework planning Sraegical Level Inpu : Timeable parameers (ravel ime, sopping ime) Inpu : Time dependen passenger flow Line Planning Our inegraed model Tacic Level Train imeabling Rolling sock usage Oupu : Near opimal imeable and rolling sock usage plan Crew schedules Real ime managemen Operaional Level FIGURE Framework of inegraed URT planning mehodology
0 0 0 We inend o address he following specific objecives:. Analyze ime-dependen passenger flow, rain imeabling and rolling sock scheduling in URT sysems and discusses heir ineracions among hem. The paper develops an opimizaion mehodology ha enables he inegraion of rain imeabling, line planning, and rolling sock scheduling.. Include boh loop and linear URT lines. In order o formulae an inegraed opimizaion model, his paper proposes a general rain flow model ha can be applied o all ypes of URT lines.. Formulae a bi-level programming model for inegraed rain imeabling and rolling sock scheduling. The upper-level model opimizes rain frequency and rain imeables; minimize passengers waiing imes and operaion coss. The lower-level model schedules rolling sock o minimize he number of infeasible rain pahs and proposes a SA-based heurisic algorihm o solve he model.. Illusraes he use of an inegraed opimizaion model and he SA-based algorihm o improve he imeable of ypical real-world URT lines in Beijing, China. The remainder of his paper proceeds as follows. Firs, we presen deailed problem descripions and assumpions for URT lines. Second, we develop an inegraed rain frequency, rain imeabling and rolling sock scheduling model. Third, a SA-based algorihm o solve he model is proposed. Subsequenly, we generae compuaional resuls from real-world insances of he Beijing URT, and demonsrae he effeciveness of our proposed model. Finally are principal conclusions and sugges possible fuure research direcions. PROBLEM ESCRIPTION Key elemens in URT A rain imeable defines rain deparure and arrival imes a each saion, which is an essenial plan for he operaion of a railway sysem. Each rain depars from a depo when service begins and reurns o a depo when service erminaes. Saions ha direcly connec o depos are of greaer imporance and are named -saions in he model. For example, in Figure, Xizhimen and ishuian are -saions for Line, Tianongyuanbei and Songjiazhuang are -saions for Line, Bagou, Chedaogou and Songjiazhuang are -saions for Line 0. Chedaogou Fengai Bagou ishuian Xizhimen Line Line Huixinxijienankou Yonghegong Chongwenmen Songjiazhuang Tianongyuanbei ongzhimen epo -saion Oher saion Line 0 Guomao FIGURE Illusraion of rail ransi nework in our cases
There are hree key elemens in URT lines: passenger flow, rain flow and rolling sock. Passenger flow is easy o undersand. And we illusrae relaionships beween rain flow and rolling sock in Figure. Each solid line represens a rain and he lines of he same color denoe rolling sock rajecories. Assume here are hree rolling socks in depo. The number of rains beween saion and saion are deermined by rain inervals (passenger flow). ue o consrains on he number of available rolling socks, some rains (denoed by solid lines) can be placed ino service, whereas oher rains (denoed by dashed lines canno. ashed lines need o be deleed in he final oupu rain imeable. Time direcion epo Saion 0 Saion Rolling sock rajecory Feasible rain Infeasible rain FIGURE Relaionship beween rain flow and rolling socks MATHEMATICAL FORMULATION AN ALGORITHM Noaions The general subscrips, inpu parameers and decision variables ha are employed in our mahemaical formulaions are lised as follows. General subscrips: C Se of saions:,,c Se of -saions:,, d, C E Se of ime inervals:,,e F G Se of rains Se of rolling socks i, i', j, j ' Saion indices, i, i', j, j' C kk, ' -saion indices, k, k ', s, r Time inerval indices,, s, r E f Trains index, f F g Rolling socks index, g G
