Optimized Skip-Stop Metro Line Operation Using Smart Card...

Research ArticleOptimized Skip-Stop Metro Line Operation UsingSmart Card Data

Peitong Zhang Zhanbo Sun and Xiaobo Liu

School of Transportation and Logistics Southwest Jiaotong University Chengdu China

Correspondence should be addressed to Xiaobo Liu xiaoboliuswjtucn

Received 4 June 2017 Accepted 14 August 2017 Published 13 November 2017

Academic Editor Giulio E Cantarella

Copyright copy 2017 Peitong Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Skip-stop operation is a low cost approach to improving the efficiency of metro operation and passenger travel experience Thispaper proposes a novel method to optimize the skip-stop scheme for bidirectional metro lines so that the average passenger traveltime can be minimized Different from the conventional ldquoABrdquo scheme the proposed Flexible Skip-Stop Scheme (FSSS) can betteraccommodate spatially and temporally varied passenger demand A genetic algorithm (GA) based approach is then developed toefficiently search for the optimal solution A case study is conducted based on a real world bidirectional metro line in ShenzhenChina using the time-dependent passenger demand extracted from smart card data It is found that the optimized skip-stopoperation is able to reduce the average passenger travel time and transit agencies may benefit from this scheme due to energyand operational cost savings Analyses are made to evaluate the effects of that fact that certain number of passengers fail to boardthe right train (due to skip operation) Results show that FSSS always outperforms the all-stop scheme even when most passengersof the skipped OD pairs are confused and cannot get on the right train

1 Introduction

The time required for stops at stations is usually a majortravel time component for urban transit systems Cortes etal [1] discussed different approaches and operation strategiesthat can reduce total passenger travel time With the goal ofreducing travel time and increasing overall transit operationefficiency skip-stop operation is a scheme that can have fleetsnot making all designated stops along a route Skip-stopoperation has been widely applied in bus transit based on theAB mode Under the AB mode operating fleets are labeledas either type A or type B stations are also predetermined astype A type B or type AB stations Type A fleets can onlystop at type A and type AB stations and type B fleets can onlystop at type B and type AB stations [2 3] Eberlein et al [4]proposed a real-time deadheading strategy to determine thedispatching time Deadheading vehicles and skip operationswere optimized tominimize the total passenger cost Sun andHickman [5] formulated the real-time skip-stop problem forbus operation and presented an enumeration method withgood computation speed Yu et al [6] studied the servicereliability of a bus route in the city of Dalian and proposed

a heuristic algorithm-based deadheading optimization strat-egy Fu et al [7] presented a new strategy to have all-stop andskip-stop trains operate in pairs The interests of operatorsand passengers were balanced in the model by using anexhaustive search procedure Skip-stop operation is appliedin practice by the Transmilenio system in Bogota and MetroRapid system in Los Angeles for bus operation and has beenproved to be highly effective according to the US NationalResearch Council [8] and Leiva et al [9]

For urban metro systems the AB mode-based skip-stopstrategy was first developed for the metro systems in Chicagocity since 1948 [10 11] It was later implemented in Philadel-phia and the City of New York [12] However due to budgetconstraints and large train headways caused by operatingdifferent types of trains it switched back to the all-stop modeafter several years AB mode has also been implementedin the metro system of Santiago Chile since 2007 whichhas demonstrated a significant benefit to passengers andtransit operators [10] Lee et al [13] and Jamili and Aghaee[14] reported that under skip-stop operation passengersexperienced shorter travel time and operators benefited fromthe scheme by having reduced fleet size and operational costs

HindawiJournal of Advanced TransportationVolume 2017 Article ID 3097681 17 pageshttpsdoiorg10115520173097681

2 Journal of Advanced Transportation

In addition skip-stop operation has been commonly used asa recovery strategy in case of special events such as demandsurges and major disruptions according to Cao et al [15]

With respect to the use of skip-stop metro operationthere are some challenges that have not been fully addressedin the existing literature The first is that since overtakingis prohibited on single-track metro lines a larger departureheadway is needed for the case that the target train travelsfaster (due to skip operation) than the previous train Thismay also reduce the line capacity and cause some difficultyin timetable design The second challenge is brought bythe infrastructure constraints at the terminal stations wheretrains need to wait andor be stored at the turnaroundterminal before serving the backward directionTherefore thesafe and efficient operation at the turnaround terminals needsto be properly considered The third challenge is the preciseestimation of dynamic passenger demand pattern whichsignificantly affects the design of skip-stop scheme Lastbut not least skip-stop operation usually leads to passengerconfusion As a result certain number of passengers may failto board the right train

To better address the first challenge recent researchinterest has shifted from the conventional AB skip-stopmode to Flexible Skip-Stop Schemes (FSSS) A summaryand comparison of the existing skip-stop schemes for metrooperation is provided in Table 1 Sun andHickman [5] Zhenget al [16] Freyss et al [10] Lee et al [13] and Cao et al[15] used different approaches to optimize AB skip-stopscheme for a single direction urban transit line based onassumed origin-destination (OD) information Sogin et al[17] developed a mixed integer program that could be solvedby commercial solvers for small scale problems A geneticalgorithm (GA) based solution approach was developed tosolve larger scale problems with more stations and trainsconsideredNiu et al [19] optimized a skip-stop scheme basedon predetermined train stop plans a mixed integer nonlinearprogrammingmodel was constructed and solved by a GAMSsolver in their paper Wang et al [18] developed optimizationmodels that offer real-time operationswith Flexible Skip-StopSchemes that is trains are allowed to adjust their skip-stoppatterns in real-time to minimize passenger travel time andenergy consumption Jamili and Aghaee [14] built a modelto determine the ldquouncertain skip patternsrdquo which means thattrains can flexibly skip or stop at stations along the line so thatthe train operation speed is optimized

Another often overlooked challenge is the operationat turnaround terminals The only work that considersturnaround operation is Wang et al [18] which treated theturnaround terminal as an intermediate station The authorsbelieve it is more realistic to consider turnaround operationwith departure time optimization and storage capacity Thisenables more flexible bidirectional operation that can lead tomore efficient line utilization For the third challenge recentresearch has shown that transit smart card data can be utilizedto better interpret the temporal and spatial dynamics ofpassenger demand [20 21] Niu et al [19] used time-varyingpassenger demand obtained from the high-speed rail systemFor metro operation researchers (eg Sogin et al [17] Wanget al [18] Jamili and Aghaee [14]) usually assume uniform

passenger arrival rates Little work has been done usingdynamic passenger demand from real operation data Withrespect to the last challenge although it is widely recognizedas a major drawback of skip-stop operation the effect of ldquofailto board the right trainrdquo has often been overlooked and littlerelevant analysis has been reported in literature

In this paper we develop a FSSS for bidirectional metrooperation during off-peak periods The problem is formu-lated as a mixed integer nonlinear programming (MINLP)model that minimizes the average passenger travel time Theproposed scheme is able to consider the train operation atthe turnaround terminal with departure time optimizationand storage capacityThe departure times at the two terminalstations are constrained by three types of departure intervalsthat is the minimum safety headway headway that meetsthe passenger demand and the turnaround headway A GA-based solution approach is then developed to efficiently solvethe problemTheproposed scheme is tested using a realmetroline in Shenzhen China The case study considers time-dependent demand information inferred from the smart carddata of the metro system Compared to the all-stop schemewe find that the proposed FSSS can reduce the averagepassenger travel time Transit agencies may also benefit fromthis scheme due to energy and operational cost savings Forthe cases that certain percentage of passengers fail to boardthe right train results show that FSSS always outperforms theall-stop scheme even when most passengers of the skippedOD pairs are confused and cannot get on the right train Theremainder of this paper is organized as follows Section 2presents the FSSS model formulation Section 3 proposes aGA-based solution approach to efficiently solve the modelThe experiment and numerical results are summarized inSection 4 followed by the concluding remarks in Section 5

2 FSSS Timetabling

In this section we first introduce the notations followed bya brief description of bidirectional train operation problemWe then introduce the proposed FSSS model formulation

21 Notations

119896 train index119894 119895 station indices119892 time period index

Input Parameters119873119904 number of stations of a one-way line119873tr number of trains119897 length of each time period119905119894tr section travel time between stations 119894 and 119894 + 1 inseconds119905119894dw dwell time at station 119894 in seconds119903119894119895119892 time-dependent passengerrsquos OD from stations 119894 to119895 during period 119892 in people per second120575 time loss during a stop in seconds

Journal of Advanced Transportation 3

Table 1 Existing skip-stop schemes for metro operation

Publication Skip-stopmode

Input datatypesource

Singlebidirectionalline Objective Model Solution

approach Results

Suh et al[12] Fixed AB Static OD Single

Maximize thetotal time savingof a skip-stop

schedule in bothpeak and

off-peak period

Trainoperationsimulationmodel

Calculate thetotal travel timeof five passengerboarding types

Per the specificscenario theskip-stop

operation resultsin less total

travel saving inthe peak periodcompared to theoff-peak period

Zheng et al[16] Fixed AB Static OD with

13 stations SingleMinimize totalpassenger travel

time

A 0-1 integerprogrammingmodel by

analyzing thepassenger

traveling time

Tabu search

It is better toassign a type ABtrain between atype A train andtype B train

Freyss et al[10] Fixed AB

Assumed staticOD from

Chilean metrodata

Single

Minimize thecost function

that iscomprised oftotal passengertransfer timewaiting timeand total

traction andenergy cost

Consider fivepassengertypes underdifferent

combination ofA or B or AB

stations

Continuousapproximation

approach

Short lines withfewer stations

are lessfavorable forskip-stop

operation largerbenefit can beobtained in the

lines withsmaller

minimumheadway

Lee et al[13] Fixed AB

Manipulatedstatic OD basedon boarding andalighting ratiofrom the SeoulMetro line

SingleMinimize thepassenger travel

time

Consider threetypes (A Band AB) of

passenger ODand collisionconstraints infour scenarios

Geneticalgorithm

17ndash20 traveltime saving forpassengers and

reducedoperational costfor the operator

Cao et al[15] Fixed AB

Simulated staticOD based onBeijing Metro

data

Single

Biobjectiveformulation thatminimizes thepassenger

waiting timeand trip time

A 0-1 mixedinteger modelthat considersthe constraints

of twosuccessivetrains trainoperationsimulation

Tabu search

Smallerheadway leads

to betterpassenger


Sogin et al[17] Dynamic

Simulated staticOD usinggravity

modelChicagoMetro lines

SingleMinimize thetotal passengertravel time

A mixedinteger

formulationthat considersthe tradeoffbetweenpassenger

travel time andservice

frequencycaused byskip-stop

GAMS andCPLEX forsmall scaleproblemsgenetic

algorithm forlarge-scaleproblems

Skip-stopoperation can

reducepassenger traveltime by 95


Table 1 Continued




approach Results

Wang et al[18] Dynamic

Static OD basedon real data

from the BeijingMetro

(Yizhuang) Line

Bidirectionalconsider turnaround

terminal as anintermediate stationwithout storage

capacity

Minimize thetotal passengertravel time and

energyconsumption

Consider theimpact ofboardingpassenger

number on thetrain dwell

time and trainutilization atthe departureterminal

Bilevel mixedinteger

programmingmodel with

rolling horizon

Time andenergy savingup to 15

Niu et al[19]

Predeterminedskip-stop

Time-dependent

demand for ninestations of theShanghai-Hangzhou

High-Speed RailLine in China

SingleMinimize thetotal passengerwaiting time

A unifiedquadraticinteger


linearconstraintsconsider safeoperation andtrain capacity

GAMS solver

The model canbe solved bystandard

commercialoptimizationpackages with

goodcomputation

time

Jamili andAghaee [14]

Uncertainskip-stop

Static OD basedon an Iranianmetro line


A nonlinearprogrammingmodel thatconsidersboarding

passengers andtrain capacity

Decomposition-based algorithmand simulatedannealing (SA)

algorithm

Trip time isreduced from923 minutes to866 minutesthe average

speed increasesby 34 kmh

Ourapproach Flexible

Time-dependentarrival rate

based on smartcard data fromthe ShenzhenMetro line

Bidirectionalconsider departure

time optimization andstorage capacity at theturnaround terminal

Minimize theaverage travel

time

Three types ofheadways areconsidered

train operationsimulationturnaroundoperation

Geneticalgorithm

See numericalsection

119868min minimum headway for safety operation inseconds119868119889 headway that meets the passenger demand inseconds119868119905 minimum turnaround headway in seconds119868119901 maximum acceptable passenger waiting time inseconds119888 number of lines that allow trains to be stored at theturnaround terminalSN set contains all the stations along a line119899ps number of the solutions selected at the first stepof the GA119903119888 crossover rate119903119898 mutation rate119872 maximum iteration number119899119888 number of initial chromosomes119903119891 confusion rate

Variables

119886119896119894 Operation state of train 119896 at the station 119894 it equals1 if train 119896 stops at station 119894 otherwise it is zero119889119896119894 departure time of train 119896 at station 119894 in seconds

