TRAIN TIMETABLE AND ROUTE GENERATION USING A CONSTRAINT-BASED APPROACH

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1 Proceedigs of the Teth Iteratioal Coferece o Idustrial & Egieerig Applicatios of Artificial Itelligece & Expert Systems, Atlata, Jue, 997. TRAIN TIMETABLE AND ROUTE GENERATION USING A CONSTRAINT-BASED APPROACH Ho Wai Chu Departmet of Electroic Egieerig, City Uiversity of Hog Kog Tat Chee Aveue, Kowloo, Hog Kog eehwchu@cityu.edu.hk ABSTRACT This paper describes research i modellig trai timetable ad route geeratio as a costraitsatisfactio problem (CSP). The key objective of this research is to desig a costrait-based schedulig algorithm that ca be used to geerate a trai timetable give headway requiremets at differet times of the day. The key costrait is to avoid track circuit or route cotetios while maximisig trai utilisatio. The objective of the schedulig algorithm is to determie how service levels ca be icreased without jeopardisig passeger safety. This research ivestigated traffic at a trai termius where two types of trais are competig for the use of the same tracks; trais that are reversig ad trais that are beig dispatched from the depot. The cotetio problem is particularly serious durig the rush hour trai build-up. The curret timetable ad trai routig are geerated usig two separate rule-based systems. However, due to the complexity of costraits ivolved, the curret systems caot geerate a pla that ca meet the desired service levels. INTRODUCTION This paper describes a costrait-based model ad algorithm for a trai timetable ad route geeratio program. This program determies how trais ca be efficietly dispatched from a depot to meet rush hour service requiremets while satisfyig resource costraits. Our research focused maily o geeratig a timetable ad routig for the buildig up of trai services. However, it ca also be applied to the reverse case of breakig dow the service after rush hour eds. The research was performed usig data from oe of the world s busiest subway systems. The costraitbased algorithm described i this paper was tested o oe of the busiest lies withi this subway system. The subway authority curretly uses a rulebased expert system to geerate the trai timetable based o headway requiremets, i.e., the time betwee the arrival of two cosecutive trais. The routig is the geerated usig a separate semiautomatic system which is based o heuristics. This routig program determies how each trai should travel to get to its destiatio. Subway trais ormally travel i a cyclic maer from oe fial termius to other ad back. The key problem is what happes whe the trai reaches the fial termius ad eeds to reverse while at the same time trais are beig dispatched from the depot to the termius. There is a tremedous amout of cotetio for the same set of tracks durig the rush hours. The timetable ad route geeratio is curretly performed by separate systems. The timetable geeratio system must therefore esure there is eough slack or buffer for the routig system to be able to geerate the routes. This extra time buffer causes iefficiecy. The route geeratio program, o the other had, is cofied to work withi the timetable that is give. The subway authority caot see how service levels ca be icreased without icreasig the efficiecy of the schedulig algorithm. This paper proposes a costrait-based algorithm that combies timetable ad route geeratio ito oe process. Headway costraits for timetable geeratio are cosidered at the same time as resource costraits for route geeratio. Our research focuses o desigig ad developig a costrait-based resource allocatio algorithm [DUNC94, PUGE94b] that ca assist the huma scheduler by geeratig differet what-if scearios or timetables based o differet sets of criteria, such as headway requiremets ad trai speeds. The resources to be allocated i this study are the track circuits ad the routes.