Parameers: o Number of passengers who leave saion i for saion j a ime i, j q Passengers volume beween saion i and saion j a ime i, j max L The maximum number of passengers for a rain exp L Expeced number of passengers for a rain d( f ) The origin saion of rain pah, d( f ) h Minimum headway (ime inerval) beween wo consecuive rain pahs a a saion num G k The maximum rolling socks of depo which conneced o -saion k dg ( ) epo which conneced o origin saion ha rolling sock g sars o carry ou a ask e i, j Running ime beween saion i and saion j max e run The maximum running ime for a rolling sock in a day max e dwell The maximum waiing ime in a -saion for a rolling sock min e sop The minimum sopping ime in he depo for a rolling sock w i Waiing passengers of saion i a ime ecision variables: u ( f ) i v ( f ) i s, x ( ) kk, ' g s, ykk, ( g ) s, zkk, ( g ), if rain f depars from saion i a ime 0, oherwise, if rain f arrives a saion i a ime 0, oherwise, if rolling sock g depars from -saion k a ime and arrives a -saion k ' a ime s 0, oherwise, if rolling sock g wais a -saion k from ime o ime s 0, oherwise and s, if rolling sock g sops a depo conneced o -saion k from ime o ime s 0, oherwise General rain flow model for urban ransi lines Each rail passenger rip consiss of an origin saion, a desinaion saion and ravel ime. For URT sysems, he number of passengers who ravel beween wo saions is imporan. Thus, we need o calculae his number using he number of passengers for each origin-desinaion marix. e ', q i i i, i o i ', j ' E, i C () j ' [ i, c] i ' [, i]
0 Using equaion, we can obain he passenger volume beween wo adjacen saions. The passenger volume may differ among various secions. We use a reference value o replace he real value, as shown in Equaion. Based on he secion-specific passenger flow beween wo adjacen saions, we can obain he maximum passenger volume of some successive secions. e,,, =max{,, ii e,,, i j qi j qi i qi i q j, j } E, i, j C () In a URT nework, he line opology srucure can be divided ino wo main caegories: linear lines and loop lines. ifferen ypes of lines have differen rolling sock rajecories. Figure (a) shows he opologies of six common ypes of rail ransi lines. Each line may have a single depo, such as ype, ype and ype, or muliple depos, such as ype, ype and ype. In ypes and, depos are conneced o origin or erminal saions; in ypes and, depos are conneced o inermediae saions. For mahemaical modelling purposes, we modify linear lines o loop lines by regarding racks in differen direcions as virually differen saions. Thus, we can ransfer linear lines ino a looped line opology wih one or more depos. A rolling sock depars from a depo, akes some loops along he saions and reurns o he origin depo, as shown in Figure (b). In his example, ype has one depo, ype has wo depos and ype has hree depos. Muliple-depo lines rajecories mus be coordinaed o saisfy minimum rain headway consrains a -saions. One depo muliple depos ' c-' ' c-' c - c c- c c- c c- c Origin Looped line Origin Looped line Liner line i c- c Origin Topology ' i' c-' c i Looped line c- Topology Topology Topology i Origin j c- c ' i' c-' j' c i j c- Looped line Loop line i i c c- c c- Topology Topology epo -saion Oher saion (a) Origin opologies and modified opologies of URT lines Type Type Type i i 0 c c c j epo -saion Oher saion Trajecory of rolling sock (b) Trajecories of rolling socks FIGURE Illusraion for opological srucure of URT lines and rolling sock arrangemen
