Metro Timetabling for Time-Varying Passenger Demand and...

Research ArticleMetro Timetabling for Time-Varying PassengerDemand and Congestion at Stations

Keping Li1 Hangfei Huang 1 and Paul Schonfeld2

1State Key Laboratory of Rail Traffic Control and Safety Beijing Jiaotong University Beijing 100044 China2Department of Civil amp Environmental Engineering University of Maryland College Park MD 20742 USA

Correspondence should be addressed to Hangfei Huang 14114200bjtueducn

Received 7 November 2017 Revised 4 March 2018 Accepted 11 March 2018 Published 3 July 2018

Academic Editor Giulio E Cantarella

Copyright copy 2018 Keping Li et alThis is an open access article distributed under the Creative CommonsAttribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

For the train timetabling problem (TTP) in a metro system the operator-oriented and passenger-oriented objectives are bothimportant but partly conflictingThis paper aims to minimize both objectives by considering frequency (in the line planning stage)and train cost (in the vehicle scheduling stage) Time-varying passenger demand and train capacity are considered in a nonsmoothnonconvex programming model which is transformed into a mixed integer programming model with a discrete time-space graph(DTSG) A novel dwell time determining process considering congestion at stations is proposed which turns the dwell times intodependent variables In the solution approach we decompose the TTP into a subproblem for optimizing segment travel times (OST)and a subproblem for optimizing departure headways from the shunting yard (OH) Branch-and-bound and frequency determiningalgorithms are designed to solve OST A novel rolling optimization algorithm is designed to solve OHThe numerical experimentsinclude case studies on a short metro line and Beijing Metro Line 4 as well as sensitivity analyses The results demonstrate thepredictive ability of the model verify the effectiveness and efficiency of the proposed approach and provide practical insights fordifferent scenarios which can be used for decision-making support in daily operations

1 Introduction

With increasing concerns about urban congestion and cli-mate change urban metro rail transportation receives in-creasing attention due to its high capacity punctuality andsustainability Metro timetabling is the problem of assigningprecise utilization times for infrastructure resources to everytrain in the metro system [1] In large cities a metro systemis often the busiest public transportation system To meettravel demand and passenger satisfaction a timetable shouldbe as compact and flexible as possible Meanwhile sincemany metro systems need government subsidies to coveroperating expenses the planners focus more on operatingcost than on passenger factors when designing a timetable[2] This motivates some recent studies to combine thesetwo conflicting objectives in the train timetabling problem[3 4]

Two types of timetables are used in metro systems cyclicand noncyclic In a cyclic timetable the departure times oftrains are scheduled in equally spaced cycles (eg a half hour

cycle) In a railway system the cyclic timetables are assumedto be preferred by both operators and passengers becausesuch timetables are easy to operate and remember Howeverin a metro system passengers usually arrive randomly atstations andwait for the next available train without checkingprecise timetables beforehand Besides as pointed out byRobenek et al [5] a cyclic timetable provides an inefficientoperation as there is a mismatch between the supply (deter-mined by the timetables) and the demand (characterized bythe time-varying demand) On the other hand a noncyclictimetable imposes no special rules on the departure timesof trains [6] The noncyclic timetables are more flexibleregarding time-varying passenger demand especially whenthe demand is large [2] Thus the noncyclic timetablingproblem is worth exploring [2 3 7]

The planning process in public transportation can be gen-erally split into three stages line planning timetabling andvehicle scheduling [8]The line planning stage determines theroutestation layouts and frequency Timetabling is based onthe output of line planning afterwhich the vehicle scheduling

HindawiJournal of Advanced TransportationVolume 2018 Article ID 3690603 26 pageshttpsdoiorg10115520183690603

2 Journal of Advanced Transportation

can be designed It is pointed out by Schobel [8] that goingthrough all these stages sequentially (ie independently)leads to unsatisfactory solutions Since they are interrelatedimportant factors andobjectives at the line planning stage andthe vehicle scheduling stage should be accounted for in thetimetabling problem

This paper proposes a rolling optimization algorithmto obtain noncyclic timetables considering time-varyingpassenger demand and the effects of congestion at sta-tions It integrates the objectives in the line planning (fre-quency) timetabling (conflicting objectives including pas-senger waitin-vehicle time and energy) and vehicle schedul-ing (train cost) The main contributions of this paper are asfollows

(1) This paper specifically models the dynamic evolutionof passenger loads on trains at each station by consideringpassenger arrival rates limited train capacity and actualpassenger alightingboarding rates associated with conges-tion This is an extension of existing studies [9] ParticularlyNiu and Zhou [10] and Niu et al [11] propose the conceptof ldquoeffective loading timerdquo that represents the actual timeinterval within which the arriving passengers can boarda train The detailed boarding and alighting processes arenot discussed in their model Wang et al [3] set a lowerbound (which accounts for details including the number ofboardingalighting passengers and their boardingalightingrates) for dwell times but consider the dwell times as vari-ables to be optimized rather than parameters that dependon arrivaldeparture times and passenger demand Impactsof gradually increasing passenger demand are analyzed inRobenek et al [2] however inmodelling they donot considerthe effects of passenger congestion

(2) This paper integrates objectives from different plan-ning stages in the train timetabling problem for which solu-tion approaches are scarce [8] Particularly Schobel [8] con-siders the line planning timetabling and vehicle schedulingin an integratedwaywith passenger- and cost-oriented objec-tives However since his eigenmodel is general and resultingapproaches are generic important objectives described inthis paper are not considered in his work that is passengerwait time dwell time and train capacity These objectives areincluded in [2 5] whereas the frequency and dwell times arefixed in their studies Besides energy is not considered in[2 5]

(3) This paper proposes a novel and effective solutionapproach which decomposes the master timetabling modelinto two subproblems The first subproblem is solved bybranch-and-bound and frequency determining algorithmsand the second one is solved by a rolling optimizationalgorithm which optimizes the departure headways fromthe shunting yard (which are equivalent to arrival times oftrains at their first station) and considers interdependentvariables in each rolling stepThe design of decision variablesin this paper is similar to [2 5] but the models and solutionapproaches are different Compared to Wang et al [3] whichaddresses the similar timetabling problem with traditionalapproaches such as sequential quadratic programming anda genetic algorithm the approach proposed in this paper

performs better computationally in terms of run time andsolution quality

The remainder of this paper is as follows Section 2reviews relevant studies on the train timetabling problem andsummarizes the research gap in the literature Section 3 firstdescribes the problem introduces formulation of differentcost functions and constructs the train timetabling modelThen the discrete time-space graph is proposed to transformthe model into a mixed integer program (MIP) Section 4discusses the solution approach for the optimization modelSection 5 presents different numerical experiments and sen-sitivity analyses on a short metro line and on Beijing MetroLine 4 to demonstrate the predictive ability of the modelas well as the effectiveness and efficiency of the proposedapproach Section 6 summarizes the works done in this paperand presents the limitations and potential topics in futurestudies

2 Literature Review

The train timetabling problem (TTP) aims to schedule trainsto transport passengers (or goods) without conflicts byspecifying train arrival and departure times at stations Itis interrelated with other planning stages that is the lineplanning stage and the vehicle scheduling stage We organizethe related literature from the perspectives of operator-oriented objectives passenger-oriented objectives and theirintegration

21 Operator-Oriented Objectives In the literature operator-oriented objectives can be found only for the noncyclicversion of the TTP [2] Most studies focus on the operationindicators for example energy [17 18] train delay [19ndash21]and train travel time [22ndash24]

The total train delay and train travel time are usuallyconsidered in the TTP for railway networks which aims tofind a feasible timetable by minimizing the profit loss result-ing from changes to the idealplanned timetable Energyis the focus and main cost for metro system operatorsMost energy is consumed in train operations [25 26] andthus obtaining an energy-efficient train timetable receivesconsiderable attention [17 18 27] Note that energy is affectedby the travel time in the segment and the segment travel timeis determined by the arrival and departure times at adjacentstations that is the timetable Hence the energy-efficienttrain timetabling problem may be extended to include otherobjectives For example Yang et al [23] consider both energyand train travel time as optimization objectives Some recentstudies also account for regenerative braking Yang et al[28] develop a scheduling approach to coordinate arrivalsand departures of trains within the same electricity supplyzones thereby effectively recovering the regenerative energyGenerally these studies focus solely on the timetabling stagewhere frequency (in line planning) is fixed and train cost (in-vehicle scheduling) is not considered

The efficient use of railway rolling stock (vehicle schedul-ing) is an important objective pursued by a railway agencyor company because of intensive capital investment in rollingstock [29] To this end Lai et al [29] develop an optimization

Journal of Advanced Transportation 3

model to improve the efficiency of rolling stock usage consid-ering necessary regulations and practical constraints wherea hybrid heuristic process is designed to improve solutionquality and efficiency Haahr et al [27] use CPLEX and acolumn and row generation approach to assign rolling stockunits to timetable services in passenger railways preparedaily schedules and check their real-time applicability bytesting different disruption scenarios These vehicle schedul-ing studies require the trips as input data Based on Schobel[8] in a metro system every trip describes the operation ofa train between the start and end time of the line at its firstand its last stations (given from the timetable) The objectivefunction then aims at minimizing the number of trains andthe costs for their movements

Particularly Schmid and Ehmke [30] and Schobel [8]demonstrate that the integration of timetabling and vehiclescheduling is more beneficial than the sequential plan-ning process However the studies on the integration oftimetabling and vehicle scheduling usually adopt relativelygeneral and generic models and the TTP model is oftensimplified [31] For example Cadarso and Marin [32] focuson shunting operations and their timetable is calculated byreadjusting frequencies whereas specific details such as thedwell times and segment travel times are not consideredThismotivates us to consider train cost in the timetabling stagewhile focusing on the specified TTP formulation

In addition the conflict-free timetable is also pursued bythe operators [33 34] It is worthmentioning that the discretetime-space graph proposed in Caprara et al [6] is a directedmultigraph in which nodes correspond to arrivalsdeparturesat a certain station and at a given time stamp This graph iswidely used to formulate the TTP and derive different integerprogramming models that correspond to specific objectivesFor example Caprara et al [6] use the graph to derive aninteger linear programming (ILP) model with Lagrangianrelaxation which is embedded within a heuristic algorithmCacchiani et al [12] use the graph to formulate an ILP withlinear programming (LP) relaxation Cacchiani et al [35] usethe time-space graph-based LP relaxation of an ILP to derivea dual bound in the TTP for a set of stations in an urban areainterconnected by tracks thus aiming to resolve the conflictsand evaluate the capacity saturation

22 Passenger-Oriented Objectives For a metro system pas-senger-oriented timetables that consider reliability andreduction of passenger time are most desirable [11] In somestudies the passenger demand is considered stable [36]Assuming that passengers prefer easily memorable timeta-bles such timetables are usually cyclic They are designedon the basis of a period event scheduling problem (PESP)aiming to minimize passenger travel and wait time [15 37]and to maximize network stability [38] However as men-tioned above metro passengers arrive randomly and do notremember the timetable Besides cyclic timetables are lessflexible than noncyclic ones in accounting for the passengerdemand [2]

Recent studies consider time-varying demands due to thegrowing concerns for service level and congestion at stationsSuch timetables are noncyclic with objectives of improving

passenger satisfaction [3 7 10] Particularly Barrena et al [7]propose two nonlinear programming (NP) formulations togeneralize noncyclic train timetables on a single line whichare solved by a fast adaptive large neighborhood searchmeta-heuristic The objective is to minimize passenger wait timesat stations Niu et al [11] construct mathematically rigorousand algorithmically tractable nonlinear mixed integer pro-gramming (MIP) models for both real-time scheduling andmedium-term planning applications to jointly synchronizeeffective passenger loading time windows and train arrivaland departure times at each station Their work aims tominimize the total waiting times of passengers at stationsRobenek et al [5] define a timetable as a set of departuretimes of every train from its origin station on every lineand consider four attributes in passenger satisfaction the in-vehicle-time the waiting time at transfers the number oftransfers and the schedule passenger delay They describethe hybrid timetables through additional constraints that areimposed on the original passenger centric train timetablingproblem [2] so that the passengers would obtain the samelevel of service as a cyclic timetable with more flexibility Aspecifically defined simulated annealing heuristic is proposedto solve the problem Luan et al [16] integrate the TTPand preventive maintenance time slots (PMTSs) planningproblem on a general railway network They propose anMILP model to jointly optimize train routes orders andpassing times at each station and PMTSs The objective isto minimize the sum of the absolute arrival time deviationsof real trains at destinations between the ideal and actualtimetables A Lagrangian-based label correcting algorithmis designed for solving the time-dependent least cost pathproblem

In addition the delay management (DM) problem deter-mines whether trains should wait for a delayed feeder train orshould depart on time while considering passenger-orientedobjectives [13] Some recent studies integrate themacroscopicDM with the microscopic TTP [39] Particularly Dollevoetet al [14] formulate an integer programming (IP) modeland propose an iterative optimization approach to solvesuch a bilevel problem that the macroscopic level is thedelay management and the microscopic level is the traintimetabling The optimization approach repeats a processthat uses DM to find a solution and uses TTP to validateit until a feasible solution is found It should be noted thatthe graph-based models are adopted in their work whichsimplify the formulations the TTP is formulated with thealternative graph [40] and the DM is formulated with anevent-activity network [15]

23 Integrated Objectives Since the operator- and passenger-oriented objectives are partly conflicting focusing on onlyone side would yield undesirable solutions for example thebest possible service for passengers may also be the mostcostly alternative for the operator [2]ThusmanyTTP studiesintegrate these objectives jointly [22 41]

Wang et al [3] aim to minimize passenger travel timeand energy in their nonsmooth nonconvex programming(NSNCP) model They propose an event-driven modelthat involves arrival and departure events and passenger


arrival rates change events and use nonlinear programmingapproaches and evolutionary algorithms to solve the prob-lem

Qi et al [42] focus on a line planning problem withbudget constraints and evaluate the station layout with traintimetable indicatorsThe objectives include constructing costfor additional tracks total travel time and network capacityA mixed integer linear programming (MILP) model is for-mulated to address the TTP and then a bilevel programming(BP) model is designed to address the integrated problemBothCPLEX and a local search-based heuristic are developedto solve the models

Robenek et al [2] model the PCTTP as an MILPmodel with objectives of maximizing the operatorrsquos profitwhile maintaining passenger satisfaction The model usesthe output of the line planning problem as the input of thetrain timetabling problem to address the gap found betweenthem CPLEX is used to construct a Pareto Frontier of thecontradictory objectives

Li et al [43] combine dynamic train regulation andpassenger flow control to minimize the timetable and theheadway deviations for metro lines thereby reducing theoperator profit loss and the passenger delay The modelpredictive control and a quadratic programming algorithmare proposed to solve the problem and stability analysis isconducted to verify the system convergence

However the above studies do not consider the impactsof frequency Schobel [44] reviews some approaches tomodeland solve the line planning problem which demonstratethe impacts of frequency to both operator- and passenger-oriented objectives Particularly Samanta and Jha [45] con-sider different objectives in the line planning stage whichinclude minimizing user cost operator cost and locationcost and maximizing the ridership or the service coverageof the line The objectives are separately formulated with NPmodels and a GA is developed to optimize the objectivefunctions subject to constraints on the number and spacing ofstations Fu et al [46] propose a BP model to determine linefrequencies and the stopping patterns for a mix of fast andslow train lines with the objectives of minimizing passengertravel time and the number of transfers and maximizingtrain capacity occupancy The heuristics solve the problemby combining additional conflicting demand-supply factorsCascetta and Coppola [47] specify both the frequency-based assignment models and the timetable-based modelsin the demand forecasting to analyze their differences inmodal split estimates and flows on individual trains Theresults show that when passenger demand is time-varyingor the timetable is noncyclic the difference is significantThus the integration of frequency and timetabling is worthinvestigating

24 Research Gap Table 1 highlights some publicationsrelated to the TTP All the graph-based models use integerprograms including the model in this paper Althoughdifferent objectives are considered very few studies integratethe objectives in all three stages Particularly Wang et al [3]only consider the objectives in one stage that is timetablingstage Robenek et al [2] integrate passenger satisfaction and

operating profit (including train cost) but do not consider fre-quency and energy Schobel [8] takes into account objectivesin all stages with an iterating framework however crucialfactors such as passenger wait time dwell time and traincapacity are missing Kroon et al [15] integrate the objectivesin the TTP and vehicle scheduling but omit energy

Apart from the studies focusing on the delay manage-ment most TTP-related studies optimize arrival and depar-ture times of trains at each station that is the dwell times areoptimized in the solution approach However the dwell timesare heavily affected by the number of boarding and alightingpassengers and their boarding and alighting rates associatedwith congestion at stations Thus they should be consideredas parameters that are dependent on other decision variablesthat is the departure times from the first station [2 5]and the segment travel times In this paper we accountfor these characteristics and introduce specific equations todetermine the dwell times in different scenarios Such aprocess for determining dwell times has not been found in theliterature

Finally heuristics and exact approaches are designedto find solutions based on the properties of the modelsParticularly rolling horizon [3] and iterative optimization[8 14] are used to solve problems that have similar propertiesto our proposedmodelHowever the boundary constraints ofthe rolling horizon limit the solution flexibility of our modelthe iterative optimization considers different stageswith sameimportance while this paper focuses only on the timetablingstage Besides the algorithms used in the rolling horizon anditerative optimization are relatively generic [3 8] Thus amore specific algorithm is desirable for better solutions Inthis regard we develop a novel rolling optimization approachto solve the model which combines the benefits of rollinghorizon and iterative optimization

3 Model Formulation

In this section we present a mixed integer programmingformulation for the TTP It is a multiobjective optimizationproblem The weighted-sum scalarization of different objec-tive functions in terms of generalized monetary cost is usedto resolve the tradeoffs among them [8]

31 Problem Formulation Table 2 summarizes the parame-ters and variables used in this paper Some are also explainedfurther in the text The modelrsquos simplifying assumptions arelisted below and also explained in the problem description

Assumption 1 Within one period trains travel through eachsegment with the same speed profile that is the travel timesfor different trains in a segment are equal

Assumption 2 We assume that the last train arrives at the firststation at the end time of the period to ensure that no late-arriving passengers will be left behind

Assumption 3 Passenger arrival rate 120582119894(119905) should not exceedtotal passenger boarding rate 120577119887 sdot119899119889 If 120582119894(119905) gt 120577119887 sdot119899119889 the queueof waiting passengers increases even if a train is dwelling at


