Determining routes for an auxiliary fleet in presence of emergencies on a railway line

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1 Determg rotes for a alary fleet presee of emergees o a ralay le Fraso A. Ortega 1 Ja A. Mesa 2 Mgel A. ozo Uersty of Selle Sella Spa {reos@s.es mesa@s.es mgelpozo@s.es} Abstrat. Trasportato o-demad represets a mode of trasport hh a set of sers formlates trasport reests betee pp ad delery pots hh mst be sered by meas of ehles of a ge apaty. Whe the tems to be trasported are people t s eessary to smltaeosly osder a mmzato of operatg osts (perspete of trasportato systems maagers) ad a redto of the oeees for the ser. A epeted falre of the plaed shedlg the trasport system (e.g. a falre the ege of a tra) may ase a sere dsrpto to a large mber of passegers ho old rema solated at org statos. The reslt of that stato old be a sattered dstrbted poplato pots th heterogeeos demad terms of destatos ad spef reremets of sere. I ths paper a model of trasportato o-demad s formlated to address a dee hh happes a trast le (metro tras) forg to reshedle the sere sh trast le. The goal s to mmze the total ost assoated to the delay epereed by sers ho hae to be sered by a alary fleet to atted ths dee. eyords: Ralays dsrptos reshedlg. 1 Itrodto The maagemet of terrptos ad delays s a ral tas the otrol of operatos for ay pbl trasport ompay. I the settg of rapd trast systems le metro ad loal tras a epeted falre may leae thot sere to a large mber of solated sers at ther org statos. Moreoer the la of formato abot ths dee a lead to matag a reglar rate of ser arrals to the statos ad theg the lst of atteded sers. rodg a protool ase of otgees the system s a top of terest profsely reogzed the spealzed lteratre. The hgh ost oled the mplemetato of a emergey system s of partlar terest so effey rtera mst be osdered the desg. I that sese mathematal models ad optmzato methods are essetal tools for deso-mag prodg ade to mmze the egate oseees ased sh as the oerall delay of the sers ad the eoom mpat to the system. I metro ad sbrba systems a despread prate to address the orree of dets or maor dsrptos (temporarly eteded) s to establsh speal bs seres that operate the affeted setos (Ortega et al. 2008). Ths tehe s eessary he t s mpossble to absorb the mpat th the same etor or th omplemetary spportg etors (Cepeda et al. 2006). Hoeer despte ts dobted terest for operators the aalytal perspete of sh problem has ot reeed sffet atteto the lteratre (Arrola et al. 2009). The proso of a alary fleet th seeral ehles of dfferet apates a atted to a heterogeeos trasport demad betee dfferet orgs ad destatos. Moreoer the effey of the alary rotes a be reased by ldg epress seres omttg stops he a ehle s rg empty (deadheadg) or mag that a ehle hages ts trael dreto before reahg ts termal stato (short-trg). The smltaeos applato of these strateges alog of the terary of a trast le reslts asymmetr rotes formg short yles (Trah et al. 2007) th the possble lso of termedate epress seres (Mesa et al. 2009). Fgre 1 shos a eample of ombed strateges here the hte dots date omtted stops alog the le r