0 0 0 Bi-Level Mahemaical Model We use a bi-level programming model o describe he problem for URT. The upper level model is used o opimize rain imeables, including minimizing waiing imes for passengers and reducing operaing coss for URT sysem. The objecive of he lower level model is o schedule rolling sock and minimize infeasible rains. When he soluions of he upper level model and he lower level model are feasible, he problem will be solved. The upper level model In he upper level model, we replace he waiing imes of each passenger wih he number of queuing passengers in each ime inerval., are weighs for he cos of passengers and he operaion cos. If decision makers wan o save passengers waiing ime and improve service levels, he value of / should be larger; if decision makers wan o reduce he rain operaing coss and improve operaing income, he value of / should be smaller. obj _ up Min ( obj _ w obj _ u) () obj _ w wk () k E obj _ u u ( f ) k () f F k E ) Passengers flow consrains The number of passengers who wai a ime is he number of waiing passengers a he prior ime plus he number of arrival passengers a ime minus he number of deparure passengers a ime max. uring sauraion, he number of deparing passengers is equal o L, and max he number of deparing passengers in non-sauraion is less han L. Considering cases of sauraion and non-sauraion, we use greaer han in equaion (), equaion () and equaion (). wk 0 E, k () max wk qk, k ' uk ( f ) L k, k ' () f F w w q, ' u ( f ) L E / {}, k, k ' () max k k k k k f F ) Train flow consrains Two rains ha ravel in he same direcion canno depar from a saion a he same ime, and a reasonable ime inerval beween he wo rains is needed (ypically, he minimum headway is based on he shores braking disance beween he rains given he rain ypes and he signaling sysems). The inerval beween wo consecuive rain deparures from he same saion i mus be greaer han or equal o he minimum deparure headway h. u k ( f ) r E, k () f F [ r, r h) k f F [ r, r h) v ( f ) r E, k Train flow f depars from -saion k a ime and i arrives a -saion k ' a ime k ' d( f ), i depars from -saion k ' a ime e kk, ' e, ' ( ) kk k k' ( ),, ', (0) e kk, '. If u f v f E k k f F () v ( f ) u ( f ) E, k /{ d( f )}, f F () k k For all rain pahs, he number of deparure imes and arrival imes are equal. vk( f ) uk( f ) f F () k E k E
0 0 0 0 The lower level model The lower level model is used o schedule rolling sock based on rain flows. The objecive funcion is o minimize he number of infeasible rain pahs. obj _ low Min obj _ x (), obj _ x [ u ( ) s, '( )] k f xk k g uk ( f ), f F, g G () k E k ' s E ) Flow conservaion consrain For any space-ime node ( k),, he number of ouflows is no more han one., s, s, s [ xk, k '( g) yk, k ( g) zk, k ( g)] E, k, g G () s E k ' For any space-ime node ( k),, he subraced difference beween ouflows and inflows should be equal o or - if he node is a source or a sink node, respecively; oherwise, i should be equal o 0., s, s, s s, s, s, [ xk, k '( g) yk, k ( g) zk, k ( g)] [ xk ', k ( g) yk, k ( g) zk, k ( g)] s E k ' s E k ', k d( g) e, k d( g) E, k, g G 0 oherwise ) Train flow consrains For any space-ime node ( k),, he number of rain rajecories is less han he number of rain pahs., s xk, k '( g) uk ( f ) E, k, d( f ) d( g) () k ' s E ) Rolling sock consrains For any space-ime node ( k),, rains canno be overaken in a saion, and only one rain a a ime may remain in a saion. r, s r, xk, k '( g) yk, k ( g) r E, k () g G k ' s E g G E max For any space-ime node ( k),, a rain canno run more han e run per day s, max [ rk, k ' xk, k '( g)] erun g G (0) k k ' E s E max For any space-ime node ( k),, a rain canno wai in a -saion for more han e dwell each ime. s, max yk, k ( g) edwell r E, k, g G () E max s [ r, r Gdwell ] min For any space-ime node ( k),, a rain canno sop in a depo for less han e sop for one ime. s, zkk, ( g) 0 k, g G min () E s [, esop ) SIMULATE ANNEALING ALGORITHM To solve he model, we ried commercial solvers such as GAMS and CPLEX, and we also designed our own algorihm. Because of he scale of he problem, we selec a simulaedannealing-based algorihm o solve our proposed models. Is evaluaion funcion is obj _ A obj _ w obj _ u obj _ x, where is he weigh for he number of infeasible rain pahs. In he early sage of ieraion, obj _ x has lile influence on he objecive funcion and reduces he limiaion of feasible soluion. In he lae sage of ieraion, obj _ x has grea influence on he objecive funcion, his can ensure he rolling sock usage plan. Figure shows he framework for he simulaed annealing algorihm. ()
