Train Scheduling Networks under Time Duration Uncertainty

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1 Preprnts of the 9th World Congress The Internatonal Federaton of Automatc Control Tran chedulng Networs under Tme Duraton Uncertanty Adnen El Amraou and Khaled Mesghoun Laboratore d Automatque, Géne Informatque et gnal (CNR-LAGI-UMR 89 Ecole Centrale de Llle (EC-Llle Cté centfque, 5965 Vlleneuve d Ascq Cedex, France (adnen.elamraou@ec-llle.fr, elamraou.adnen@gmal.com, haled.mesghoun@ec-llle.fr Abstract: In ths paper, we consder a ralway traffc schedulng problem. We am to fnd a schedule for a tran schedulng networs where tme duraton uncertantes are consdered. Ths problem was ntensvely studed wth mxed lnear models where trans movng duraton are determnstc. In ths paper, we formulate the problem as a classcal one wth scenaro-based stochastc programmng tang expected values as obectve functons. Then, a new crteron s proposed to quantfy schedulng robustness n the face of uncertanty. Besdes, addtonal constrants were ntroduced to mae the model feasble when unexpected event occurs durng schedulng executon. Keywords: tochastc programmng, optmzaton, robustness, ralway traffc schedulng, tme duraton uncertanty.. INTRODUCTION Ths paper deals wth the tran schedulng networs problems. It conssts n fndng the arrval and the departure tmes of the lnes at certan stages of the networ. Dependng on requred obectves, these stages can be referred to publc staton and/or swtches. nce 87s, and more specfcally snce the frst tran schedule conference n Germany, tran schedulng plannng problems have been wdely studed (Carey and Locwood (995; Cordeau et al. (998 and several programmng model have been proposed despte ther NP-had characterstc (Assad, (980; Nachtgall and Voget (997. Ths category of schedulng problems can be shared nto two classes, as descrbed as follows: tatc or Predctve problem It conssts frstly on allocatng resources to all trans n all routes. Then, the tran sequencng entrans the prespecfcaton of the arrval and departure order of trans at statons. Fnally, a tme-table s then resulted. Ths class ams on the mnmzaton of the maespan or the cycle tme or on the maxmzaton of the traffc frequency (Harrod (0; Hggns et al, (996; Kraay and Harer (995. Dynamc or Reactve problem It nvolves when the tran planned schedule cannot be respected due to a dsturbance handlng actvty. In ths case a new tmetable should be found whle all the problem constrants are respected. Generally, the obectve functon conssts on the mnmzaton of tran delays (Narayanaswam and Rangara (03. The focus of ths paper s to present a new model whch can be useful for the two problem classes smultaneously. The orgnalty s of ths approach consst on the followng: Due to tran travellng duraton s uncertantes, we present a handlng scenaros approach. Where addtonal crteron s consdered to fnd the most robust schedule. We defne new varables and constrants to control the tran speed and the tran watng tme on statons n order to remove a dsrupton, f any. Most prevous models are handlng ether predctve tran schedulng problem or reactve one. If unexpected event happens, a frst schedule soluton can be determned usng ths frst problem class models. Ths can be reached by nstantatng nown decson varables. Nevertheless, ths soluton s very smple and cannot, at any way, guarantee the soluton performance. Besdes, weather condtons can requre on trans to reduce ther speeds on some tracs. In fact, a wheels sldng or a wheels satng can happen due to the snow or to the tree leaves on the ral n autumn generally. o, ths could lead to several perturbatons n arrval and departure tmes of the tmetablng passenger trans. Ths problem has become recurrent n Europe at the approach of wnter holdays and Chrstmas whle a large number of people tae the tran to travel. Whch mae ral transport less compettve compared to other means of travel (ar and ground transportaton. Followng to these ntroductory remars, ecton s devoted to the problem statement. ecton 3 dscusses the mathematcal problem formulaton. The problem solvng methodology as well as the new model extensons are presented n ecton 4 and 5, respectvely. Copyrght 04 IFAC 876