Table 1 Summary of publications related to the TTP

Work Model Objectives Decision variable Solution approachCaprara et al [6] ILP Maximize profit Arrivaldeparture times Lagrangian-based heuristic

Cacchiani et al [12] ILP Maximize profit A full timetable of a trainColumn generationdecomposition and

heuristic

Dollevoet et al [13] IP Minimize total passenger delay If a connection ismaintainedused Modified Dijkstra

Niu and Zhou [10] IP Minimize passenger wait time Arrivaldeparture times GABarrena et al [7] LP Minimize passenger wait time Arrivaldeparture times Branch and bound

Dollevoet et al [14] IPMinimize (1) passenger arrivaltimes (2)maximize consecutive

delay

Connection andarrivaldeparture times

Iterative optimizationapproach

Kroon et al [15] MIPMinimize the number of trainspassenger transfer time and total

travel timeArrivaldeparture times CPLEX

Niu et al [11] MIP Minimize passenger wait time Arrivaldeparture times GAMS

Wang et al [3] NSNCP Minimize passenger travel timeand energy Arrivaldeparture times Rolling horizon with SQP

and GA

Robenek et al [2] MILPMaximize operating profit while

maintaining passengersatisfaction

Departure times at firststation CPLEX

Robenek et al [5] MILP Maximize passenger satisfaction Departure times at firststation

Simulated Annealing (SA)heuristic

Schobel [8] EigenmodelPassenger- and operator-orientedobjectives in all three planning

stages

Frequency AD times tripassignment Iterating algorithm

Luan et al [16] MILP Minimize absolute arrival timedeviations AD times route selection Lagrangian relaxation

Yin et al [4] MILP Minimize passenger wait timeand energy AD times train control Lagrangian-based heuristic

This paper MIP Minimize passenger wait andtravel time energy and train cost

Segment travel times andarrival times at the first

station

Branch and bound Rollingoptimization

the station In practice passenger entrance flow controllowast istaken to avoid such a case

119877119890119898119886119903119896lowast At most busy metro stations in China passengerentrance flow control is adopted during peak hours to ensureoperation safety and thus the passenger arrival rate is arestricted number which is expected not to exceed passengerboarding rate Assumption 3 is introduced for consistencywith practical experience

Assumption 4 All passengers in train 119896 who have reachedtheir destination are assumed to exit the train at its arrivaltime of station 119894 that is 1199050119896119894 All waiting passengers at station119894 able to board train 119896 are assumed to board the train at itsdeparture time that is 119905119887119896119894 In other words the number of in-vehicle passengers 119875V

119896119905 changes only at the arrivaldeparturetimes of train 119896 at stations

This paper focuses on a bidirectional metro line with asingle track as shown in Figure 1 Properties of such a metro

line are described in [48]The timetabling problem is studiedwithin a given periodThe time-varying passenger demand isgiven as input and a series of trains (whose frequency is to beoptimized) are scheduled to transport the passengers

The arrival times at the first station are decision variablesthat must be optimized The departure times at the first sta-tion are determined by the dwell times As mentioned earlierthis paper obtains the dwell times based on trainsrsquo conditionswhen they arrive at stations In this regard optimizing arrivaltimes at the first station is equivalent to optimizing departuretimes at the first station as in [2 5] For the second station thearrival times are determined by the segment travel time in thefirst segment Practically for the metro operators uniformsegment travel times are preferredwithin one period becausethey not only make the system more stable and easier tomanage but also simplify synchronized passenger transferswith connecting transit lines Assumption 1 is introducedto consider this preference Thus within one period thesegment travel time in each segment needs to be optimizedonly once (for all trains within this period)The speed profile


Table 2 Parameters and variables

Name Description Units PurposeI Set of stations - SetK Set of trains - Set

U119894Set of time stamps in which trains can arrive at

station 119894 - Set

W119894Set of time stamps in which trains departure from


120579 Shunting yard of the metro line - Symbol119894 119895 Station index - Symbol119868 Total number of stations - Symbol119896 Train index - Symbol

119870 Total number of trains within a givenperiodFrequency - Symbol

119905 Time index - Symbol119906119896 Arrival node of train 119896 - Symbol119908119896 Departure node of train 119896 - Symbol120587(119906119896) Time stamp associated with node 119906119896 (or 119908119896) - Symbol

Δ(119906119896 1199081198961015840 ) Time duration between two nodes 119906119896 and 1199081198961015840(119896 = 1198961015840 is allowed) - Symbol

119877119896 Path of train 119896 - Symbol119879 Time duration - Symbol

lfloor119886rfloor The largest integer that is smaller than realnumber 119886 - Symbol

lceil119886rceil The smallest integer that is larger than realnumber 119886 - Symbol997888

b A vector (represented by any bold alphabet) - Symbol

MAmatrix (represented by any double-struck

alphabet) - Symbol

M119886119887119909119910

A sub-matrix whose elements correspond to thosein matrixM from rows 119886 to 119887 and from columns

from 119909 to 119910 - Symbol

M Sum of all (numerical) elements in matrixM - Symbol119888119890 Average energy cost yenKwh Parameter119888119900 Average train operating cost yen (trainsdotkm) Parameter119888pt Average passenger travel time cost yen(passengersdothour) Parameter119888pw Average passenger wait time cost yen(passengersdothour) Parameter119888tra Average capital cost of one train yen(trainsdothour) Parameter119862total Total cost of the whole system yen Parameter119862119890total Total cost of energy consumption yen Parameter119862119901total Total cost of passenger time yen Parameter119862tratotal Total cost of train (rolling stock) yen Parameter119889119894 Maximum allowable dwell time at station 119894 Second Parameter119889119894 Minimum allowable dwell time at station 119894 Second Parameter

119890119894 Energy consumption per unit mass for one trainin segment 119894 KwhKg Parameter

119891119905 Tractive force per unit mass ms2 Parameter119891119903 Running resistance per unit mass ms2 Parameter119865119904 Fleet size (the number of trains) Trains Parameterℎmin Minimum headway Second Parameter119872119896 Net weight of a train Kgtrain Parameter119872119901 Average weight of a passenger Kgpassenger Parameter119899119889 Number of doors per train - Parameter119873119896 Capacity of train 119896 Passengers Parameter


Table 2 Continued

Name Description Units Purpose119875119886119896119894 Number of passengers alighting train 119896 at station 119894 Passengers Parameter119875119887119896119894 Number of passengers boarding train 119896 at station 119894 Passengers Parameter

119875119887119896(119894119895) Number of passengers boarding train 119896 at station 119894with destination 119895 Passengers Parameter

119875119894(1199051 1199052) Number of passengers arriving at station 119894 during[1199051 1199052] Passengers Parameter

119875(119894119895) (1199051 1199052) Number of passengers arriving at station 119894 withdestination 119895 during [1199051 1199052] Passengers Parameter

119875119908119894119905 Number of passengers waiting at station 119894 at time 119905 Passengers Parameter119875V119896119905 Number of passengers in train 119896 at time 119905 Passengers Parameter119904119897(119894) Length of segment 119894 (between station 119894 and 119894 + 1) Meter Parameter119904max119905 (119894) Maximum segment travel time for segment 119894 Second Parameter119904min119905 (119894) Minimum segment travel time for segment 119894 Second Parameter1199050119896119894 Arrival time of train 119896 at station 119894 Second Parameter119905119886119896119894 Alighting end time for train 119896 at station 119894 Second Parameter

119905119887119896119894 Boarding end time for train 119896 at station119894Departure time Second Parameter

119905119901end End time of passenger demand for a period Second Parameter119905119905end End time of a complete timetable for a period Second Parameter

119879119896roundRound trip time (total travel time from the

departure from the shunting yard to the arrival atthe shunting yard) of train 119896 Second Parameter

119879pt Total passenger travel time Second Parameter119879pw Total passenger wait time Second Parameter119879turn Turnaround time Second ParameterV119896(119905) Speed of train 119896 at time 119905 ms Parameter119909119896(119905) Position of train 119896 at time 119905 Meter Parameter

120591119894119895(119905) Passenger arrival rate at station 119894 with destination119895 at time 119905 Passengerssecond Parameter

120582119894(119905) Passenger arrival rate at station 119894 at time 119905 Passengerssecond Parameter120577119886 Passenger alighting rate Passengerssecond Parameter120577119887 Passenger boarding rate Passengerssecond Parameter120594119909 Parameter for energy approximation - Parameter120594119910 Parameter for energy approximation - Parameter119904119905(119894) Segment travel time of segment 119894 Second Decision11990501198961 Arrival time of train 119896 at the first station Second Decision

Station 1 Station 2 Station I

Station 2I

Platform Platform Platform

Down-direction

Up-direction

Turnaournd TurnaourndSegment 2

Station I + 1Station 2I minus 1

Segment 2I minus 2

Segment I minus 1Segment 1

Segment 2I minus 1 Segment I + 1

middot middot middot



Figure 1 A bidirectional metro line with 2I stations


Time

Space

NotesAvailable arrivaldeparture node Node that a train will visit

Available path with different timePath that a train will travel onShunting yard

t 2t 3t 4t 5t 6t 7t 8t 9t 10t 11t 12t middot middot middot


Station 1 (U1)

Station 1 (W1)

Station 2 (U2)

Station 2 (W2)

Station 3 (U3)

Station 3 (W3)

Station 4 (U4)

Station 4 (W4)







Figure 2 Representation of discrete time-space graph

optimization is outside the scope of this paperThe departuretimes at the second station are also determined by the dwelltime determining process

Analogously for subsequent train arrivals anddeparturesthe arrival times are determined by the previous segmenttravel time and the departure times are determined by thedwell times Thus within a period the arrival times at thefirst station and the segment travel times determine the traintimetable

To represent the TTP a directed multigraph called dis-crete time-space graph (DTSG) is proposed [6] As shown inFigure 2 120579 represents the shunting yard of the metro line andU119894 (W119894) corresponds to the set of time stamps inwhich trainscan arrive at (depart from) station 119894 The nodes in U119894 (W119894)are called arrival (departure) nodes Unlike the original graphthat is designed for a railway network [6 12] this paperconsiders a single-track metro line and thus the constraintsare different and implicitly imposed in the definition of thestructure of the DTSG

The continuous time is discretized into evenly spacedtime intervals (1 second in this paper) and the period extendsfrom 0 to 119905119905end Note that the values of 119905119905end and 119905119901end aredifferent 119905119901end is the last time stamp of the passenger demandThe periodmentioned above refers to the one extending from0 to 119905119901end Assumption 2 imposes that the last train departsfrom the first station at time 119905119901end However if the DTSG onlyconsiders the period from 0 to 119905119901end the arrivaldepartures of

the last train at subsequent stations are not considered andthe passenger-oriented objectives are hard to formulateThusthe DTSG considers the period from 0 to 119905119905end The relationbetween 119905119905end and 119905119901end is set as

119905119905end = 119905119901end +2119868minus1sum119894=1

119904119905 (119894) + 2119868sum119894=1

119889119894 (1)

where 119904119905(119894) is the segment travel time of segment 119894 and 119889119894is the maximum dwell time of station 119894 This setting ensuresthat even the slowest train can be completely included in theDTSG

The construction of the DTSG is noncyclicThe objectivefunctions of passenger time energy and train cost areassociated with the chosen arcs and are described below

32 Passenger Time

321 Passenger Arriving Alighting and Boarding

(1) Time-Varying Demand Information The passenger de-mand at each station can be expressed by 120591119894119895(119905) which isdefined as

120591119894119895 (119905) ge 0 If 1 le 119894 lt 119895 le 119868 or 119868 le 119894 lt 119895 le 2119868120591119894119895 (119905) = 0 Otherwise

forall119894 119895 isin I(2)


Passenger arrival rates at stations are 120582119894(119905) = sum2119868119895=119894+1 120591119894119895(119905)forall119894 119895 isin I In this paper demands are regarded as determin-istic values which are given by 120591119894119895(119905) Thus the number ofarriving passengers at station 119894with destination 119895 during timeinterval [1199051 1199052] is

119875(119894119895) (1199051 1199052) =1199052sum119905=1199051

120591119894119895 (119905) forall119894 119895 isin I (3)

The number of arriving passengers at station 119894within timeinterval [1199051 1199052] is the sumof arriving passengerswith differentdestinations

119875119894 (1199051 1199052) = 2119868sum119895=119894+1

119875(119894119895) (1199051 1199052) = 2119868sum119895=119894+1

1199052sum119905=1199051

120591119894119895 (119905)

= 1199052sum119905=1199051

120582119894 (119905) forall119894 119895 isin I(4)

(2) Determination of Dwell Arcs Letting 119899119889 be the number ofdoors per train the average number of alighting (boarding)passengers per door when train 119896 arrives at (departs from)station 119894 is119875119886119896119894119899119889 (119875119887119896119894119899119889) In addition we denote the averagenumber of boarding passengers per second (boarding rate) as120577119887 and alighting rate as 120577119886 which vary with passenger densityin trains For simplicity we consider that boarding passengerswait until all alighting passengers get off the train that is theboarding process starts at time 119905119886119896119894

The dwell time is the sum of alighting and boardingtimes The calculation of alighting time is solely based on thenumber of alighting passengers

119905119886119896119894 = 1199050119896119894 + 119875119886119896119894(119899119889 sdot 120577119886) forall119896 isin K 119894 isin I (5)

The calculation of boarding time is slightlymore complexFigure 3 shows the relations of train remaining capacitywaiting passengers boarding rate and maxmin dwell timesThe horizontal axis represents time and the vertical axisrepresents the number of passengers These four factorsdetermine the boarding time for train 119896 at station 119894 and inFigure 3 they are represented by (1) remaining capacity oftrain 119896 at station 119894 119873119896 minus 119875V

119896119905119886119896119894 a horizontal line (2) number

of waiting passengers at station 119894 at time 119905 a curve 119875119908119894119905(3) maximum allowable number of boarding passengers adiagonal function119891119896(119905)with the slope of 119899119889 sdot 120577119887 (4)minimumand maximum dwell times at station 119894 that is 1199050119896119894 + 119889119894 and1199050119896119894 + 119889119894 In Figure 3 the diagonal 119875 = 119891119896(119905) intersects thehorizontal line 119875 = 119873119896 minus 119875V

119896119905119886119896119894at time 1199051198961198941 and intersects the

curve 119875 = 119875119908119894119905 at time 1199051198961198942 The curve 119875 = 119875119908119894119905 intersects thehorizontal line119875 = 119873119896minus119875V

119896119905119886119896119894at time 1199051198961198943 Note that the time is

discrete andAssumption 3 ensures that the slope of119875 = 119891119896(119905)always exceeds the slope of 119875 = 119875119908119894119905 Thus we have

1199051198961198941 = max 119905 | (119905 minus 119905119886119896119894) sdot 119899119889 sdot 120577119887 le 119873119896 minus 119875V119896119905119886119896119894

forall119896 isin K 119894 isin I (6)

Time

Passengers

Remaining Capacity

Pw

it

Pwit

t0 ki+di

t0 ki+di

tki

3tki

1tki

2ta ki

Nk minus Pkt

fk(t)

tki

3 tki

2

Figure 3 Relations of remaining capacity waiting passengersboarding rate and maxmin dwell times

1199051198961198942 = max 119905 | (119905 minus 119905119886119896119894) sdot 119899119889 sdot 120577119887 le 119875119908119894119905 forall119896 isin K 119894 isin I (7)

1199051198961198943 = max 119905 | 119875119908119894119905 le 119873119896 minus 119875V119896119905119886119896119894 forall119896 isin K 119894 isin I (8)

Practically in an oversaturated scenario a train departsfrom a station as soon as the number of in-train passengersreaches its capacity Based on this strategy (6) shows thatthe boarding passengers who arrive after 1199051198961198941 exceed traincapacity and they will be left behind (7) shows that before1199051198961198942 there are passengers waiting at station and after 1199051198961198942 allpassengers board the train (8) points that the train is full at1199051198961198943 and passengers arriving after 1199051198961198943 should wait for the nexttrain

The departure time of train 119896 at station 119894 (119905119887119896119894) is deter-mined based on the relations among 1199051198961198941 1199051198961198942 1199051198961198943 1199050119896119894 + 119889119894and 1199050119896119894 + 119889119894 Based on Assumption 3 we consider two cases(1) 1199051198961198941 gt 1199051198961198942 and (2) 1199051198961198941 le 1199051198961198942 Note that 119889119894 and 119889119894 should beconsidered in both cases In case (1) we can further derivethat 1199051198961198943 gt 1199051198961198941 gt 1199051198961198942 (which is the solid curve in Figure 3)where the number of boarding passengers per second isdetermined by passenger arrival rate Thus the departuretime is determined as 119905119887119896119894 = min1199050119896119894 + 119889119894max1199051198961198942 1199050119896119894 +119889119894 1199051198961198943 In case (2) we can further derive that 1199051198961198943 le 1199051198961198941 le 1199051198961198942(which is the dashed curve in Figure 3) where the numberof boarding passengers per second is determined by theactual passenger boarding rate Thus the departure time isdetermined as 119905119887119896119894 = min1199050119896119894 + 119889119894 1199051198961198941 The departure time oftrain 119896 at station 119894 is expressed as

119905119887119896119894=

min 1199050119896119894 + 119889119894max 1199051198961198942 1199050119896119894 + 119889119894 1199051198961198943 1199051198961198941 gt 1199051198961198942min 1199050119896119894 + 119889119894 1199051198961198941 1199051198961198941 le 1199051198961198942

(9)

Finally we consider the relations between alightingboarding rates and congestion at stations Here the conges-tion is described by the number of in-vehicle passengers


Practically the greater the passenger density in a train is theslower the passengers can get offon the trainHere we designa monotonically decreasing function 120588(119875V

119896119905) le 1 with respectto the number of in-vehicle passengers to approximate actualalightingboarding rates according to the passenger density intrains For simplicity in this paper the value of 120588(119875V

119896119905) duringthe dwell process is fixed and is determined by the numberof in-vehicle passengers at its arrival time That is 120577119886 and 120577119887remain constant during the dwell process

120577119886 = 120588 (119875V1198961199050119896119894) sdot 120577119886ideal

120577119887 = 120588 (119875V1198961199050119896119894) sdot 120577119887ideal

forall119896 isin K 119894 isin I(10)

where 120577119886ideal and 120577119887ideal are the ideal alighting and boardingrates in the uncongested condition