119888119896119894 train stopped at station 119894 prior to the arrival of train119896119862119896119894 the set of trains between trains 119896 and 119888119896119894 that stopat station 119894 and skip station 119895119894119896 the next stopped station of train 119896 after departurefrom station 119894119896119890 the next available train after train 119896 that cantransport the confused passengers alighted at 119894119896 to 119895119899119901119894119895119896 number of passengers who board train 119896 at

station 119894 with a destination 119895tw119894119895119896 total waiting time associated with passengers

119899119901119894119895119896


tv119894119895119896 total in-vehicle time associated with passengers

119899119901119894119895119896

119899119896119894pg total number of onboard passengers for train 119896 atstation 119894119905119896119894wait total waiting time for passengerswho board train119896 at station 119894119905119896119894veh total in-vehicle time for passengers who boardtrain 119896 at station 119894119899pg number of onboard passengers119905wait total passenger waiting time in seconds119905veh total passenger in-vehicle time in secondsSS the candidate skipping station set that contains thestations that can be potentially skipped119878119896 the set that records the skip-stop schemes of train119896 for the forward direction1198781015840119896 the set that records the skip-stop schemes of train119896 for the backward direction119898119896119904 starting station of arc 119904 of train 119896119899119896119904 ending station of arc 119904 of train 119896arc119898119896119904 119899119896119904 skip arc that connects two stations (119898119896119904 and119899119896119904 ) between which at least one station is skipped

Δ119905119898119896119904 119899119896119904 maximum travel time difference between 119898119896119904and 119899119896119904 Δ 119896 maximum travel time difference between train 119896minus1 and train 119896 for the forward directionΔ1015840119896 maximum travel time difference between train 119896minus1 and train 119896 for the backward direction

Outputs

119905pgwait average waiting time per passenger in minutes119905pgveh average in-vehicle time per passenger in min-utes119905pgtv average travel time per passenger which equals119905pgwait + 119905pgveh in minutes119885 objective value119899ss number of skipped stations119899bpg total number of the passengers whose in-vehicletime is reduced by skip-stop operation

22 Bidirectional Metro Operation Problem Consider a bidi-rectional metro line The system is comprised of 119873tr + 1trains and 119873119904 stations indexed from 0 to 119873tr and 1 to 119873119904respectively Train 0 is an all-stop train that departs at time0 this all-stop train is operated to ensure the system is inoperation at the beginning of the optimization period Station1 is the departure terminal and station 119873119904 is the turnaroundterminal The line has only one track in each direction asshown in Figure 1 To avoid confusion stations are denoted as1 2 119873119904 119873119904+1 2119873119904 in which stations 1 (2119873119904) indicate

departure terminal stations 119873119904 and 119873119904 + 1 represent theturnaround terminal 1 2119873119904 2 2119873119904 minus 1 119873119904 119873119904 + 1are used to represent the same stations which share the samedwell time The section travel times between two adjacentstations are assumed to be the same for both directions Inreal practices there are usually more than one turnaroundline at the turnaround terminal where trains are allowed tobe stored on the lines As illustrated in Figure 1 train 119896minus1 upto train 119896 minus 119888 minus 1 are allowed to wait at turnaround terminalbefore train 119896minus119888 departs from station119873119904 +1 A proof is givenin Appendix to show that more turnaround lines (eg 119888 ge 1)can lead to better train utilization and therefore increase theline capacity The solution of the metro operation problemis to make skipstop decisions and determine the departuretime at each station Thereby a complete operation timetablecan be obtained

The authors notice that there is a debate regardingwhether skip-stop ismore suitable for off-peak or peak periodoperation For example Niu et al [19] built mathematicalmodels to solve the rail optimization problem during peakperiod As reported in Suh et al [12] the benefits receivedin off-peak period (due to skip-stop) are more significantcompared to the peak periodThe results are largely due to thefact that there are larger passenger demands at most stationsduring the peak period The skip operation may lead toexcessive waiting time at the skipped stations and it may alsocause station congestion In this work the authors considerthe off-peak operation The train capacity is not consideredas a constraint since our early study [22] found out that trainsare far from crowded during the off-peak period

23 Assumptions Thekey assumptions used in this paper aresummarized and discussed as below

Assumption 1 The proposed method targets offlinetimetabling design and the passenger information (eg OD)is assumed to be known and can be accurately calibratedusing smartcard data collected from their day-to-day travelactivities

Assumption 2 The number of the supplied trains and thestorage capacity at the departure terminal are unlimited

Assumption 3 All stations can only accommodate one trainat a time for one direction overtaking is prohibited atany point of the line This assumption is consistent withthe common infrastructure setup at most metro systems inChina

Assumption 4 The travel time between two adjacent stationsand the dwell time at each station varies along the line butthey are assumed to be fixed for all trains The time lossassociated with a stop (due to braking and acceleration) isassumed to be a constant for all trains and all stops

Assumption 5 It is assumed that if train 119896 stops at station 119894and is about to skip station 119895 certain percentage of passengersfrom OD pairs 119894 to 119895 will mistakenly board the train (thispercentage is referred to as the confusion rate) It is assumed


1 2Departureterminal

Turnaroundterminal

2Ns 2Ns minus 1 Ns + 2 Ns + 1

Ns minus 1 Ns

Train k minus c

Train k

Train k minus c minus 1

Train k minus 1

Figure 1 Illustration of a bidirectional metro system

that these passengers shall notice their mistake (eg via trainradio broadcast) once they get on the train and they willalight at the next stopping station These passengers willthen wait at the station until being transported by the nextavailable (correct) train to their destination 11989524 Objective Function Theobjective of FSSS is to determinethe operation scheme and departure times to minimize theaverage passenger travel time including the in-vehicle traveltime and waiting timeThe operation for train 119896 at station 119894 isdefined as a binary variable 119886119896119894

119886119896119894 = 1 if train 119896 stops at the station 1198940 otherwise (1)

The total number of onboard passengers 119899pg the total passen-ger waiting time 119905wait and the total passenger in-vehicle time119905veh can be calculated using (2) in which 119899119896119894pg 119905119896119894wait and 119905119896119894vehare the number of onboard passengers the passenger waitingtime and the passenger in-vehicle time for train 119896 at station119894 respectively

119899pg =119873trsum119896=1

2119873119904sum119894=1

119899119896119894pg

119905wait =119873trsum119896=1

2119873119904sum119894=1

119905119896119894wait

119905veh =119873trsum119896=1

2119873119904sum119894=1

119905119896119894veh

(2)

We use 119889119896119894 to denote the departure time of train 119896 atstation 119894Thedwell time at station 119894 is 119905119894dw which is determinedbased on the real metro operation data The arrival time atstation 119894 can be easily acquired by subtracting 119905119894dw from 119889119896119894Variable 119888119896119894 is used to denote the train that stopped at station 119894prior to the arrival of train 119896 which can be referred to as 119888119896119894 =max1198961015840 | 1198961015840 lt 119896 119886119896119894 = 1198861198961015840 119894 = 1 according to Niu et al [19]

The time-dependent passenger OD rate between stations119894 and 119895 during time period119892 is denoted as 119903119894119895119892 and the length ofeach period is 120591 To calculate the values of 119899119896119894pg 119905119896119894wait and 119905119896119894vehtwo cases are considered see Figure 2 The first case is whentrains 119896 and 119888119896119894 depart in the same time period (ie 119889119896119894 119889119888119896119894 119894 isin[(119892 minus 1)120591 119892120591)) In this case the passenger arrival rate is

uniformThe resulting 119899119896119894pg 119905119896119894wait and 119905119896119894veh can be calculated asin (3)ndash(5) Here 119889119896119894 minus 119889119888119896119894 119894 represents the departure intervalbetween trains 119896 and 119888119896119894 at station 119894 therefore 119903119894119895119892 (119889119896119894 minus 119889119888119896119894 119894)represents the amount of passengers who arrive at station 119894and plan to travel to station 119895 during this time interval 119889119896119895 minus119889119896119894 minus 119905119895dw refers to the in-vehicle travel time for passengerstravel from 119894 to 119895 The second case is when trains 119896 and 119888119896119894depart in different time periods (ie 119889119888119896119894 119895 isin [(119892 minus 1)120591 119892120591) and119889119896119894 isin [119892120591 (119892 + 119904)120591) forall119904 ge 1) The corresponding 119899119896119894pg 119905119896119894waitand 119905119896119894veh can be calculated as in (6)ndash(8)The objective functionis then formulated to minimize the average passenger traveltime see formulation (9)

119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895119903119894119895119892 (119889119896119894 minus 119889119888119896119894 119894) (3)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (4)

119905119896119894veh =2119873119904sum119895=119894+1

119886119896119894119886119896119895119903119894119895119892 (119889119896119894 minus 119889119888119896119894 119894) (119889119896119895 minus 119889119896119894 minus 119905119895dw) (5)

119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905

+ 119903119894119895119892+119904 (119889119896119894 minus (119892 + 119904 minus 1) 120591)] (6)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (7)

119905119896119894veh =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905

+ 119903119894119895119892+119904 (119889119896119894 minus (119892 + 119904 minus 1) 120591)] (119889119896119895 minus 119889119896119894 minus 119905119895dw) (8)

min119885 = (119905wait + 119905veh)119899pg (9)

25 Model Constraints The model is subject to a few safetyoperational and infrastructure constraints Train headwayis an important concept that is directly related to operation


Time

Rate

Station i

rijg

dc i

(g minus 1) g

dki

Period ldquogrdquo

(a)

Time

Rate Period

Station i

rijg

dc i

g

dki

PeriodPeriod

(g + 1) (g + s minus 1) (g + s)

middot middot middot

(g minus 1)

ldquogrdquoldquog + 1rdquo

ldquog + srdquo

(b)

Figure 2 The departure of two consecutive trains (a) trains 119896 and 119888119896119894 depart in the same time period (b) trains 119896 and 119888119896119894 depart in differenttime periods

safety and the capacity of a metro line Three types oftrain headways are considered in this study which are theminimum safety headway 119868min the headway that meetsthe passenger demand 119868119889 and the minimum turnaroundheadway 119868119905 Different from the work in Wang et al [18] whotook the turnaround terminal as a normal station we specif-ically consider the departure time optimization and storagecapacity at the turnaround station In this subsection theheadway-related constraints are discussed below followed bythe operational constraints

251 Minimum Safety Headway Equations (10) and (11)represent that train 0 departures at time 0 after arriving atthe turnaround terminal the new departure time of train0 is calculated by adding the section travel time stationdwell time and the turnaround time For safe operation thedeparture interval of two consecutive trains should satisfyformulation (12) at all stations

11988901 = 0 (10)

1198890119873119904+1 =119873119904sum119903=1

(119905119894tr + 119905119894dw) + 119868119905 (11)

119889119896119894 minus 119889119896minus1119894 ge 119868min forall1 le 119894 le 2119873119904 1 le 119896 le 119873tr (12)

Note that skip operation leads to shorter travel time andit may cause some safety concerns An illustrative exampleis provided in Figure 3 Suppose train 1 stops at all stationsand train 2 skips station 2 and station 3 If train 2 only keepsthe minimum safety headway 119868min at the departure terminaltheminimum safety headway will not suffice at station 3Thissituation poses safety issues and requires further interventioncontrol To avoid this an extra amount of timeΔ 119896 is added to

the departure time (of train 119896) at the departure terminal seethe trajectory of train 2

To calculate the value of Δ 119896 set 119878119896 and set 1198781015840119896 are definedas the skip-stop plans of train 119896 for the forward direction andbackward direction respectively Each set is comprised of aseries of the ldquoskip arcsrdquo each ldquoskip arcrdquo connects two stationswhere train 119896 stops between which at least one station hasbeen skipped For example as shown in Figure 4 arc1198981198961 1198991198961denotes the fact that train 119896 stops at station1198981198961 = 1 and stopsagain at station 1198991198961 = 4 between which station 2 and station 3are skipped Here119898119896119904 refers to the starting station of arc 119904 119899119896119904refers to the ending station of arc 119904 The last ldquoskip arcrdquo in set119878119896 is denoted as arc119898119896119902 119899119896119902

119878119896 = arc1198981198961 1198991198961 arc119898119896119904 119899119896119904 arc119898119896119902 119899119896119902 (13)

Under this definitionsum119899119896119904119894=119898119896119904

(1minus119886119896119894) can be used to recordthe total number of skip operations of train 119896 from stations119898119896119904 to 119899119896119904 with respected to arc119898119896119904 119899119896119904 The travel time saving of

train 119896 from stations 119898119896119904 to 119899119896119904 can be expressed as sum119899119896119904119894=119898119896119904

(1 minus119886119896119894)(119905119894dw+120575) where 119905119894dw is the dwell time at station 119894 and 120575 is thetime loss due to the acceleration and deceleration associatedwith a stop at a station Therefore the maximum travel timedifference between train 119896 minus 1 and train 119896 from stations 119898119896119904and 119899119896119904 denoted as Δ119905119898119896119904 119899119896119904 can be represented as

Δ119905119898119896119904 119899119896119904 =119899119896119904sum119894=119898119896119904

(1 minus 119886119896119894) (119905119894dw + 120575)