2 . PROBLEM TATEMENT In ths study, we consder the sngle trac, b-drectonal ralway traffc. Trans have to travel n two drectons: from rght to left (RtoL and from left to rght (LtoR (called also nomnal drecton n lterature (Abbas-Tur et al, 0. Each rght to left drecton tran s travellng, as soon as possble from the startng staton (staton, then t s vstng successvely m- statons, numbered from to m-, before arrvng to the end staton (staton m. Whle each left to rght drecton tran has to start by the fnal staton, and then t s vstng the statons: m-, m-, and successvely, before reachng the startng staton. We call sngle trac (or segment the slce of the lne confned between any two statons. In general, on each tran staton several tracs (called bloc are avalable to allow overtaes and crossngs. One of the man specfctes of such a system s that the average tran travellng duratons values are nown and any delay can mae a networ dsrupton. Moreover, tracs are the most crtcal resource of such lnes. Besdes, there are no multple-tracs between statons and each staton can receve more than one tran at the same tme. Fg. shows an example of a lne layout wth sngle trac and b-drectonal tran movements. Fg.. Example of a sngle trac, bdrectonal ralway Ths problem can be consdered as a ob-shop schedulng problem wth very specfc constrants, where each segment s consdered as a machne and each tran as a ob. The constrants we consder here are the followng ones: (C Each trac can receve smultaneously ether RtoL trans or LtoR trans. (C In each staton, trans must reman at least a lower duraton and at most an upper duraton. These duratons can vary from one staton to another due to staton passengers frequency. (C3 Between two successve trans movng on the same drecton, a mnmum safety tme duraton s requred. (C4 tatons 3 m- m LtoR drecton RtoL drecton In each meetng staton, mnmum meetng tme duraton has to be ensured between the arrval and the departure of trans movng n dfferent drecton. Defntely, passages have to be allowed to change from trans. The studed problem requres two dstnct but dependent decsons to be made: ( schedulng decson-sequence, n whch trans have to move (prorty to move, and ( staton watng decson (real watng tme on statons. The strong dependence between these two decsons maes the problem very hard to model. Yet, gettng motvated by prevous researches usng mxed nteger lnear programmng methods, we elaborate a new model; whch can solve ths tran schedulng problem to optmalty. In ths programmng model, we are loong for optmzng the smultaneous travellng duratons of several trans movng n dfferent drectons through a sngle lne. Besdes, t can be also useful to fnd a soluton when a dsturbance occurs. In general, the problem soluton s presented usng graphc tmetable as shown n Fg.. Fg.. Graphc tmetable Ths fgure llustrates an example of a sngle trac lne layout wth three statons and three trans (two movng from left to rght and one from rght to left. lash lnes represents the tran moves whle horzontal lnes show the watng tmes on statons. In ths example, only one tran meetng s carred out on staton between tran 3 and tran. After the passage of the tran, tran 3 reaches staton, wat there untl the trac becomes avalable and the mnmum meetng tme taes n. Then, t leaves staton to staton. Whle tran spends the lower bound of ts requred watng tme on staton and go to staton 3. In the followng secton, we formally descrbe the problem and formulate t as a mxed nteger lnear programmng model. 3. MATHEMATICAL FORMULATION A defnton of notatons used n ths paper s necessary n order to descrbe the elaborated models. o, let s defne the followng parameters and varables: Problem Parameters. n total number of trans.. n trans that move n the LtoR drecton. 3. n trans that move n the RtoL drecton. 4. m total number of statons. NOTE : Trac Trac taton Wantng tme on staton Tran Tran 3 Tran - tatons are ndexed from to m. Tme - To smplfy the followng notaton, we denote by a tran movng n LtoR drecton and by a tran movng n RtoL drecton. 8763