(3) Passenger Alighting and Boarding Once the dwell arcs aredetermined the alighting and boarding times for train 119896 atstation 119894 are determined Based on the results of dwell arcsthree cases should be considered (1) 1199051198961198941 lt 1199051198961198942 (2) 1199051198961198941 ge 1199051198961198942and 1199051198961198942 le 1199050119896119894 + 119889119894 (3) 1199051198961198941 ge 1199051198961198942 and 1199051198961198942 gt 1199050119896119894 + 119889119894 In case (1)waiting passengers board the train at rate 120577119887 In case (2) actualpassenger boarding rate is equal to the passenger arrival rateIn case (3) the actual boarding rate is 120577119887Thus the number ofboarding passengers for train 119896 at station 119894 is119875119887119896119894=

119875119908119894119905119887119896119894

if 1199051198961198941 ge 1199051198961198942 1199051198961198942 le 1199050119896119894 + 119889119894(119905119887119896119894 minus 119905119886119896119894) sdot 120577119887 sdot 119899119889 otherwise


Wang et al [3] point out that in a busy metro linepassengers with different destinations are well mixed at eachstationThus the ratio between the numbers of boarding pas-sengers with destination 119895 and of total boarding passengers isequivalent to the ratio between 120591119894119895(119905) and 120582119894(119905)

119875119887119896(119894119895) =sum119905119887119896119894119905=119905119887119896minus1119894

120591119894119895 (119905)sum119905119887119896119894119905=119905119887119896minus1119894

120582119894 (119905)sdot 119875119887119896119894

forall119896 isin K 119895 isin I1 119894 = 1 2 119895 minus 1(12)

Thenumber of alighting passengers from train 119896 at station119895 consists of all previously boarding passengers in train 119896withdestination 119895119875119886119896119895 =

119895minus1sum119894=1

119875119887119896(119894119895) forall119896 isin K 119895 = 2 119868 119868 + 2 2119868 (13)

There are no alighting passengers at the first station ineach direction (station 1 and 119868 + 1)

322 Passenger Wait and Travel Time

(1) PassengerWait Time Total passenger wait time at a stationcan be calculated by summing up the products of the cumu-lative number of passengers and their time Let 119896last be theindex of last train that has departed from station 119894 beforetime 119905 that is 119905119887119896last 119894 le 119905 lt 119905119887119896last+1 119894 The cumulative numberof passengers at station 119894 at time 119905 is the difference betweencumulative numbers of arrival passengers and boardingpassengers within the period [0 119905]

119875119908119894119905 = 119875119894 (0 119905) minus119896lastsum119896=1

119875119887119896119894 forall119894 isin I (14)

Since the continuous time is discretized into 1 secondintervals the passenger wait time between times 119905 and 119905 + 1is 119875119908119894119905 sdot (119905 + 1 minus 119905) = 119875119908119894119905 Thus the total passenger wait timeis

119879119901119908 = sum119894isinI

119905119905endsum119905=0

119875119908119894119905 (15)

(2) Passenger Travel Time The number of in-train passengersin train 119896 at time 119905 is the difference between cumulativenumbers of boarding passengers and alighting passengersLet 119894last be the index of the last station that train 119896 has visitedbefore time 119905 We consider two cases (1) train 119896 is dwellingat station 119894last that is 1199050119896119894last le 119905 lt 119905119887119896119894last (2) train 119896 is in thesegment between station 119894last and 119894last + 1 that is 119905119887119896119894last le 119905 lt1199050119896119894last+1 Based on Assumption 4 the boarding passengers atstation 119894last are not considered in case (1) but are included incase (2) Thus the number of in-vehicle passengers in train 119896at time 119905 is

119875V119896119905 =

119894lastminus1sum119894=1

119875119887119896119894 minus119894lastsum119894=1

119875119886119896119894 1199050119896119894last le 119905 lt 119905119887119896119894last 119894lastsum119894=1

(119875119887119896119894 minus 119875119886119896119894) 119905119887119896119894last le 119905 lt 1199050119896119894last+1forall119896 isin K

(16)

Analogously the total passenger travel time is

119879pt = sum119896isinK

119905119905endsum119905=0

119875V119896119905 (17)

Finally the total cost of passenger time is formulated as

119862119901total = (119888pw sdot 119879pw)3600 + (119888pt sdot 119879pt)3600 (18)

The cost unit is yuan (yen)33 Energy In general for each segment 119894 with givensegment travel time 119904119905(119894) and segment length 119904119897(119894) the speedprofile determines energy use [49]Thus given a speed profile


generating method the energy per unit mass of differenttrains in segment 119894 is the sameWe denote the energy per unitmass for one train in segment 119894 as 119890119894 Its calculation usuallyconsiders the following train dynamic equations [48]

119889V119896 (119905)119889119905 = 119891119905 minus 119891119903119889119909119896 (119905)119889119905 = V119896 (119905) forall119905 isin [119905119887119896119894 1199050119896119894+1] 119894 119894 + 1 isin I

119891119903 = 120572 sdot V1198962 (119905) + 120573 sdot V119896 (119905) + 120574V119896 (1199050119896119894) = V119896 (119905119887119896119894) = 0

(19)

The first two equations are train motion equations whereV119896(119905) and 119909119896(119905) respectively represent traveling speed anddistanceThe third one is the Davis resistance function whereDavis parameters are 120572 120573 and 120574 The last one requires thatevery train stops at each station An optimal train controlstrategy can be obtained based on (19) in which energy perunit mass is

119890119894 = int1199050119896119894+1

119905119887119896119894

119891119905 sdot V119896 (119905) 119889119905 forall119894 119894 + 1 isin I 119896 isin K (20)

However in (20) the relation between energy and deci-sion variable 119904119905(119894) is unclear It is also difficult to integratethe optimal train control with other objective functions [18]Since the speed profile optimization is not considered forsimplicity we use the method in [50] to generate practicalspeed profiles Based on these speed profiles we then adopta linear piecewise curve to approximate the energy per unitmass of train 119896 in segment 119894 [48]

119890119894 = (120594119909 sdot 119904119905 (119894) + 120594119910) sdot 119904119897 (119894) forall119894 isin I (21)

We introduce here an estimation procedure for the valuesof 120594119909 and 120594119910 which is based on mean squared error (MSE)First we set several feasible values (eg 119911 = 1 2 100) of119904119897(119911) and 119904119905(119911) Then we obtain optimal speed profiles basedon (19) and (20) thereby obtaining the optimal energy withrespect to 119904119897(119911) and 119904119905(119911) Finally we define the optimal energyper unit mass as 119890119911 and the estimated energy per unit massobtained with (21) as 119890119911 The MSE is formulated as

MSE (120594119909 120594119910) = 1100100sum119911=1

(119890119911 minus 119890119911)2 (22)

The optimal values of 120594119909 and 120594119910 can be obtained whenMSE(120594119909 120594119910) reaches its minimum and are found as follows

120597MSE (120594119909 120594119910)120597120594119909 = 0120597MSE (120594119909 120594119910)120597120594119910 = 0

(23)

With 120594119909lowast and 120594119910lowast obtained the total energy can be cal-culated as

119864total = 2119868minus1sum119894=1

119870sum119896=1

[(120594119909lowast sdot 119904119905 (119894) + 120594119910lowast) sdot 119904119897 (119894)]sdot (119872119896 +119872119901 sdot 119875V

119896119905119887119896119894)

(24)

Thus the total cost of energy consumption is

119862119890total = 119888119890 sdot 119864total = 119888119890sdot 2119868minus1sum119894=1

119870sum119896=1


119896119905119887119896119894)

(25)

Note that when all tracks are straight and level 120594119909 and 120594119910suffice for estimating energy However if grades and curvesare considered we may have 119890119894 = (120594119894119909 sdot 119904119905(119894) + 120594119894119910) sdot 119904119897(119894) forall119894 isinI where 120594119894119909 and 120594119894119910 are specified parameters to describethe characteristics in segment 119894 The parameter estimationprocedure is the same

34 Train Cost In this paper the train cost is estimatedwith empirical equations with respect to the interest ratetrain price and average round trip time where long-termrequirements (eg the consideration of the highest peakduring a year and demand evolution over the planninghorizon) are not considered

Train cost consists of train capital cost and operating cost(excluding energy) Train capital cost is related to the priceand economic life of a train as well as interest rate Let119862annualbe the annual cost per train 119862train be the current price pertrain 119903 be the interest rate and 119879ls be the lifespan of a trainThe relations among these four factors can be expressed as

119862annual = 119862train sdot 119903 sdot (1 + 119903)119879ls(1 + 119903)119879ls minus 1 (26)

The capital cost per train hour 119888tra can be then calculatedby dividing the annual cost 119862annual by the product of averageworking hours per day (119863work) and working days per year(119884work)

119888tra = 119862annual119863work sdot 119884work (27)

In addition a train can be reassigned after it finishes around trip and returns to the shunting yard and the roundtrip time 119879119896round can be obtained as

119879119896round = 1199051198871198962119868 minus 11990501198961 + 119879turn forall119896 isin K (28)

Thus the actual period for train running issum119870119896=1 119879119896round (sum119870119896=1 119879119896round le 119905119905end) The estimated fleet size 119865119904within period [0 sum119870119896=1 119879119896round] depends on the round triptime 119879119896round and the departure headway from the shunting


yard (which is equivalent to the arrival time of train 119896 at thefirst station 11990501198961)

119865119904 = (sum119870119896=1 119879119896round) 119870[sum119870119896=2 (11990501198961 minus 1199050119896minus11)] (119870 minus 1) (29)

where the term (sum119870119896=1 119879119896round)119870 represents the average roundtrip time and the term [sum119870119896=2 (11990501198961 minus 1199050119896minus11)](119870minus1) representsthe average headway The total cost of trains is formulatedas

119862tratotal = 119888tra sdot 119865119904 sdot (sum

119870119896=1 119879119896round3600 ) + 119888119900 sdot 119870

sdot sum2119868minus1119894=1 119904119897 (119894)1000 (30)

The first term 119888tra sdot 119865119904 sdot (sum119870119896=1 119879119896round3600) representsthe capital cost of 119865119904 trains for the actual running period[0 sum119870119896=1 119879119896round] The second term 119888119900 sdot 119870 sdotsum2119868minus1119894=1 119904119897(119894) representsthe operating cost with frequency 119870 (trains) on a line withthe length of the sum of 2119868 minus 1 segments (Km) Both decisionvariables (segment travel times and arrival times at the firststation) affect the train cost

35 Mixed Integer Programming Formulation The optimiza-tion model aims to minimize the weighted-sum of differentcost functions so that the conflicting objectives are general-ized in terms of monetary cost In this paper the weights ofdifferent functions are reflected by their average costs thatis 119888119890 119888119900 119888pt 119888pw and 119888tra For example an operator-orientedtimetable (neglecting passenger satisfaction) can be obtainedby setting 119888pt = 0 and 119888pw = 0 a passenger-oriented timetablecan be obtained by setting 119888119890 = 119888119900 = 119888tra = 0 The weightsfor integrated objectives can be set similarly with larger costcoefficients for larger weights The model is formulated asfollows

Minimize

119862total = 119862119901total + 119862119890total + 119862tratotal (31)

Subject to

11990501198701 = 119905119901end (32)

119904119905 (119868) = 119879turn (33)

119904min119905 (119894) le 119904119905 (119894) le 119904max

119905 (119894) forall119894 isin I (34)

119905119887119896119894 minus 119905119887119896minus1119894 ge ℎmin forall119896 = 2 119870 119894 isin I (35)

119875V119896119905 le 119873119896 forall119896 isin K 119905 isin [0 119905119905end] (36)

This model is a nonsmooth nonconvex programmingmodel where the nonsmoothness is caused by the limitedtrain capacity and the nonconvexity is caused by the deter-mination of dwell times [3] Constraint (32) is equivalentto Assumption 2 Constraint (33) considers the turnaround

from station 119868 to 119868 + 1 as traveling in segment 119868 withfixed travel time 119879turn Constraint (34) ensures that thesegment travel time should be optimized within a reasonablerange [119904min

119905 (119894) 119904max119905 (119894)] Constraint (35) imposes the mini-

mum headway constraint for both departures and arrivals(based on Assumption 1) Constraint (36) requires thatthe number of in-train passengers must not exceed traincapacity

In Caprara et al [6] the time-space graph is used toderive an integer linear programming model thereby allow-ing a considerably faster solution Analogously this papertransforms the nonsmooth nonconvex programming modelinto a mixed integer programming (MIP) model based onthe DTSG First we construct two (2119868 times 119905119905end) matrixes U

andW respectively representing the selection of arrival anddeparture nodes in the DTSG

U119894119905 = 1 if 119905 = 120587 (119906119896) 119906119896 isin U119894 119896 isin K0 otherwise

W119894119905 = 1 if 119905 = 120587 (119908119896) 119908119896 isin W119894 119896 isin K0 otherwise

forall119894 isin I 119905 isin [0 119905119905end]

(37)

where U119894119905 = 1 (W119894119905 = 1) if a train arrives at (departs from)station 119894 at time 119905 As mentioned earlier the value of U1119905 isdetermined by 120587(119906119896) = 11990501198961 for forall119906119896 isin U1 119896 isin K Thevalue ofW119894119905 is determined by U119894119905 and the dwell arcs that isΔ(119906119896 119908119896) = 119905119887119896119894 minus 1199050119896119894 for forall119906119896 isin U119894 119908119896 isin W119894 119894 isin I 119896 isin KThe value of U119894+1119905 is determined by W119894119905 and the travel timein segment 119894 that is Δ(119908119896 119906119896) = 119904119905(119894) for forall119908119896 isin W119894 119906119896 isinU119894+1 and 119894 isin I2119868 119896 isin K Thus the decision variablesdetermine U andW

Based on the structures of U and W as well as Assump-tion 4 we introduce two (2119868 times 119905119905end) matrixes PA and PBrespectively representing passenger alighting and boardinginformation (number and time)

PA119894119905 = 119875119886119896119894 119905 = 120587 (119906119896) 119906119896 isin U119894 119896 isin K0 otherwise

PB119894119905 = 119875119887119896119894 119905 = 120587 (119908119896) 119908119896 isin W119894 119896 isin K0 otherwise


(38)

The cumulative numbers of waiting passengers Q (2119868 times119905119905end) and in-vehicle passengers V (119870 times 119905119905end) can be con-structed based on PA and PB

Q119894119905 = 119876119894119905 minus 10038171003817100381710038171003817PB1198941119905

10038171003817100381710038171003817 forall119894 isin I 119905 isin [0 119905119905end] (39)


V119896119905

=

V119896119905minus1 + 119875119887119896119894 if 119905 = 120587 (119908119896) 119908119896 isin W119894 119894 isin IV119896119905minus1 minus 119875119886119896119894 if 119905 = 120587 (119906119896) 119906119896 isin U119894 119894 isin IV119896119905minus1 otherwise

forall119896 isin K 119905 isin [1 119905119905end]

(40)

In (39)119876119894119905 = 1198761198940+sum119905119902=0 120582119894(119905) represents the total waitingpassengers at station 119894 where 1198761198940 is the number of waitingpassengers at the start of this period According to thedefinition of cumulative passengers we have Q119894119905 = 119875119908119894119905Thus the total passenger wait time can be expressed by Q 119879pw = sum119894isinIsum119905119905end119905=0 119875119908119894119905 = sum119894isinIsum119905119905end119905=0 Q119894119905 = Q (since Q119894119905 ge0) In (40) the initial number of in-vehicle passengers iszero that is V1198960 = 0 The number of in-vehicle passengerschanges only at arrivaldeparture times which is consistentwith Assumption 4 Analogously the total passenger traveltime can be expressed as 119879pt = V Thus the total cost forpassenger time can be rewritten as

119862119901total = 13600 (119888pw sdot Q + 119888pt sdot V) (41)

For energy cost the energy per unitmass in segment 119894 canbe rewritten as

119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W

119894 119906119896 isin U119894+1 forall119894 isin I 2119868 (42)

The total energy can be rewritten as

119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0

120601119905 sdot [(120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894)]sdot (119872119896 +119872119901 sdot V119896119905) 119908119896 isin W

119894 119906119896 isin U119894+1

(43)

where 120601119905 = 1 if 119905 = 120587(119908119896) otherwise 120601119905 = 0 The total energycost can be rewritten correspondingly

For train cost the round trip time 119879119896round can be rewrittenas

119879119896round = Δ (119906119896 119908119896) + 119879turn119906119896 isin U

1 119908119896 isin W2119868 forall119896 isin K (44)

The departure headway from the shunting yard can berewritten as

11990501198961 minus 1199050119896minus11 = Δ (119906119896minus1 119906119896) 119906119896minus1 119906119896 isin U

1 forall119896 isin K 1 (45)

The total cost of trains can be rewritten correspondinglyFinally the last departing train constraint (32) is equiva-

lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)

The turnaround constraint (33) is equivalent to

Δ (119908119896 119906119896) = 119879turn 119908119896 isin W119868 119906119896 isin U

119868+1 forall119896 isin K (47)

The segment travel time window constraint (34) is equiv-alent to

119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W

119894minus1 119906119896 isin U119894 forall119896 isin K 119894 isin I 1 (48)

The minimum headway constraint (35) is equivalent to

Δ (119908119896minus1 119908119896) ge ℎmin119908119896minus1 119908119896 isin W

119894 119896 isin K 1 119894 isin I (49)

The train capacity constraint (36) is equivalent to

V119896119905 le 119873119896 forall119896 isin K 119905 isin [0 119905119905end] (50)

The selection of arcs determines the arrival nodes 119906119896 anddeparture nodes 119908119896 thereby determining every element inthe optimization model Thus the model is transformed intoan MIP The next section discusses how its solution can befound

4 Solution Approach

In this section we decompose the TTP into two subproblemsThe first subproblem optimizes the segment travel times(OST) aiming at minimizing the costs for energy and pas-senger segment travel times A branch-and-bound algorithmand a frequency determining algorithm are introduced tosolve the OST The second subproblem optimizes departureheadways from the shunting yard (OH) aiming to minimizepassenger wait time dwell time and train cost A novel rollingoptimization algorithm is designed to solve the OH

41 Decomposition The rationale of the decomposition isexplained as follows The objectives in the optimizationmodel include passenger wait time (at their original stations)passenger travel time (segment travel times and station dwelltimes) energy and train cost Based on (43) the total energyis related to the energy per unit mass and the number ofpassengers in each segmentThe energy per unitmass is solelydetermined by segment travel times 119904119905(119894) Since passengersboard at their origins and alight at their destinations regard-less of which train they take the sum of in-vehicle passengersin one segment 119894 is determined by passenger demand In thisregard the total energy can be expressed as