2 6 th Iteratoal Coferee o Iformato Systems Logsts ad Spply Cha ILS Coferee 2016 Je 1 4 Bordea Frae Fgre 1: Asymmetr short yle th epress seres a trast le I ths or a adaptato of the trasport o-demad (TOD) o as "p ad Delery" problems (D) s deeloped to maage dsrptos the sere th a pbl trasport system. 2 p ad delery problems the o-demad trasport Ds (Desalers et al 2002 Berbegla et al 2007) are a type of problems hh ossts of determg ehle rotes here a set of obets or people hae to be trasported betee erta orgdestato pars. Ds hae bee speally stded the last years ad are maly related to logsts ad the trasportato o-demad for passegers (Cordea 2006 Melahrods et al Jørgese et al. 2007). The ma dfferee ompared to other Ds appled to trasport of obets s that the TOD the mmzato of operatoal osts mst share mportae th redg ser oeees. Therefore der ofltg obetes (e.g. to mamze the mber of total sered reests erss to mmze the operatoal osts) the hma perspete (mmze ser s dsomfort) healy eghs o the model ether the ombed formlato of the global obete or ther mportae the lst of restrtos o the feasblty of soltos. Oe of the prortes the alty of sere lamed by the ser s to restrt the tme teral hh eah pot of pg p mst be sted. Moreoer frot of hat happes may Ds assoated th the trasport of obets (sh as small mal olmes) the ehle apaty s a prorty he t omes to trasport passegers ho trael th ther lggage tems. A ell-o eample of TOD s the pbl trasport o demad rral areas hose profle of potetal passegers are elderly poplato dsabled people or hldre stdets ad here the mplemetato of a fed trast le of pbl harater s ot feasble or epese. Hoeer the eoom ad soal terest of TOD a be ot lmted to trobleshoot sere to small ommtes or passegers th speal reremets. A approprate lsterg of passegers grops (Dmas et al Ioahm et al. 1995) ad the orporato of trasfer odes the model (Mtroć-Mć ad Laporte 2006 Cortés et al. 2009) allo to trasform the problem to a real ase of plag a mass trast system. Althogh there hae bee proposed aros eat methods for solg TOD problems th mltple ehles (see Berbegla et al. 2007) hoeer the hgh omplety of the problem aot be aoded ad the solto mst be lmted to small-saled searos. Ths has led to the eed of prodg effet algorthms to sole large-saled staes. The pratal maagemet of o-demad trasportato oles deso mag 3 areas: lsterg of reests rotg of ehles ad shedlg of lers. Clsterg reests s deoted to reate grops of pettos th a stato to be sered by the same ehle. The gropg s made aordg to rtera of pp tme ad promty to the destatos. Oe formed these lsters the rotg stage ossts of dedg hat order the orrespodg odes ll be sted for the aalable ehles. Fally the shedlg phase ll determe the eat tmes to st eah ode. Eah of these phases s strogly led to the other so that a good solto ll rere a smltaeos mmzato of them. The a formlato s proposed to determe the optmal sere rered to meet th ehles of dfferet sze ad org a set of reests here the sal trasport sere that rs alog a trast le has bee aelled

3 6 th Iteratoal Coferee o Iformato Systems Logsts ad Spply Cha ILS Coferee 2016 Je 1 4 Bordea Frae 3 Formlato Let ( A ) G be a dreted graph here the erte set s omposed of three sbsets O D. Set O represets odes assoated to tal ad fal depots 1 D 1 2 s the ode set s the ode set of pp pots of the reests ad of destato pots of those reests. A pot of a pp reest ( ) ll hae assoated aother ode of delery reest labeled by. Reests from N sers are oetrated statos of set S that ollet grops of passegers shg to trael to other statos of S. The affeted le r had shedled a tmetable ge adae hee e deote the desred tme hh the reest shold omplete ther orey. We deote as follos the follog dees ad parameters: detfer of aalable ehle for attedg the emergey ( ) t a Q e l L f r d trael tme of ehle betee odes ad. If trael tmes rred he ehles moe from ode to ode are ot depedg o tra type the t t. tme of sere / atg rered by the ehle at ode eght / load assoated to the ode mamm load alloable for ehle earlest tme for pg reest at ode latest tme for pg reest at ode mamm alloable tme for a reest delay fed ost for retg ehle fed ost of trasport assoated to ehle per t of tme fed ost per t of tme de to delay respose to a reest fed ost of ompesato for eah passeger de ot atted ther reest Addtoally the model ses a set of bary arables to establsh the rotg throghot edges the graph that are sed by eah ehle ad to sets of arables to respetely otrol the startg tme of sere ad the load o eah erte: bary arable hh eals 1 oly f the ehle rosses the edge ( ) tme he ehle begs ts sere at erte load of ehle he goes ot from erte The measres tae frot of dets a be of to types: - Use a resere fleet of ehles smlar to hh sffered the dee (e tra ts) or dfferet (bses tas). - Reoe the se of rollg sto ad proeed to ompesate passegers affeted by the det. I ths paper both strateges are preseted the same model to ealate the effeteess of possble measres to be mplemeted. There are ost- ad passeger-oreted ompoets all models for the le plag problem. I or model ost-oreted ompoets are the fed ost for retg ehle ad the fed ost of trasport assoated to ehle per t of tme. O the other had passeger-oreted osts are the fed ost per t of tme de to delay respose to a reest ad the fed ost of ompesato for eah passeger - 3 -

