research article scheduling additional train unit services on rail … · 2019. 7. 31. ·...

Research ArticleScheduling Additional Train Unit Services on Rail Transit Lines

Zhibin Jiang1 Yuyan Tan2 and Oumlzguumlr YalccedilJnkaya3

1 School of Transportation Engineering Key Laboratory of Road and Traffic Engineering of theMinistry of Education Tongji University4800 Caorsquoan Road Shanghai 201804 China

2 Institute of Railway Systems Engineering and Traffic Safety Technical University of Braunschweig Pockelsstraszlige 338106 Braunschweig Germany

3Department of Industrial Engineering Dokuz Eylul University 35160 Buca-Izmir Turkey

Correspondence should be addressed to Yuyan Tan ytantu-bsde

Received 4 June 2014 Accepted 31 July 2014 Published 25 September 2014

Academic Editor Wuhong Wang

Copyright copy 2014 Zhibin Jiang 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

This paper deals with the problem of scheduling additional train unit (TU) services in a double parallel rail transit line and amixedinteger programming (MIP) model is formulated for integration strategies of new trains connected by TUs with the objective ofobtaining higher frequencies in some special sections and special time periods due to mass passenger volumes We took timetablescheduling and TUs scheduling as an integrated optimization model with two objectives minimizing travel times of additionaltrains and minimizing shifts of initial trains We illustrated our model using computational experiments drawn from the real railtransit line 16 in Shanghai and reached results which show that rail transit agencies can obtain a reasonable new timetable fordifferent managerial goals in a matter of seconds so the model is well suited to be used in daily operations

1 Introduction

Transit scheduling is the processes of computing the fre-quency of services the number of required vehicles thetiming of their travel and other related operating elementsThe outcomes of scheduling include graphical and numericalschedules for operators and supervisors timetables for thepublic and operating data for a line [1] The rail transittimetable is aimed to meet the passenger demand whichvaries during the hours of a day the day of a week from oneseason to another and so forth [2] On rail transit lines due tothe high frequencies and strict stock capacities in terminalsthe timetable scheduling and the TUs scheduling should beconsidered simultaneously Inserting some new train servicesinto an initial timetable is one of the important methods inthe process of redeveloping a timetable

The primary motivation of this research based on addi-tional demands occurrence in the rail transit lines of Shang-hai These additional demands causing timetabling prob-lems have been determined by the Shanghai ShentongMetroOperation Company which is the responsible authority forthe daily operations The authority thinks it is an important

problem and needed to be solved more efficiently accuratelyand fast Up toMarch 2014 there have been 14 rail transit lines(with an operating route length of 538 kilometers and 329stations) operated in Shanghai On a normal weekday morethan 8 million people use the Shanghai rail transit networkPlanning of the rail transit operations primarily concerns thetimetable and two other main resources the rolling stocksand the crews Planning of these resources undergoes twomain phases (tactical and short-term planning) before theactual operationThe planning horizon in tactical planning isfrom one month up to one year The steps conducted duringthis planning phase are constructing several initial timetables(for working days weekend days holidays etc) which satisfydifferent service demands and allocating the rolling stocksand the crews to the initial timetables On the other hand theshort-term planning phase refers to planning taskswith a timehorizon of a fewdays up to onemonth In this phase the initialplans are adapted to the demands of the corresponding daysSpecial holidays and events that attract a lot of people suchas exhibitions concerts and major sports events generallyrequire an offered capacity in different times and positionsConsequently some train services are required to be inserted

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2014 Article ID 954356 13 pageshttpdxdoiorg1011552014954356

2 Mathematical Problems in Engineering

to improve the capacity of some special sections with timewindowsThemost commonway is inserting additional trainservices into the initial timetables

The problem which is called scheduling additional trainunit services (SATUS) problem is a problem in which newtrains connected by a number of TUs start their trip from adepot or reversing tracks after collectively visiting a numberof routes they return to the starting points The SATUSproblem is a complicated one because the efficient circulationof TUs is an important consideration for operators of railtransit trains additionally the large number of trips linksand paths to be considered rapidly increases the numberof variables and constraints in any model developed TheSATUS problem is not a real-time rescheduling problemsince themain difference between it and timetable reschedul-ing in short-term planning or in disruption managementis the absence of uncertainty and the fact that the latter ismuch less time-critical while the first is often thought as atemporary redevelopment strategy of an initial timetable

This paper deals with the problem of SATUS in a doubleparallel rail transit line and a MIP model is formulated forintegration strategies of new trains connected by TUs withthe objective of obtaining higher frequencies in some specialsections and special time periods due to mass passengervolumes

The study contributes a number of new features tocapture the influence of specific elements that have not beenconsidered in studies on the SATUS problem in the relatedliterature First of all the approach decides on timetablesand TUs schedules using an integrated optimization modelaccording to sections and turnback capacities Second amaximum deviation for arrival or departure times of trainsin an initial timetable all-station-stopping policy and expressservice strategy linking orders and time windows of newinserted trains are also considered Finally thismodel has twoobjectives minimizing travel times of additional trains andminimizing shifts of initial trains

The paper is organized as follows In Section 2 a shortreview related to the SATUS problem is provided Afterthat Section 3 introduces a brief summary of the relevantconcepts in the model description A MIP model includingsets parameters decision variables and objective functionsis presented in Section 4 Section 5 illustrates the proposedmodel with an example The conclusions and future studiesare summarized in Section 6