119864total = 119872119896 sdot 119870 sdot 2119868minus1sum119894=1

119890119894 +119872119901

sdot 2119868minus1sum119894=1

2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)

Therefore the total energy cost depends only on segmenttravel times 119904119905(119894) The passenger segment travel times alsodepend only on segment travel times


On the other hand passenger wait time dwell time andtrain cost depend on the train arrival and departure timesat stations Based on Assumption 1 train arrivaldeparturetimes at stations are affected by the departure headways fromthe shunting yardThus it is reasonable to use segment traveltimes as input and consider the costs for passenger wait timedwell time and trains as functions of the departure headways

In general the OST subproblem is formulated as followsMinimize

119862OSTtotal = 2119868minus1sum

119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894

+ (119888119890 sdot 119872119901 sdot 119890119894 + 119888pt sdot 119904119905 (119894)) sdot [[2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)

Subject to constraints (47) and (48)It should be noted that since segment travel times are

given inputs minimizing the cost of passenger dwell time isequivalent to minimizing the cost of total travel time Sincewe already obtain the in-train passenger matrix V there isno need to further derive dwell times at stations Thus theobjectives in the OH subproblem include the total passengertime cost (both Q and V) instead of costs of passenger waitand dwell timesTheOH subproblem is formulated as follows(with segment travel times as input)

Minimize

119862OHtotal = 119862119901total + 119862tra

total (53)

Subject to constraints (46) (49) and (50)

42 Solution Approach for OST The solution approach forOST consists of branch-and-bound (BampB) and frequencydetermining algorithms On one hand the BampB algorithmneeds frequency as input On the other hand the frequencydetermining algorithm needs segment travel times obtainedby the BampB to evaluate the performance Thus we iterativelyevaluate the performance with different frequencies (eachfrequency is used as input for BampB to find a solution ofsegment travel times) and choose the one with the bestperformance The segment travel times are then determinedcorrespondingly

421 Branch-and-Bound (BampB) Algorithm The OST sub-problem is an integer linear programming problem with 2119868 minus1 decision variables (119904119905(119894) for 119894 = 1 2 2119868 minus 1) Herewe use a generic BampB which is widely used to solve linearprogramming problems [7] Generally a BampB needs an inputmodel formulated as follows

min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)

where 997888c is the coefficient vector and 997888x 119879 is the decisionvariable vector Both 997888c and 997888x 119879 have 2119868 minus 1 elements A andAeq are coefficient matrixes and

997888b119879

and997888997888beq119879

are constantvectors representing the bounds HereA is a (4119868minus2)times(2119868minus1)matrix andAeq is a 1times(2119868minus1)matrix Correspondingly

997888b119879

has 2119868 minus 1 elements and997888997888beq119879

has one element We define avector with 2119868 minus 1 elements as

997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)

The decision variable vector can be represented as 997888x 119894 =119904119905(119894) for 119894 = 1 2 2119868 minus 1 where 997888x 119894 is the 119894-th element inthe vector 997888x We then defineℸ119894 = sum2119868119895=119894+1sum119894119903=1sum119905119901end119905=0 120591119903119895(119905) for119894 = 1 2 2119868minus1 as the total number of passengers that travelthrough segment 119894 Based on (52) 997888c 119894 can be represented as

997888c 119894 = 119888119890 sdot 119872119896 sdot 119870 sdot 120594119909 sdot 119904119897 (119894) + ℸ119894sdot (119888119890 sdot 119872119901 sdot 120594119909 sdot 119904119897 (119894) + 119888pt)

119894 = 1 2 2119868 minus 1(57)

Constraints (55) are represented as follows Based onconstraint (47) we have Aeq = 997888

120598 I and997888997888beq = 119879turn Based

on constraint (48) we haveA119894 = minus997888120598 i for 119894 = 1 2 2119868 minus 1A119894 = 997888120598 i for 119894 = 2119868 2119868 + 1 4119868 minus 2 where A119894 represents

the 119894-th row of matrix A Analogously997888b 119894 = minus119904min

119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max

119905 (119894) for 119894 = 2119868 2119868 + 1 4119868 minus 2Since we use the standard BampB algorithm [40] the detailedalgorithm description is not shown here

422 Frequency Determining Algorithm Here we considercyclic timetables (train departure headways from the shunt-ing yard are constant eg 200 s) to discuss the frequencybecause they have following properties (1) The cyclictimetable with minimum cost (ie the best cyclic timetable)can be considered as an upper bound of the noncyclictimetable (2) If passenger demand is stable the best cyclictimetable is a good approximation to an optimal noncyclicone (3)The cyclic timetables are easy to obtain

First the range of frequency should be estimated Wedefine the peak passenger volume in one segment as 119902peakwhich is the maximum value among the numbers of passen-gers traveling through the segment that is 119902peak = max(ℸ119894 |119894 = 1 2 2119868 minus 1) The minimum frequency 119870min isdetermined by the train capacity 119873119896 and 119902peak 119870min =lceil119902peak119873119896rceil The maximum frequency 119870max is determined bythe minimum headway 119870max = lfloor119905119901endℎminrfloor + 1

Second for each frequency 119870 isin [119870min 119870max] we use 119870as input of the BampB to obtain the segment travel times Sincethe frequency for a cyclic timetable determines the departureheadways from the shunting yard (ℎ = 119905119901end(119870minus1) where ℎ isthe departure headway) the cyclic timetable that correspondsto frequency 119870 can be obtained Then the total cost of thetimetable can be calculated with (52)


Finally we obtain different cyclic timetableswith differentcosts each corresponding to a frequency 119870 isin [119870min 119870max]We select the timetable with minimum cost as the best cyclictimetable and its corresponding segment travel times as theOST solution within this period The detailed algorithm is asshown in Algorithm 1

43 Solution Approach for OH Here we design a rollingoptimization (RO) algorithm that optimizes the arrival timeat the first station for one train at a timeThe main differencebetween the RO and the rolling horizon (RH) approach is inthe RH the bound conditions (ie the start and end timesof each horizon) require some of the original variables tohave fixed known values [3] while in the RO each decisionvariablemay still be optimizedThus the rolling optimizationis more flexible

431 The Rolling Optimization (RO) Algorithm Based on(53) the objectives include the total costs of passenger timeand trains To obtain these costs we need the segment traveltimes as input Here instead of considering the whole periodwe consider a much shorter period 119905119901 le ℎmin + sum2119868minus1119894=1 119904119905(119894) +sum2119868119894=1 119889119894 eg 600 s for solving the OST For example foreach train 119896 ge 2 its previous train arrives at the firststation at time 1199050119896minus11 (the first train arrives at time 0) andwe consider the period [1199050119896minus11 1199050119896minus11 + 119905119901] to solve the OST(the constraint of 119905119901 ensures that 1199050119896minus11 + 119905119901 le 119905119905end) In thisway the key ideas of ldquoiterative optimizationrdquo (the frequenciessegment travel times and headways are interdependent) andldquorolling horizonrdquo (the smaller period studied in each step) areadopted and future passenger demand is considered

Then we discuss the costs for one train 119896 ge 2 if its arrivaltime at the first station is given Since each train is scheduledin chronological order the waiting passenger matrix Q

and the in-train passenger matrix V can be constructedchronologically For each train 119896 ge 2 the departure time of itspreceding train at the first station is 119905119887119896minus11 and train 119896 departsfrom the first station at time 1199051198871198961We define the passenger waittime with respect to train 119896 as 119879pw(119896) which is formulatedas

119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)

Analogously we define the passenger travel time for train119896 as 119879pt(119896) and we obtain

119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)

The estimated fleet size of train 119896 (119865119896119904 ) depends on itsround trip time (119879119896round) and the departure headway from theshunting yard (11990501198961 minus 1199050119896minus11)

119865119896119904 = 119879119896round11990501198961minus 1199050119896minus11

forall119896 isin K (60)

Based on (18) (30) (53) (58) (59) and (60) we definethe total costs in the OH for train 119896 as 119862OH

total(119896) which can beexpressed as

119862OHtotal (119896)= 119888pw sdot 119879pw (119896) + 119888pt sdot 119879pt (119896) + 119888tra sdot 119865119896119904 sdot 119879119896round3600 + 119888119900sdot sum2119868minus1119894=1 119904119897 (119894)1000 forall119896 isin K

(61)

The value of 119862OHtotal(119896) is a function of the arrival time

for train 119896 at the first station Since the arrival time for theprevious train is given we consider the value of 119862OH

total(119896) asa function of the headway ℎ between train 119896 minus 1 and train 119896(ℎ = 11990501198961 minus 1199050119896minus11) for consistency with other TTP studies

Next we discuss how the optimal value for ℎ can beobtained Based on (58) (59) and (60) both the passengertime and train cost depend on passenger demand Sincethe passenger demand is time-varying the relation between119862OHtotal(119896) and ℎ cannot be described with a fixed functionThe expected objective function of OH is shown in

Figure 4 Its value first decreases (the headway starts fromℎmin) to a minimum value and then increases monotonicallyWe specify that this function reaches its minimum value atheadway ℎlowast Since the DTSG divided the continuous timeinto seconds we iteratively calculate the objective function ofOHwith respect to ℎ (starting from ℎ = ℎmin and ℎ = ℎ+1 ineach following step) until the optimal headway ℎlowast is foundor the total cost increases asℎ increases (in case thatℎlowast cannotbe reached) It should be noted that sometimes a headwayis not feasible due to the model constraints (eg a shortheadway might violate the minimum headway constraint)In such cases we define the value of 119862OH

total(119896) to be +infin if itcorresponds to an infeasible headway

Finally for each train 119896 ge 2 we can obtain its optimalheadway ℎlowast (between train 119896 minus 1 and train 119896) By rollingoptimization of departure headways for trains the timetablecan be obtained Let 997888H be the vector of optimal headways forall trains 119896 ge 2 The detailed rolling optimization (RO) algo-rithm is described in Algorithm 2 The arrival and departuretimes of the first train should be obtained beforehand and119905011 = 0432 Other Approaches As pointed out by Wang et al[3] other approaches such as a pattern search [51] or agenetic algorithm [36] can be applied to solve the TTPHere we briefly describe the generic structures of these twoapproaches

A genetic algorithm (GA) is an iterative heuristic thatseeks an optimal individual among a population of solutionsin each generation and uses selection crossover mutationand possibly other operations to obtain new generationsSuch process is repeated until the termination condition issatisfied Since the size of the chromosome should be presetthe frequency is considered here a given and unchangeableinput Then the segment travel times can be obtained with


Step 1 Estimate the frequency range ie [KminKmax] Set the initial frequency of the cyclic timetable as K = Kminthe best frequency found during the process as Kbest = 0 and the minimum total cost found during the process asTCmin = +infin Go to Step 2

Step 2 For frequency K obtain the segment travel times through the BampB algorithm and obtain the uniform departureheadways from the shunting yard h = tpend(K minus 1) Go to Step 3

Step 3 Calculate total cost TCK corresponding to the frequency K If TCK lt TCmin then set TCmin = TCK and Kbest = KGo to Step 4

Step 4 If K = Kmax then go to Step 5 Otherwise set K = K + 1 and return to Step 2Step 5 Output Kbest and TCmin The segment travel times within this period are the solution of the BampB algorithm that

corresponds to frequency Kbest

Algorithm 1 Frequency determining algorithm

Step 1 Set 997888H = (0 0 0 0) Set the period tp for OST solution approach Set k = 2 h = hmin and go to Step 2Step 2 Obtain the segment travel times for train k by OST solution approach Obtain the arrival and departure times

at stations for train k Go to Step 3Step 3 Calculate the value of COH

total(k) with respect to h and calculate the value of COHtotal1015840(k) with respect to h1015840 = h + 1

Go to Step 4Step 4 If COH

total1015840(k) gt COH

total(k) then go to Step 5 Otherwise set h = h1015840 and return to Step 2Step 5 Set the optimal headway for train k as hlowast = h and set 997888H(k minus 1) = hlowast Go to Step 6Step 6 If 997888H ge tpend then the rolling optimization is finished and thus the timetable for the period [0 ttend] can be obtained

Otherwise set k = k + 1 h = hmin and return to Step 2

Algorithm 2 Rolling optimization algorithm

Cost

Headway

Total costTime costTrain cost

ℎＧＣＨ ℎlowastℎlowast

ℎＧＣＨ

Figure 4 Total cost for train k with respect to headway

the given frequency An individual consists of headways forall trains (119870 minus 1 headways) and its objective function iscalculated based on (31) The initial individuals are generatedas the cyclic headways Random values are then added orsubtracted to the headways considering constraints (46) and(49) The details of standard GA [3 36] are not shown here

Compared with the proposed solution approach (RO)in this paper generic GA (GGA) has following properties(1) In terms of input both GGA and RO require passengerdemand as input However GGA requires fixed frequencyas input while RO determines optimal frequency during theprocess (2) In terms of optimization process GGA optimizes

the solution (of all trains) in a random direction while ROoptimizes one train at a time (3) In terms of computationcomplexity GGA calculates the objective function of thewhole system for each chromosome while RO calculates theobjective of one train at each step These differences resultin different computation performances which will be furtherdemonstrated in the numerical experiments

For the numerical experiments we set the parametervalues in the generic GA as follows The population size is20 themaximum iteration number is 1000 the crossover rateis 07 and the mutation rate is 01 The algorithm terminatesif the maximum number of iterations is reached or the bestfitness value in this generation is the same as in the previous100 generations ahead

The pattern search (PS) algorithm is proposed to solveunconstrained optimization problems [52] which varies onevariable (headway) at a time by steps of the same magnitudeWhen no such increase or decrease in any one variable fur-ther decreases the objective function it decreases the step sizeand repeats the process until the steps are deemed sufficientlysmall Here the unconstrained objective is constructed withthe Augmented Lagrangian (AL) method which adds theconstraints as penalty terms to the objective function Thepenalty term is the product of a Lagrangian multiplier anda constraint equation (eg from 119892(119909) ge 0 to 119892(119909) minus 1199102 = 0where 119910 is a new variable) Let the number of constraints be119898 Set the multiplier that corresponds to the 119895-th constraintas 120596119895 and set a large enough parameter 120590 By minimizing theobjective function with respect to 119910 (so that the intermediate


Table 3 Parameter values in the experiments

Parameter ValueMk (Kg) 199 times 105119872119901 (Kg) 75Nk (Passenger) 1290119888119890 (yenKwh) 079co (yen (trainsdotKm)) 20119888pt(yen(passengersdothour)) 10cpw(yen(passengersdothour)) 20119888tra (yen(trainsdothour)) 800hmin (s) 100119891119903 (ms) 136times10minus4 sdotV1198962 (119905)+145times10minus2 sdotV119896 (119905)+008120594x minus1944 times 10minus4120594119910 04176

variable 119910 does not influence the objective function) we canobtain the typical form of the penalty term as

12120590119898sum119895=1

[max (0 120596119895 minus 120590 sdot 119892119895 (119909))]2 minus 1205961198952 (62)

The parameter value120596119895 is optimized in each iterative stepuntil the termination condition is satisfied In each step of ALwe use the PS to find a minimum value of the unconstrainedobjective function In this way we integrate AL and PS anddefine such approach as AL + PS The same drawback existsthat the frequency is preset and considered as input Theinitial solution is the cyclic headways For the numericalexperiments we set the values of parameters in theAL+PS asfollowsThe initial step size for PS is 32 sThe speeding factorfor PS is 1 The shrinking factor for PS is 05 The allowanceerror for both PS and AL is 1 s (since the continuous time isdiscretized into seconds) The augmented factor for AL is 12The large enough parameter 120590 in AL is 200

5 Numerical Experiments

In this section we present several numerical experiments totest the effectiveness and efficiency of the proposed solutionapproach All experiments are performed with Matlab on aPC with 2GHz Intel Core7 and 8GB memory Parameters inthe examples are shown in Table 3 We adopt here the sameenergy calculation method as in [48] To save space we donot show the detailed calculations and directly provide theestimated values of parameters 120594119909 and 120594119910 in Table 3

51 A Small Case Study Here we consider a short metro lineas shown in Figure 5The segment lengths are 1800m 1600mand 2000m For simplicity the minimum and maximumdwell times for each station are all set as 30 s and 90 srespectively

The baseline passenger demand is shown in Table 4The ldquoBoarding (Alighting) Sumrdquo represents the number of

1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m

Figure 5 Representation of a short metro line

Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25

Figure 6 Passenger demand distributions in different cases

total boarding (alighting) passengers per second at a stationFour cases are considered here As shown in Figure 6 thecurves represent demand variance over time and ldquomul-tiplierrdquo represents the ratio between actual and baselinepassenger demand For example in case 1 during period[1200 2400] (s) the multiplier is 16 and passenger demandfrom station 2 to station 3 at 119905 isin [1200 2400] is 096 (06times16)passengerssecond The four cases have different passengerdemand distributions

The computational results of different cases are shown inTable 5 and Figure 7 Specifically in Table 5 the noncyclicresult obtained by RO is denoted as119873119870=119870119873 which schedules119870119873 trains within a given time duration The cyclic resultobtained by Algorithm 1 is denoted as 119880119870=119870119880 which sched-ules 119870119880 trains within a given time duration Particularlythe best cyclic result is denoted as 119880best

119870=119870119880 The GGA result

is denoted as GA119870=119870119873 which uses frequency 119870119873 fromRO result as input The objectives of different results arecalculated including passenger wait time (119879pw) passengertravel time (119879pt) total energy (119864total) fleet size (119865119904) and totalcost (119862total) Figure 7 specifically shows the numerical resultsin passenger satisfaction

Table 5 and Figure 7 demonstrate that under time-varying passenger demand noncyclic timetables obtainedby RO outperform the other timetables while GGA resultsoutperform cyclic ones obtained by Algorithm 1 This resultverifies that the noncyclic timetable improves passenger satis-faction Particularly the passenger wait time (119879pw) passengertravel time (119879pt) and total cost (119862total) have their lowestvalues in the RO results which demonstrates that the RO


Table 4 Baseline passenger demands for a short metro line (passengers per second)

Station index 1 2 3 4 5 6 7 8 Boarding sum1 0 03 03 045 0 0 0 0 1052 0 0 06 03 0 0 0 0 093 0 0 0 03 0 0 0 0 034 0 0 0 0 0 0 0 0 05 0 0 0 0 0 045 075 03 156 0 0 0 0 0 0 06 03 097 0 0 0 0 0 0 0 045 0458 0 0 0 0 0 0 0 0 0Alighting sum 0 03 09 105 0 045 135 105 -