Terminal station 1

Station 2

Station 3

Terminalstation 4

TimeTrain 1 Train 2

Imin

Imin

Train 2lowast

Δ2

Figure 3 The illustration of the method to obtain the train safetyheadway

minus 119899119896119904sum119894=119898119896119904

(1 minus 119886119896minus1119894) (119905119894dw + 120575)

= 119899119896119904sum119894=119898119896119904

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) (14)

The next step is to calculate the maximum traveltime difference between train 119896 minus 1 and train 119896 at anypoint along the metro line which can be expressed asmaxsum119904119906=1sum119899119896119906119894=119898119896119906(119886119896minus1119894 minus 119886119896119894)(119905119894dw + 120575) | 119904 = 1 119902 Inthe cases that this maximum travel time difference is largerthan zero (ie train 119896 travels faster than train 119896 minus 1) anextra amount of time is added to the departure time of train119896 to avoid collision In the case that the maximum traveltime difference is smaller than zero no adjustment is neededsee formulation (15) Similarly we use Δ1015840119896 to represent themaximum travel time difference for the backward direction

Δ 119896 = maxmax

119904sum119906=1

119899119896119906sum119894=119898119896119906

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) | 119904 = 1 119902

0

(15)

To ensure minimum safety headway between two con-secutive trains at any point of the line the departure timeof train 119896 is constrained at terminal stations 1 and 119873119904 + 1 byformulation (16a) and formulation (16b) respectively

1198891198961 minus 119889119896minus11 ge 119868min + Δ 119896 forall1 le 119896 le 119873tr (16a)

119889119896119873119904+1 minus 119889119896minus1119873119904+1 ge 119868min + Δ1015840119896 forall1 le 119896 le 119873tr (16b)

252 Headway That Meets the Passenger Demand A strictconstraint that satisfies the passenger demand should take the

form of constraints (17a) and (17b) Here 119868119889 is the headwaythat meets the passenger demand

1198891198961 le 119896119868119889 forall1 le 119896 le 119873tr (17a)

119889119896119873119904+1 le 1198890119873119904+1 + 119896119868119889 forall1 le 119896 le 119873tr (17b)

This constraint however is too strict for skip-stop oper-ation In the cases that the safety operation headway is largerthan the demand headway (due to skip operation) we cannotguarantee that the demand headway is satisfied at all timesTherefore we consider constraints (18a) and (18b) as a com-promise to constraints (17a) and (17b)

1198891198961 le max 119896119868119889 119868min + Δ 119896 + 119889119896minus11 forall1 le 119896 le 119873tr (18a)

119889119896119873119904+1 le max 1198890119873119904+1 + 119896119868119889 119868min + Δ 119896 + 119889119896minus1119873119904+1 forall1 le 119896 le 119873tr (18b)

253 Turnaround Headway In our model we specificallyconsider 119888 (119888 ge 1) turnaround lines at the turnaround ter-minal Under the infrastructure geometry train 119896 minus 119888 shoulddepart from station119873119904 +1 before train 119896 departs from station119873119904 That is formulations (18a) and (18b) should be satisfiedHere 119868119905 refers to the minimum turnaround headway at theturnaround station The formulation can better utilize thestorage capacity at the turnaround station

119889119896119873119904 + 119868119905 minus 119889119896minus119888119873119904+1 ge 0 forall119888 le 119896 le 119873tr (19)

254 Operational Constraint To avoid intolerable waitingtime at the trip origins we assume that each viable ODpair should be served in an acceptable time period which ismarked as 119868119901 that is constraint (20) needs to be satisfiedHere 119909 = max1198961015840 | 1198961015840 lt 119896 119886119909119894119886119909119895119899119901119894119895119909 gt 0and119886119896119894119886119896119895119899119901119894119895119896 gt 0forall1 le 119896 le 119873tr 0 le 119894 le 119873119904 119894 lt 119895 le 119873119904

119889119896119894 minus 119889119909119894 le 119868119901 (20)

In practice some stations such as terminal stations andtransfer stations should not be skipped For this we furtherdefine a set SS that contains the stations that can be skippedFor the stations that do not belong to this set they should notbe skipped see constraint (21) Here SS cup SS = SN where SNis the set that contains all metro stations

119886119896119894 = 0 forall119894 isin SS (21)

3 Solution Approach

In the previous section the FSSS model is formulated as amixed integer nonlinear programming (MINLP) problemwhich is comprised of the objective function (9) constraint(1) constraint (10)-constraint (11) constraints (16a) and (16b)and constraints (18a) and (18b)ndashconstraint (21) Note that themodel requires departure time optimization at each stationwhich imposes high computational complexity To simplify


4

Skip the stationStop at the station

2 31

arcm1 n

1

mk1 nk

1

arc arcm

n

mk2 nk

2

m n

mkq nk

q

Ns minus 2 Ns minus 1 Ns

Figure 4 Definition of 119878119896 and ldquoskip arcsrdquo

the problem we apply a rule-based simulation ((22)ndash(24))that generates departure times to satisfy constraints (16a)and (16b) and constraints (18a) and (18b)-constraint (19) Asimilar rule-based computation of train departure times wasimplemented by Salzborn [23]

First we define the terminal departure time as (22) thedeparture time at the terminal departure takes the maximum

allowed value that satisfies the safety and passenger demandconstraints therefore it should result in the smallest requirednumber of trains Similarly the departure time at theturnaround terminal should take (23) Equation (24) is thenused to update the departure times at the intermediatestations

1198891198961 = 119889119896minus11 +max 119896119868119889 minus 119889119896minus11 119868min + Δ 119896 +max 119889119896minus119888119873119904+1 minus 119868119905 minus 119889119896119873119904 0 forall119888 le 119896 le 119873tr (22)

119889119896119873119904+1 = max 119889119896119873119904 + 119868119905 119889119896minus1119873119904+1 + 119868min + Δ1015840119896 forall1 le 119896 le 119873tr (23)

119889119896119894 =

1198891198961 + 119894sumV=0

(119905119894tr + 119886119896119894119905119894dw minus (V minusVsum0

119886119896119894)120575) 119894 le 119873119904 forall1 le 119896 le 119873tr

119889119896119873119904+1 +119894sum

V=119873119904+1(119905119894tr + 119886119896119894119905119894dw minus (V minus

Vsum0

119886119896119894)120575) 119894 gt 119873119904 forall1 le 119896 le 119873tr(24)

It is particularly important to note that if the turnaroundterminal is considered as an intermediate station (as in mostexisting literature) an extra amount of the maximum traveltime difference which takes the form of maxΔ 119896 Δ1015840119896 needsto be added to the departure terminal In the cases that Δ1015840119896 islarger than Δ 119896 the departure time at the departure terminalcan be expressed by (25) And this leads to delayed departuretime compared to (22) which results in line capacity loss

1198891198961 = 119889119896minus11 +max 119896119868119889 minus 119889119896minus11 119868min + Δ1015840119896+max 119889119896minus119888119873119904+1 minus 119868119905 minus 119889119896119873119904 0

forall119888 le 119896 le 119873tr

(25)

Furthermore it is widely known that NP-hard problemssuffer from the ldquoCurse of Dimensionalityrdquo separating theforward and backward operation scheme can significantlyreduce the dimension of the problem that is the computa-tional complexity reduces from 119874(22119873119904119873tr) to 119874(2119873119904119873tr+1)

Using the simulation scheme the departure times can bedetermined and the problem is thus reduced to the optimiza-tion of the binary variables 119886119896119894 The problem is NP-hardand an exhaustive search of the skip-stop schemes has anexponential time complexity119874(2119873119904119873tr) To avoid enumeratingall possible outcomes we develop a genetic algorithm (GA)approach to guide the searching process GA is a metaheuris-tic method that imitates the process of natural evolution

GA is motivated by the principles of natural selection andsurvival-of-the-fittest individuals [24] It is proved to beefficient in solving NP-hard optimization problems in thetransportation field [25ndash28] This method was also used byLee et al [13] to find a solution to the AB mode-based skip-stop operation problem In their work gene is defined as thestation type and it can take the choices of A B or AB Thetime complexity of their problem is therefore 3119873119904 Comparedto Lee et al [13] the proposed FSSS model offers flexibleschemes and it can further consider the operation at theturnaround terminal Realistic operational and infrastructureconstraints are also considered Therefore the proposedmethod is different from the one in Lee et al [13] The solu-tion procedure of the developed GA algorithm is shown inFigure 5

In our solution approach a gene is defined as the opera-tion choice at the specific station (0 or 1 which denotes skipor stop) and therefore a chromosome is defined as a binarystring that embodies 119886119896119894 | 119894 isin SS And this ensures thatconstraint (21) is satisfied throughout the searching processIn the initialization step a number of 119899119888 chromosomes arerandomly generated Next we check the feasibility of thegenerated chromosomes using constraint (20) An arbitrarilylarge objective (fitness) value is assigned to the infeasiblechromosomes to penalize the infeasible predecessors for thefeasible predecessors their fitness values need to be evaluatedusing the objective functionWe then apply the GA operators


Operationparameters O-D

Initialization

Stop criteria

Penalize

Yes

NoFeasiblilityYes

Generate the next generation

No

Genetic algorithm

Model output

Chromosome list

Evaluate the objective value viaEqu (14)ndash(16a) and (16b)

Figure 5 A GA-based solution approach

to the list of chromosomes to generate a list of new solutionsThe GA operators include (i) a selection operator whichis used to select 119899ps fitter solutions based on the RouletteWheel Selection Algorithm [29] (ii) a crossover operatorwhich is used to generate the child solutions from the selectedsolutions through a recombination process (with crossoverrate 119903119888 see Back [30]) (iii) a mutation operator which uses asmall probability (ie 119903119898) to model the genetic mutation pro-cess We further interpret the binary strings as Gray-codedintegers which is a proven practice that can be used to avoidthe hamming cliff problem (see Back [30]) We then checkthe generated new solutions to see if they satisfy the stoppingcriteria that is the number of iterations exceeds a predefinedvalue (119872) or convergence is reached If so the final solu-tion is recorded otherwise return to check the feasibilityReaders can refer to Jong [24] for more detailed informationof GA

4 Case Study and Numerical Results

41 Data Processing and Experiment Setup Smart card datacollected from urban transit systems have been widely usedin transportation applications Pelletier et al [31] reviewed the

application of smart card data in public transit field and cate-gorized the data usage into three levels the strategic level forlong-term planning the tactical level for service adjustmentand the operational level for performance measurementMunizaga and Palma [32] proposed a method to estimatemetro passenger OD using GPS data and smart card data inSantiago Hong et al [33] developed an approach that canmatch passenger trips to trains Zhang et al [22] formulateda model to optimize the urban rail operation based on thedetailed passenger transaction data during weekdays It isrevealed by the literature that smart card data can betterreflect the passenger OD pattern

The proposed FSSSmodel was tested using the smart carddata and operation parameters froma real world bidirectionalmetro line in Shenzhen China The metro system is shownin Figure 6 We selected Line 1 (with 28 intermediate stationsand two terminals highlighted in green) to test the proposedFSSS model

The dynamic passenger OD rates were obtained from thesmart card data using the following steps (i) a transfer rulewas defined based on the least number of stations that is fora trip that starts and ends at different lines the trip will usethe transfer station that yields the least number of stations


Dafen

Danzhutou

Mumianwan

Buji

Changelong

Xiashuijing

ShangshuijingYangmeiBantianWuheMinzhi

ShenzhenNorthStation

Shangtang

Baishilong

Minle

Shangmeilin

LianhuaNorth

Futian

LianhuaWest

Civic Center

Lianhuacun

GangxiaNorth

HuaqiangNorth

Yannan

Hongling

Jingtian

XiangmeiNorth

Xiangmi

Qiaoxiang

AutuoHill

OCT OiaochengEast

Zhuzilin Chegongmiao

Xiangmihu GangxiaConvention amp

Exhibition CenterShopping

Park

Shixia Fumin

huaqiangRd

ScienceMuseum

Luohu

guomao

GrandTheater

Hubei Huangbeiling

Xinxiu

Yijing

Tairsquoan

Buxin

LaojieShaibu

Cuizhu

Tianbei

Shuibel

Caopu

Baigelong

Departureterminal

ChildrenrsquosPalace

Hongshan

ChanglingpiTanglangUniversity

TownLiuxian

dongXingdong Xili

Shenzhen

HonglangNorth

Turnaroundterminal

Lingzhi

Fanshen

Baohua

Linhai Qianhiwan

Xinrsquoan

Liyumen Daxin TaoyuaShenzhenUniversity

Hi-TechPark

QiaochengNorth

Baishizhou

Hongshuwan

Keyuan

Houhai

Denglian

Shenkang

Windowof theWorld

Pingzhou

Xixiang

Gushu

hourui

AirportEast

BaorsquoanCenter

BaorsquoanStadium

Figure 6 Shenzhen Metro map

during the trip (ii) the transfer rule was applied to the recordsthat either start or terminate at some station of Line 1 (iii)the origins and destinations of all the Line 1 trips were thenidentified (iv) by aggregating the OD data for the weekdayoff-peak period (1330ndash1700) during onemonth the dynamicpassenger OD rates (119903119894119895119892 ) were calculated for each time period119892 which was set to be 15 minutes in the experiment