4 - 4-6 th Iteratoal Coferee o Iformato Systems Logsts ad Spply Cha ILS Coferee 2016 Je 1 4 Bordea Frae de to ot atted ther reest. A albrato of these parameters old be rered for a better adst of the reslts. The proposed goal s to mmze the osts dered from sg the aalable fleet (ret ad trasportato osts) assmed by the system operator ad smltaeosly to mmze the oeee osts to the ser (formlated terms of delay ost ad ompesato). Ths short lst of rtera s addtely epressed by meas of: r f ) ( Mmze ) ( 1 d Sbet to ostrats: D 1 (1) D 1 (2) D (3) 0 (4) D 0 (5) t a (6) l e (7) L D (8) (9) Q Q m 0 m a (10) 1 0 (11) 0 (12) Costrats (1) - (5) fore the feasblty for the osdered rotes (ostrat blo of rotg) hle restrtos (6) - (8) otrol the feasblty of that sere a temporal otet (ostrat blo of shedlg). Costrats (9) ad (10) rere that the load restrtos that ere establshed o ehles aot be olated. If e aalyze the proded model e a see that ostrats (1) - (5) together th restrto (11) orrespod to a lassal ehle Rotg roblem (R). The lso of restrtos (9) ad (10) leads to a apatated erso of the R. Fally oe restrtos (6) - (8) are added to the formlato the model leads to a arat of the Capatated ehle Rotg roblem th Tme Wdos here learty property has dsappeared bease bary arables of rotg appear mltplyg to other arables that otrol the tme of sere ad the ehle apaty both obete fto ad ostrats (6) - (9). Se the resltg optmzato model s olear ad t s based o a set of dsrete depedet arables the problem s N-hard ad to sole sh heretly tratable problem o largesze real staes some algorthm mst be spefally deeloped for that prpose. Spefally the algorthm proded Mesa et al. (2014) allos to amlate at eah ode the pealzato osts by delay ad ompesato de to ot hag atteded th the orrespodg pp

5 6 th Iteratoal Coferee o Iformato Systems Logsts ad Spply Cha ILS Coferee 2016 Je 1 4 Bordea Frae reest. I ths ay the mber of reests a be reded by osderg lsters of reest hh share a ommo preferred departre tme. Let be the e set of lstered reests that are detfed by meas of de eghted by fator ad that hae assoated a preferred teral for the departre tme e l. The follog rote desrbes ho set of lstered reests a be ostrted: For eah ddal reest e mst he f there already ests a lstered reest ' sh that e l L ad addtoally ealtes Q are held for '. I that ase reest s aggregated to lstered reest ' otherse a e lster of reests s talzed startg from tem. Net order to herstally sole or model e apply the follog proedre: 1. Sole the model (1)-(12) for arables (the rotg sb-problem) by osderg the set of lstered reests stead of. Let ( ) be the retred soltos. 2. Sole the model (1)-(12) for arables ad (the shedlg sb-problem th apates) by osderg the tal set of reests ad matag arables fed ales obtaed from the preos step. Let ( ) soltos that determe the tmetables of le rs. 3. Sole the model (1)-(12) for arables ales eal to ad he arables ad ( ) respetely. Let ( ) ad th be the sets of are fed tag be the retred set of soltos of le rs. 4. Repeat steps 2-3 tl the mber of teratos reahes a pre-defed mber. 4 Model applato Sppose a trast le rg betee for ralay statos. There are s org-destato (OD) pars to dstrbte the trps orrespodg to the 300 sers of ths le. I ths searo assme that tra 1 of Fgre 2 (left polygoal) sffers a det ad aot operate ormally leag a set of sers thot trasport sere alog the le. The aboe model as programmed GAMS ad ge the small sze of the epermet the omptato tme has bee osdered as ot releat

6 6 th Iteratoal Coferee o Iformato Systems Logsts ad Spply Cha ILS Coferee 2016 Je 1 4 Bordea Frae Fgre 2: Dfferet optos of shedlg respose to a dee o tra. The remag polygoal les of Fgre 2 sho dagrams of possble tmetables to be mplemeted by spportg desos that atted the dee sffered by tra 1 by meas of ehles loated at depot S2. The llstrated atos orrespod to the strateges otled Seto 1. Tra 2 repeats the terary of the aelled le r 1. ehle 3 (3) stops at statos S2 ad S3 ad maes a shorttrg before retrg to depot S2. Tra 4 maes a epress sere betee statos S2 ad S4 before retrg to the depot. ehle 5 rs mode deadheadg betee statos S1 ad S2 ad oers trps betee grop of statos S1-S3 S1-S4 S1-S3-S4 ad S1-S2. Table 1 smmarzes the reslts obtaed dfferet searos determed by the aalable mber () of rese ehles thot load restrtos ad by dfferet ales assged to the for types of osts osdered. f Table 1: Optmm reslts a otet of absee of load restrtos. d Ret Delay Comp. Users Ob. Tras I Table 2 the optmm reslts aheed he there s a restrto of mamm apaty of 60 sers per ehle are eposed. I otrast to the preos searo t s possble to obta a optmal reslt or model hose aser s "ot serg ad to demfy" he the lmted apaty of the ehles fores to do may ad ostly shpmets (ase 70). I the opposte sde he the shppg ost s lo (ase 20) t s also possble to reommed that destatos are sered by meas of epress seres tag the shortest possble trael tmes (Fgre 3). f Table 2: Optmm reslts a otet that ldes load restrtos d Ret Delay Comp. Users Ob. Tras Not serg see Fgre 3-6 -