Table 5 Comparison between cyclic and non-cyclic timetables

Time duration (s) Timetable type Tpw (106) Tpt (106) Etotal (103) Fs Ctotal (104) Gap ()0-10800 Non-cyclic 119873119870=60 72330 14384 18069 7 12320 -

Case 1Non-cyclic 119866119860119870=60 72786 14402 18069 7 12329 007Cyclic 119880best

119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2

Non-cyclic 119866119860119870=24 25489 49758 71411 6 44954 304Cyclic 119880best

119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221

Case 1 Case 2 Case 3 Case 4

Numerical results of passenger costs

RO resultGGA result

Best cyclic result

Figure 7 Passenger costs for different results in different cases

method is effective in solving the proposed train timetablingproblem In case 1 the result of 119880119870=60 is infeasible becauseit violates the minimum headway constraint at intermediatestations (the dwell times are different from 119880119870=60 to 119873119870=60)and thus we have 119862total = +infin It is interesting that 119873119870=60and 119880119870=60 require the same energy and fleet size This result

verifies the reasonableness of model decomposition in thispaper from two perspectives (1) When all passengers reachtheir destinations energy is solely determined by segmenttravel times (they are the same in119873119870=60 and119880119870=60) (2)Withgiven segment travel times the estimated fleet size is solelydetermined by headways Similar results can be observed inother cases

The gaps in Table 5 shows that case 2 has the largest gapcase 3 and 4 have similar gaps and case 1 has the smallest gapNote that in case 2 passenger demand changes drasticallyin cases 3 and 4 demand patterns are quite similar in case1 within certain subperiods passenger demand is the samewhich is the steadiest among all cases Therefore we canconclude that as the passenger demand varies more signif-icantly over time the gap between the cyclic and noncyclictimetables increases

Figure 8 illustrates the obtained train timetables in differ-ent cases where the departure headways vary in accordancewith the fluctuating passenger demand For example in cases2 3 and 4 the headways are small in the peak period and


Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)

(a) Obtained timetable for case 1 (119873119870=60)

Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)

(b) Obtained timetable for case 2 (119873119870=24)

Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)

(c) Obtained timetable for case 3 (119873119870=45)

Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8

(d) Obtained timetable for case 4 (119873119870=60)

Figure 8 Obtained timetables for different cases

are larger before and after the peak period In case 1 thesmall headways occur during subperiods [1200 2400] (s) and[7200 8400] (s) where the multipliers of passenger demandare largest (16 and 18) In Figure 8 the circles represent thedwell times that exceed the minimum dwell time (ie 30 s)We observe that the excess dwell times (circles) mostly occurat station 7 This is consistent with the OD pairs we presentin Table 4 Specifically the difference between the cumulativeboarding sum and the cumulative alighting sum reaches itsmaximum value at station 7 (15 + 09 minus 045 = 195) In thisregard the alighting and boarding rates are smallest at station7 In addition the total number of alighting and boardingpassengers is largest (ie 18 passengers per second) at station7 and thus the excess dwell times here are reasonable

Moreover despite the symmetry of passenger demanddistributions the headways at the ends of periods are largerthan at their starts This can be explained by that passengerdemand increases at the start and decreases to zero atthe end which leads to different cost changes Specificallythe passenger time cost increases more significantly whendemand increases while the train cost does not change withdemand Thus the headway tends to be smaller at the startof periods than at the ends In fact it is often seen inactual operations that the headways between the last trainsat the end of day exceed those between the first trains in themorning

Finally Table 6 shows the computational performancesfor different methods including RO method GGA and

Table 6 Computational performances for different methods

RO method Generic GA AL + PS(Case 1) Time (s) 2255 82994 10800Total cost (yen) 1232 times 104 1233 times 104 1242 times 104(Case 2) Time (s) 861 46582 10800Total cost (yen) 43586 times 104 44954 times 104 44872 times 104(Case 3) Time (s) 1206 50093 10800Total cost (yen) 88022 times 104 89245 times 104 89567 times 104(Case 4) Time (s) 240717 82291 10800Total cost (yen) 11621 times 104 11854 times 104 11774 times 104

AL + PS method As described above GGA and AL + PSrequire a preset fixed frequency and thus we use the samefrequency from RO result 119870119873 as input We terminate thealgorithm when its computation time reaches 3 hours andoutput the best solution obtained by then

Table 6 demonstrates that RO is very efficient obtaininga timetable very fast In comparison GGA takes more than15 hours to obtain a timetable and AL + PS takes morethan 3 hours in all cases In each case the timetable obtainedwith the RO method has the lowest total cost which furtherverifies its effectiveness The GGA timetables perform betterthan AL + PS in cases 1 and 3 but are outperformed in cases2 and 4


Table 7 Real-world infrastructure data of Beijing Metro Line 4

Station Minimum dwelltime (s)

Maximum dwelltime (s) Segment Minimum travel

time (s)Maximum travel

time (s)Segment length

(m)148 AHQB 30 90 1 124 170 1363247 BGM 30 90 2 114 156 1251346 XY 30 90 3 152 209 1672445 YMY 30 90 4 118 162 1295544 BDDM 30 90 5 81 111 887643 ZGC 30 90 6 82 113 900742 HDHZ 30 90 7 97 133 1063841 RMDX 30 90 8 96 131 1051940 WGC 30 90 9 151 207 16581039 NL 30 90 10 138 190 15171138 BJZ 30 90 11 131 180 14411237 XZM 30 90 12 93 128 10251336 XJK 30 90 13 100 138 11001435 PAL 30 90 14 100 138 11001534 XS 30 90 15 79 109 8691633 LJHT 30 90 16 92 126 10111732 XD 30 90 17 74 102 8151831 XWM 30 90 18 105 144 11521930 CSK 30 90 19 109 150 12002029 TRT 30 90 20 149 205 16432128 BJSRS 30 90 21 135 185 14802227 MJP 30 90 22 75 103 8272326 JMX 30 90 23 90 124 9892425 GYXQ 30 90 - - - -

Combining all the computational results in this casestudy we can conclude some disadvantages of GGA andAL +PS compared to RO (1) Although the timetables obtained bythem are noncyclic train departure headways in their resultsdiffer greatly from one another without clear correspondenceto the passenger demand In comparison RO timetables aremore regular (headways change gradually) and capture wellthe passenger demand patterns (2) Due to the computationcomplexity the computation time for runningGGAandAL+PS once is far greater than RO (3)They both require a presetfrequency as input If we do not know the best frequencybeforehand which is true in most real-world scenarios wemust run these methods with different frequencies and selectthe best solution thereby further increasing substantiallycomputation time

In general the RO method is superior to GGA and AL+ PS in obtaining a noncyclic timetable with time-varyingpassenger demand

52 Case Study of Beijing Metro Line 4 This section aimsto test the applicability and effectiveness of RO in a real-world case study based onBeijingrsquosMetro Line 4 (BML4)Theinfrastructure information and some operational parametersof BML4 are shown in Table 7 The turnaround time is 120 sTo satisfy the large passenger demand we set train capacity

24 20 2416

Pass

enge

r dem

and

20Station index

12 16

Station index128 84 40 0

0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)

Figure 9 Baseline passenger demand for BML4

as 119873119896 = 2160 passengers which is a maximum overloadedcapacity If not mentioned the other parameters are the sameas in Table 3 The baseline passenger demand data are basedon the research conducted by Duan et al [53] which arepresented in Figure 9 where 119911(119909 119910) forall119909 lt 119910 representsmorning peak demand and 119911(119909 119910) forall119909 gt 119910 representsevening peak demand

Here we consider three types of scenarios whose pas-senger demands are large and unsteady which are difficultto manage (a) period of 3 hours (ie 10800 s) includingmorning peak (b) period of 3 hours including evening peakand (c) a shopping festival hosted in the shopping centers nextto the XD station (whose index is 841) Representations of


Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r

Case a-1 amp bCase a-2

Case a-3Case c-1 amp c-2

0

05

1

15

2

Figure 10 Passenger demand patterns in different cases

passenger demands are the same as in the small case study(as shown in Figure 10) the actual passenger demand isthe product of baseline passenger demand (Figure 9) andthe multipliers (Figure 10) To test RO with more instanceswe design three subcases of scenario (a) with different peakmultipliers and two subcases of scenario (c) without thespecial event (c-1) or with it (c-2) In case (c-2) the passengerdemand at station XD is tripled

It should be noted that the real operation data of BML4 isdynamic and confidential Besides the operator today prefersto schedule trains uniformly Therefore in this experimentwe use cyclic timetables with best performance to approxi-mate the real-world timetablesThe notations are the same asin the small case study

Table 8 and Figure 11 show the computational resultsin different cases Note that for a major metro line suchas BML4 the computation times of GGA and AL + PSare extremely large and in many cases they do not yielda satisfactory solution within a reasonable time Thus wedo not present their results in this experiment The resultsof 119880119870=46 in case (a-1) 119880119870=54 in case (a-2) and 119880119870=54 incase (a-3) are infeasible and thus their total costs are +infinFigure 11 shows that considerable costs are reduced by thenoncyclic timetables obtained by RO Interestingly in thecases where cyclic timetables are feasible that is cases (a-1)(b) (c-1) and (c-2) the frequency of the noncyclic timetableis always larger than the best cyclic ones This is because thepassenger demand in the real-world fluctuates so drasticallythat cyclic timetables cannot capture its dynamic patternsand theminimumheadway constraintmakes it hard to obtainfeasible cyclic timetables In this regard adding more trainsin a cyclic timetable may be either uneconomic (ie theincreased train operating cost exceeds the reduced passengertime value) or infeasible

In comparison the noncyclic timetables obtained by ROrespond well to demand changes For example in case (a-1) the result of 119873119870=46 substantially reduces passenger waittime compared to 119880best

119870=44 and 119880119870=46 The adaptation fortime-varying demand is further demonstrated in cases (c-1)and (c-2) In case (c-1) the demand distribution pattern is

Case a-1 Case b Case c-1 Case c-2

Numerical results of total costs

RO resultBest cyclic result

Figure 11 Total costs for different results in different cases

similar to cases (a-1) and (b) and the cost reduction of thenoncyclic timetable is closer to the other cases (3608) Incase (c-2) the special event leads to a more heterogeneousdemand distribution and the cost reduction reaches 5335Besides the computation times (shown below each case) areacceptable for such a major and busy metro line

However in cases (a-2) and (a-3) the passenger waittimes for noncyclic timetables are larger than for cyclic onesThis occurs because the RO method considers the objectivefunction that corresponds to an infeasible headway as +infinWith a large demand the first departure headway that isfeasible may greatly exceed ℎlowast (as can be seen in Figure 4)whereas the cyclic timetable does not consider the feasibilityand divides headways uniformly In other words the increaseof departure headways (leading to larger 119879pw) is due tosafety issues which is consistent with the actual operationalrequirements that put safety ahead of costs In addition withparameters given in this experiment the largest frequency forBML4 is119870119873 = 54 as can be concluded by comparing cases (a-2) and (a-3) Therefore real-time dispatching and passengerflow control methods should be applied with passengerdemand at or above the levels in case (a-2)

Figures 12 13 and 14 illustrate the noncyclic timetablesobtained in case (a-1) case (b) and case (c-2) respectivelyThe circles here represent the occurrences of maximum dwelltimes These figures illustrate that the headways vary inaccordance with demand patterns which is consistent withthe results in the small case study Particularly in Figure 14the dense headways in the middle are set to transport manypassengers due to the special event This shows the flexibilityof noncyclic timetables in special scenarios

In addition from the distribution of the circles we cansee that case (a-1) is the busiest scenario among these figuresThemany circles in case (a-1) help explain the results in cases(a-2) and (a-3) (which have larger demands than case (a-1))With so many trains taking maximum dwell times to loadpassengers real-time dispatching and passenger flow controlmethods are necessary In case (b) the maximum dwelltimes mostly occur at stations 7 11 and 12 Specifically theneighborhood of station 7 (HDHZ) includes famous schoolsfashion shopping streets and dense office buildings Theneighborhood of station 11 (BJZ) includes a major bus hubthe Beijing Zoo and one of the largest wholesalemarketsTheneighborhood of station 12 (XZM) which has transfers withtwo other busy metro lines includes a railway station large


Table 8 Computational results for BML4

Scenario Tpw (108) Tpt (108) Etotal (104) Fs Ctotal (106) Gap ()Case (a-1) Non-cyclic 119873119870=46 38665 30555 74898 31 31788 -

3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038

Cyclic 119880119870=46 49805 30985 74898 31 +infin -Case (a-2) Non-cyclic 119873119870=54 11426 43898 92418 37 77869 -1544 s Cyclic 119880119870=54 10653 44060 92418 37 +infin -Case (a-3) Non-cyclic 119873119870=54 19568 47953 94566 38 12426 -1133 s Cyclic 119880119870=54 18891 4816 94958 38 +infin -Case (b) Non-cyclic 119873119870=38 22416 21501 59624 25 19902 -3932 s Cyclic 119880best

119870=30 44170 21632 49379 21 31748 3731Case (c-1) Non-cyclic 119873119870=30 05482 10256 43421 19 07004 -6848 s Cyclic 119880best


119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)

Figure 12 Timetable for morning peak (case a-1)

shoppingmalls and office buildingsThey are well known forthe large numbers of passengers during peak hours Thus wefind that the model is reasonable and consistent with real-world scenarios Analogously in case (c-2) the circles aremostly seen at stations near the station XD where the specialevent is held

It should be noted that the last two trains have anunusually small headway in case (b) The reason is that weassume that the last train must arrive at the first station atthe end of the period It is expected that if there is stilldemand after 10800 s the headways will be optimized by ourapproaches

53 Additional Experiments In this section we implementthree sets of additional experiments to investigate the rela-tions between the values of costs and the objectives Thefirst experiment tests the sensitivity of passenger time andpeak frequency to unit passenger time cost based on thedemand in case (b) of BML4 study In practice the unit timevalue of passenger travel time is often estimated at half ofpassenger wait time Thus we set 119888pt = 05 sdot 119888pw and obtainthe results that correspond to different values of 119888pw as shownin Figure 15 Results show that as time value increases totalpassenger wait time tends to decrease while the frequencywithin the peak hour tends to increase They demonstrate

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)

Figure 13 Timetable for evening peak (case b)

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)

Figure 14 Timetable for a special event (case c-2)

the tradeoffs between passenger-oriented objectives (servicelevel) and operator-oriented objectives (train cost) It may beexpected that if the average value of time continues growingthe system manager should add more trains to improvepassenger satisfaction

In addition neither passenger wait time nor peak hourfrequency changes are strictly monotonic This occurs be-cause the minimum headway constraints would somehowlimit the flexibility of the RO method especially based on amajor metro line (as described above) where very different


times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020

Unit passenger wait time cost (yuanhour)

Passenger wait timePassenger travel time

(a) Passenger time with different unit time costs

Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20

Within peak time period [3600 s 7200 s]

20 30 40 50 75 100 1500Unit passenger wait time cost (yuanhour)

(b) Peak hour frequency with different unit time costs

Figure 15 Sensitivity to unit passenger time cost

times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)

05 1 15 2 25 30Unit energy cost (yuanKwh)


(a) Passenger time with different unit energy costs

Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200

15 25 50 75 1005Unit energy cost (yuanKwh)

(b) Round trip time with different unit time costs

Figure 16 Sensitivity to unit energy cost

distributions of passenger demand lead to varying dwelltimes Thus in the real-world operations safety issues maylimit improvements in service level The timetables respondwell to different values of time

Besides we find that total passenger travel time barelychanges In fact all the segment travel times reach minimumvaluesThe reason will be specifically discussed in the secondexperiment

The second experiment tests how unit energy cost affectspassenger time and train round trip time based on thepassenger demand in case (c-1) of BML4 Figure 16(a)demonstrates that when unit energy cost changes within apractical range neither passenger wait time nor travel timeis affected The minimum segment travel times are obtainedin all cases This can be seen in daily operations that withlarge enough passenger demand the metro operators wouldrun trains at the fastest speed that is withminimum segmenttravel times

To extend the discussion we run more experiments toshow the relations between larger unit energy costs andround trip time and obtain the results shown in Figure 16(b)Although such large unit energy costs are unrealistic we cansee that the round trip time increases that is the averagespeed decreases as energy cost increases In other words asthe ratio between unit energy cost and passenger demandincreases round trip time increases This result is consistentwith the practical operation strategy that trains should runfaster withmore passengers and slowerwhen they carry fewerpassengers (where the unit energy cost is fixed and thus theratio depends on the passenger demand)

The third experiment aims to test the sensitivity of thepassenger wait time and the total cost to train capital costbased on the passenger demand in case (c-1) Figure 17illustrates that with the increased train capital cost thepassenger wait time and total cost tend to increase Thetradeoffs between passenger wait time and train cost that is


Total cost (times106 yuan)Passenger wait time (times108 s)

200 400 600 800 1000 14000 1200

Unit train capital cost (yuanmiddottrainminus1middothourminus1)

175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n

Figure 17 Sensitivity to train capital cost

the passenger-oriented and operator-oriented objectives isagain demonstrated The fluctuation of passenger wait timeis caused by the minimum headway constraints as explainedfor the first experiment In addition the monotonicallyincreasing total cost demonstrates that technologies whichincrease the trainrsquos economic life help reduce total cost in ametro system

Generally we obtain reasonable and practical results withthe proposed solution approach These results are consistentwith practical experiences and can be used as decision-making support in daily operations Since similar results areobtained with different passenger demand patterns and ondifferent metro lines the predictive ability of the model isdemonstrated The effectiveness of the proposed approach isfurther verified

6 Conclusions

This paper investigates the train timetabling problem (TTP)with time-varying passenger demand and accomplishes thefollowing

(1) It accounts for both the passenger-oriented andoperator-oriented objectives Factors in the line planningstage the timetabling stage and the vehicle scheduling stageare integrated in our model and solution approach Thenoncyclic timetable obtained in this paper outperforms thecyclic timetable the noncyclic timetables obtained by thegeneric GA and a classic direct optimization approach

(2) The dwell time determining process is specificallydesigned to consider both congestion at stations and passen-ger demand Besides limited train capacity and varying pas-senger alightingboarding rates are accounted for Thus thenoncyclic timetable obtained in this paper ismore practical indepicting passenger loading evolution For this reason sometimetables with otherwise good properties would violate theminimum headway constraint at intermediate stations andthus be infeasible Passenger flow control methods and real-time dispatching can be employed to make these timetablesapplicable