Iniialize Simulaed Annealing Algorihm upper level model Train imeable lower level model Rolling sock scheduling Inegraed rain imeable and rolling socks usage plan 0 0 0 FIGURE Framework of he simulaed annealing algorihm We provide deailed procedures for he simulaed annealing algorihm o solve he model: Sep. Iniialize parameers and variables. Inpu dae of line and passenger volume, iniialize parameers of he algorihm, and iniialize variables: ui( f ), vi( f ) and w i. Sep. Obain an iniial feasible soluion. We use a blank rain imeable as he iniial feasible soluion: A. Sep. Updae he curren feasible soluion, A '. We add and reduce rains based on a small probabiliy o obain a new feasible soluion. Sep. Compue an evaluaion funcion for he new feasible soluion, A '. Compue obj _ w ', obj _ u '. Compue obj _ x '. We use a neighborhood search mehod o solve obj _ x '. Calculae he evaluaion funcion: obj _ A' obj _ w' obj _ u' obj _ x'. Calculae he difference: obj obj _ A' obj _ A Sep. Updae he curren bes soluion: eermine wheher he new soluion A ' is acceped based on he Meropolis-Hasings algorihm. If acceped, A A' Sep. Sop or coninue. Two rules can end he algorihm: ) Ieraion number reaches he maximum limi, ) Sable ieraion number reaches he maximum limi and he number of infeasible rain pahs equals 0. If one rule is saisfied, we proceed o Sep ; oherwise, we proceed o Sep. Sep. Updae emperaure We updae he emperaure using a mehod of wo sage-exponenial decline, which can ensure efficiency and accuracy. Sep. Schedule rolling sock. We use a neighborhood search mehod o schedule rolling sock. Sep. Presen he oupu. CASE STUY: LINE OF BEIING RAIL TRANSIT NETWORK We use a real-world insance (Subway Line ) from Beijing s rail ransi nework o demonsrae he applicaion of our opimizaion model and algorihm. The scheduling algorihms are
implemened in Microsof Visual Sudio 00 on Windows OS. All experimens are conduced on a PC wih an Inel Core uo. GHz CPU and GB RAM. Table shows he parameers of he algorihm. d is he number of depos and b represens he number of ime inervals of min.( For example, when b, a ime inerval is 0 seconds; when b, a ime inerval is seconds.) The ime span for he rain operaion considered in his paper is 0 hours (,00 mins) from :00 am o :00 am he nex day. The ime inerval of passenger flow is from :00 am o :00 pm for a oal ime of hours (,00 mins). TABLE Parameers definiion in our case Parameers Value Meaning Number of ime inervals,00 b Number of ime inervals Weigh of passenger waiing ime exp b L Weigh of rain frequency 00,000 d Weigh of infeasible rains M 0 00,000 b d Iniial emperaure rise 0. Heaing coefficien 0. Cooling coefficien N 0 0,000 b (d+) Maximum number of ieraions sop N N / 00 Ieraion number of erminaing algorihm for 0 no longer improvemen N N /0,000 Ieraion number in iniial sage 0 N N / 00 Ieraion number in medium sage 0 upae N N / 00 Ieraion number for emperaure o updae 0 rise N N / Ieraion number for emperaure o increase 0 change p 0.0/b/d Updae probabiliy in iniial sage change p 0.00/b/d Updae probabiliy in medium sage change p 0.00/b/d Updae probabiliy in laer sage max e run 0 b Maximum running ime max e dwell b Maximum dwelling ime in -saions min e sop 0 b Minimum sopping ime in depos 0 Table (a) liss he daa of Beijing rail ransi lines. Line of he Beijing railway ransi nework is a loop line. There is only one depo (refer o green line in Figure ) ha connecs wih Xizhimen saion and ishuian saion. We consider Xizhimen saion as a -saion in his case. The minimum deparure headway beween wo consecuive rain pahs is minues. In his case,