2 Our costrait-based schedulig algorithm performs three mai schedulig tasks - () geerate a trai timetable, (2) geerate the route sequece each trai must travel, ad (3) determie the travel time withi turaroud ad headshut iterlockig area. The most importat cocer for the subway authority is the ability to icrease service levels durig rush hours. The schedulig algorithm must be able to geerate a timetable ad route sequece that ca maximise the utilisatio of the track circuit resources i order to meet the service level demads. PHYSICAL CONSTRAINTS Figure is a simplified diagram of the track circuits ad sigals withi the turaroud ad headshut iterlockig area that are used withi the study. There are two platforms at the fial termius -- a Arrival Platform for trais arrivig from the other termius ad the Departure Platform for trais travellig to the other termius. The study oly ivolved this termius sice there is o depot at the other termius ad hece o track-circuit resource cotetio. The track circuits withi the turaroud ad headshut iterlockig area allow trais to reverse from the Arrival Platform to the Departure Platform. Subway trais ca travel i both directios ad have a driver compartmet at each eds of the trai. As part of the reversig process, the trai driver must walk from oe ed of the trai to the other. I some cases, there may be a additioal driver at the other ed to reduce the amout of time eeded to reverse a trai. While trais are usig the tracks to reverse, other trais may be dispatched from the depot through Depot Track ad Depot Track 2. Furthermore, all tracks withi the turaroud ad headshut iterlockig area ca be used for trai travel i either directios. The tracks i the turaroud ad headshut iterlockig area are divided ito routes. Routes are further divided ito track circuits. Figure shows the ames of the track circuits. A route is defied to be a set of track circuits betwee two sigals. Sice trais ca travel i either directio o the same tracks, there are sigals for both directio of travel. The relevat sigal is o the right had side of the directio of trai travel. Arrival S8 S2 Headshut T7 T9 T T3 T5 T7 T2 T23 T25 T27 T29 S5 T9 22 T S9 Crossover Tracks T2 T23 S23 24 a S0 S2 S6 2 T8 S4 Headshut 2 T26b T26a 24 b 24a T22 T20 T8 T6 T4 T2 T0 T8 T6 T4 T2 Departure Depot Track Depot Track 2 S32 S35 T3 T38 S S7 T4 S2 T36 T0 S25 Track Circuits ad Sigals Icluded i Study Fig. Track circuits ad sigals withi the turaroud ad headshut iterlockig area. Track Circuits. For the purpose of our research, a track circuit is the smallest piece of railway track that ca be uiquely idetified. However, the legth of each track circuit may be differet. The state of a track circuit may either be dow idicatig that a trai is curretly over this piece of track circuit or up idicatig that the track circuit is free. The track circuits i Figure are draw roughly to proportios. A total of 33 track circuits are icluded ito the study. The same track umbers are used to idetify tracks brachig out from track circuits, such as the crossover tracks, ad their paret braches.

3 Sigals. Figure illustrates all the sigals that were used i our research. There are a total of 6 sigals that will defie the startig poit for 42 possible routes that a trai may take. However, several routes ed at the Departure Platform, which does ot have a sigal after track circuit T26b. For uiformity, the Departure Platform will be cosidered as a pseudo-sigal. Routes. For this study, a route is defied as a sequece of track circuits that starts ad eds at a sigal (icludig the pseudosigal). Before a trai eters a route, it must first call ad set the route. Settig a route reserves a sectio of track so that o other trai will use the same track resource. A set of Boolea equatios or costraits defies whe a route ca be set. The equatios esure that the route is safe to eter. Our schedulig system differetiates routes that spa two immediate sigals as basic routes. Routes that are composed of more tha oe basic routes are called composite routes. The schedulig algorithm cosiders a total of 22 basic routes ad 20 composite routes. To travel from a start sigal to the target destiatio sigal may require the trai to traverse a sequece of basic routes. This sequece is referred to as a route sequece. Trai Legths. To determie track circuit occupacy, the legth of the trai is eeded. The legth of the trais used i this research is 77 metres from axle to axle, while the ed to ed legth is 82.5 metres. TIMING CONSTRAINTS Several differet types of timig data were used by our schedulig algorithm. This icludes the ru times of the trai, the sigal timig, the time to chage the crew, ad the trasmissio time. Ru Times. Oly the omial ru times are used i this study. The schedulig algorithm used the ru times to determie whe routes will be freed up for other trais to use. The ru times are defied as the travel time eeded betwee ay two sigals without stoppig. Target Arrival Time Sigal clearig & trasmissio delay Start Wait time Reverse time Ru time Ru time Safety margi A Route Sequece Diagram Usig Our Graphic Notatio Fig. 2 The represetatio of a route sequece ad its timig. Sigal Timig. It takes about 7 secods to set a route if poit movemet is required. It takes about 3 secods to set a route if poit movemet is ot required. Double-eded Timig. Double-eded crew is the case where there is a driver at both eds of a trai. This techique is used to reduce the time eeded to reverse a trai. Normally, to reverse a trai, the driver should stop the trai ad walk from the oe ed to the other to restart the trai. The time required for the whole process is aroud 4 miutes. With double-eded crew, the additioal driver will restart the trai after the curret driver stops it. Reversig, i this case, oly takes about 5 secods to fiish but requires a larger crew size to operate. Trasmissio Delay. A 2-secod trasmissio delay from issuig the route