3 5. Uptd, Lwtd The upper and lower bounds travellng tme of a LtoR tran to run through the trac between the statons and Uptd, Lwtd The upper and lower bounds travellng tme of a RtoL tran to run through the trac between the statons and Upwt, Lwwt The upper and lower bounds watng tme of LtoR tran on staton. 8. Upwt, Lwwt The upper and lower bounds watng duraton of RtoL tran on staton. 9. fte, fte The safety tme duratons between,, the arrval of two trans ( and of the same drecton to staton : , f ( s < s 0 otherwse E, f ( e < e 0 otherwse, f ( s < s 0 otherwse E, f ( e < e 0 otherwse, f ( s < s 0 otherwse 0.. fte = e e fte = e e ;,, fts, fts The safety tme duratons between,, the departure of two trans ( and of the same drecton from staton : Mmt Mmt, Mmt,,, Mmt,, fts = s s fts = s s ;,, The mnmum duratons for the meetng of two trans ( and on staton.. M Very bg number (+. 3. Dd, Dd Date when a dsturbance occurs n the networ for tran on or after vstng staton δ, δ Dsturbance duraton. Problem Decson Varables 5. Rtd, 6. Rwt, 7. s 8. s 9. e, e 0. Tgs,. Tgw, Rtd Rwt Tgs Real travellng tme of a LtoR and a RtoL tran from staton to the followng one. Real watng tme of a LtoR and a RtoL tran on staton. tart movng tme of a tran from staton to staton +. tart movng tme of a tran from staton to staton -. End movng tme of a tran from staton. Term gan speed from staton -. Tgw Term gan watng on staton. 7. E, f ( e < e 0 otherwse NOTE : - We assume that trans travel on a sngle trac lne layout n b-drectonal movements. In addton, all trans have to pass through all statons. Moreover, re-routng s not allowed and safetes as well as lower tme duratons have to be respected. - It s mportant to menton that the followng mathematcal formulaton s a tran movement s model: where decson varables defne the startng dates of all the tran moves from each staton and ther end dates ( s, s, e + and e. NOTE 3: We denote by ɶ x the new decson varable value of x when a regulaton s performed after a dsturbance. In other words ɶ x s the new value of x after a new schedulng. Problem Formulaton Mnmze : C = e m + e,n,n max, m,, n and, n s Rtd e ( + = ( s Rtd e + + = (3 (4 Lwtd Rtd Uptd (5 Lwtd Rtd Uptd + (6 e Rwt s (7 Lwwt Rwt Upwt (8 Lwwt Rwt Upwt + +. (9,, s fts s M 8764

4 s fts s M + + (. (0,, e fte e E M + +. (,, e fte e E M + + (. (,, s fts s M + +. (3,, s fts s M + + (. (4,, e fte e E M + +. (5,, e fte e E M + + (. (6,, s Mmt e M + +. (7,, e Mmt s M + + (. (8,, e Mmt s E M + +. (9,, s Mmt e E M + + (. (0,, = E, (, = E, (, = E, (3, e = s (4 m m s = e (5 ; E ; ; E ; ; E 0, (6,,,,,, In ths model, the obectve functon conssts n mnmzng the maespan C max. Ths varable, as gven here, can be defned as the total tme needed for all trans to reach ther termnals. Constrants (-(3 guarantee the fact that: before arrvng to the destnaton staton each tran has to spend ts requred travellng tme (.e. the most frequent duraton. Ths duraton must be confned wthn lower and upper bounds as traduced by constrants (4-(5. Constrant (6 defnes the statons watng dates that must be respected. These duratons have to be bounded, as defned by constrants (7-(8. Before startng to move, each tran must be sure that the mnmum safety tme duraton between hm and the prevous one usng the same trac s mantaned. Ths statement s translated by constrants (9-( for LtoR trans and by constrants (3- (6 for RtoL trans. Whle constrants (7-(0 