0 b, d. We consider he clockwise direcion of his line in our (model/parameer- use one). Table (b) shows passenger demand in workdays. TABLE Inpu daa of Beijing rail ransi Line (a) Basic daa of line Number of Number of Number of Running ime Line lengh Line depos rolling socks saions (minues/lap) (km/lap). (b) Passenger demand of line Scenarios :00~ :00~ :00~ :00~ :00~ :00~ :00~ :00 :00 :00 :00 :00 :00 :00 Workday(persons/min) 0 0 00 00 00 00 0 Weekend(persons/min) 0 00 0 00 00 00 0 Resuls of Line and analysis. Figure (a) shows he resuls of passenger flow during workdays. We couned he average number and he maximum number of waiing passengers in 0 minues. Operaors can make reasonable decisions based on passenger daa. The number of waiing passengers reaches he maximum value of 0 beween :0am and :00am. The rain imeable of Beijing rail ransi Line for workdays is shown in Figure (b). ue o larger passenger flows, he rain pahs on he rain diagram are more inensive, and rain operaing frequency reaches is maximum limi during peak hours (:00-:00, and :00-:00). Fewer rains operae during off-peak hours. The soluion no only avoids wasing resources and underuilizaion of ranspor capaciy, bu also decreases he cos of rain operaion. The densiy of he rain pahs adequaely reflecs he change in raffic flow. 0 (a) Passenger flow of Beijing rail ransi Line in workdays
0 (b) Train imeable of Beijing rail ransi Line in workdays FIGURE Oupus of Beijing rail ransi Line in workdays Table shows he rolling sock schedule of he depo ha is conneced o Xizhimen saion for workdays. The firs column represens he rolling socks number. The second column shows oal running ime for each rolling sock. The hird column represens he imeable of every rolling sock. For example, [ ] means ha for he firs rolling sock, i begins is firs service a he h minue, and ends a he h minue (service ime inerval is minues). Similarly, is las service sars a he nd minue and ends a he 00 h minue. As menioned before, he ime span of rolling sock operaion is 00min. The resuls reveal ha he proposed model and algorihm applies o rail ransi ring line of muli-depos and can ge an approximae opimal soluion. NO. TABLE Rolling sock schedule of Beijing rail ransi Line in workdays Running Time (min) 0 0 0 0 Rolling socks scheduling [ ][ ][ ][ ][ ][ ][ ][ 0][ ][ ][ ][ ][ ][ 0][ 00] [ ][ 0][0 0][0 ][ ][ 0][ ][ ][0 ][ ][ ][ ][ 0][0 0] [ ][ ][ 0][0 ][ ][ ][ 0][ ][ ][ 0][0 0][0 ][ ][ 0] [ ][ ][ ][ 0][ ][ ][ ][ ][ 0][ ][ ][ 0][ 0][0 ][0 0] [ ][ ][ ][ ][ ][ ][ 0][0 ][ ][0 ][ ][ ][ ][ 0] [ ][00 ][ ][ ][0 ][ ][ 00][0 ][0 ][ ][ 0][ ][ ][ ][0 ] [ 0][0 ][ 0][0 ][ ][ ][ ][ ][ ][ ][ ][ ][0 ][ ][ 00] [ ][ ][ ][ ][ ][ ][ 0][0 ][ 0][0 ][ ][ ][ 0] [ ][ 0][0 ][ 0][0 ][0 ][ ][
0 0 0 0 0 0 ][0 ][ ][0 ][00 0][0 0] [ ][ ][ ][ ][ ][ ][ ][ ][ ][ 0][0 0][0 ] [ 0][0 ][ ][ 0][0 ][ 0][ 0][0 ][ ][ ][ ][ ][ ][ ] [ ][ ][ ][ ][ ][ 0][0 ][ ][ ][ ][ ][0 ][ ][ 0][ ] [0 ][ ][ ][ ][ ][ 0][0 ][ 00][0 ][0 ][ ][ ][ ][ ][ 00] [ ][ ][ ][ ][ ][ ][ ][ 0][0 ][ ][ 0][ ][ ][ ] [ 0][ ][ ][ ][ ][ 0][ ][ ][ ][ ][ ][ 0][0 0] [ ][ ][ ][ ][ ][ ][ ][ 0][0 ][ 0][0 ][ ][ ][ ] [ ][ ][ ][ ][ ][ 0][ ][ ][ ][ 0][ 0][0 ][ 00][00 0] [ ][ ][ ][ ][ ][ 0][0 ][ ][ ][ ][ ] [ ][ ][ ][0 ][ ][ ][ ][ 0][0 0] [ ][ ][ ][ ][ ][ ][ ][ ][ ][ 0] [ ][00 ][ ][ 0][0 ][ ][ 0] 0 [0 0][0 ][ ][00 ][ ][ ][ ][ ][ ][ ][ 0][0 ][ ][ ][ 0] [ 0][0 ][0 ][ 0][0 ][ ][ 0] 0 Efficiency analysis of he algorihm [ ][ ][0 0][0 ][ ][0 ][ ][ ][ ][ ][ ][ ][ ][ 0][ 0] [ ][ ][ ][ ][ ][ ][ ][ 0][0 ][ 0][0 ][ 0][0 0] We also apply a similar opimizaion approach o Line and Line 0 in Beijing. To evaluae he efficiency of he proposed algorihm, we ry o use wo service packages (GAMS and ILOG CPLEX) o solve he model and compare he resuls of differen mehods. ue o a large number of variables, we figure ou he upper model using GAMS because CPLEX is no suiable o solve he complee model. We only lis he calculaion resuls of he upper model by GAMS, CPLEX and SA ( =0 ). We employ he calculaion resuls of GAMS as a benchmark and make comparaive analysis. Table liss he comparison of he differen solving mehods. Three cases