4 settig commad (by computer or operator) to actuatio of the route settig activity is required. Figure 2 is a Gatt chart that shows the type of timig iformatio that is represeted ad used by our schedulig algorithm. Each row, alog the vertical axis, represets oe particular route ad the differet types of timig ivolved whe a trai travels i that route. The complete sequece of rows represets the sequece of routes a trai takes from the start sigal, e.g. the depot, to the ed sigal, e.g. the Departure Platform. THE CONSTRAINT-BASED MODEL AND ALGORITHM To solve this problem, we represeted the timetable ad route geeratio problem as a object-orieted [LEPA93] costrait-satisfactio problem (CSP) [KUMA92]. Although CSP or costrait-programmig has a relatively log history [STEE80], with costrait laguage extesios foud i Prolog [COLM90, VANH89], ad Lisp [SISK93], it is oly recetly that costrait-programmig became more popular with the availability of the ILOG s C++ class libraries [PUGE94a]. This library provided a very efficiet ad clea implemetatio of costrait-based programmig features i a covetioal laguage. Our schedulig algorithm was implemeted usig the ILOG C++ class libraries. Timetable _etries TableEtry _time Costraied Variable _route_asg Route Assigmet _route Costraied Variable TraiActiv ity _start _route _delay Costraied Variable Costraied Variable Fig. 3 The UML class diagram of the costraied variables. I geeral, ay schedulig ad resource allocatio problems ca be formulated as a costrait-satisfactio problem (CSP) which ivolves the assigmet of values to variables subjected to a set of costraits. CSP ca be defied as cosistig of a fiite set of variables v, v 2,..., v, a set of domais d, d 2,..., d, ad a set of costrait relatios c, c 2,..., c m. Each d i defies a fiite set of values (or solutios) that variable v i may be assiged. A costrait c j specifies the cosistet or icosistet choices amog variables ad is defied as a subset of the Cartesia product: c j d x d 2 x... x d. The goal of a CSP algorithm is to fid oe tuple from d x d 2 x... x d such that assigmets of values to variables satisfy all costraits simultaeously. Whe the trai timetable ad route geeratio problem is formulated as a CSP, the problem is represeted as a set of costraied variables. The first type of costraied variable represets etries i a trai timetable. There are two timetables i our problem a timetable for the Arrival Platform (T) ad a timetable for the Departure Platform (T2). Each timetable is a list of costraied variables. The domai of the variables will deped o the desired headway values requested by the user. Each timetable etry also cotais a route assigmet. Each route assigmet represets a sequece of trai activity. Each trai activity represets the selectio of a particular route, at a particular start time, with a particular delay. The route, start time, ad delay are all represeted as costraied variables. The domai of the route variable is all routes that ca start from the curret locatio of a trai. The domai of the start time is related to the previous route s ed time. The domai of the delay represets the amout of time a trai might eed to wait at a sigal. Figure 3 is a simplified Uified Modellig Laguage (UML) class diagram that shows the essetial classes i our desig. This costrait-based formulatio was desiged from the requiremet that the iput to the schedulig algorithm will be a table of desired headway values at the Departure Platform. Therefore the schedule algorithm will schedule a time for the arrival of a trai at the Departure Platform. However, we will eed to work backward to figure whe the trai departs from the depot ad how log the trai waits at each itermediate sigal. O the other had, this trai might be a reversig trai from the Arrival Platform. I this case, the schedulig algorithm works forward from whe the trai leaves the Arrival Platform util it reaches the Departure Platform. This combiatio of searchig backwards ad forwards at the same time complicates the algorithm, but is a ecessity give the subway authority s iput requiremet. Figure 4 illustrates this combied search. Although the search is a combied backward ad forward search, the schedulig of the timetable