This metro line currently operates under an all-stopschemeThe round trip travel time is about 1388minutes119873trwas set to be 10 The minimum safety headway the demandheadway and the minimum turnaround headway were set tobe 2 minutes 6 minutes and 3 minutes respectively Accord-ing to the real world practice of Chicago Metro [11] themaximum acceptable waiting time 119868119901 was 12 minutes duringthe peak hours As our study targets off-peak operation 119868119901was set to be 15 minutesThe time loss 120575was calculated basedon the reachable maximum speed 119881max using formulation(26) Here 120572 is the maximum acceleration rate and 120573 isthe maximum deceleration rate Parameters 119881max 120572 and 120573were set to be 80 kmh 08ms2 and 1ms2 respectively thecalculated 120575 was 25 seconds using (26) Further discussionsof this formulation can be found in Zheng et al [16]

120575 = (120572 + 120573)119881max432120572120573 (26)

The off-peak boarding and alighting counts at eachstation are shown in Figure 7(a) It is found that the number ofpassengers is relatively low at station 21 station 22 and station23 An example of the time-dependent passenger demandfrom station 8 to station 15 is illustrated in Figure 7(b) Thesection travel time between two consecutive stations and thedwell time at each station are shown in Table 2 The dwelltime at transfer stations 3 4 8 9 15 24 is 45 seconds whichis higher compared to the 35-second dwell time at otherintermediate stationsTheminimum turnaround timewas setto be 120 seconds The section travel times and dwell times

were set according to the real world operating conditionsParameter 119888 was set to be 2 which means that one train isallowed to be stored at the turnaround line

The all-stop operation scheme was simulated by settingparameter 119886119896119894 to 1 for all the trains and stations We then useformulation (1)ndashformulation (11) and formulation (22)ndashfor-mulation (24) to calculate the performance of the systemunder the all-stop scheme It is found that the average waitingtime 119905pgwait and the average in-vehicle time 119905pgtv are 30 minutesand 135 minutes respectively The average travel time underthe all-stop scheme is 165 minutes The total number ofonboard passengers 119899pg is 2614642 Skip-Stop Optimization Results The proposed FSSSmodel is then solved using the GA-based solution approachThere are fivemajor parameters used inGA including the sizeof the chromosome list (119899119888) populationrsquos size (119899ps) maximumnumber of iterations (119872) crossover rate (119903119888) and mutationrate (119903119898) The first three parameters were set to be 200 500and 2000 respectively The crossover rate and the mutationrate are particularly important and are often found to affectthe modeling performance (Back [30]) In the experimentthese parameters were set to be 08 and 0005 The selectionof parameters is based on a standard sensitivity analysisapproach details are not presented here

For the FSSS we manually set SS to include 20 lowdemand stations that is SS = 10 13 14 18 21 22 23 2728 29 32 33 34 38 39 40 43 47 48 51The correspondingchromosome length is 200 (20 binary variables multiple 10trains) The problem is solved on a computer with 20GBRAMon a 20GHz Intel Core i7 processorTheGA runtime isaround 2 hours However as the FSSS model is proposed foroffline operations the computation time is not a big concernBy solving theMINLP the optimized FSSS is listed in Table 3only for the stations that belong to set SS For the stationsthat belong to SS they cannot be skipped It is found that thestations that have the lower passenger demand (eg station


0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns

Station (Ns + 1) minus 2Ns

(a) Passenger demand at each station

050 15 2 25 3 351Arriving rate (passenger per minutes)

123456789

1011121314

Tim

e per

iodg

(b) Time-dependent passenger OD rates from station 8 to station 15

Figure 7

Table 2 Train operation parameters

Station 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15119905119894tr (seconds) 0 95 68 85 103 77 145 95 62 132 97 93 122 84 85119905119894dw (seconds) 35 35 45 45 35 35 35 45 45 35 35 35 35 35 45Station 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30119905119894tr (seconds) 69 101 90 171 83 90 78 100 81 72 104 84 204 210 156119905119894dw (seconds) 35 35 35 35 35 35 35 35 45 35 35 35 35 35 35

10 station 14 station 21ndashstation 23 station 38ndashstation 40and station 51) are frequently skipped under the optimizedscheme A total number of 35 stations are skipped whichcan lead to energy and operational cost savings for operatingagencies The performance of FSSS is then compared to theall-stop operation as shown in Table 3

The results present inTable 4 show that the objective valueof FSSS decreases by 5 from 990 seconds to 940 secondswhich indicates a travel time saving of about 50 secondsper passenger It is found that the average in-vehicle time isreduced by 65 with slightly increased (less than 4 seconds)passenger waiting time As indicated by the value of 119899bpgabout 269 of total passengers (6604 out of 24576) benefitedfrom FSSS Compared to the all-stop operation the numberof onboard passengers under FSSS slightly decreases by 6This is mainly because the last train skips some stations andsomepassengers are not served by this trainThese passengersare assumed to be picked up by the subsequent trains [18]

The timetables of the two operation schemes are illus-trated in Figure 8 The light line denotes the all-stop opera-tion while the dark line represents the FSSS operation It isfound that FSSS has a slightly higher travel speed (the darklines travel faster than the light lines) In specific the averageround trip travel time under FSSS is 1357 minutes whichis more than 3 minutes shorter compared to the all-stopoperation The average departure interval is about 6 minuteswhich is similar to the all-stop operation

Figure 9 indicates the seat occupancy and maximum sec-tion passenger load under the two schemes Seat occupancyis defined as the passenger load over the seat numbers It is

found that compared with all-stop scheme passengers whoboard the skip-stop trains have a better chance to find emptyseats meaning FSSS is more favorable to the passengers as itenhances their level of service By analyzing the maximumsection passenger load we find that the highest passengerload under FSSS is 558 which is far below the train capacity(1860) Therefore FSSS does not cause congestion on thetrains

43 Analysis of the Passenger Confusion Rate It is widelyrecognized that the major problem in applying skip-stopoperation is to keep the passengers informed regarding theskip-stop plans The solution to it relies on informationdissemination such as radio broadcast in stationstrainsNonetheless the authors believe the effect of certain percent-age of passengers being confused by the skip operation andfailing to find the right train is worth studying The percent-age is referred to as the passenger confusion rate hereafter inthe paper As noted in Assumption 5 we assume that theseconfused passengers shall notice their mistake (eg via trainradio broadcast) once they board the train and they will getoff at the next stopping station They will then wait at thestation until being transported by the next available (correct)train to their destination Variable 119903119891 is introduced here todenote the confusion rate and a sensitivity analysis was con-ducted to see the effects of 119903119891 on the modeling performanceThe variable 119903119891 was tested by taking values from 0 to 100with an increment of 20

The calculation of the boarded passengers (on train 119896which stops at station 119894 with a destination 119895) can be divided


Table 3 Optimized chromosome

Train S10 S13 S14 S18 S21 S22 S23 S27 S28 S29 S32 S33 S34 S38 S39 S40 S43 S47 S48 S510 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 11 1 1 1 1 0 1 0 1 1 1 1 1 1 1 0 0 1 1 1 02 1 1 1 1 1 0 1 1 1 1 1 1 1 1 1 1 1 1 1 13 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 1 1 1 1 04 0 1 1 1 1 0 1 1 1 1 1 1 1 0 1 0 1 1 1 15 1 1 1 1 0 1 0 1 1 1 1 1 1 1 1 1 1 1 1 16 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 1 0 1 07 0 1 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 18 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0 0 1 0 1 19 0 1 0 1 1 1 1 1 0 0 1 1 1 1 1 1 1 1 1 110 0 1 0 0 1 1 0 0 0 0 1 1 1 1 1 1 1 1 1 1Skipped times 4 0 3 1 3 3 3 1 2 2 0 0 0 2 4 4 0 0 0 3

Table 4 Performance of all-stop operation and FSSS

Operation scheme 119905pgwait (min) 119905pgveh (min) 119905pgtv (min) 119899pg 119885 119899bpg 119899ssAll-stop 298 1350 1648 26146 990 0 0FSSS 304 1262 1566 24576 940 6604 21

+006 (2) minus088 (65) minus082 (5) minus1570 (6) minus50 (5) mdash mdash

into two cases Case 1 is that train 119896 will stop at station 119895therefore passengers (from OD 119894 to 119895) board the right trainIn Case 2 train 119896 will skip station 119895 people who board thetrain are the confused passengers Further define set 119862119896119894 torepresent the set of trains between trains 119888119896119894 and 119896 that stop atstation 119894 and skip station 119895 that is119862119896119894 = 119909 | 119886119909119894 = 1and119886119909119895 = 0119888119896119894 le 119909 lt 119896 the cardinality of the set is denoted as 119899 119888ℎ is usedto denote the value of the ℎ-th element in set 119862119896119894

For Case 1 the number of onboard passengers fromstation 119894 to station 119895 via train 119896 can be calculated using (27)The corresponding passenger waiting time tw119894119895

119896and passen-

ger in-vehicle time tv119894119895119896can be calculated using (28) and (29)

respectively

np119894119895119896= 119899minus1sumℎ=1

(1 minus 119903119891) lowast 119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894) + 119903119894119895119892 (119889119896119894minus 119889119888119899 119894)

(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2

+ 119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894) (119889119896119894 minus 119889119888ℎ+1119894)] + 119903119894119895119892 (119889119896119894minus 119889119888119899 119894)2

(28)

tv119894119895119896= np119894119895119896lowast (119889119896119895 minus 119889119896119894 minus 119905119895dw) (29)

For Case 2 the number of confused passengers can becalculated using (30) They will get off at the nearest stopdenoted as 119894119896 = min1198941015840 | 1198941015840 gt 119894 1198861198961198941015840 = 1 after departurefrom station 119894 If 119894119896 lt 119895 (destination has not been passed)

these passengers shall wait until 119896119890 = min1198961015840 | 1198961015840 gt119896 1198861198961015840 1198941198961198861198961015840 119895 = 1 arrives If 119894119896 gt 119895 these passengers need totake a train in the opposite direction noted as 119896119890 = min1198961015840 |1198891198961015840 (2119873119904+1minus119894119896) gt 119889119896119894119896 and 1198861198961015840 (2119873119904+1minus119894119896)1198861198961015840 (2119873119904+1minus119895) = 1 where 119873119904is the total number of stations The associated waiting timecan be obtained using (31) and the in-vehicle time can becalculated using (32) for 119894119896 lt 119895 and (33) for 119894119896 gt 119895np119894119895119896= 119903119894119895119892 (119889119896119894 minus 119889119888119899 119894) lowast 119903119891 (30)

tw119894119895119896

= 12 (119889119896119894 minus 119889119888119899 119894) np119894119895119896 + (119889119896119890 119894119896 minus 119889119896119894119896 + tw119894119896) np119894119895119896 (31)

tv119894119895119896

= (119889119896119894119896 minus 119889119896119894 minus tw119894119896) np119894119895119896+ (119889119896119890 119895 minus 119889119896119890 119894119896 minus tw119895) np119894119895119896

(32)

tv119894119895119896

= (119889119896119894119896 minus 119889119896119894 minus tw119894119896) np119894119895119896+ (119889119896119890 (2119873119904+1minus119895) minus 119889119896119890 (2119873119904+1minus119894119896) minus tw(2119873119904+1minus119895)) np119894119895119896

(33)

By taking the summation of the above variables for all ODpairs of all trains the total boarded passenger number totalwaiting time and in-vehicle time could be calculated and theobjective value could then be obtained using (9) The resultsare provided in Table 5

It is observed that as the confusion rate increases (egpoorer information dissemination) the number of confused


Table 5 Sensitive analysis of the passenger confusion rate

Schedulingschemes 119903119891 Number of

confusedpassengers

Ave waiting timeof con-

fusedunconfusedpassengers

Ave in-vehicletime of con-




119905pgwait (min) 119905pgveh (min) 119905pgtv (min)

All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29

Figure 8 Timetables of all-stop scheme and FSSS

passengers increases proportionally However this numberonly represents a small proportion of the some 25000passengers in total This is because only the OD pairs withskipped destinations are affected and the skipped stationshave comparably low volume For the confused passengers itis found that their in-vehicle time and especially their waitingtime increase The increased in-vehicle time is because somepassengers missed their destination and have to take thetrains to the inverse direction (as in Case 2 119894119896 gt 119895) Andthe increased waiting time is mainly because the correct trainhas skipped station 119894119896 and the confused passengers are notable to get on the correct train at that station Per the specificexperiment we found that FSSS always outperforms the skip-stop scheme even when all the passengers of the skippedOD pairs get confused (ie 119903119891 = 1) Therefore the systemperformance is stable under FSSS