(3) Based on a general scheduling scheme the TTP isdecomposed into theOST andOHsubproblemsThe solution

approach includes a branch-and-bound (BampB) algorithma frequency determining algorithm and a novel rollingoptimization (RO) method Then computational resultsbased on a short metro line and the Beijing Metro Line 4are obtained They verify the effectiveness and efficiency ofthe proposed solution approach and demonstrate that thenoncyclic timetables obtained in this paper respond well topassenger demand characteristics Since the results are basedon different cases with different passenger distributions thepredictive ability of the proposed solution approach can alsobe demonstrated

(4) Additional experiments are presented whose resultsdemonstrate the following points (a) As passenger timevalues increase passenger wait time may be further reducedby scheduling more trains or by shifting departure timesfor riders (b) Average train speed decreases as passengerdemand increases due to longer dwell times at stations Thisdwell time effect outweighs the speed increase justified bymore riders per train (c) Longer train economic life not onlyreduces total system cost but also indirectly improves theservice level

This paper has some limitations Specifically the linearpiecewise approximation simplifies the energy calculationAssumption 1 is consistent with a common strategy inpractice but does not account for the relevant effects ofdriver behaviors Assumption 4 may be extended to a moreaccuratemodel which distributes the passenger boarding andalighting over the station dwell times A more systematiccomparison with real data of BML4 in the case study shouldbe conducted to fully validate RO method

These drawbacks can be addressed by a more complexmodel which considers driver behaviors (by estimating themagnitude of their relevance) line layouts and physics ofreal train run process as well as simultaneously optimizingdeparture headways and segment travel times under contin-uous time Future studies can investigate the improvementof our solution approach to reduce the computation timeneeded for such a complex model Besides it would also beinteresting to adjust the modelrsquos parameters with respect todifferent real-world operation data to obtain more realistictimetables


Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This work was supported by National Natural ScienceFoundation of China (Grant no U1434209) NationalKey Research and Development Program of China(2017YFB1201105) and Research Foundation of State KeyLaboratory of Railway Traffic Control and Safety BeijingJiaotong University (Grant no RCS2018ZZ003)

References

[1] S Burggraeve S H Bull P Vansteenwegen and R M LusbyldquoIntegrating robust timetabling in line plan optimization forrailway systemsrdquo Transportation Research Part C EmergingTechnologies vol 77 pp 134ndash160 2017

[2] T Robenek YMaknoon S S Azadeh J Chen andM BierlaireldquoPassenger centric train timetabling problemrdquo TransportationResearch Part B Methodological vol 89 pp 107ndash126 2016

[3] Y Wang T Tang B Ning T J J van den Boom and B DeSchutter ldquoPassenger-demands-oriented train scheduling for anurban rail transit networkrdquo Transportation Research Part CEmerging Technologies vol 60 pp 1ndash23 2015

[4] J Yin L Yang T Tang Z Gao and B Ran ldquoDynamic pas-senger demand oriented metro train scheduling with energy-efficiency and waiting time minimization Mixed-integer linearprogramming approachesrdquo Transportation Research Part BMethodological vol 97 pp 182ndash213 2017

[5] T Robenek S Sharif Azadeh Y Maknoon and M BierlaireldquoHybrid cyclicity Combining the benefits of cyclic and non-cyclic timetablesrdquo Transportation Research Part C EmergingTechnologies vol 75 pp 228ndash253 2017

[6] A CapraraM Fischetti and P Toth ldquoModeling and solving thetrain timetabling problemrdquo Operations Research vol 50 no 5pp 851ndash861 2002

[7] E Barrena D Canca L C Coelho and G Laporte ldquoSingle-line rail rapid transit timetabling under dynamic passengerdemandrdquo Transportation Research Part B Methodological vol70 pp 134ndash150 2014

[8] A Schobel ldquoAn eigenmodel for iterative line planning time-tabling and vehicle scheduling in public transportationrdquo Trans-portation Research Part C Emerging Technologies vol 74 pp348ndash365 2017

[9] A Berbey R Galan J D Sanz Bobi and R Caballero ldquoA fuzzylogic approach to modelling the passengersrsquo flow and dwellingtimerdquo in Proceedings of the 18th International Conference onUrban Transport and the Environment UT12 pp 359ndash369Spain May 2012

[10] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research Part C Emerging Technologies vol 36 pp212ndash230 2013

[11] H Niu X Zhou and R Gao ldquoTrain scheduling for minimizingpassenger waiting time with time-dependent demand and skip-stop patterns nonlinear integer programming models withlinear constraintsrdquoTransportation Research Part BMethodolog-ical vol 76 pp 117ndash135 2015

[12] V Cacchiani A Caprara and P Toth ldquoA column generationapproach to train timetabling on a corridorrdquo 4OR vol 6 no2 pp 125ndash142 2008

[13] T Dollevoet D Huisman M Schmidt and A Schobel ldquoDelaymanagement with rerouting of passengersrdquo TransportationScience vol 46 no 1 pp 74ndash89 2012

[14] T Dollevoet F Corman A DrsquoAriano and D Huisman ldquoAniterative optimization framework for delay management andtrain schedulingrdquo Flexible Services and Manufacturing Journalvol 26 no 4 pp 490ndash515 2014

[15] L G Kroon LW P Peeters J CWagenaar and R A ZuidwijkldquoFlexible connections in PESP models for cyclic passengerrailway timetablingrdquo Transportation Science vol 48 no 1 pp136ndash154 2014

[16] X Luan J Miao L Meng F Corman and G Lodewijks ldquoIn-tegrated optimization on train scheduling and preventivemain-tenance time slots planningrdquo Transportation Research Part CEmerging Technologies vol 80 pp 329ndash359 2017

[17] S Su X Li T Tang and Z Gao ldquoA subway train timetable opti-mization approach based on energy-efficient operation strat-egyrdquo IEEE Transactions on Intelligent Transportation Systemsvol 14 no 2 pp 883ndash893 2013

[18] X Li and K Lo Hong ldquoAn energy-efficient scheduling andspeed control approach for metro rail operationsrdquo Transporta-tion Research Part B Methodological vol 64 pp 73ndash89 2014

[19] A Higgins E Kozan and L Ferreira ldquoOptimal scheduling oftrains on a single line trackrdquo Transportation Research Part BMethodological vol 30 no 2 pp 147ndash161 1996

[20] JWang Y Yu R Kang and JWang ldquoA novel space-time-speedmethod for increasing the passing capacity with safety guaran-teed of railway stationrdquo Journal of Advanced Transportation vol2017 Article ID 6381718 2017

[21] X Xu K Li and L Yang ldquoScheduling heterogeneous train traf-fic on double tracks with efficient dispatching rulesrdquo Transpor-tation Research Part B Methodological vol 78 pp 364ndash3842015

[22] X Zhou and M Zhong ldquoSingle-track train timetabling withguaranteed optimality branch-and-bound algorithms withenhanced lower boundsrdquo Transportation Research Part BMethodological vol 41 no 3 pp 320ndash341 2007

[23] L Yang K Li Z Gao and X Li ldquoOptimizing trains movementon a railway networkrdquo Omega vol 40 no 5 pp 619ndash633 2012

[24] S Yang K Yang Z Gao L Yang and J Shi ldquoLast-Train Time-tabling under Transfer Demand Uncertainty Mean-VarianceModel andHeuristic Solutionrdquo Journal of Advanced Transporta-tion vol 2017 pp 1ndash13 2017

[25] L Yang Y Zhang S Li and Y Gao ldquoA two-stage stochasticoptimization model for the transfer activity choice in metronetworksrdquo Transportation Research Part B Methodological vol83 pp 271ndash297 2016

[26] G Malavasi P Palleschi and S Ricci ldquoDriving and operationstrategies for traction-energy saving in mass rapid transitsystemsrdquo Proceedings of the Institution of Mechanical EngineersPart F Journal of Rail and Rapid Transit vol 225 no 5 pp 475ndash482 2011

[27] J T Haahr J C Wagenaar L P Veelenturf and L G Kroon ldquoAcomparison of two exact methods for passenger railway rollingstock (re)schedulingrdquo Transportation Research Part E Logisticsand Transportation Review vol 91 pp 15ndash32 2016

[28] X Yang A Chen X Li B Ning and T Tang ldquoAn energy-efficient scheduling approach to improve the utilization of


regenerative energy formetro systemsrdquoTransportation ResearchPart C Emerging Technologies vol 57 pp 13ndash29 2015

[29] Y-C Lai D-C Fan andK-LHuang ldquoOptimizing rolling stockassignment and maintenance plan for passenger railway opera-tionsrdquoComputers amp Industrial Engineering vol 85 pp 284ndash2952015

[30] V Schmid and J F Ehmke ldquoIntegrated timetabling and vehiclescheduling with balanced departure timesrdquo OR Spectrum vol37 no 4 pp 903ndash928 2015

[31] V Guihaire and J K Hao ldquoTransit network design and schedul-ing a global reviewrdquo Transportation Research Part A Policy andPractice vol 42 no 10 pp 1251ndash1273 2008

[32] L Cadarso and A Marin ldquoIntegration of timetable planningand rolling stock in rapid transit networksrdquo Annals of Opera-tions Research vol 199 pp 113ndash135 2012

[33] J MMera E CarabanoM Soler and E Castellote ldquoIncreasingmetro line capacity by optimisation of track circuit in a speedcode Automatic Train Protection systemrdquo Proceedings of theInstitution of Mechanical Engineers Part F Journal of Rail andRapid Transit vol 230 no 1 pp 165ndash180 2016

[34] S Chiusolo A Dicembre S Ricci and F Sorace ldquoAutomationof high density metro lines Rome line a case studyrdquo inProceedings of the 13th International Conference on AutomatedPeople Movers and Transit Systems 2011 From People Movers toFully Automated Urban Mass Transit pp 491ndash501 France May2011

[35] V Cacchiani F Furini and M P Kidd ldquoApproaches to a real-world Train Timetabling Problem in a railway noderdquo OMEGA- The International Journal of Management Science vol 58 pp97ndash110 2016

[36] X Xu K Li and X Li ldquoA multi-objective subway timetableoptimization approach with minimum passenger time andenergy consumptionrdquo Journal of Advanced Transportation vol50 no 1 pp 69ndash95 2016

[37] L G Kroon and LW P Peeters ldquoA variable trip time model forcyclic railway timetablingrdquoTransportation Science vol 37 no 2pp 198ndash212 2003

[38] D Sparing and R M P Goverde ldquoA cycle time optimizationmodel for generating stable periodic railway timetablesrdquo Trans-portation Research Part B Methodological vol 98 pp 198ndash2232017

[39] F Corman A DrsquoAriano A D Marra D Pacciarelli and MSama ldquoIntegrating train scheduling and delay management inreal-time railway traffic controlrdquo Transportation Research PartE Logistics amp Transportation Review 2016

[40] ADrsquoArianoD Pacciarelli andM Pranzo ldquoAbranch and boundalgorithm for scheduling trains in a railway networkrdquo EuropeanJournal of Operational Research vol 183 no 2 pp 643ndash6572007

[41] K Ghoseiri F Szidarovszky and M J Asgharpour ldquoA multi-objective train scheduling model and solutionrdquo TransportationResearch Part B Methodological vol 38 no 10 pp 927ndash9522004

[42] J Qi L Yang Y Gao S Li and Z Gao ldquoIntegrated multi-track station layout design and train scheduling models onrailway corridorsrdquo Transportation Research Part C EmergingTechnologies vol 69 pp 91ndash119 2016

[43] S Li M M Dessouky L Yang and Z Gao ldquoJoint optimaltrain regulation and passenger flow control strategy for high-frequencymetro linesrdquoTransportationResearch Part BMethod-ological vol 99 pp 113ndash137 2017

[44] A Schobel ldquoLine planning in public transportationmodels andmethodsrdquo OR Spectrum vol 34 no 3 pp 491ndash510 2012

[45] S Samanta and M K Jha ldquoModeling a rail transit alignmentconsidering different objectivesrdquo Transportation Research PartA Policy and Practice vol 45 no 1 pp 31ndash45 2011

[46] H L Fu L Nie L Y Meng B R Sperry and Z H He ldquoAhierarchical line planning approach for a large-scale high speedrail network the China caserdquo Transportation Research Part APolicy and Practice vol 75 pp 61ndash83 2015

[47] E Cascetta and P Coppola ldquoAssessment of schedule-basedand frequency-based assignmentmodels for strategic and oper-ational planning of high-speed rail servicesrdquo TransportationResearch Part A Policy and Practice vol 84 pp 93ndash108 2016

[48] J Yin T Tang L Yang Z Gao and B Ran ldquoEnergy-efficientmetro train reschedulingwith uncertain time-variant passengerdemands an approximate dynamic programming approachrdquoTransportation Research Part B Methodological vol 91 pp 178ndash210 2016

[49] A P Cucala A Fernandez C Sicre andMDomınguez ldquoFuzzyoptimal schedule of high speed train operation to minimizeenergy consumption with uncertain delays and drivers behav-ioral responserdquo Engineering Applications of Artificial Intelli-gence vol 25 no 8 pp 1548ndash1557 2012

[50] H Huang K Li and Y Wang ldquoA simulation method for ana-lyzing and evaluating rail system performance based on speedprofilerdquo Journal of Systems Science and Systems Engineering

[51] R Hooke and R Jeeves ldquoDirect search solution of numericaland statistical problemsrdquo Journal of the ACM vol 8 pp 212ndash229 1961

[52] W C Davidon ldquoVariable metric method for minimizationrdquoSIAM Journal on Optimization vol 1 no 1 pp 1ndash17 1991

[53] W Duan Y Chen and J Lai Passenger flow characteristics inBeijing Subway Line 4 Urban Rapid Rail Transit vol 26 43-4626 2013

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can be designed It is pointed out by Schobel [8] that goingthrough all these stages sequentially (ie independently)leads to unsatisfactory solutions Since they are interrelatedimportant factors andobjectives at the line planning stage andthe vehicle scheduling stage should be accounted for in thetimetabling problem

This paper proposes a rolling optimization algorithmto obtain noncyclic timetables considering time-varyingpassenger demand and the effects of congestion at sta-tions It integrates the objectives in the line planning (fre-quency) timetabling (conflicting objectives including pas-senger waitin-vehicle time and energy) and vehicle schedul-ing (train cost) The main contributions of this paper are asfollows

(1) This paper specifically models the dynamic evolutionof passenger loads on trains at each station by consideringpassenger arrival rates limited train capacity and actualpassenger alightingboarding rates associated with conges-tion This is an extension of existing studies [9] ParticularlyNiu and Zhou [10] and Niu et al [11] propose the conceptof ldquoeffective loading timerdquo that represents the actual timeinterval within which the arriving passengers can boarda train The detailed boarding and alighting processes arenot discussed in their model Wang et al [3] set a lowerbound (which accounts for details including the number ofboardingalighting passengers and their boardingalightingrates) for dwell times but consider the dwell times as vari-ables to be optimized rather than parameters that dependon arrivaldeparture times and passenger demand Impactsof gradually increasing passenger demand are analyzed inRobenek et al [2] however inmodelling they donot considerthe effects of passenger congestion

(2) This paper integrates objectives from different plan-ning stages in the train timetabling problem for which solu-tion approaches are scarce [8] Particularly Schobel [8] con-siders the line planning timetabling and vehicle schedulingin an integratedwaywith passenger- and cost-oriented objec-tives However since his eigenmodel is general and resultingapproaches are generic important objectives described inthis paper are not considered in his work that is passengerwait time dwell time and train capacity These objectives areincluded in [2 5] whereas the frequency and dwell times arefixed in their studies Besides energy is not considered in[2 5]

(3) This paper proposes a novel and effective solutionapproach which decomposes the master timetabling modelinto two subproblems The first subproblem is solved bybranch-and-bound and frequency determining algorithmsand the second one is solved by a rolling optimizationalgorithm which optimizes the departure headways fromthe shunting yard (which are equivalent to arrival times oftrains at their first station) and considers interdependentvariables in each rolling stepThe design of decision variablesin this paper is similar to [2 5] but the models and solutionapproaches are different Compared to Wang et al [3] whichaddresses the similar timetabling problem with traditionalapproaches such as sequential quadratic programming anda genetic algorithm the approach proposed in this paper

performs better computationally in terms of run time andsolution quality

The remainder of this paper is as follows Section 2reviews relevant studies on the train timetabling problem andsummarizes the research gap in the literature Section 3 firstdescribes the problem introduces formulation of differentcost functions and constructs the train timetabling modelThen the discrete time-space graph is proposed to transformthe model into a mixed integer program (MIP) Section 4discusses the solution approach for the optimization modelSection 5 presents different numerical experiments and sen-sitivity analyses on a short metro line and on Beijing MetroLine 4 to demonstrate the predictive ability of the modelas well as the effectiveness and efficiency of the proposedapproach Section 6 summarizes the works done in this paperand presents the limitations and potential topics in futurestudies

2 Literature Review

The train timetabling problem (TTP) aims to schedule trainsto transport passengers (or goods) without conflicts byspecifying train arrival and departure times at stations Itis interrelated with other planning stages that is the lineplanning stage and the vehicle scheduling stage We organizethe related literature from the perspectives of operator-oriented objectives passenger-oriented objectives and theirintegration

21 Operator-Oriented Objectives In the literature operator-oriented objectives can be found only for the noncyclicversion of the TTP [2] Most studies focus on the operationindicators for example energy [17 18] train delay [19ndash21]and train travel time [22ndash24]

The total train delay and train travel time are usuallyconsidered in the TTP for railway networks which aims tofind a feasible timetable by minimizing the profit loss result-ing from changes to the idealplanned timetable Energyis the focus and main cost for metro system operatorsMost energy is consumed in train operations [25 26] andthus obtaining an energy-efficient train timetable receivesconsiderable attention [17 18 27] Note that energy is affectedby the travel time in the segment and the segment travel timeis determined by the arrival and departure times at adjacentstations that is the timetable Hence the energy-efficienttrain timetabling problem may be extended to include otherobjectives For example Yang et al [23] consider both energyand train travel time as optimization objectives Some recentstudies also account for regenerative braking Yang et al[28] develop a scheduling approach to coordinate arrivalsand departures of trains within the same electricity supplyzones thereby effectively recovering the regenerative energyGenerally these studies focus solely on the timetabling stagewhere frequency (in line planning) is fixed and train cost (in-vehicle scheduling) is not considered

The efficient use of railway rolling stock (vehicle schedul-ing) is an important objective pursued by a railway agencyor company because of intensive capital investment in rollingstock [29] To this end Lai et al [29] develop an optimization





















3 Model Formulation










heuristic




delay







and GA







stages






station

























alphabet) - Symbol

M119886119887119909119910







Table 2 Continued














Station 2I


Down-direction

Up-direction



Segment 2I minus 2








Time

Space





Station 1 (U1)