5 etries is a forward process; timetable etries are istatiated from the earliest etries first. I other words, the timetable is geerated from the first trai to arrive at the T2 Departure Platform to the last trai of the desired schedulig period. Durig the costrait-based search, the algorithm predicts whe a trai leavig the Departure Platform will retur back to the Arrival Platform from the other termius usig omial turaroud times. I additio, the algorithm merges the time a trai leaves the Arrival Platform with the timetable etry for the arrival of that same trai at the Departure Platform. This mergig operatio is the major source of backtrackig for the schedulig algorithm. The time a trai arrives at the Departure Platform from the Arrival Platform might ot match exactly the headway required for the Departure Platform ad the trai must wait iside the turaroud area for the ext departure time slot. For each time value that the algorithm assigs to the timetable, the algorithm also selects a route sequece that will lead the trai to arrive/depart at the desired time. The route sequece with the shortest total rutime that does ot iterfere with ay other previously made route assigmets will be selected first. For each potetial route sequece that is selected, the algorithm also determies how much time the trai should wait at each sigal for sigal clearace. The miimum amout of waitig time will be used. This is equivalet to stretchig the route sequece as show i Figure 5 (the arrows highlight the stretchig actio ad ot the temporal evolutio). these times are scheduled Search forward from T T Search backwards from T2 T2 T2 departure time must match T2 timetable Fig. 4 The combied backward ad forward search. Achor at desired arrival time Stretch route sequece "Stretchig" a Potetial Route Sequece Fig. 5 Stretchig a route sequece by adjustig the waitig time.

6 Fig. 6 The graphic simulator used to display the schedulig results. If the proposed route sequece geerated by the schedulig algorithm does ot coflict with ay other previously assiged routes, the that route will be assiged to the trai. O the other had, if o feasible routes ca be foud, the algorithm will try to adjust the waitig times at the sigals. If o solutio ca be foud, the algorithm the adjusts the arrival/departure times at the statio. If still o solutio ca be foud, the the algorithm tries to assig a differet route sequece to the previous timetable etry. This udoig of previously made assigmets is performed automatically by the backtrackig mechaism provided by costrait programmig. This backtrackig will cotiue util a solutio is foud or whe all possible solutios have bee tried. Oce the timetables ad route sequeces have bee geerated, graphic simulatio software is used to display ad visualise the resultig schedule (see Figure 6). The costrait-based approach, proposed i this paper, was able to geerate timetables ad route sequeces withi a reasoable time because of costrait propagatio. Costrait propagatio elimiated ivalid choices before they ca be selected for search. I the case of timetable geeratio, routes that coflict with assiged routes will be elimiated ad timetable etries that violated headway requiremets will ot be selected. The costrait-based approach makes use of arc cosistecy [MACK77] ad costrait propagatio [WALT72] to reduce the domai size of each costraied variable before search. Smaller domai size meas smaller ad more focused search space. RESULTS FROM TEST CASES The costrait-based schedulig algorithm was tested o may test cases. The algorithm was desiged to miimise chages i the desired headway value as much as possible. However, miimisig headway chages i early morig forces the headway at the fial target to fluctuate slightly. The followig table lists test cases that achieved the desired 05-secod headway values, i.e., roughly 34 trais per hour. Curretly, the trais operate at roughly 2 or 3-secod headway values. Also listed is the umber of trais that were dispatched from 6am to 8:40am, the umber of trais that had a headway value of exactly 05 or 06 secods, ad the average deviatio from 05 secods withi the 7:40am to 8:40am peak. ADVANTAGES OF CONSTRAINT PROPAGATION To illustrate the problem complexity, for the 3 hours of morig trai dispatch there are aroud 70 choice poits for just the timetable geeratio - each with potetially over 00 possible values. For route selectio, there are a total of 30 route sequece combiatios withi the iterlockig ad headshut area, each with a average of 4 routes, each route with possibly 60 timig variatios. The total complexity of just the morig dispatchig is far too much for ay rule-based approach. Table. Summary of Testig Results Test Case No Total trais sice 6am No. of 05 or 06 trais Avg. Deviatio from 05 sec As a example, Test Case 5 was produced with the followig iput headway table.