5 Conclusion

In this study we developed a FSSS approach for bidirec-tional metro line operation during off-peak periods to better

accommodate the spatially and temporally varied passengerdemand The problem was formulated as a mixed integernonlinear programming (MINLP) model that minimizes theaverage passenger travel time The proposed scheme is ableto optimize the skip-stop scheme and it also models the traindeparture time at the departure and turnaround terminalsThe departure intervals are constrained by three types ofheadways that is theminimum safety headway headway thatmeets the passenger demand and the minimum turnaroundheadway A GA-based solution approach was then developedto efficiently solve the problem The proposed scheme wastested using the smart card data collected at a real metro linein Shenzhen China Time-dependent passenger demandswere considered in the experiment and the passenger con-fusion rate was also analyzed in the case study part

The model performance was assessed with respect to thepassenger benefits and the operational benefits Overall theFSSS is effective in timetable design It was found that theFSSS is able to reduce the average passenger travel time by082 minutes which corresponds to 5 travel time savingAbout 269 of total passengers benefit from the skip-stop


scheme It was also observed that the average round trip traveltime under the skip-stop scheme is 1357 minutes whichis more than 3 minutes shorter compared to the all-stopoperation The average train departure interval is 6 minuteswhich is the same compared to the all-stop scheme From thetimetable it was found that 35 stations are skipped Transitagencies may benefit from this scheme due to the savings ofenergy and operational costsThrough the sensitivity analysisof passenger confusion rate we found that the proposedscheme always outperforms the all-stop scheme even whenmost passengers of the skipped OD pairs are confused andfail to get on the right train

This paper mainly considers the skip-stop operationduring off-peak scenarios inwhich the demands for some sta-tions are quite low As shown by the results the skip operationdoes not lead to excessive waiting time at these stations Forpeak-hour operations since the passenger demands at moststations are relatively large the skip operation may lead tolonger waiting time at the skipped stations In extreme casesthe number of waiting passengers may exceed the designcapacity of the station In future work the tradeoff betweenin-vehicle travel time and passenger waiting time needs to beproperly leveragedThe FSSSmodel currently requires a rule-based simulation to calculate the train departure time Underthe rule-based simulation the terminal departure timesneed to be adjusted by the maximum travel time differencebetween two consecutive trains This adjustment howevermay correspond to suboptimal solutions since departurechoice can be made more wisely at each station includingthe intermediate stations The GA-based solution approachrequires a predetermined set (SS) that specifies the stationsthat can be skipped This set needs to be determined usinga more rigorous approach The authors are also interested inextending the current methodology to optimize the skip-stopoperation for a network-widemetro systemMore results andfindings will be provided in our subsequent studies

Appendix

We consider 119888 turnaround lines that can be utilized to storetrains at the turnaround terminal Belowwe prove that undercertain circumstances more turnaround lines (ie 119888 gt 1) canlead to reduced departure time at the turnaround station

Let us first consider single turnaround line (ie 119888 = 1)According to constraint (19) the following relationships hold

When 119888 = 1 and 119888 le 119909 le 119873tr minus 119896119889119896+1119873119904 minus 119889119896119873119904+1 ge 119868119905119889119896+2119873119904 minus 119889119896+1119873119904+1 ge 119868119905

119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)

For train 119896 + 119909 formulation (A2) also holds

119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10

Max section load under FSSSMax section load under all-stop schemeSeat occup under FSSSSeat occup under all-stop scheme

0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load

Figure 9 Seat occupancy and onboard passengers per train

By summing up the formulations in (A1) and (A2) above itcan be shown that

119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)

Based on constraint (12) (A4) can be derived

119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)

Therefore given the turnaround departure time of train 119896 theturnaround departure time of train 119896 + 119909 should satisfy

119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge max 2119909119868119905 119909119868minforall119888 = 1 and 119888 le 119909 le 119873tr minus 119896 (A5)

For the cases with more than one turnaround line similarderivations are made as follows

When 119888 gt 1 and 119888 le 119909 le 119873tr minus 119896 we have119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119868119905 (A6)

Since (A4) also holds for this case the turnaround departuretime of train 119896 + 119909 should satisfy (A6)

119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge max 2119868119905 119909119868minforall119888 gt 1 and 119888 le 119909 le 119873tr minus 119896 (A7)

Since 119909119868min ge 2119868119905 ge 119868min is usually true in real practices thefollowing relationships hold

119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905forall119888 = 1 and 119888 le 119909 le 119873tr minus 119896

119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min

forall119888 gt 1 and 119888 le 119909 le 119873tr minus 119896(A8)


Since 2119909119868119905 ge 119909119868min it can be concluded that the turnarounddeparture time can be reduced if they are more than oneturnaround line and 2119868119905 ge 119868min

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

This research was financially supported by the National Nat-ural Science Foundation of China under Grant no 71671147

References

[1] C E Cortes V Burgos and R Fernandez ldquoModelling passen-gers buses and stops in traffic microsimulation Review andextensionsrdquo Journal of Advanced Transportation vol 44 no 2pp 72ndash88 2010

[2] Vuchic V R ldquoSkip-stop operation high speed with good areacoveragerdquo Revue de IrsquoUITP vol 2 pp 105ndash128 1976 (Frenchand German)

[3] V R Vuchic Urban transit Operations Planning and Eco-nomics John Wiley Sons Hoboken NJ USA 2005

[4] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquoTransportationResearch Part BMethodological vol 32no 2 pp 77ndash100 1998

[5] A Sun and M Hickman ldquoThe real-time stop-skipping prob-lemrdquo Journal of Intelligent Transportation Systems TechnologyPlanning and Operations vol 9 no 2 pp 91ndash109 2005

[6] B Yu Z Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portation Research Part A Policy and Practice vol 46 no 8 pp1265ndash1279 2012

[7] L Fu Q Liu and P Calamai ldquoReal-time optimization modelfor dynamic scheduling of transit operationsrdquo TransportationResearch Record Journal of the Transportation Research Boardvol 1857 pp 48ndash55 2003

[8] National Research Council TCRP Report 100 Transit Capacityand Quality of Service Manual Transportation Research BoardWashington DC USA 2nd edition 2003

[9] C Leiva J C Munoz R Giesen and H Larrain ldquoDesign oflimited-stop services for an urban bus corridor with capacityconstraintsrdquo Transportation Research Part B Methodologicalvol 44 no 10 pp 1186ndash1201 2010

[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch Part C Emerging Technologies vol 36 pp 419ndash4332013

[11] Chicago-Lorg AB Skip-Stop Express Service httpswwwchicago-lorgoperationslinesroute opsA-Bhtml

[12] W Suh K-S Chon and S-M Rhee ldquoEffect of skip-stop policyon a Korean subway systemrdquo Transportation Research RecordJournal of the Transportation Research Board vol 1793 pp 33ndash39 2002

[13] Y-J Lee S Shariat and K Choi ldquoOptimizing skip-stop railtransit stopping strategy using a genetic algorithmrdquo Journal ofPublic Transportation vol 17 no 2 pp 135ndash164 2014

[14] A Jamili and M P Aghaee ldquoRobust stop-skipping patternsin urban railway operations under traffic alteration situationrdquo

Transportation Research Part C Emerging Technologies vol 61pp 63ndash74 2015

[15] Z Cao Z Yuan and D Li ldquoEstimation method for a skip-stopoperation strategy for urban rail transit in Chinardquo Journal ofModern Transportation vol 22 no 3 pp 174ndash182 2014

[16] L Zheng R Song S W He and H D Li ldquoOptimizationmodel and algorithm of skip stop strategy for urban rail transitrdquoJournal of the China Railway Society vol 6 pp 135ndash164 2009

[17] S L Sogin B M Caughron and S G Chadwick ldquoOptimizingskip stop service in passenger rail transportationrdquo in Proceed-ings of the 2012 Joint Rail Conference JRC 2012 pp 501ndash512 April2012

[18] Y Wang B De Schutter T J J van den Boom B Ning andT Tang ldquoEfficient Bilevel approach for urban rail transit opera-tionwith stop-skippingrdquo IEEE Transactions on Intelligent Trans-portation Systems vol 15 no 6 pp 2658ndash2670 2014

[19] 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

[20] B Agard C Morency and M Trepanier ldquoMining public trans-port user behaviour from smart card datardquo IFAC ProceedingsVolumes vol 39 no 3 pp 399ndash404 2006

[21] M Bagchi and P R White ldquoThe potential of public transportsmart card datardquo Transport Policy vol 12 no 5 pp 464ndash4742005

[22] P Zhang X Liu and M Chen ldquoOptimal train schedulingunder a flexible skip-stop scheme for urban rail transit based onsmartcard datardquo in Proceedings of the 2016 IEEE InternationalConference on Intelligent Rail Transportation ICIRT 2016 pp13ndash22 August 2016

[23] F JM Salzborn ldquoTimetables for a suburban rail transit systemrdquoTransportation Science vol 3 no 4 pp 297ndash316 1969

[24] J C Jong Optimizing highway alignments with genetic algo-rithms [Doctoral dissertation] Department of Civil Engineer-ing University of Maryland College Park MD USA 1998

[25] D Arenas R Chevrier S Hanafi and J Rodriguez ldquoSolvingthe train timetabling problem A mathematical model and agenetic algorithm solution approachrdquo in Proceedings of theIn Proceedings of the 6th International Conference on RailwayOperations Modelling and Analysis Tokyo Japan 2015

[26] YHuang XMa S Su andT Tang ldquoOptimization of train oper-ation in multiple interstations with multi-population geneticalgorithmrdquo Energies vol 8 no 12 pp 14311ndash14329 2015

[27] Y Yin ldquoGenetic-algorithms-based approach for bilevel pro-gramming modelsrdquo Journal of Transportation Engineering vol126 no 2 pp 115ndash119 2000

[28] Y Yin ldquoMulti-objective bilevel optimization for transportationplanning and management problemsrdquo Journal of AdvancedTransportation vol 36 no 1 pp 93ndash105 2002

[29] D E Goldberg and K Deb ldquoA comparative analysis of selectionschemes used in genetic algorithmsrdquo in Foundations of GeneticAlgorithms pp 69ndash93 Morgan Kaufmann 1991

[30] T Back ldquoOptimal Mutation Rates in Genetic Searchrdquo in Pro-ceedings of the In Proceedings of the 5th International Conferenceon Genetic Algorithms pp 2ndash8 1993

[31] M-P Pelletier M Trepanier and C Morency ldquoSmart carddata use in public transit a literature reviewrdquo TransportationResearch Part C Emerging Technologies vol 19 no 4 pp 557ndash568 2011


[32] M A Munizaga and C Palma ldquoEstimation of a disaggregatemultimodal public transport Origin-Destination matrix frompassive smartcard data from Santiago Chilerdquo TransportationResearch Part C Emerging Technologies vol 24 pp 9ndash18 2012

[33] S-P Hong Y-H Min M-J Park K M Kim and S M OhldquoPrecise estimation of connections of metro passengers fromSmart Card datardquo Transportation vol 43 no 5 pp 749ndash7692016

RoboticsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


Active and Passive Electronic Components

Control Scienceand Engineering

Journal of


International Journal of

RotatingMachinery


Hindawi Publishing Corporation httpwwwhindawicom

Journal of

Volume 201

Submit your manuscripts athttpswwwhindawicom

VLSI Design



Shock and Vibration


Civil EngineeringAdvances in

Acoustics and VibrationAdvances in



Electrical and Computer Engineering

Journal of

Advances inOptoElectronics


Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

SensorsJournal of


Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Chemical EngineeringInternational Journal of Antennas and

Propagation




Navigation and Observation



DistributedSensor Networks



In addition skip-stop operation has been commonly used asa recovery strategy in case of special events such as demandsurges and major disruptions according to Cao et al [15]

With respect to the use of skip-stop metro operationthere are some challenges that have not been fully addressedin the existing literature The first is that since overtakingis prohibited on single-track metro lines a larger departureheadway is needed for the case that the target train travelsfaster (due to skip operation) than the previous train Thismay also reduce the line capacity and cause some difficultyin timetable design The second challenge is brought bythe infrastructure constraints at the terminal stations wheretrains need to wait andor be stored at the turnaroundterminal before serving the backward directionTherefore thesafe and efficient operation at the turnaround terminals needsto be properly considered The third challenge is the preciseestimation of dynamic passenger demand pattern whichsignificantly affects the design of skip-stop scheme Lastbut not least skip-stop operation usually leads to passengerconfusion As a result certain number of passengers may failto board the right train

To better address the first challenge recent researchinterest has shifted from the conventional AB skip-stopmode to Flexible Skip-Stop Schemes (FSSS) A summaryand comparison of the existing skip-stop schemes for metrooperation is provided in Table 1 Sun andHickman [5] Zhenget al [16] Freyss et al [10] Lee et al [13] and Cao et al[15] used different approaches to optimize AB skip-stopscheme for a single direction urban transit line based onassumed origin-destination (OD) information Sogin et al[17] developed a mixed integer program that could be solvedby commercial solvers for small scale problems A geneticalgorithm (GA) based solution approach was developed tosolve larger scale problems with more stations and trainsconsideredNiu et al [19] optimized a skip-stop scheme basedon predetermined train stop plans a mixed integer nonlinearprogrammingmodel was constructed and solved by a GAMSsolver in their paper Wang et al [18] developed optimizationmodels that offer real-time operationswith Flexible Skip-StopSchemes that is trains are allowed to adjust their skip-stoppatterns in real-time to minimize passenger travel time andenergy consumption Jamili and Aghaee [14] built a modelto determine the ldquouncertain skip patternsrdquo which means thattrains can flexibly skip or stop at stations along the line so thatthe train operation speed is optimized