Station 1 (W1)

Station 2 (U2)

Station 2 (W2)

Station 3 (U3)

Station 3 (W3)

Station 4 (U4)

Station 4 (W4)













119905119905end = 119905119901end +2119868minus1sum119894=1

119904119905 (119894) + 2119868sum119894=1

119889119894 (1)



32 Passenger Time




forall119894 119895 isin I(2)



119875(119894119895) (1199051 1199052) =1199052sum119905=1199051

120591119894119895 (119905) forall119894 119895 isin I (3)


119875119894 (1199051 1199052) = 2119868sum119895=119894+1

119875(119894119895) (1199051 1199052) = 2119868sum119895=119894+1

1199052sum119905=1199051

120591119894119895 (119905)

= 1199052sum119905=1199051

120582119894 (119905) forall119894 119895 isin I(4)













Time

Passengers

Remaining Capacity

Pw

it

Pwit

t0 ki+di

t0 ki+di

tki

3tki

1tki

2ta ki

Nk minus Pkt

fk(t)

tki

3 tki

2






119905119887119896119894=

min 1199050119896119894 + 119889119894max 1199051198961198942 1199050119896119894 + 119889119894 1199051198961198943 1199051198961198941 gt 1199051198961198942min 1199050119896119894 + 119889119894 1199051198961198941 1199051198961198941 le 1199051198961198942

(9)






120577119886 = 120588 (119875V1198961199050119896119894) sdot 120577119886ideal

120577119887 = 120588 (119875V1198961199050119896119894) sdot 120577119887ideal




119875119908119894119905119887119896119894




119875119887119896(119894119895) =sum119905119887119896119894119905=119905119887119896minus1119894

120591119894119895 (119905)sum119905119887119896119894119905=119905119887119896minus1119894

120582119894 (119905)sdot 119875119887119896119894



119895minus1sum119894=1

119875119887119896(119894119895) forall119896 isin K 119895 = 2 119868 119868 + 2 2119868 (13)




119875119908119894119905 = 119875119894 (0 119905) minus119896lastsum119896=1

119875119887119896119894 forall119894 isin I (14)


119879119901119908 = sum119894isinI

119905119905endsum119905=0

119875119908119894119905 (15)


119875V119896119905 =


119875119887119896119894 minus119894lastsum119894=1



(16)



119905119905endsum119905=0

119875V119896119905 (17)







119891119903 = 120572 sdot V1198962 (119905) + 120573 sdot V119896 (119905) + 120574V119896 (1199050119896119894) = V119896 (119905119887119896119894) = 0

(19)


119890119894 = int1199050119896119894+1

119905119887119896119894





MSE (120594119909 120594119910) = 1100100sum119911=1

(119890119911 minus 119890119911)2 (22)


120597MSE (120594119909 120594119910)120597120594119909 = 0120597MSE (120594119909 120594119910)120597120594119910 = 0

(23)


119864total = 2119868minus1sum119894=1

119870sum119896=1


119896119905119887119896119894)

(24)



119870sum119896=1


119896119905119887119896119894)

(25)















119870119896=1 119879119896round3600 ) + 119888119900 sdot 119870

sdot sum2119868minus1119894=1 119904119897 (119894)1000 (30)



Minimize


Subject to

11990501198701 = 119905119901end (32)

119904119905 (119868) = 119879turn (33)

119904min119905 (119894) le 119904119905 (119894) le 119904max

119905 (119894) forall119894 isin I (34)












(37)






(38)


Q119894119905 = 119876119894119905 minus 10038171003817100381710038171003817PB1198941119905



V119896119905

=



(40)




119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W



119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0


119894 119906119896 isin U119894+1

(43)







1 forall119896 isin K 1 (45)


lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)



119868+1 forall119896 isin K (47)


119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W




119894 119896 isin K 1 119894 isin I (49)




4 Solution Approach




119890119894 +119872119901


2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






















3 Model Formulation










heuristic




delay







and GA







stages






station

























alphabet) - Symbol

M119886119887119909119910







Table 2 Continued














Station 2I


Down-direction

Up-direction



Segment 2I minus 2








Time

Space





Station 1 (U1)

Station 1 (W1)

Station 2 (U2)

Station 2 (W2)

Station 3 (U3)

Station 3 (W3)

Station 4 (U4)

Station 4 (W4)













119905119905end = 119905119901end +2119868minus1sum119894=1

119904119905 (119894) + 2119868sum119894=1

119889119894 (1)



32 Passenger Time




forall119894 119895 isin I(2)



119875(119894119895) (1199051 1199052) =1199052sum119905=1199051

120591119894119895 (119905) forall119894 119895 isin I (3)


119875119894 (1199051 1199052) = 2119868sum119895=119894+1

119875(119894119895) (1199051 1199052) = 2119868sum119895=119894+1

1199052sum119905=1199051

120591119894119895 (119905)

= 1199052sum119905=1199051

120582119894 (119905) forall119894 119895 isin I(4)













Time

Passengers

Remaining Capacity

Pw

it

Pwit

t0 ki+di

t0 ki+di

tki

3tki

1tki

2ta ki

Nk minus Pkt

fk(t)

tki

3 tki

2






119905119887119896119894=

min 1199050119896119894 + 119889119894max 1199051198961198942 1199050119896119894 + 119889119894 1199051198961198943 1199051198961198941 gt 1199051198961198942min 1199050119896119894 + 119889119894 1199051198961198941 1199051198961198941 le 1199051198961198942

(9)






120577119886 = 120588 (119875V1198961199050119896119894) sdot 120577119886ideal

120577119887 = 120588 (119875V1198961199050119896119894) sdot 120577119887ideal




119875119908119894119905119887119896119894




119875119887119896(119894119895) =sum119905119887119896119894119905=119905119887119896minus1119894

120591119894119895 (119905)sum119905119887119896119894119905=119905119887119896minus1119894

120582119894 (119905)sdot 119875119887119896119894



119895minus1sum119894=1

119875119887119896(119894119895) forall119896 isin K 119895 = 2 119868 119868 + 2 2119868 (13)




119875119908119894119905 = 119875119894 (0 119905) minus119896lastsum119896=1

119875119887119896119894 forall119894 isin I (14)


119879119901119908 = sum119894isinI

119905119905endsum119905=0

119875119908119894119905 (15)


119875V119896119905 =


119875119887119896119894 minus119894lastsum119894=1



(16)



119905119905endsum119905=0

119875V119896119905 (17)







119891119903 = 120572 sdot V1198962 (119905) + 120573 sdot V119896 (119905) + 120574V119896 (1199050119896119894) = V119896 (119905119887119896119894) = 0

(19)


119890119894 = int1199050119896119894+1

119905119887119896119894





MSE (120594119909 120594119910) = 1100100sum119911=1

(119890119911 minus 119890119911)2 (22)


120597MSE (120594119909 120594119910)120597120594119909 = 0120597MSE (120594119909 120594119910)120597120594119910 = 0

(23)


119864total = 2119868minus1sum119894=1

119870sum119896=1


119896119905119887119896119894)

(24)



119870sum119896=1


119896119905119887119896119894)

(25)















119870119896=1 119879119896round3600 ) + 119888119900 sdot 119870

sdot sum2119868minus1119894=1 119904119897 (119894)1000 (30)



Minimize


Subject to

11990501198701 = 119905119901end (32)

119904119905 (119868) = 119879turn (33)

119904min119905 (119894) le 119904119905 (119894) le 119904max

119905 (119894) forall119894 isin I (34)












(37)






(38)


Q119894119905 = 119876119894119905 minus 10038171003817100381710038171003817PB1198941119905



V119896119905

=



(40)




119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W



119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0


119894 119906119896 isin U119894+1

(43)







1 forall119896 isin K 1 (45)


lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)



119868+1 forall119896 isin K (47)


119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W




119894 119896 isin K 1 119894 isin I (49)




4 Solution Approach




119890119894 +119872119901


2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






heuristic




delay







and GA







stages






station

























alphabet) - Symbol

M119886119887119909119910







Table 2 Continued














Station 2I


Down-direction

Up-direction



Segment 2I minus 2








Time

Space





Station 1 (U1)

Station 1 (W1)

Station 2 (U2)

Station 2 (W2)

Station 3 (U3)

Station 3 (W3)

Station 4 (U4)

Station 4 (W4)













119905119905end = 119905119901end +2119868minus1sum119894=1

119904119905 (119894) + 2119868sum119894=1

119889119894 (1)



32 Passenger Time




forall119894 119895 isin I(2)



119875(119894119895) (1199051 1199052) =1199052sum119905=1199051

120591119894119895 (119905) forall119894 119895 isin I (3)


119875119894 (1199051 1199052) = 2119868sum119895=119894+1

119875(119894119895) (1199051 1199052) = 2119868sum119895=119894+1

1199052sum119905=1199051

120591119894119895 (119905)

= 1199052sum119905=1199051

120582119894 (119905) forall119894 119895 isin I(4)













Time

Passengers

Remaining Capacity

Pw

it

Pwit

t0 ki+di

t0 ki+di

tki

3tki

1tki

2ta ki

Nk minus Pkt

fk(t)

tki

3 tki

2






119905119887119896119894=

min 1199050119896119894 + 119889119894max 1199051198961198942 1199050119896119894 + 119889119894 1199051198961198943 1199051198961198941 gt 1199051198961198942min 1199050119896119894 + 119889119894 1199051198961198941 1199051198961198941 le 1199051198961198942

(9)






120577119886 = 120588 (119875V1198961199050119896119894) sdot 120577119886ideal

120577119887 = 120588 (119875V1198961199050119896119894) sdot 120577119887ideal




119875119908119894119905119887119896119894




119875119887119896(119894119895) =sum119905119887119896119894119905=119905119887119896minus1119894

120591119894119895 (119905)sum119905119887119896119894119905=119905119887119896minus1119894

120582119894 (119905)sdot 119875119887119896119894



119895minus1sum119894=1

119875119887119896(119894119895) forall119896 isin K 119895 = 2 119868 119868 + 2 2119868 (13)




119875119908119894119905 = 119875119894 (0 119905) minus119896lastsum119896=1

119875119887119896119894 forall119894 isin I (14)


119879119901119908 = sum119894isinI

119905119905endsum119905=0

119875119908119894119905 (15)


119875V119896119905 =


119875119887119896119894 minus119894lastsum119894=1



(16)



119905119905endsum119905=0

119875V119896119905 (17)







119891119903 = 120572 sdot V1198962 (119905) + 120573 sdot V119896 (119905) + 120574V119896 (1199050119896119894) = V119896 (119905119887119896119894) = 0

(19)


119890119894 = int1199050119896119894+1

119905119887119896119894





MSE (120594119909 120594119910) = 1100100sum119911=1

(119890119911 minus 119890119911)2 (22)


120597MSE (120594119909 120594119910)120597120594119909 = 0120597MSE (120594119909 120594119910)120597120594119910 = 0

(23)


119864total = 2119868minus1sum119894=1

119870sum119896=1


119896119905119887119896119894)

(24)



119870sum119896=1


119896119905119887119896119894)

(25)















119870119896=1 119879119896round3600 ) + 119888119900 sdot 119870

sdot sum2119868minus1119894=1 119904119897 (119894)1000 (30)



Minimize


Subject to

11990501198701 = 119905119901end (32)

119904119905 (119868) = 119879turn (33)

119904min119905 (119894) le 119904119905 (119894) le 119904max

119905 (119894) forall119894 isin I (34)












(37)






(38)


Q119894119905 = 119876119894119905 minus 10038171003817100381710038171003817PB1198941119905



V119896119905

=



(40)




119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W



119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0


119894 119906119896 isin U119894+1

(43)







1 forall119896 isin K 1 (45)


lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)



119868+1 forall119896 isin K (47)


119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W




119894 119896 isin K 1 119894 isin I (49)




4 Solution Approach




119890119894 +119872119901


2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


















alphabet) - Symbol

M119886119887119909119910







Table 2 Continued














Station 2I


Down-direction

Up-direction



Segment 2I minus 2








Time

Space





Station 1 (U1)

Station 1 (W1)

Station 2 (U2)

Station 2 (W2)

Station 3 (U3)

Station 3 (W3)

Station 4 (U4)

Station 4 (W4)













119905119905end = 119905119901end +2119868minus1sum119894=1

119904119905 (119894) + 2119868sum119894=1

119889119894 (1)



32 Passenger Time




forall119894 119895 isin I(2)



119875(119894119895) (1199051 1199052) =1199052sum119905=1199051

120591119894119895 (119905) forall119894 119895 isin I (3)


119875119894 (1199051 1199052) = 2119868sum119895=119894+1

119875(119894119895) (1199051 1199052) = 2119868sum119895=119894+1

1199052sum119905=1199051

120591119894119895 (119905)

= 1199052sum119905=1199051

120582119894 (119905) forall119894 119895 isin I(4)













Time

Passengers

Remaining Capacity

Pw

it

Pwit

t0 ki+di

t0 ki+di

tki

3tki

1tki

2ta ki

Nk minus Pkt

fk(t)

tki

3 tki

2






119905119887119896119894=

min 1199050119896119894 + 119889119894max 1199051198961198942 1199050119896119894 + 119889119894 1199051198961198943 1199051198961198941 gt 1199051198961198942min 1199050119896119894 + 119889119894 1199051198961198941 1199051198961198941 le 1199051198961198942

(9)






120577119886 = 120588 (119875V1198961199050119896119894) sdot 120577119886ideal

120577119887 = 120588 (119875V1198961199050119896119894) sdot 120577119887ideal




119875119908119894119905119887119896119894




119875119887119896(119894119895) =sum119905119887119896119894119905=119905119887119896minus1119894

120591119894119895 (119905)sum119905119887119896119894119905=119905119887119896minus1119894

120582119894 (119905)sdot 119875119887119896119894



119895minus1sum119894=1

119875119887119896(119894119895) forall119896 isin K 119895 = 2 119868 119868 + 2 2119868 (13)




119875119908119894119905 = 119875119894 (0 119905) minus119896lastsum119896=1

119875119887119896119894 forall119894 isin I (14)


119879119901119908 = sum119894isinI

119905119905endsum119905=0

119875119908119894119905 (15)


119875V119896119905 =


119875119887119896119894 minus119894lastsum119894=1



(16)



119905119905endsum119905=0

119875V119896119905 (17)







119891119903 = 120572 sdot V1198962 (119905) + 120573 sdot V119896 (119905) + 120574V119896 (1199050119896119894) = V119896 (119905119887119896119894) = 0

(19)


119890119894 = int1199050119896119894+1

119905119887119896119894





MSE (120594119909 120594119910) = 1100100sum119911=1

(119890119911 minus 119890119911)2 (22)


120597MSE (120594119909 120594119910)120597120594119909 = 0120597MSE (120594119909 120594119910)120597120594119910 = 0

(23)


119864total = 2119868minus1sum119894=1

119870sum119896=1


119896119905119887119896119894)

(24)



119870sum119896=1


119896119905119887119896119894)

(25)















119870119896=1 119879119896round3600 ) + 119888119900 sdot 119870

sdot sum2119868minus1119894=1 119904119897 (119894)1000 (30)



Minimize


Subject to

11990501198701 = 119905119901end (32)

119904119905 (119868) = 119879turn (33)

119904min119905 (119894) le 119904119905 (119894) le 119904max

119905 (119894) forall119894 isin I (34)












(37)






(38)


Q119894119905 = 119876119894119905 minus 10038171003817100381710038171003817PB1198941119905



V119896119905

=



(40)




119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W



119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0


119894 119906119896 isin U119894+1

(43)







1 forall119896 isin K 1 (45)


lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)



119868+1 forall119896 isin K (47)


119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W




119894 119896 isin K 1 119894 isin I (49)




4 Solution Approach




119890119894 +119872119901


2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Table 2 Continued














Station 2I


Down-direction

Up-direction



Segment 2I minus 2








Time

Space





Station 1 (U1)

Station 1 (W1)

Station 2 (U2)

Station 2 (W2)

Station 3 (U3)

Station 3 (W3)

Station 4 (U4)

Station 4 (W4)













119905119905end = 119905119901end +2119868minus1sum119894=1

119904119905 (119894) + 2119868sum119894=1

119889119894 (1)



32 Passenger Time




forall119894 119895 isin I(2)



119875(119894119895) (1199051 1199052) =1199052sum119905=1199051

120591119894119895 (119905) forall119894 119895 isin I (3)


119875119894 (1199051 1199052) = 2119868sum119895=119894+1

119875(119894119895) (1199051 1199052) = 2119868sum119895=119894+1

1199052sum119905=1199051

120591119894119895 (119905)

= 1199052sum119905=1199051

120582119894 (119905) forall119894 119895 isin I(4)













Time

Passengers

Remaining Capacity

Pw

it

Pwit

t0 ki+di

t0 ki+di

tki

3tki

1tki

2ta ki

Nk minus Pkt

fk(t)

tki

3 tki

2






119905119887119896119894=

min 1199050119896119894 + 119889119894max 1199051198961198942 1199050119896119894 + 119889119894 1199051198961198943 1199051198961198941 gt 1199051198961198942min 1199050119896119894 + 119889119894 1199051198961198941 1199051198961198941 le 1199051198961198942

(9)






120577119886 = 120588 (119875V1198961199050119896119894) sdot 120577119886ideal

120577119887 = 120588 (119875V1198961199050119896119894) sdot 120577119887ideal




119875119908119894119905119887119896119894




119875119887119896(119894119895) =sum119905119887119896119894119905=119905119887119896minus1119894

120591119894119895 (119905)sum119905119887119896119894119905=119905119887119896minus1119894

120582119894 (119905)sdot 119875119887119896119894



119895minus1sum119894=1

119875119887119896(119894119895) forall119896 isin K 119895 = 2 119868 119868 + 2 2119868 (13)




119875119908119894119905 = 119875119894 (0 119905) minus119896lastsum119896=1

119875119887119896119894 forall119894 isin I (14)


119879119901119908 = sum119894isinI

119905119905endsum119905=0

119875119908119894119905 (15)


119875V119896119905 =


119875119887119896119894 minus119894lastsum119894=1



(16)



119905119905endsum119905=0

119875V119896119905 (17)







119891119903 = 120572 sdot V1198962 (119905) + 120573 sdot V119896 (119905) + 120574V119896 (1199050119896119894) = V119896 (119905119887119896119894) = 0

(19)