are used to ensure the mnmum requred tme duraton at a staton between the arrval and the departure of two trans movng n dfferent drectons. Constrants (-(3 defne the precedence rule: f a tran leaves frst a staton, t must reach frst the destnaton staton (before tran +. Consstency constrants are gven by (4 and (5, whle bnary decson varables are defned by (6. Problem Complexty The complexty of ths problem s related to the number of trans, drectons and statons (or/and tracs. For ths proposed schedulng model, there are a total of m(7n n + 4n + 4 n + n + n constrants. Besdes, there are four types of decsons varables: a total of 4 m( n + n + nteger varables and 6mnn bnary varables to fnd. Problem Formulaton lmts Ths model can be used n general to fnd a tmetablng for a sngle trac lne layout, wth b-drectonal tran movements problem. Nevertheless, due to some unexpected events, tran traffc can be dsturbed. Consequently, a new, quc and robust schedule soluton should be found. For ths am, we propose: frstly, a new solvng methodology for the problem on hand, usng metrcs that taes nto account the schedulng characterstcs under tran transportaton tme uncertanty and we llustrate t by an example. econdly, an evoluton of the prevous model s performed to mae t able to fnd effcent schedule soluton, when an unforeseen event happens. Ths am can be acheved usng two flexbltes provded by such problems: speed tran control and tran watng tme control. These studes wll be presented consecutvely, n the followng sectons. 4. PROBLEM OLVING METHODOLOGY Gettng motvated by such transportaton schedulng problems and prevous robustness studes n optmzaton lterature and more precsely n ob-shop schedulng problems (Hardng and Floudas (997; Vn and Ierapetrtou (00, we propose a scenaro based stochastc problems methodology. The robustness, as defned for same problems, measures the reslence of the schedule obectve to vary under uncertan parameters and dsruptve events. Moreover, as explaned above, the weather can have a bg mpact on the tran tme travellng duratons and consequently on the dsturbance networ. Therefore, the tran travellng duratons can be assumed to vary by P% about ther nomnal values. Realstcally, ths probablty may be calculated for each lne trac and each perod of the year, on the bass of the hstorc of ralways transportaton companes le NCF n France. Hence, the performance of the schedule can be evaluated n terms of the maespan requred whle random transportaton duratons are satsfed. The proposed robustness evaluaton methodology can be resumed n the followng steps: tep. Consder a random of the tran transportaton tmes n the freedom degree nterval. tep. olve the determnstc problem, for all the consdered scenaros, wth the followng new obectve functon: ps. Cmax (7 s s Where: - p s s the probablty of a scenaro s. - Cmax s s the maespan of a scenaro s. 8765

5 tep 3. olve the same problem as n tep, usng the followng functon as obectve: ps. Cmaxs + ps. ( Cmaxs ( ps. Cmaxs s s s (8 tep and tep can be consdered enough to fnd the best soluton under the consdered uncertantes whle soluton stablty s not guaranteed. That s why, on step 3, we use the standard devaton, as a metrc for robustness evaluaton, to determne the more reslent schedule. Illustratve Example In ths example three trans must travel on a lne wth 4 statons ncludng the startng and the endng statons. Tran and travel on LtoR drecton, whle tran 3 travels on RtoL drecton. The travellng tme duratons are assumed to vary from 0% to 5% about ther nomnal values. These dstrbutons are gven n table (see step. Table. Tran travellng tme duratons dstrbuton Tran Trac Travellng tme / Probablty 45 / / / / / / / / / / / / / / / / The parameter Probablty n ths table could defne the frequency of spendng these tme duratons over a perod of tme. Ths probablty can vary from one season to another due for example to