7 6 th Iteratoal Coferee o Iformato Systems Logsts ad Spply Cha ILS Coferee 2016 Je 1 4 Bordea Frae Fgre 3: Optmm ehle shedlg respose to falre of tra 1 for searo of last ro Table 2 5 Colso I ths or a optmzato model has bee formlated to sole a rotg problem ase of a emergey stato ased by the falre of a tra that rs alog a le. The respose ato s based o the aalablty of spport ehles th dfferet apates ad dfferet osts preosly establshed. The proposed model has bee appled to a smple ase order to mae a sestty aalyss for the parameters prodg formato o the total operator osts ad o the mber of sered sers. The sze of the epermet has alloed to obtag eat soltos shog the strog depedee of the reslts th respet to the parameter ales a sse of partlar terest the ealato of real otets. I ase of large-sze real searos herst proedres may be rered for rmetg the dobted omptatoal omplety of the model. Aoledgemets Ths researh or as partally spported by the Spash Mstero de Cea e Ioaó der grats MTM C02-01 ad MTM by the Eellee program of the Adalsa Goermet der grat FQM-241 ad FQM-5849 ad fally by the FEDER fds of the Eropea Uo. Referees 1. Arrola R. Ortega F.A. ozo M.A.: Aplaó de los roblemas p ad Delery e la Gestó de Emergeas de Sstema de Ferroarrl. Atas de las II Joradas sobre Modelos de Optmzaó Aplados a la lafaó Robsta y la Gestó de los seros de trasporte e aso de Emergea (MORE 2009) Sella España. ISBN-13: (2009) 2. Berbegla G. Cordea J.-F. Grbosaa I. Laporte G.: Stat pp ad delery problems: A lassfato sheme ad srey. TO (2007) 3. Cepeda M. Comett R. Flora M.: A freey-based assgmet model for ogested trast etors th strt apaty ostrats: haraterzato ad omptato of elbra. Trasportato Researh B (2006) 4. Cordea J.-F.: A Brah-ad-t algorthm for the dal-a-rde problem. Operatos Researh (2006) 5. Cortés C.E. Matamala M. Cotardo C.: The pp-ad-delery problem th trasfers: formlato ad a brah-ad-t solto method. Eropea Joral of Operatoal Researh (2010) 6. Desalers G. Desrosers J. Erdma A. Solomo M.M. Soms F.: R th pp ad delery. I: Toth. go D. (Eds.) The ehle Rotg roblem. SIAM Moographs o Dsrete Mathemats ad Applatos (2002) 7. Dmas Y. Desrosers J. Soms F.: Large sale mlt-ehle dal-a-rde problems. Les Cahers d GERAD G HEC Motréal (1989) 8. Ioahm I. Desrosers J. Dmas Y. Solomo M. M.: A reest lsterg algorthm for door-todoor hadapped trasportato. Trasportato See (1995) - 7 -

8 6 th Iteratoal Coferee o Iformato Systems Logsts ad Spply Cha ILS Coferee 2016 Je 1 4 Bordea Frae 9. Jørgese R. M. Larse J. Bergsdottr. B.: Solg the dal-a-rde problem sg geet algorthms. The Joral of the Operatoal Researh Soety (2007) 10. Melahrods E. Ilha A. B. M H.: A dal-a-rde problem for let trasportato a healthare orgazato. Compters & Operatos Researh (2007) 11. Mesa J.A. Ortega F.A. ozo M.A.: Effete Alloato of Fleet Freees by Redg Itermedate Stops ad Short Trg Trast Systems. R.. Aha et al. (Eds): Robst ad Ole Large-Sale Optmzato. LNCS (2009) 12. Mesa J.A. Ortega F.A. ozo M.A.: Loatg optmal tmetables ad ehle shedles a trast le. Aals of Operatos Researh (2014) 13. Mtroć-Mć S. Laporte G.: Watg strateges for the dyam pp ad delery problem th tme dos. Trasportato Researh B (2004) 14. Ortega F.A. Mesa J.A. ozo M.A.: Robstez Ate Ideas e el Dseño Topológo de Redes Metropoltaas de Trasporte úblo F.A. Ortega J.A. Mesa M.A. ozo. Atas del III Cogreso de Igeería del Trasporte. A Corña España. ISBN: (2008) 15. Trah A. Cortés C.E. Jara-Díaz S.: Estratega Itegrada de Asgaó de Flota e Corredor de Trasporte úblo. Atas del XIII Cogreso Chleo de Igeería de Trasporte Satago Chle (2007)