119890119894 = int1199050119896119894+1

119905119887119896119894





MSE (120594119909 120594119910) = 1100100sum119911=1

(119890119911 minus 119890119911)2 (22)


120597MSE (120594119909 120594119910)120597120594119909 = 0120597MSE (120594119909 120594119910)120597120594119910 = 0

(23)


119864total = 2119868minus1sum119894=1

119870sum119896=1


119896119905119887119896119894)

(24)



119870sum119896=1


119896119905119887119896119894)

(25)















119870119896=1 119879119896round3600 ) + 119888119900 sdot 119870

sdot sum2119868minus1119894=1 119904119897 (119894)1000 (30)



Minimize


Subject to

11990501198701 = 119905119901end (32)

119904119905 (119868) = 119879turn (33)

119904min119905 (119894) le 119904119905 (119894) le 119904max

119905 (119894) forall119894 isin I (34)












(37)






(38)


Q119894119905 = 119876119894119905 minus 10038171003817100381710038171003817PB1198941119905



V119896119905

=



(40)




119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W



119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0


119894 119906119896 isin U119894+1

(43)







1 forall119896 isin K 1 (45)


lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)



119868+1 forall119896 isin K (47)


119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W




119894 119896 isin K 1 119894 isin I (49)




4 Solution Approach




119890119894 +119872119901


2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Time

Space





Station 1 (U1)

Station 1 (W1)

Station 2 (U2)

Station 2 (W2)

Station 3 (U3)

Station 3 (W3)

Station 4 (U4)

Station 4 (W4)













119905119905end = 119905119901end +2119868minus1sum119894=1

119904119905 (119894) + 2119868sum119894=1

119889119894 (1)



32 Passenger Time




forall119894 119895 isin I(2)



119875(119894119895) (1199051 1199052) =1199052sum119905=1199051

120591119894119895 (119905) forall119894 119895 isin I (3)


119875119894 (1199051 1199052) = 2119868sum119895=119894+1

119875(119894119895) (1199051 1199052) = 2119868sum119895=119894+1

1199052sum119905=1199051

120591119894119895 (119905)

= 1199052sum119905=1199051

120582119894 (119905) forall119894 119895 isin I(4)













Time

Passengers

Remaining Capacity

Pw

it

Pwit

t0 ki+di

t0 ki+di

tki

3tki

1tki

2ta ki

Nk minus Pkt

fk(t)

tki

3 tki

2






119905119887119896119894=

min 1199050119896119894 + 119889119894max 1199051198961198942 1199050119896119894 + 119889119894 1199051198961198943 1199051198961198941 gt 1199051198961198942min 1199050119896119894 + 119889119894 1199051198961198941 1199051198961198941 le 1199051198961198942

(9)






120577119886 = 120588 (119875V1198961199050119896119894) sdot 120577119886ideal

120577119887 = 120588 (119875V1198961199050119896119894) sdot 120577119887ideal




119875119908119894119905119887119896119894




119875119887119896(119894119895) =sum119905119887119896119894119905=119905119887119896minus1119894

120591119894119895 (119905)sum119905119887119896119894119905=119905119887119896minus1119894

120582119894 (119905)sdot 119875119887119896119894



119895minus1sum119894=1

119875119887119896(119894119895) forall119896 isin K 119895 = 2 119868 119868 + 2 2119868 (13)




119875119908119894119905 = 119875119894 (0 119905) minus119896lastsum119896=1

119875119887119896119894 forall119894 isin I (14)


119879119901119908 = sum119894isinI

119905119905endsum119905=0

119875119908119894119905 (15)


119875V119896119905 =


119875119887119896119894 minus119894lastsum119894=1



(16)



119905119905endsum119905=0

119875V119896119905 (17)







119891119903 = 120572 sdot V1198962 (119905) + 120573 sdot V119896 (119905) + 120574V119896 (1199050119896119894) = V119896 (119905119887119896119894) = 0

(19)


119890119894 = int1199050119896119894+1

119905119887119896119894





MSE (120594119909 120594119910) = 1100100sum119911=1

(119890119911 minus 119890119911)2 (22)


120597MSE (120594119909 120594119910)120597120594119909 = 0120597MSE (120594119909 120594119910)120597120594119910 = 0

(23)


119864total = 2119868minus1sum119894=1

119870sum119896=1


119896119905119887119896119894)

(24)



119870sum119896=1


119896119905119887119896119894)

(25)















119870119896=1 119879119896round3600 ) + 119888119900 sdot 119870

sdot sum2119868minus1119894=1 119904119897 (119894)1000 (30)



Minimize


Subject to

11990501198701 = 119905119901end (32)

119904119905 (119868) = 119879turn (33)

119904min119905 (119894) le 119904119905 (119894) le 119904max

119905 (119894) forall119894 isin I (34)












(37)






(38)


Q119894119905 = 119876119894119905 minus 10038171003817100381710038171003817PB1198941119905



V119896119905

=



(40)




119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W



119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0


119894 119906119896 isin U119894+1

(43)







1 forall119896 isin K 1 (45)


lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)



119868+1 forall119896 isin K (47)


119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W




119894 119896 isin K 1 119894 isin I (49)




4 Solution Approach




119890119894 +119872119901


2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




119875(119894119895) (1199051 1199052) =1199052sum119905=1199051

120591119894119895 (119905) forall119894 119895 isin I (3)


119875119894 (1199051 1199052) = 2119868sum119895=119894+1

119875(119894119895) (1199051 1199052) = 2119868sum119895=119894+1

1199052sum119905=1199051

120591119894119895 (119905)

= 1199052sum119905=1199051

120582119894 (119905) forall119894 119895 isin I(4)













Time

Passengers

Remaining Capacity

Pw

it

Pwit

t0 ki+di

t0 ki+di

tki

3tki

1tki

2ta ki

Nk minus Pkt

fk(t)

tki

3 tki

2






119905119887119896119894=

min 1199050119896119894 + 119889119894max 1199051198961198942 1199050119896119894 + 119889119894 1199051198961198943 1199051198961198941 gt 1199051198961198942min 1199050119896119894 + 119889119894 1199051198961198941 1199051198961198941 le 1199051198961198942

(9)






120577119886 = 120588 (119875V1198961199050119896119894) sdot 120577119886ideal

120577119887 = 120588 (119875V1198961199050119896119894) sdot 120577119887ideal




119875119908119894119905119887119896119894




119875119887119896(119894119895) =sum119905119887119896119894119905=119905119887119896minus1119894

120591119894119895 (119905)sum119905119887119896119894119905=119905119887119896minus1119894

120582119894 (119905)sdot 119875119887119896119894



119895minus1sum119894=1

119875119887119896(119894119895) forall119896 isin K 119895 = 2 119868 119868 + 2 2119868 (13)




119875119908119894119905 = 119875119894 (0 119905) minus119896lastsum119896=1

119875119887119896119894 forall119894 isin I (14)


119879119901119908 = sum119894isinI

119905119905endsum119905=0

119875119908119894119905 (15)


119875V119896119905 =


119875119887119896119894 minus119894lastsum119894=1



(16)



119905119905endsum119905=0

119875V119896119905 (17)







119891119903 = 120572 sdot V1198962 (119905) + 120573 sdot V119896 (119905) + 120574V119896 (1199050119896119894) = V119896 (119905119887119896119894) = 0

(19)


119890119894 = int1199050119896119894+1

119905119887119896119894





MSE (120594119909 120594119910) = 1100100sum119911=1

(119890119911 minus 119890119911)2 (22)


120597MSE (120594119909 120594119910)120597120594119909 = 0120597MSE (120594119909 120594119910)120597120594119910 = 0

(23)


119864total = 2119868minus1sum119894=1

119870sum119896=1


119896119905119887119896119894)

(24)



119870sum119896=1


119896119905119887119896119894)

(25)















119870119896=1 119879119896round3600 ) + 119888119900 sdot 119870

sdot sum2119868minus1119894=1 119904119897 (119894)1000 (30)



Minimize


Subject to

11990501198701 = 119905119901end (32)

119904119905 (119868) = 119879turn (33)

119904min119905 (119894) le 119904119905 (119894) le 119904max

119905 (119894) forall119894 isin I (34)












(37)






(38)


Q119894119905 = 119876119894119905 minus 10038171003817100381710038171003817PB1198941119905



V119896119905

=



(40)




119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W



119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0


119894 119906119896 isin U119894+1

(43)







1 forall119896 isin K 1 (45)


lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)



119868+1 forall119896 isin K (47)


119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W




119894 119896 isin K 1 119894 isin I (49)




4 Solution Approach




119890119894 +119872119901


2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

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ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






120577119886 = 120588 (119875V1198961199050119896119894) sdot 120577119886ideal

120577119887 = 120588 (119875V1198961199050119896119894) sdot 120577119887ideal




119875119908119894119905119887119896119894




119875119887119896(119894119895) =sum119905119887119896119894119905=119905119887119896minus1119894

120591119894119895 (119905)sum119905119887119896119894119905=119905119887119896minus1119894

120582119894 (119905)sdot 119875119887119896119894



119895minus1sum119894=1

119875119887119896(119894119895) forall119896 isin K 119895 = 2 119868 119868 + 2 2119868 (13)




119875119908119894119905 = 119875119894 (0 119905) minus119896lastsum119896=1

119875119887119896119894 forall119894 isin I (14)


119879119901119908 = sum119894isinI

119905119905endsum119905=0

119875119908119894119905 (15)


119875V119896119905 =


119875119887119896119894 minus119894lastsum119894=1



(16)



119905119905endsum119905=0

119875V119896119905 (17)







119891119903 = 120572 sdot V1198962 (119905) + 120573 sdot V119896 (119905) + 120574V119896 (1199050119896119894) = V119896 (119905119887119896119894) = 0

(19)


119890119894 = int1199050119896119894+1

119905119887119896119894





MSE (120594119909 120594119910) = 1100100sum119911=1

(119890119911 minus 119890119911)2 (22)


120597MSE (120594119909 120594119910)120597120594119909 = 0120597MSE (120594119909 120594119910)120597120594119910 = 0

(23)


119864total = 2119868minus1sum119894=1

119870sum119896=1


119896119905119887119896119894)

(24)



119870sum119896=1


119896119905119887119896119894)

(25)















119870119896=1 119879119896round3600 ) + 119888119900 sdot 119870

sdot sum2119868minus1119894=1 119904119897 (119894)1000 (30)



Minimize


Subject to

11990501198701 = 119905119901end (32)

119904119905 (119868) = 119879turn (33)

119904min119905 (119894) le 119904119905 (119894) le 119904max

119905 (119894) forall119894 isin I (34)












(37)






(38)


Q119894119905 = 119876119894119905 minus 10038171003817100381710038171003817PB1198941119905



V119896119905

=



(40)




119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W



119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0


119894 119906119896 isin U119894+1

(43)







1 forall119896 isin K 1 (45)


lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)



119868+1 forall119896 isin K (47)


119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W




119894 119896 isin K 1 119894 isin I (49)




4 Solution Approach




119890119894 +119872119901


2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





119891119903 = 120572 sdot V1198962 (119905) + 120573 sdot V119896 (119905) + 120574V119896 (1199050119896119894) = V119896 (119905119887119896119894) = 0

(19)


119890119894 = int1199050119896119894+1

119905119887119896119894





MSE (120594119909 120594119910) = 1100100sum119911=1

(119890119911 minus 119890119911)2 (22)


120597MSE (120594119909 120594119910)120597120594119909 = 0120597MSE (120594119909 120594119910)120597120594119910 = 0

(23)


119864total = 2119868minus1sum119894=1

119870sum119896=1


119896119905119887119896119894)

(24)



119870sum119896=1


119896119905119887119896119894)

(25)















119870119896=1 119879119896round3600 ) + 119888119900 sdot 119870

sdot sum2119868minus1119894=1 119904119897 (119894)1000 (30)



Minimize


Subject to

11990501198701 = 119905119901end (32)

119904119905 (119868) = 119879turn (33)

119904min119905 (119894) le 119904119905 (119894) le 119904max

119905 (119894) forall119894 isin I (34)












(37)






(38)


Q119894119905 = 119876119894119905 minus 10038171003817100381710038171003817PB1198941119905



V119896119905

=



(40)




119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W



119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0


119894 119906119896 isin U119894+1

(43)







1 forall119896 isin K 1 (45)


lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)



119868+1 forall119896 isin K (47)


119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W




119894 119896 isin K 1 119894 isin I (49)




4 Solution Approach




119890119894 +119872119901


2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia







119870119896=1 119879119896round3600 ) + 119888119900 sdot 119870

sdot sum2119868minus1119894=1 119904119897 (119894)1000 (30)



Minimize


Subject to

11990501198701 = 119905119901end (32)

119904119905 (119868) = 119879turn (33)

119904min119905 (119894) le 119904119905 (119894) le 119904max

119905 (119894) forall119894 isin I (34)












(37)






(38)


Q119894119905 = 119876119894119905 minus 10038171003817100381710038171003817PB1198941119905



V119896119905

=



(40)




119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W



119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0


119894 119906119896 isin U119894+1

(43)







1 forall119896 isin K 1 (45)


lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)



119868+1 forall119896 isin K (47)


119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W




119894 119896 isin K 1 119894 isin I (49)




4 Solution Approach




119890119894 +119872119901


2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



V119896119905

=



(40)




119890119894 = (120594119909 sdot Δ (119908119896 119906119896) + 120594119910) sdot 119904119897 (119894) 119908119896 isin W



119864total = 2119868minus1sum119894=1

119870sum119896=1

119905119905endsum119905=0


119894 119906119896 isin U119894+1

(43)







1 forall119896 isin K 1 (45)


lent to

120587 (119906119870) = 119905119905end 119906119870 isin U1 (46)



119868+1 forall119896 isin K (47)


119904min119905 (119894) le Δ (119908119896 119906119896) le 119904max

119905 (119894) 119908119896 isin W




119894 119896 isin K 1 119894 isin I (49)




4 Solution Approach




119890119894 +119872119901


2119868sum119895=119894+1

119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905) sdot 119890119894(51)






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






119894=1

119888119890 sdot 119872119896 sdot 119870 sdot 119890119894


119894sum119903=1

119905119901

endsum119905=0

120591119903119895 (119905)]](52)



Minimize


total (53)




min 997888c sdot 997888x 119879 (54)

st A sdot 997888x 119879 le 997888b119879

Aeq sdot 997888x 119879 = 997888997888beq119879(55)


997888b119879

and997888997888beq119879


997888b119879



997888120598 j = (0 0 1 0 0) 119895 = 1 2119868 minus 1 (56)



119894 = 1 2 2119868 minus 1(57)





119905 (119894) for119894 = 1 2 2119868 minus 1 997888b 119894 = 119904max











119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








119879pw (119896) = 2119868sum119894=1

1199051198871198961sum119905=119905119887119896minus11

Q119894119905 forall119896 isin K (58)


119879pt (119896) =1199051198871198962119868sum119905=11990501198961

V119896119905 forall119896 isin K (59)







(61)
























Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia

















Cost

Headway



ℎＧＣＨ











12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia






12120590119898sum119895=1







1 2 3Segment 1

4

8 7 6Segment 7

5

Segment 2 Segment 3

Segment 5Segment 6

Segment 4Turnaround

Up-direction

Down-direction

2000 m1600 m1800 m


Mul

tiplie

r2000 4000 6000 8000 10000 120000

Time (s)

Case 1Case 2

Case 3Case 4

0

05

1

15

2

25




119870=119870119880 The GGA result









119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia








119870=53 83360 14523 16148 6 12496 141Cyclic 119880119870=60 73561 14406 18069 7 +infin -

0-4800 Non-cyclic 119873119870=24 23171 49470 71411 6 43586 -

Case 2


119870=24 26102 49988 71411 6 45407 401Cyclic 119880119870=23 27619 50347 68667 6 45681 459Cyclic 119880119870=25 24989 49859 74156 7 45441 408

0-7200 Non-cyclic 119873119870=45 43948 11292 13615 8 88022 -


119870=48 43975 11319 14438 8 90243 246Cyclic 119880119870=45 47268 11417 13615 8 90345 257

0-10800 Non-cyclic 119873119870=60 60280 14301 18069 7 11621 -

Case 4


119870=60 64638 14330 18069 7 11875 214Cyclic 119880119870=59 65981 14381 17795 7 11899 234Cyclic 119880119870=61 63572 14327 18344 7 11884 221



RO resultGGA result

Best cyclic result







Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Stat

ion

inde

x

1

2

3

45

6

7

8

2000 4000 6000 8000 10000 120000Time (s)


Stat

ion

inde

x

1

2

3

45

6

7

8

1000 2000 3000 4000 5000 60000Time (s)


Stat

ion

inde

x

1

2

3

45

6

78

1000 2000 3000 4000 5000 6000 7000 80000Time (s)


Stat

ion

inde

x

2000 4000 6000 8000 10000 120000Time (s)

1

2

3

45

6

7

8




















24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia












24 20 2416

Pass

enge

r dem

and

20Station index

12 16


0 16 20St 12 16128 8

001020304

(Pas

seng

ers

Seco

nd)





Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



Time (s)

0

900

180

0

540

0

900

0

990

0

Mul

tiplie

r



0

05

1

15

2


















3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





3869 s Cyclic 119880best119870=44 53237 30943 72385 30 39925 2038




119870=25 38100 15533 39911 17 26459 5353

Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)


Stat

ion

inde

x

2000 4000 6000 8000 10000 12000 14000 16000 180000

Time (s)

1(48)2(47)3(46)4(45)5(44)6(43)7(42)8(41)9(40)

10(39)11(38)12(37)13(36)14(35)15(34)16(33)17(32)18(31)19(30)20(29)21(28)22(27)23(26)24(25)





times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia



times108

205

21

215

22

225

Pass

enge

r tim

e (s)

25 30 35 40 50 75 100 15020




Peak

hou

r fre

quen

cy (t

rain

s)

12

13

14

15

16

17

18

19

20





times108

2

202

204

206

208

21

212

214

Pass

enge

r tim

e (s)




Roun

d tr

ip ti

me (

s)

5000

5200

5400

5600

5800

6000

6200











200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia




200 400 600 800 1000 14000 1200


175

18

185

19

195

2

205

21

Obj

ectiv

e fun

ctio

n




6 Conclusions












Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia





Acknowledgments


References


























































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia































RoboticsJournal of




VLSI Design



Shock and Vibration







Journal of



Volume 2018



Volume 2018


Journal of




SensorsJournal of



RotatingMachinery





Propagation






Hindawi


Advances in

Multimedia


Metro Timetabling for Time-Varying Passenger Demand and...

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