Another often overlooked challenge is the operationat turnaround terminals The only work that considersturnaround operation is Wang et al [18] which treated theturnaround terminal as an intermediate station The authorsbelieve it is more realistic to consider turnaround operationwith departure time optimization and storage capacity Thisenables more flexible bidirectional operation that can lead tomore efficient line utilization For the third challenge recentresearch has shown that transit smart card data can be utilizedto better interpret the temporal and spatial dynamics ofpassenger demand [20 21] Niu et al [19] used time-varyingpassenger demand obtained from the high-speed rail systemFor metro operation researchers (eg Sogin et al [17] Wanget al [18] Jamili and Aghaee [14]) usually assume uniform

passenger arrival rates Little work has been done usingdynamic passenger demand from real operation data Withrespect to the last challenge although it is widely recognizedas a major drawback of skip-stop operation the effect of ldquofailto board the right trainrdquo has often been overlooked and littlerelevant analysis has been reported in literature

In this paper we develop a FSSS for bidirectional metrooperation during off-peak periods The problem is formu-lated as a mixed integer nonlinear programming (MINLP)model that minimizes the average passenger travel time Theproposed scheme is able to consider the train operation atthe turnaround terminal with departure time optimizationand storage capacityThe departure times at the two terminalstations are constrained by three types of departure intervalsthat is the minimum safety headway headway that meetsthe passenger demand and the turnaround headway A GA-based solution approach is then developed to efficiently solvethe problemTheproposed scheme is tested using a realmetroline in Shenzhen China The case study considers time-dependent demand information inferred from the smart carddata of the metro system Compared to the all-stop schemewe find that the proposed FSSS can reduce the averagepassenger travel time Transit agencies may also benefit fromthis scheme due to energy and operational cost savings Forthe cases that certain percentage of passengers fail to boardthe right train results show that FSSS always outperforms theall-stop scheme even when most passengers of the skippedOD pairs are confused and cannot get on the right train Theremainder of this paper is organized as follows Section 2presents the FSSS model formulation Section 3 proposes aGA-based solution approach to efficiently solve the modelThe experiment and numerical results are summarized inSection 4 followed by the concluding remarks in Section 5

2 FSSS Timetabling

In this section we first introduce the notations followed bya brief description of bidirectional train operation problemWe then introduce the proposed FSSS model formulation

21 Notations

119896 train index119894 119895 station indices119892 time period index

Input Parameters119873119904 number of stations of a one-way line119873tr number of trains119897 length of each time period119905119894tr section travel time between stations 119894 and 119894 + 1 inseconds119905119894dw dwell time at station 119894 in seconds119903119894119895119892 time-dependent passengerrsquos OD from stations 119894 to119895 during period 119892 in people per second120575 time loss during a stop in seconds






approach Results




off-peak period








time



traveling time

Tabu search




Chilean metrodata

Single






stations


approach




lines withsmaller

minimumheadway




time



Geneticalgorithm





data

Single





Tabu search


to betterpassenger






A mixedinteger









Table 1 Continued




approach Results




(Yizhuang) Line



capacity


energyconsumption






rolling horizon


Niu et al[19]


Time-dependent







GAMS solver



goodcomputation

time


Uncertainskip-stop






algorithm









time



Geneticalgorithm



Variables




119899119901119894119895119896



119899119901119894119895119896



Outputs













Turnaroundterminal


Ns minus 1 Ns

Train k minus c

Train k


Train k minus 1





119899pg =119873trsum119896=1

2119873119904sum119894=1

119899119896119894pg

119905wait =119873trsum119896=1

2119873119904sum119894=1

119905119896119894wait

119905veh =119873trsum119896=1

2119873119904sum119894=1

119905119896119894veh

(2)




119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895119903119894119895119892 (119889119896119894 minus 119889119888119896119894 119894) (3)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (4)

119905119896119894veh =2119873119904sum119895=119894+1


119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905

+ 119903119894119895119892+119904 (119889119896119894 minus (119892 + 119904 minus 1) 120591)] (6)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (7)

119905119896119894veh =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905


min119885 = (119905wait + 119905veh)119899pg (9)



Time

Rate

Station i

rijg

dc i

(g minus 1) g

dki

Period ldquogrdquo

(a)

Time

Rate Period

Station i

rijg

dc i

g

dki

PeriodPeriod

(g + 1) (g + s minus 1) (g + s)


(g minus 1)


ldquog + srdquo

(b)




11988901 = 0 (10)

1198890119873119904+1 =119873119904sum119903=1

(119905119894tr + 119905119894dw) + 119868119905 (11)





119878119896 = arc1198981198961 1198991198961 arc119898119896119904 119899119896119904 arc119898119896119902 119899119896119902 (13)





Δ119905119898119896119904 119899119896119904 =119899119896119904sum119894=119898119896119904

(1 minus 119886119896119894) (119905119894dw + 120575)


Terminal station 1

Station 2

Station 3

Terminalstation 4

TimeTrain 1 Train 2

Imin

Imin

Train 2lowast

Δ2


minus 119899119896119904sum119894=119898119896119904

(1 minus 119886119896minus1119894) (119905119894dw + 120575)

= 119899119896119904sum119894=119898119896119904

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) (14)


Δ 119896 = maxmax

119904sum119906=1

119899119896119906sum119894=119898119896119906

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) | 119904 = 1 119902

0

(15)














119889119896119894 minus 119889119909119894 le 119868119901 (20)


119886119896119894 = 0 forall119894 isin SS (21)

3 Solution Approach



4


2 31

arcm1 n

1

mk1 nk

1

arc arcm

n

mk2 nk

2

m n

mkq nk

q








119889119896119894 =

1198891198961 + 119894sumV=0


119886119896119894)120575) 119894 le 119873119904 forall1 le 119896 le 119873tr

119889119896119873119904+1 +119894sum


Vsum0

119886119896119894)120575) 119894 gt 119873119904 forall1 le 119896 le 119873tr(24)



forall119888 le 119896 le 119873tr

(25)







Initialization

Stop criteria

Penalize

Yes

NoFeasiblilityYes


No

Genetic algorithm

Model output

Chromosome list










Dafen

Danzhutou

Mumianwan

Buji

Changelong

Xiashuijing



Shangtang

Baishilong

Minle

Shangmeilin

LianhuaNorth

Futian

LianhuaWest

Civic Center

Lianhuacun

GangxiaNorth

HuaqiangNorth

Yannan

Hongling

Jingtian

XiangmeiNorth

Xiangmi

Qiaoxiang

AutuoHill

OCT OiaochengEast




Park

Shixia Fumin

huaqiangRd

ScienceMuseum

Luohu

guomao

GrandTheater

Hubei Huangbeiling

Xinxiu

Yijing

Tairsquoan

Buxin

LaojieShaibu

Cuizhu

Tianbei

Shuibel

Caopu

Baigelong

Departureterminal


Hongshan


TownLiuxian

dongXingdong Xili

Shenzhen

HonglangNorth

Turnaroundterminal

Lingzhi

Fanshen

Baohua

Linhai Qianhiwan

Xinrsquoan


Hi-TechPark

QiaochengNorth

Baishizhou

Hongshuwan

Keyuan

Houhai

Denglian

Shenkang

Windowof theWorld

Pingzhou

Xixiang

Gushu

hourui

AirportEast

BaorsquoanCenter

BaorsquoanStadium




120575 = (120572 + 120573)119881max432120572120573 (26)






0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns




123456789

1011121314

Tim

e per

iodg


Figure 7


















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation














approach Results




off-peak period








time



traveling time

Tabu search




Chilean metrodata

Single






stations


approach




lines withsmaller

minimumheadway




time



Geneticalgorithm





data

Single





Tabu search


to betterpassenger






A mixedinteger









Table 1 Continued




approach Results




(Yizhuang) Line



capacity


energyconsumption






rolling horizon


Niu et al[19]


Time-dependent







GAMS solver



goodcomputation

time


Uncertainskip-stop






algorithm









time



Geneticalgorithm



Variables




119899119901119894119895119896



119899119901119894119895119896



Outputs













Turnaroundterminal


Ns minus 1 Ns

Train k minus c

Train k


Train k minus 1





119899pg =119873trsum119896=1

2119873119904sum119894=1

119899119896119894pg

119905wait =119873trsum119896=1

2119873119904sum119894=1

119905119896119894wait

119905veh =119873trsum119896=1

2119873119904sum119894=1

119905119896119894veh

(2)




119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895119903119894119895119892 (119889119896119894 minus 119889119888119896119894 119894) (3)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (4)

119905119896119894veh =2119873119904sum119895=119894+1


119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905

+ 119903119894119895119892+119904 (119889119896119894 minus (119892 + 119904 minus 1) 120591)] (6)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (7)

119905119896119894veh =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905


min119885 = (119905wait + 119905veh)119899pg (9)



Time

Rate

Station i

rijg

dc i

(g minus 1) g

dki

Period ldquogrdquo

(a)

Time

Rate Period

Station i

rijg

dc i

g

dki

PeriodPeriod

(g + 1) (g + s minus 1) (g + s)


(g minus 1)


ldquog + srdquo

(b)




11988901 = 0 (10)

1198890119873119904+1 =119873119904sum119903=1

(119905119894tr + 119905119894dw) + 119868119905 (11)





119878119896 = arc1198981198961 1198991198961 arc119898119896119904 119899119896119904 arc119898119896119902 119899119896119902 (13)





Δ119905119898119896119904 119899119896119904 =119899119896119904sum119894=119898119896119904

(1 minus 119886119896119894) (119905119894dw + 120575)


Terminal station 1

Station 2

Station 3

Terminalstation 4

TimeTrain 1 Train 2

Imin

Imin

Train 2lowast

Δ2


minus 119899119896119904sum119894=119898119896119904

(1 minus 119886119896minus1119894) (119905119894dw + 120575)

= 119899119896119904sum119894=119898119896119904

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) (14)


Δ 119896 = maxmax

119904sum119906=1

119899119896119906sum119894=119898119896119906

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) | 119904 = 1 119902

0

(15)














119889119896119894 minus 119889119909119894 le 119868119901 (20)


119886119896119894 = 0 forall119894 isin SS (21)

3 Solution Approach



4


2 31

arcm1 n

1

mk1 nk

1

arc arcm

n

mk2 nk

2

m n

mkq nk

q








119889119896119894 =

1198891198961 + 119894sumV=0


119886119896119894)120575) 119894 le 119873119904 forall1 le 119896 le 119873tr

119889119896119873119904+1 +119894sum


Vsum0

119886119896119894)120575) 119894 gt 119873119904 forall1 le 119896 le 119873tr(24)



forall119888 le 119896 le 119873tr

(25)







Initialization

Stop criteria

Penalize

Yes

NoFeasiblilityYes


No

Genetic algorithm

Model output

Chromosome list










Dafen

Danzhutou

Mumianwan

Buji

Changelong

Xiashuijing



Shangtang

Baishilong

Minle

Shangmeilin

LianhuaNorth

Futian

LianhuaWest

Civic Center

Lianhuacun

GangxiaNorth

HuaqiangNorth

Yannan

Hongling

Jingtian

XiangmeiNorth

Xiangmi

Qiaoxiang

AutuoHill

OCT OiaochengEast




Park

Shixia Fumin

huaqiangRd

ScienceMuseum

Luohu

guomao

GrandTheater

Hubei Huangbeiling

Xinxiu

Yijing

Tairsquoan

Buxin

LaojieShaibu

Cuizhu

Tianbei

Shuibel

Caopu

Baigelong

Departureterminal


Hongshan


TownLiuxian

dongXingdong Xili

Shenzhen

HonglangNorth

Turnaroundterminal

Lingzhi

Fanshen

Baohua

Linhai Qianhiwan

Xinrsquoan


Hi-TechPark

QiaochengNorth

Baishizhou

Hongshuwan

Keyuan

Houhai

Denglian

Shenkang

Windowof theWorld

Pingzhou

Xixiang

Gushu

hourui

AirportEast

BaorsquoanCenter

BaorsquoanStadium




120575 = (120572 + 120573)119881max432120572120573 (26)






0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns




123456789

1011121314

Tim

e per

iodg


Figure 7


















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Table 1 Continued




approach Results




(Yizhuang) Line



capacity


energyconsumption






rolling horizon


Niu et al[19]


Time-dependent







GAMS solver



goodcomputation

time


Uncertainskip-stop






algorithm









time



Geneticalgorithm



Variables




119899119901119894119895119896



119899119901119894119895119896



Outputs













Turnaroundterminal


Ns minus 1 Ns

Train k minus c

Train k


Train k minus 1





119899pg =119873trsum119896=1

2119873119904sum119894=1

119899119896119894pg

119905wait =119873trsum119896=1

2119873119904sum119894=1

119905119896119894wait

119905veh =119873trsum119896=1

2119873119904sum119894=1

119905119896119894veh

(2)