0 are employed : Line on workdays, Line on workdays and Line 0 on workdays. For a largesized problem, our proposed algorihm ouperforms he commercial sofware. Case Line, workday Line, workday Line 0, workday Number of epo TABLE Comparison of differen solving mehods Solver Upper level model/gams Upper level model/cplex Upper level model/sa Bi-level model /SA+ neighborhood searching Upper level model/gams Upper level model/cplex Upper level model/sa Bi-level model /SA+ neighborhood searching Upper level model/gams Upper level model/cplex Upper level model/sa Bi-level model /SA+ neighborhood searching obj _ A Comparison of objec value Compuing ime(s) Comparison of compuaional ime,,0,,0 0.,,0.0,0,.0.,,, unavailable unavailable unavailable unavailable,, 0. 0.0,0,0 0. 0.0,,00, unavailable unavailable unavailable unavailable,, 0. 0.0,, 0.,00 0. The following are several observaions based on he resuls presened in Table : ) For he case of Line on workdays, he resuls of he upper level model using GAMS, CPLEX and SA are similar, bu he compuing ime by CPLEX is en imes faser han he compuing ime by GAMS, and SA is also much faser han GAMS. The compuing ime of he bi-level model by SA and neighborhood searching is slighly longer han he compuing ime of he bi-level model by GAMS and he resuls are worse han GAMS by % considering he lower level of he rolling sock schedule. ) For he large-scale problems, such as cases and, CPLEX canno obain a soluion wihin a reasonable compuing ime. In he case of Line wih a ime inerval of 0s and wo depos, GAMS can solve he model; however, he compuing ime exceeds hour. SA and neighborhood searching can obain a beer soluion wihin minues. ) In he case of Line 0 wih a ime inerval of s and depos, GAMS can barely obain a soluion wihin hours, whereas SA and neighborhood searching can obain he soluion wihin minues and seconds. The objecive funcion value is greaer han %.
0 0 From hese observaions, i appears ha SA, as a mea-heurisics, is fas bu canno guaranee global opimaliy. For a pracical, large-scale problem, SA may be a promising approach o yield adequae enough soluions (may no be perfec) given a reasonable ime span. CONCLUSIONS This paper proposes a new mahemaical model for he opimizaion of rain service plans, rain imeables and rolling sock schedules for URT. This model can simulaneously reduce passenger waiing ime and rain operaion coss while also improving he uilizaion of rain ses. Firs, we inroduce hree key elemens in URT lines and propose a general rain flow model, which can be employed across all opologies of rain lines. Then we apply a bi-level programming model o formulae he scheduling problem for URT. The upper level model opimizes rain imeables by minimizing waiing imes for passengers and operaion coss for URT sysems. The lower level model is used o schedule rolling sock by minimizing he number of infeasible rain pahs. We use a simulaed-annealing-based heurisic algorihm o solve he large-scale model. In a case sudy of he Beijing rail ransi nework, we es our opimizaion mahemaical model and algorihm. The calculaion resuls indicae ha he proposed model and algorihm can obain reasonable schedule planning. In paricular, our new, inegraed algorihm can rapidly obain a near-opimal soluion and bes uilize rolling socks, which renders i useful for complex real-world applicaions. A similar opimizaion mehodology can also be adaped for high-speed and freigh rail lines. Our fuure research will focus on hree major areas. Firs, he neighborhood searching mehod for vehicle scheduling is no a global opimizaion mehod; we aim o improve he algorihm performance. Second, he model can be exended o accoun for rain skip-sop paerns and long/shor rains. Las, we will refine and validae he proposed model wih observed daa from URT sysems in Beijing.
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