7 Table 2. The Headway Table Used for Test Case 5 Statio Start Ed Headway No Trais TSW2 6:00 6: TSW2 6:30 7: TSW2 7:00 7: TSW2 7:25 7: TSW2 7:40 8: Out of these test cases, the best result was obtaied i Test Case 2 where trais were dispatched for the 05-secod headway with a average deviatio of oly.4 secods. The curret rule-based systems ca at most dispatch up to 2- secod headway values. SYSTEM IMPLEMENTATION The object-orieted costrait-based [LEPA93] allocatio system was implemeted i C++ usig the ILOG Solver class library [PUGE94a] ad the RTL Schedulig Framework developed by Resource Techologies Limited [CHUN96a, CHUN96b]. The graphic user iterface that simulates the geerated schedule was developed usig C++ graphic compoets provided by ILOG Views. The system was developed usig platform idepedet codig ad ca execute withi Widows 95/NT or Uix eviromet. CONCLUSIONS This paper documets our research i modellig trai timetable ad route geeratio as a costraitsatisfactio problem. The costrait-based schedulig algorithm was tested usig data from oe of the busiest subway systems i the world. The results showed that the schedulig algorithm was able to geerate timetables ad routes that had a higher service level tha that was previously possible with a rule-based approach. ACKNOWLEDGEMENTS The author would like to thak the Hog Kog Mass Trasit Railway Corporatio for the cooperatio received ad for makig actual data available for this research. Part of this research was performed i cooperatio with Resource Techologies Limited ( i Hog Kog ad with assistace from ILOG ( i Sigapore. REFERENCES [CHUN96a] H.W. Chu, K.H. Pag, ad N. Lam, Cotaier Vessel Berth Allocatio with ILOG SOLVER, The Secod Iteratioal ILOG SOLVER User Coferece, Paris, July, 996. [CHUN96b] H.W. Chu, M.P. Ng, ad N. Lam, Rosterig of Equipmet Operators i a Cotaier Yard, The Secod Iteratioal ILOG SOLVER User Coferece, Paris, July, 996. [COLM90] A. Colmerauer, A Itroductio to Prolog III, Commuicatios of the ACM, 33(7), pp.69-90, 990. [DUNC94] T. Duca, Itelliget Vehicle Schedulig: Experieces with a Costrait-based Approach, ILOG Techical Report [KUMA92] V. Kumar, Algorithms for Costrait Satisfactio Problems: A Survey, I AI Magazie, 3(), pp.32-44, 992. [LEPA93] C. Le Pape, Usig Object-Orieted Costrait Programmig Tools to Implemet Flexible Easy-to-use Schedulig Systems, I Proceedigs of the NSF Workshop o Itelliget, Dyamic Schedulig for Maufacturig, Cocoa Beach, Florida, 993. [MACK77] A.K. Mackworth, Cosistecy i Networks of Relatios, I Artificial Itelligece, 8, pp.99-8, 977. [PUGE94a] J.-F. Puget, A C++ Implemetatio of CLP, I ILOG Solver Collected Papers, ILOG SA, Frace, 994. [PUGE94b] J.-F. Puget, Object-Orieted Costrait Programmig for Trasportatio Problems, I ILOG Solver Collected Papers, ILOG SA, Frace, 994. [SISK93] J.M. Siskid ad D.A. McAllester, Nodetermiistic Lisp as a Substrate for Costrait Logic Programmig, I Proceedigs of the Eleveth Natioal Coferece o Artificial Itelligece, Washigto, DC, pp.33-38, July, 993.

8 [STEE80] G.L. Steele Jr., The Defiitio ad Implemetatio of a Computer Programmig Laguage Based o Costraits, Ph.D. Thesis, MIT, 980. [VANH89] P. Va Heteryck, Costrait Satisfactio i Logic Programmig, MIT Press, 989. [WALT72] D.L. Waltz, Uderstadig Lie Drawigs of Scees with Shadows, I The Psychology of Computer Visio, McGraw-Hill, pp.9-9, 975.

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