119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895119903119894119895119892 (119889119896119894 minus 119889119888119896119894 119894) (3)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (4)

119905119896119894veh =2119873119904sum119895=119894+1


119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905

+ 119903119894119895119892+119904 (119889119896119894 minus (119892 + 119904 minus 1) 120591)] (6)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (7)

119905119896119894veh =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905


min119885 = (119905wait + 119905veh)119899pg (9)



Time

Rate

Station i

rijg

dc i

(g minus 1) g

dki

Period ldquogrdquo

(a)

Time

Rate Period

Station i

rijg

dc i

g

dki

PeriodPeriod

(g + 1) (g + s minus 1) (g + s)


(g minus 1)


ldquog + srdquo

(b)




11988901 = 0 (10)

1198890119873119904+1 =119873119904sum119903=1

(119905119894tr + 119905119894dw) + 119868119905 (11)





119878119896 = arc1198981198961 1198991198961 arc119898119896119904 119899119896119904 arc119898119896119902 119899119896119902 (13)





Δ119905119898119896119904 119899119896119904 =119899119896119904sum119894=119898119896119904

(1 minus 119886119896119894) (119905119894dw + 120575)


Terminal station 1

Station 2

Station 3

Terminalstation 4

TimeTrain 1 Train 2

Imin

Imin

Train 2lowast

Δ2


minus 119899119896119904sum119894=119898119896119904

(1 minus 119886119896minus1119894) (119905119894dw + 120575)

= 119899119896119904sum119894=119898119896119904

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) (14)


Δ 119896 = maxmax

119904sum119906=1

119899119896119906sum119894=119898119896119906

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) | 119904 = 1 119902

0

(15)














119889119896119894 minus 119889119909119894 le 119868119901 (20)


119886119896119894 = 0 forall119894 isin SS (21)

3 Solution Approach



4


2 31

arcm1 n

1

mk1 nk

1

arc arcm

n

mk2 nk

2

m n

mkq nk

q








119889119896119894 =

1198891198961 + 119894sumV=0


119886119896119894)120575) 119894 le 119873119904 forall1 le 119896 le 119873tr

119889119896119873119904+1 +119894sum


Vsum0

119886119896119894)120575) 119894 gt 119873119904 forall1 le 119896 le 119873tr(24)



forall119888 le 119896 le 119873tr

(25)







Initialization

Stop criteria

Penalize

Yes

NoFeasiblilityYes


No

Genetic algorithm

Model output

Chromosome list










Dafen

Danzhutou

Mumianwan

Buji

Changelong

Xiashuijing



Shangtang

Baishilong

Minle

Shangmeilin

LianhuaNorth

Futian

LianhuaWest

Civic Center

Lianhuacun

GangxiaNorth

HuaqiangNorth

Yannan

Hongling

Jingtian

XiangmeiNorth

Xiangmi

Qiaoxiang

AutuoHill

OCT OiaochengEast




Park

Shixia Fumin

huaqiangRd

ScienceMuseum

Luohu

guomao

GrandTheater

Hubei Huangbeiling

Xinxiu

Yijing

Tairsquoan

Buxin

LaojieShaibu

Cuizhu

Tianbei

Shuibel

Caopu

Baigelong

Departureterminal


Hongshan


TownLiuxian

dongXingdong Xili

Shenzhen

HonglangNorth

Turnaroundterminal

Lingzhi

Fanshen

Baohua

Linhai Qianhiwan

Xinrsquoan


Hi-TechPark

QiaochengNorth

Baishizhou

Hongshuwan

Keyuan

Houhai

Denglian

Shenkang

Windowof theWorld

Pingzhou

Xixiang

Gushu

hourui

AirportEast

BaorsquoanCenter

BaorsquoanStadium




120575 = (120572 + 120573)119881max432120572120573 (26)






0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns




123456789

1011121314

Tim

e per

iodg


Figure 7


















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











119899119901119894119895119896



Outputs













Turnaroundterminal


Ns minus 1 Ns

Train k minus c

Train k


Train k minus 1





119899pg =119873trsum119896=1

2119873119904sum119894=1

119899119896119894pg

119905wait =119873trsum119896=1

2119873119904sum119894=1

119905119896119894wait

119905veh =119873trsum119896=1

2119873119904sum119894=1

119905119896119894veh

(2)




119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895119903119894119895119892 (119889119896119894 minus 119889119888119896119894 119894) (3)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (4)

119905119896119894veh =2119873119904sum119895=119894+1


119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905

+ 119903119894119895119892+119904 (119889119896119894 minus (119892 + 119904 minus 1) 120591)] (6)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (7)

119905119896119894veh =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905


min119885 = (119905wait + 119905veh)119899pg (9)



Time

Rate

Station i

rijg

dc i

(g minus 1) g

dki

Period ldquogrdquo

(a)

Time

Rate Period

Station i

rijg

dc i

g

dki

PeriodPeriod

(g + 1) (g + s minus 1) (g + s)


(g minus 1)


ldquog + srdquo

(b)




11988901 = 0 (10)

1198890119873119904+1 =119873119904sum119903=1

(119905119894tr + 119905119894dw) + 119868119905 (11)





119878119896 = arc1198981198961 1198991198961 arc119898119896119904 119899119896119904 arc119898119896119902 119899119896119902 (13)





Δ119905119898119896119904 119899119896119904 =119899119896119904sum119894=119898119896119904

(1 minus 119886119896119894) (119905119894dw + 120575)


Terminal station 1

Station 2

Station 3

Terminalstation 4

TimeTrain 1 Train 2

Imin

Imin

Train 2lowast

Δ2


minus 119899119896119904sum119894=119898119896119904

(1 minus 119886119896minus1119894) (119905119894dw + 120575)

= 119899119896119904sum119894=119898119896119904

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) (14)


Δ 119896 = maxmax

119904sum119906=1

119899119896119906sum119894=119898119896119906

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) | 119904 = 1 119902

0

(15)














119889119896119894 minus 119889119909119894 le 119868119901 (20)


119886119896119894 = 0 forall119894 isin SS (21)

3 Solution Approach



4


2 31

arcm1 n

1

mk1 nk

1

arc arcm

n

mk2 nk

2

m n

mkq nk

q








119889119896119894 =

1198891198961 + 119894sumV=0


119886119896119894)120575) 119894 le 119873119904 forall1 le 119896 le 119873tr

119889119896119873119904+1 +119894sum


Vsum0

119886119896119894)120575) 119894 gt 119873119904 forall1 le 119896 le 119873tr(24)



forall119888 le 119896 le 119873tr

(25)







Initialization

Stop criteria

Penalize

Yes

NoFeasiblilityYes


No

Genetic algorithm

Model output

Chromosome list










Dafen

Danzhutou

Mumianwan

Buji

Changelong

Xiashuijing



Shangtang

Baishilong

Minle

Shangmeilin

LianhuaNorth

Futian

LianhuaWest

Civic Center

Lianhuacun

GangxiaNorth

HuaqiangNorth

Yannan

Hongling

Jingtian

XiangmeiNorth

Xiangmi

Qiaoxiang

AutuoHill

OCT OiaochengEast




Park

Shixia Fumin

huaqiangRd

ScienceMuseum

Luohu

guomao

GrandTheater

Hubei Huangbeiling

Xinxiu

Yijing

Tairsquoan

Buxin

LaojieShaibu

Cuizhu

Tianbei

Shuibel

Caopu

Baigelong

Departureterminal


Hongshan


TownLiuxian

dongXingdong Xili

Shenzhen

HonglangNorth

Turnaroundterminal

Lingzhi

Fanshen

Baohua

Linhai Qianhiwan

Xinrsquoan


Hi-TechPark

QiaochengNorth

Baishizhou

Hongshuwan

Keyuan

Houhai

Denglian

Shenkang

Windowof theWorld

Pingzhou

Xixiang

Gushu

hourui

AirportEast

BaorsquoanCenter

BaorsquoanStadium




120575 = (120572 + 120573)119881max432120572120573 (26)






0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns




123456789

1011121314

Tim

e per

iodg


Figure 7


















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Turnaroundterminal


Ns minus 1 Ns

Train k minus c

Train k


Train k minus 1





119899pg =119873trsum119896=1

2119873119904sum119894=1

119899119896119894pg

119905wait =119873trsum119896=1

2119873119904sum119894=1

119905119896119894wait

119905veh =119873trsum119896=1

2119873119904sum119894=1

119905119896119894veh

(2)




119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895119903119894119895119892 (119889119896119894 minus 119889119888119896119894 119894) (3)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (4)

119905119896119894veh =2119873119904sum119895=119894+1


119899119896119894pg =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905

+ 119903119894119895119892+119904 (119889119896119894 minus (119892 + 119904 minus 1) 120591)] (6)

119905119896119894wait = 12119899119896119894pg (119889119896119894 minus 119889119888119896119894 119894)2 (7)

119905119896119894veh =2119873119904sum119895=119894+1

119886119896119894119886119896119895 [119903119894119895119892 (119892120591 minus 119889119888119896119894 119894) +119892+119904minus1sum119905=119892+1

120591119903119894119895119905


min119885 = (119905wait + 119905veh)119899pg (9)



Time

Rate

Station i

rijg

dc i

(g minus 1) g

dki

Period ldquogrdquo

(a)

Time

Rate Period

Station i

rijg

dc i

g

dki

PeriodPeriod

(g + 1) (g + s minus 1) (g + s)


(g minus 1)


ldquog + srdquo

(b)




11988901 = 0 (10)

1198890119873119904+1 =119873119904sum119903=1

(119905119894tr + 119905119894dw) + 119868119905 (11)





119878119896 = arc1198981198961 1198991198961 arc119898119896119904 119899119896119904 arc119898119896119902 119899119896119902 (13)





Δ119905119898119896119904 119899119896119904 =119899119896119904sum119894=119898119896119904

(1 minus 119886119896119894) (119905119894dw + 120575)


Terminal station 1

Station 2

Station 3

Terminalstation 4

TimeTrain 1 Train 2

Imin

Imin

Train 2lowast

Δ2


minus 119899119896119904sum119894=119898119896119904

(1 minus 119886119896minus1119894) (119905119894dw + 120575)

= 119899119896119904sum119894=119898119896119904

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) (14)


Δ 119896 = maxmax

119904sum119906=1

119899119896119906sum119894=119898119896119906

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) | 119904 = 1 119902

0

(15)














119889119896119894 minus 119889119909119894 le 119868119901 (20)


119886119896119894 = 0 forall119894 isin SS (21)

3 Solution Approach



4


2 31

arcm1 n

1

mk1 nk

1

arc arcm

n

mk2 nk

2

m n

mkq nk

q








119889119896119894 =

1198891198961 + 119894sumV=0


119886119896119894)120575) 119894 le 119873119904 forall1 le 119896 le 119873tr

119889119896119873119904+1 +119894sum


Vsum0

119886119896119894)120575) 119894 gt 119873119904 forall1 le 119896 le 119873tr(24)



forall119888 le 119896 le 119873tr

(25)







Initialization

Stop criteria

Penalize

Yes

NoFeasiblilityYes


No

Genetic algorithm

Model output

Chromosome list










Dafen

Danzhutou

Mumianwan

Buji

Changelong

Xiashuijing



Shangtang

Baishilong

Minle

Shangmeilin

LianhuaNorth

Futian

LianhuaWest

Civic Center

Lianhuacun

GangxiaNorth

HuaqiangNorth

Yannan

Hongling

Jingtian

XiangmeiNorth

Xiangmi

Qiaoxiang

AutuoHill

OCT OiaochengEast




Park

Shixia Fumin

huaqiangRd

ScienceMuseum

Luohu

guomao

GrandTheater

Hubei Huangbeiling

Xinxiu

Yijing

Tairsquoan

Buxin

LaojieShaibu

Cuizhu

Tianbei

Shuibel

Caopu

Baigelong

Departureterminal


Hongshan


TownLiuxian

dongXingdong Xili

Shenzhen

HonglangNorth

Turnaroundterminal

Lingzhi

Fanshen

Baohua

Linhai Qianhiwan

Xinrsquoan


Hi-TechPark

QiaochengNorth

Baishizhou

Hongshuwan

Keyuan

Houhai

Denglian

Shenkang

Windowof theWorld

Pingzhou

Xixiang

Gushu

hourui

AirportEast

BaorsquoanCenter

BaorsquoanStadium




120575 = (120572 + 120573)119881max432120572120573 (26)






0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns




123456789

1011121314

Tim

e per

iodg


Figure 7


















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Time

Rate

Station i

rijg

dc i

(g minus 1) g

dki

Period ldquogrdquo

(a)

Time

Rate Period

Station i

rijg

dc i

g

dki

PeriodPeriod

(g + 1) (g + s minus 1) (g + s)


(g minus 1)


ldquog + srdquo

(b)




11988901 = 0 (10)

1198890119873119904+1 =119873119904sum119903=1

(119905119894tr + 119905119894dw) + 119868119905 (11)





