S S symmetry Article A Multiobjective Integer Linear Programming Model for the Cross-Track Line Planning Problem in the Chinese High-Speed Railway Network Peiwen Han, Lei Nie *, Huiling Fu, Yantao Gong and Gang Wang School of Traffic and Transportation, Beijing Jiaotong University, Beijing 100044, China; [email protected] (P.H.); [email protected] (H.F.); [email protected] (Y.G.); [email protected] (G.W.) * Correspondence: [email protected]; Tel.: +86-010-51688173 Received: 1 April 2019; Accepted: 9 May 2019; Published: 15 May 2019 Abstract: In China, cross-track high-speed trains (CTHSTs) play an important role in railway passenger transportation, with an increasing number of cross-track passengers sourced from the expansion of high-speed railway (HSR) network. The CTHST generally has long travel times, so running CTHSTs is not beneficial for train rescheduling work and plan’s periodicity in the periodic operation context. Thus, the main challenge in cross-track line planning is looking for a symmetry point between passenger transportation and disadvantages of running CTHSTs, which are two conflicting aspects. In this study, we developed a multiobjective integer programming model to produce a balanced cross-track line plan by combining individual-track high-speed trains (ITHSTs) into CTHSTs, which is a discrete optimization problem. This strikes a balance among four goals: the periodicity of the line plan, CTHST quantity, CTHST mileage, and CTHST stops in the context of periodic operation, while satisfying the constraints of passenger demand and the number of available ITHSTs. Numerical experiments are conducted based on a real-world network and optimal solutions were quickly obtained. We analyzed impacts of each goal and parameter on the result and influencing factors of computation. Comparisons with existing methods and real-life plans were also presented to show improvements made by proposed model. Keywords: China’s high-speed railway; cross-track passenger; cross-track line plan; multiobjective integer linear programming 1. Introduction Since the late 2000s, China’s high-speed railway (HSR) has experienced large growth, and its network has expanded from a single HSR line to multiple HSR lines forming an intersected network. By the end of 2015, China completed the construction of the HSR network, which is mainly formed by Four Vertical and Four Horizontal HSR lines (FVFH-HSR, i.e., four north–south HSR lines and four east–west HSR lines), with a total mileage of 19,000 km. By 2020, this mileage will exceed 30,000 km. With the quick expansion of the HSR network, the number of cross-track passengers, who pass through more than one individual railway line along their trips (e.g., passenger 3 in Figure1a, whose origin station is on Track A–B and destination station is on Track C–D), is rapidly increasing. In this case, determining how to rationally create a reasonable line plan (i.e., a set of train paths together with their frequencies) to serve this important group of passengers is urgently required, but this is a complex problem since there are many aspects needed to be considered in this issue: passenger demand, periodicity of the line plan, train scheduling/rescheduling work, and so on. A good cross-track line plan needs to strike a balance among all aspects. Symmetry 2019, 11, 670; doi:10.3390/sym11050670 www.mdpi.com/journal/symmetry Symmetry 2019, 11, 670 2 of 36 SymmetrySymmetry 2019 2019, 11, 11, 670, 670 22 of of 36 36 Railway Stations Railway Stations RailwayRailway Stations Stations 1 C 1 1 C 1 TracksTracks C TracksTracks C PassengersPassengers TrainsTrains PassengerPassenger Routes Routes TrainTrain Paths Paths 3 3 3 3 EE EE AA BB AA BB 2 2 2 2 DD DD (a()a ) ((bb)) FigureFigureFigure 1. 1. 1.The The The illustrationillustration illustration of of ( a()a )the the individual-track individual-track individual-track passenger passenger and and and cross-track cross-track cross-track passenger passenger passenger and and and ( (bb) () thebthe) the individual-trackindividual-trackindividual-track high-speedhigh-speed high-speed train train (ITHST) (ITHST) and and and cross-track cross-trackcross-track high-speed high-speed train train train (CTHST). (CTHST). (CTHST). AA railwayA railway railway networknetwork network isis is composed composed of of of many manymany tracks trackstracks (in (in other other words, words, words, railway railway railway lines). lines). lines). Figure Figure Figure 22 shows2shows shows aa HSR aHSR HSR network network network whichwhich which isis is thethe the mainmain main framework framework of ofof Ch China’sChina’sina’s HSR HSR network network network inin in thethe the yearyear year ofof of 2015.2015. 