weather condtons. The number of consdered scenaros s 96 (*******3*. And the remanng problem parameters used for smulatons are gven as follows: Lwwt = 5 ; fte = ; fts = ; Mmt = 3.,,, By adoptng (7 as obectve functon of the model, the maespan of all scenaros vares from 48 to 453 t.u. and the most expected values of the maespan s 445 t.u. Ths maespan s defned n lterature as the determnstc value of the maespan (Vn and Ierapetrtou (00. It determnes the most lely schedule to be followed, where probablstc travellng tme duratons are consdered. Besdes, 44,5 t.u. s the average maespan over all scenaros. taton taton To quantfy robustness, we use the standard devaton metrc and we ntegrate t n our obectve functon as gven n (8. mulaton results show that the most robust schedule s for the maespan of 446 t.u. Graphc tmetable of the maespans 445 t.u. and 446 t.u. are reported on fgures 3 and 4, respectvely. Moreover, we use drect graph models to characterze the dfference between slght and heavy robustness (see Fg. 5. In fgure 5, we model prorty movng constrants for the above two schedules. As we can notce, schedule presents more precedence constrants than schedule. In fact, n schedule, tran leaves staton 3 before the arrval of tran. Moreover, the safety tme duraton for the startng move of tran s acheved before t arrves to the departure staton whch maes t free to move at any tme. Thus, schedule s a slght robust schedule whereas schedule s a heavy robust one. 5. NEW MODEL EXTENION uch as several prevous approaches (Cordeau et al. (998, the proposed lnear programmng model of sectons 3 s ncapable to face dsruptve events when there s a perturbaton on the tran networ. Thus, n ths secton we propose to mae t able to fnd new feasble and performng schedules f conflcts occur due to one or several delays. For ths am, we propose to add new constrants to solve these nconsstences. Two alternatves solutons are possble: The frst one conssts on the propagaton of the delay on all the over tran travellng dates. For example, f a dsturbance δ happens at Dd for a tran. All the transportaton movements scheduled before Dd are not altered. Despte the others, they have to be delayed by δ. Ths soluton can be appled but the optmalty s not guaranteed. Moreover, the propagaton of the delay can have a snowball effect. Therefore, the tran delays and the refund fees of ralway companes wll ncrease drastcally. The second soluton conssts on applyng a regulaton polcy, where all the startng move dates of all the trans are updated. To get ths soluton, we apply a tran speed control and a tran watng tme control strateges. These strateges can be traduced n our lnear programmng model by the followng constrants: s + Rtd + δ Tgs = e (9 chedule (C max= taton 4 chedule (C max= taton Tme Fg. 3. Graphc tmetable (C max= 445 t.u. taton Tme Fg. 4. Graphc tmetable (C max= 446 t.u. Fg. 5. chedules robustness analyss 8766

6 0 Tgs Rtd Lwtd (30 s + + Rtd Tgs = e (3 0 Tgs Rtd Lwtd (3 e Rwt Tgw s + (33 0 Tgw Rwt Lwwt (34 Constrants (9, (3 and (33 have to be used nstead of constrants (, (3 and (6. And new tme wndows constrants ((30, (3 and (34 have to be added. Let s detal and analyss these new constrants. We assume that a dsrupton happens when a tran leaves a staton -, then the arrval date of ths tran at staton wll be equal to: s + Rtd + δ = e n spte of equalty (. Moreover, due to ground topology and thans to tran technology evoluton, a frst control strategy can be appled. In fact, each tran can ncrease ts speed on some segments of ts course. Generally, ths speed s not determnstc but confned on a speed nterval: lmted by a lower and an upper bound. o, usng ths problem specfcty, we ntroduce a new parameter called term gan speed ( Tgs to determne the gan that t can be reached when the tran speeds are well controlled. Consequently, equaltes (9 and (3 have to be used nstead of constrants ( and (3. Besdes, tran speed lmts have to be respected. For ths am, we ntroduce the new constrants (30 and (3. The second control strategy proposed n our lnear programmng model s the watng tme control technque. In fact, for ths class of schedulng problem, a watng tme s defned on each tran staton. Ths tme duraton can be decreased, to absorb the tme dsrupton, provded that a mnmum watng duraton on each staton s respected. These statements are traduced by constrants (33 and (34. To llustrate the effectveness of our new model, we propose the followng example. Illustratve Example Let s consder the same example as n secton 3. However, the followng parameters are consdered: 0 Rtd 5 ; 5 Rwt 5 ; fte = ; fts = ;,, Mmt = ; Rtd = 5 ;0 Rtd 5 ; 5 Rwt 5 3, ; fte = ; fts = ; Mmt = 3 ; 0 Tgs,,, ; 0 Tgs Let s defne the followng cases study. Case. No dsturbance. Usng the elaborated lnear programmng model, whle dsrupton and gans are omtted, the optmal soluton s obtaned for a C max of 47 t.u. as reported on fgure 6. Case. A dsturbance of 5 t.u. for tran 3 at tme Dd=0. Thans to the speed and watng tme control strateges, the optmal soluton remans the same (see Fg. 6 Case 3. Moreover the departure of tran s delayed to 5. As llustrated on fgure 6, only the tran where dsturbance has occurred was delayed. But, we gan about,3% on the C max, whch cannot be neglected, compared to the soluton where control strateges are not appled. In concluson, on these cases study, we llustrate by a scholar example the effectveness of the proposed control strateges to fnd a performng schedule soluton when an expected event happens. Fg. 6. Graphc tmetable of cases, and 3 6. CONCLUION In the am to optmze a tran traffc schedulng problem under tme travellng uncertanty, we have proposed a scenaro based stochastc problems methodology to solve the problem. Moreover we have defned new obectve functon crtera to quantfy the schedule robustness. Besdes, we have extended the proposed lnear programmng model to solve the problem on hand where unexpected event happens. For ths am, we have developed two control strateges based on tran speed and watng tme duraton on statons. Future wor s to extend the elaborated model to more complex lne confguratons. Acnowledgements: Ths research was supported under a grant from the ANR Proect: "PERFECT". REFERENCE Abbas-Tur, A., Grunder, O. and El Moudn, A. (0. equence Optmzaton for Tmetablng Hgh Frequency Passenger Trans, IEEE Jont Ral Conference, Phladelpha, Pennsylvana. Assad, A. A. (980. Models for Ral Transportaton. Transportaton Research - Part A, 4, Carey, M. and Locwood, D. (995. A Model, Algorthms and trategy for Tran Pathng, Journal of the Operatons Reasearch ocety, 46, Chandesrs, M. and Labousse, V. (003. tatstcal Methods for the Evaluaton of Tmetable Robustness, World Congress on Ralway Research. Cordeau, J.F., Toth, P. and Vgo, D. (998. A survey of optmzaton models for tran routng and schedulng, Transportaton cence, 3, Hardng,.T. and Floudas, C.A. (997. Global Optmzaton n Multproduct and Multpurpose Batch Desgn under Uncertanty, Ind. Eng. Chem. Res., 36, Harrod,. (0. Modelng Networ Transton Constrants wth Hypergraphs, Transportaton cence, 45 (, Hggns, A., Kozan, E. and Ferrera, L. (996. Optmal chedulng of Trans on a ngle Lne Trac. Transportaton Research - Part B, 30(, Kraay, D.R and Harer, P.T. (995. Real-Tme chedulng of Freght Ralroads. Transportaton Research - Part B, 9 (3, 3-9. Nachtgall, K. and Voget,. (997. Mnmzng Watng Tmes n Integrated Fxed Interval Tmetables by upgradng Ralways Tracs, European Journal of Operatonnal Research., 03, Narayanaswam,. and Rangara, N. (03. Modellng dsruptons and resolvng conflcts optmally n a ralway schedule. Computers & Industral Engneerng, 64, Vn, J.P. and Ierapetrtou, G. (00. Robust hort-term chedulng of Mutproduct Batch Plants under Demand Uncertanty, Ind. Eng. Chem. Res., 40,