119878119896 = arc1198981198961 1198991198961 arc119898119896119904 119899119896119904 arc119898119896119902 119899119896119902 (13)





Δ119905119898119896119904 119899119896119904 =119899119896119904sum119894=119898119896119904

(1 minus 119886119896119894) (119905119894dw + 120575)


Terminal station 1

Station 2

Station 3

Terminalstation 4

TimeTrain 1 Train 2

Imin

Imin

Train 2lowast

Δ2


minus 119899119896119904sum119894=119898119896119904

(1 minus 119886119896minus1119894) (119905119894dw + 120575)

= 119899119896119904sum119894=119898119896119904

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) (14)


Δ 119896 = maxmax

119904sum119906=1

119899119896119906sum119894=119898119896119906

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) | 119904 = 1 119902

0

(15)














119889119896119894 minus 119889119909119894 le 119868119901 (20)


119886119896119894 = 0 forall119894 isin SS (21)

3 Solution Approach



4


2 31

arcm1 n

1

mk1 nk

1

arc arcm

n

mk2 nk

2

m n

mkq nk

q








119889119896119894 =

1198891198961 + 119894sumV=0


119886119896119894)120575) 119894 le 119873119904 forall1 le 119896 le 119873tr

119889119896119873119904+1 +119894sum


Vsum0

119886119896119894)120575) 119894 gt 119873119904 forall1 le 119896 le 119873tr(24)



forall119888 le 119896 le 119873tr

(25)







Initialization

Stop criteria

Penalize

Yes

NoFeasiblilityYes


No

Genetic algorithm

Model output

Chromosome list










Dafen

Danzhutou

Mumianwan

Buji

Changelong

Xiashuijing



Shangtang

Baishilong

Minle

Shangmeilin

LianhuaNorth

Futian

LianhuaWest

Civic Center

Lianhuacun

GangxiaNorth

HuaqiangNorth

Yannan

Hongling

Jingtian

XiangmeiNorth

Xiangmi

Qiaoxiang

AutuoHill

OCT OiaochengEast




Park

Shixia Fumin

huaqiangRd

ScienceMuseum

Luohu

guomao

GrandTheater

Hubei Huangbeiling

Xinxiu

Yijing

Tairsquoan

Buxin

LaojieShaibu

Cuizhu

Tianbei

Shuibel

Caopu

Baigelong

Departureterminal


Hongshan


TownLiuxian

dongXingdong Xili

Shenzhen

HonglangNorth

Turnaroundterminal

Lingzhi

Fanshen

Baohua

Linhai Qianhiwan

Xinrsquoan


Hi-TechPark

QiaochengNorth

Baishizhou

Hongshuwan

Keyuan

Houhai

Denglian

Shenkang

Windowof theWorld

Pingzhou

Xixiang

Gushu

hourui

AirportEast

BaorsquoanCenter

BaorsquoanStadium




120575 = (120572 + 120573)119881max432120572120573 (26)






0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns




123456789

1011121314

Tim

e per

iodg


Figure 7


















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Terminal station 1

Station 2

Station 3

Terminalstation 4

TimeTrain 1 Train 2

Imin

Imin

Train 2lowast

Δ2


minus 119899119896119904sum119894=119898119896119904

(1 minus 119886119896minus1119894) (119905119894dw + 120575)

= 119899119896119904sum119894=119898119896119904

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) (14)


Δ 119896 = maxmax

119904sum119906=1

119899119896119906sum119894=119898119896119906

(119886119896minus1119894 minus 119886119896119894) (119905119894dw + 120575) | 119904 = 1 119902

0

(15)














119889119896119894 minus 119889119909119894 le 119868119901 (20)


119886119896119894 = 0 forall119894 isin SS (21)

3 Solution Approach



4


2 31

arcm1 n

1

mk1 nk

1

arc arcm

n

mk2 nk

2

m n

mkq nk

q








119889119896119894 =

1198891198961 + 119894sumV=0


119886119896119894)120575) 119894 le 119873119904 forall1 le 119896 le 119873tr

119889119896119873119904+1 +119894sum


Vsum0

119886119896119894)120575) 119894 gt 119873119904 forall1 le 119896 le 119873tr(24)



forall119888 le 119896 le 119873tr

(25)







Initialization

Stop criteria

Penalize

Yes

NoFeasiblilityYes


No

Genetic algorithm

Model output

Chromosome list










Dafen

Danzhutou

Mumianwan

Buji

Changelong

Xiashuijing



Shangtang

Baishilong

Minle

Shangmeilin

LianhuaNorth

Futian

LianhuaWest

Civic Center

Lianhuacun

GangxiaNorth

HuaqiangNorth

Yannan

Hongling

Jingtian

XiangmeiNorth

Xiangmi

Qiaoxiang

AutuoHill

OCT OiaochengEast




Park

Shixia Fumin

huaqiangRd

ScienceMuseum

Luohu

guomao

GrandTheater

Hubei Huangbeiling

Xinxiu

Yijing

Tairsquoan

Buxin

LaojieShaibu

Cuizhu

Tianbei

Shuibel

Caopu

Baigelong

Departureterminal


Hongshan


TownLiuxian

dongXingdong Xili

Shenzhen

HonglangNorth

Turnaroundterminal

Lingzhi

Fanshen

Baohua

Linhai Qianhiwan

Xinrsquoan


Hi-TechPark

QiaochengNorth

Baishizhou

Hongshuwan

Keyuan

Houhai

Denglian

Shenkang

Windowof theWorld

Pingzhou

Xixiang

Gushu

hourui

AirportEast

BaorsquoanCenter

BaorsquoanStadium




120575 = (120572 + 120573)119881max432120572120573 (26)






0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns




123456789

1011121314

Tim

e per

iodg


Figure 7


















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










4


2 31

arcm1 n

1

mk1 nk

1

arc arcm

n

mk2 nk

2

m n

mkq nk

q








119889119896119894 =

1198891198961 + 119894sumV=0


119886119896119894)120575) 119894 le 119873119904 forall1 le 119896 le 119873tr

119889119896119873119904+1 +119894sum


Vsum0

119886119896119894)120575) 119894 gt 119873119904 forall1 le 119896 le 119873tr(24)



forall119888 le 119896 le 119873tr

(25)







Initialization

Stop criteria

Penalize

Yes

NoFeasiblilityYes


No

Genetic algorithm

Model output

Chromosome list










Dafen

Danzhutou

Mumianwan

Buji

Changelong

Xiashuijing



Shangtang

Baishilong

Minle

Shangmeilin

LianhuaNorth

Futian

LianhuaWest

Civic Center

Lianhuacun

GangxiaNorth

HuaqiangNorth

Yannan

Hongling

Jingtian

XiangmeiNorth

Xiangmi

Qiaoxiang

AutuoHill

OCT OiaochengEast




Park

Shixia Fumin

huaqiangRd

ScienceMuseum

Luohu

guomao

GrandTheater

Hubei Huangbeiling

Xinxiu

Yijing

Tairsquoan

Buxin

LaojieShaibu

Cuizhu

Tianbei

Shuibel

Caopu

Baigelong

Departureterminal


Hongshan


TownLiuxian

dongXingdong Xili

Shenzhen

HonglangNorth

Turnaroundterminal

Lingzhi

Fanshen

Baohua

Linhai Qianhiwan

Xinrsquoan


Hi-TechPark

QiaochengNorth

Baishizhou

Hongshuwan

Keyuan

Houhai

Denglian

Shenkang

Windowof theWorld

Pingzhou

Xixiang

Gushu

hourui

AirportEast

BaorsquoanCenter

BaorsquoanStadium




120575 = (120572 + 120573)119881max432120572120573 (26)






0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns




123456789

1011121314

Tim

e per

iodg


Figure 7


















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation











Initialization

Stop criteria

Penalize

Yes

NoFeasiblilityYes


No

Genetic algorithm

Model output

Chromosome list










Dafen

Danzhutou

Mumianwan

Buji

Changelong

Xiashuijing



Shangtang

Baishilong

Minle

Shangmeilin

LianhuaNorth

Futian

LianhuaWest

Civic Center

Lianhuacun

GangxiaNorth

HuaqiangNorth

Yannan

Hongling

Jingtian

XiangmeiNorth

Xiangmi

Qiaoxiang

AutuoHill

OCT OiaochengEast




Park

Shixia Fumin

huaqiangRd

ScienceMuseum

Luohu

guomao

GrandTheater

Hubei Huangbeiling

Xinxiu

Yijing

Tairsquoan

Buxin

LaojieShaibu

Cuizhu

Tianbei

Shuibel

Caopu

Baigelong

Departureterminal


Hongshan


TownLiuxian

dongXingdong Xili

Shenzhen

HonglangNorth

Turnaroundterminal

Lingzhi

Fanshen

Baohua

Linhai Qianhiwan

Xinrsquoan


Hi-TechPark

QiaochengNorth

Baishizhou

Hongshuwan

Keyuan

Houhai

Denglian

Shenkang

Windowof theWorld

Pingzhou

Xixiang

Gushu

hourui

AirportEast

BaorsquoanCenter

BaorsquoanStadium




120575 = (120572 + 120573)119881max432120572120573 (26)






0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns




123456789

1011121314

Tim

e per

iodg


Figure 7


















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










Dafen

Danzhutou

Mumianwan

Buji

Changelong

Xiashuijing



Shangtang

Baishilong

Minle

Shangmeilin

LianhuaNorth

Futian

LianhuaWest

Civic Center

Lianhuacun

GangxiaNorth

HuaqiangNorth

Yannan

Hongling

Jingtian

XiangmeiNorth

Xiangmi

Qiaoxiang

AutuoHill

OCT OiaochengEast




Park

Shixia Fumin

huaqiangRd

ScienceMuseum

Luohu

guomao

GrandTheater

Hubei Huangbeiling

Xinxiu

Yijing

Tairsquoan

Buxin

LaojieShaibu

Cuizhu

Tianbei

Shuibel

Caopu

Baigelong

Departureterminal


Hongshan


TownLiuxian

dongXingdong Xili

Shenzhen

HonglangNorth

Turnaroundterminal

Lingzhi

Fanshen

Baohua

Linhai Qianhiwan

Xinrsquoan


Hi-TechPark

QiaochengNorth

Baishizhou

Hongshuwan

Keyuan

Houhai

Denglian

Shenkang

Windowof theWorld

Pingzhou

Xixiang

Gushu

hourui

AirportEast

BaorsquoanCenter

BaorsquoanStadium




120575 = (120572 + 120573)119881max432120572120573 (26)






0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns




123456789

1011121314

Tim

e per

iodg


Figure 7


















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation










0

1000

2000

3000

4000

5000

6000

7000

Dem

and

(pas

seng

ers)

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 301

Station

Station 1 minus Ns




123456789

1011121314

Tim

e per

iodg


Figure 7


















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation

















119896and passen-


respectively

np119894119895119896= 119899minus1sumℎ=1


(27)

tw119894119895119896= 119899minus2sumℎ=1

(1 minus 119903119891) [12119903119894119895119892 (119889119888ℎ+1119894 minus 119889119888ℎ 119894)2


(28)




tw119894119895119896


tv119894119895119896


(32)

tv119894119895119896


(33)






confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












confusedpassengers








All-stop 0 0 NA NA NA 298 1350 1648

FSSS

0 0 NA NA NA 304 1262 156602 79

575442 874819 14491261

305 1263 156804 158 306 1267 157306 234 308 1270 157808 312 309 1274 158310 390 310 1277 1587

000

6

001

2

001

8

002

4

003

0

003

6

004

2

004

8

005

4

010

6

011

2

011

8

012

4

013

0

013

6

014

2

014

8

015

4

010

0

020

6

021

2

021

8

022

4

023

0

023

6

024

2

024

8

025

4

020

0

030

6

031

2

031

8

032

4

020

0

Stat

ion

All-stop schemeFSSS

1

3

5

7

9

11

13

15

17

19

21

23

25

27

29



5 Conclusion







Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












Appendix




119889119896+119909119873119904 minus 119889119896+119909minus1119873119904+1 ge 119868119905

(A1)


119889119896+119909119873119904+1 minus 119889119896+119909119873119904 ge 119868119905 (A2)

Trai

n 1

Trai

n 2

Trai

n 3

Trai

n 4

Trai

n 5

Trai

n 6

Trai

n 7

Trai

n 8

Trai

n 9

Trai

n 10


0

05

1

15

Seat

occ

upan

cy ra

te

01002003004005006007008009001000

Max

imum

sect

ion

load



119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 2119909119868119905 (A3)


119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min (A4)









119889119896+119909119873119904+1 minus 119889119896119873119904+1 ge 119909119868min






Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation













Acknowledgments


References




































RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation












RoboticsJournal of





Journal of



RotatingMachinery



Journal of

Volume 201


VLSI Design



Shock and Vibration







Journal of



Volume 2014


SensorsJournal of





Propagation









Optimized Skip-Stop Metro Line Operation Using Smart Card...

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Transcript of Optimized Skip-Stop Metro Line Operation Using Smart Card...