2015. InIn Inthethe the figure,figure,figure, therethere there areare are eighteight eight trackstracks tracks interlacing interlacing at atat intermediate intermediateintermediate stations stations (e.g.,(e.g., (e.g., tracktrack track BJW—SZNBJW—SZN BJW—SZN andand and tracktrack track BaJS—XZE)BaJS—XZE)BaJS—XZE) or or or connectingconnecting connecting at at terminal terminal stations stations stations (e.g., (e.g.,(e.g., track track BJS—SHHQ BJS—SHHQ and and and track track track SHHQ—CSS). SHHQ—CSS). SHHQ—CSS). As As Asfor for for thethethe passengers passengers passengers travellingtravelling travelling inin in this this network, network, there therethere are areare two two kinds kinds of of of passengers passengers passengers according according according to to tothe the the number number number ofof tracksof tracks tracks covered covered covered byby by passenger’spassenger’s passenger’s travel: travel: the thethe individual-track individual-trackindividual-track passengers passengers passengers and and and cross-track cross-track cross-track passengers.passengers. passengers. TheTheThe former former former one one one representsrepresents represents passengerspassengers passengers traveling traveling on onon only onlyonly one one track track track (in (in (in other other other words words words a a railway arailway railway line) line) line) and and and latter one denotes passengers traveling on two or more tracks. According to the similar classification latterlatter one one denotes denotes passengerspassengers traveling on on two two or or mo morere tracks. tracks. According According to tothe the similar similar classification classification rule,rule, trains trains can can be be classified classified by by individual-track individual-track tr trainain (ITT) (ITT) and and cross-track cross-track train train (CTT). (CTT). An An illustrative illustrative rule, trains can be classified by individual-track train (ITT) and cross-track train (CTT). An illustrative exampleexample is is given given in in Figure Figure 1. 1. There There are are five five railway railway stations stations labeled labeled A, A, B, B, C, C, D, D, and and E. E. Track Track A–B A–B and and example is given in Figure1. There are five railway stations labeled A, B, C, D, and E. Track A–B and TrackTrack C–D C–D are are two two tracks tracks interlacing interlacing at at Station Station E E (which (which is is named named the the crossing crossing station station in in this this paper). paper). Track C–D are two tracks interlacing at Station E (which is named the crossing station in this paper). In InIn this this simple simple network, network, trains trains 1 1and and 2 2are are ITTs ITTs running running on on only only one one track,track, whilewhile traintrain 33 isis aa CTTCTT this simple network, trains 1 and 2 are ITTs running on only one track, while train 3 is a CTT because it becausebecause it itruns runs on on both both track track A–B A–B and and C–D. C–D. By By the the way, way, the the ITTs ITTs and and CTTsCTTs runningrunning onon HSRHSR areare runs on both track A–B and C–D. By the way, the ITTs and CTTs running on HSR are individual-track individual-trackindividual-track high-speedhigh-speed trainstrains (ITHSTs)(ITHSTs) andand cross-trackcross-track high-speedhigh-speed trainstrains (CTHSTs)(CTHSTs) high-speed trains (ITHSTs) and cross-track high-speed trains (CTHSTs) respectively. respectively.respectively. BJWBJW BJSBJS TJTJ SJZSJZ JNW QD TerminalTerminal Stations Stations of of Tracks Tracks JNW QD Important Intermediate Stations Important Intermediate Stations ZZE ZZE BaJS XAN XZE TracksTracks BaJS XAN XZE BBS BBS NJSNJS HF CDE HF CDE WH WH CQN SHHQ CQN SHHQ HZE HZE NB CSS NB CSS NCW KMS GYN NCW KMS GYN FZS FZS GZS GZS SZN XMN SZN XMN Figure 2. The high-speed railway (HSR) network for cross-track line planning among multiple tracks. FigureFigure 2. 2. TheThe high-speedhigh-speed railway (HSR) (HSR) network network for for cross-track cross-track line line planning planning among among multiple multiple tracks. tracks. The station names are presented in an abbreviated manner. TheThe station station namesnames areare presentedpresented in an abbreviated manner. manner. Symmetry 2019, 11, 670 3 of 36 Symmetry 2019, 11, 670 3 of 36 Correspondingly, therethere areare twotwo kindskinds ofof waysways toto transporttransport cross-trackcross-track passengers.passengers. The firstfirst one is transfertransfer pattern. In In this this way, way, passengers need to take more than one ITTs viavia transferstransfers at crossingcrossing stations (e.g., Station B inin FigureFigure3 3).). AndAnd thethe otherother oneone isis directdirect patternpattern providedprovided byby CTHSTs. CTHSTs. ByBy taking CTHSTs,CTHSTs, cross-track cross-track passengers passengers do do not not need need to maketo make any any transfer transfer during during their their trips. trips. These These two patternstwo patterns are shown are shown in Figure in Figure3. 3. Station A Track 1Station B Station B Track 2 Station C transfer Transfer Pattern Train a Train b by ITTs Direct Pattern Train c by a CTHST Figure 3. TheThe illustration illustration of of a atransfer transfer pa patternttern by by individual-track individual-track trains trains (ITTs) (ITTs) and and a direct a direct pattern pattern by byCTHSTs.