2 Literature Review

The SATUS problem is related to a variety of topics in theliterature The first and foremost is railway transportationTrain scheduling rescheduling and routing problems havehad a great deal of attention in recent years There are twomain timetable variants One of the variants is the periodic(or cyclic) timetable that is repeated every given time periodfor example every hour with only slight differences betweenpeak hours and off-peak hours The other variant is thenonperiodic timetable which allows following the passengerdemands with the frequencies of the trains In both cases

timetables are usually repeated every day although theremaybe differences between weekdays and weekend

Cacchiani et al [3 4] gave a detailed review of theliterature on timetable scheduling The timetable schedulingproblem in a rail transit system in which TUs crews andpassengers are incorporated into a single planning frame-work science is complex various constraints and objectivesshould be considered simultaneously Due to its importanceand complexity which have been acknowledged in variouspublications topics related to this issue have attracted consid-erable attention in the literature A multiphase semiregulartimetable which divides a day into several time periods andeven applies the vehicle-departing interval for each periodmay somehow help to accommodate peak-hour demandwhile maintaining a certain level of service for passengersboarding at nonpeak hours Guihaire and Hao [5] presenteda global review of the crucial strategic and tactical steps oftransit planning and also discussed the scheduling problemwith phase regular for a transit corridor Ceder [6] provided acomprehensive modeling framework for determining vehicledeparture time with either even headways or even averageloads with a special focus on smoothing the transitionsbetween time periods These studies provide useful methodsfor optimizing frequency for a particular time period while aunified framework is critically needed for scheduling meth-ods that can consider uneven headways and time-dependentdemand patterns Jiang et al [7 8] presented a computationaltimetable scheduling method in rail transit line with mul-tiroutes or circle route and a timetable designing softwarenamed Train Plan Maker (TPM) was developed and appliedbymanymetro operation companies in China Niu and Zhou[9] focused on optimizing a passenger train timetable ina heavily congested urban rail corridor A binary integerprogramming model incorporated with passenger loadingand departure events was constructed to provide a theoreticdescription for the problem under consideration Freyss et al[10] focused on the skip-stop operation for rail transit linesusing a single one-way track and the system was modeled byusing a continuous approximation approach

Once the timetable scheduling has been defined therolling stock and TUs assignments must be done An integerprogramming model was considered by Alfieri et al [11] todetermine the rolling stock circulation for multiple rollingstock types on a single line and on a single day and thismodelwas extended by Fioole et al [12] by including combining andsplitting trains as it happens at several locations in the Dutchtimetables Cadarso et al [13] studied the disruptionmanage-ment problem of rapid transit rail networks Besides optimiz-ing timetable and rolling stock schedules they explicitly dealtwith the effects of disruption on the passenger demandsTheyproposed a two-step approach that combines an integratedoptimization model (for the timetable and the rolling stock)with a model for the behaviors of passengers Lin and Kwan[14] proposed a two-phase approach for the TUs schedulingproblemThefirst phase assigned and sequenced train trips toTUs temporarily ignoring some station infrastructure detailswhich was modeled as an integer fixed-charge multicom-modity flow (FCMF) problem The second phase focused onsatisfying the remaining station detailed requirements which

Mathematical Problems in Engineering 3

was modeled as a multidimensional matching problem witha mixed integer linear programming (MILP) formulationEberlein et al [15] studied a real-time deadheading problemin transit operations control Haghani and Banihashemi [16]proposed an innovative multiple depot vehicle schedulingwith route time constraints (MDVSRTC) model to solve bustransit vehicle scheduling problems After that they derived asingle depot vehicle scheduling with route time constraints(SDVSRTC) model to solve the same problem [17] Yu etal [18] presented a partway deadheading strategy for transitoperations to improve transit service of the peak directions oftransit routes

Inserting additional train into an existing timetable isa common technique used in railway systems Burdett andKozan [19] considered techniques for scheduling additionaltrain services integrated into current timetables and involvinggeneral time window constraints fixed operations mainte-nance activities and periods of section unavailability Flieret al [20] addressed the recurring problem of adding a trainpath that is a schedule for a single train in terms of trackallocation in space and time to a given dense timetable on acorridor which is an important subnetwork in form of a pathbetween two major stations

The SATUS problem includes the timetable schedulingand the TUs circulation problems therefore it is usuallymuchmore complex and difficult to solve than the models dealingwith a single phase Cadarso and Marın [2] proposed anintegrated MIP model to adapt the frequencies in a timetabletogether with rolling stock circulation in order to deal withincreased passenger demands and traffic congestion in a rapidtransit network They also took into account the shuntingof rolling stocks in depots Canca et al [21] proposed atactical model to determine optimal policies of short-turningand nonstopping at certain stations considering differentobjectives such as minimizing the passenger overload andpreserving certain level of quality of service

Our study contributes a number of new features tocapture the influence of specific elements which have notbeen studied in the related literature as given in previoussection

3 Problem Description

In this section the SATUS problem in rail transit linesis described in detail Firstly the rail transit line and theroutes are introduced After that we describe the timetableand the TUs circulation problems Then headway and traintraveling times are introduced and finally how the capacityof turnback operation is modeled has been explained

31 Rail Transit Network and Route The rail transit line withbranch linking depots is considered to be a simple networkwith a collection of stations and sections as illustrated inFigure 1 A rail transit network119866 is defined by a set of stations119878 that are connected to each other by a set of sections 119861 Therail transit line in the model consists of parallel double lineswhere trains follow a loop running from a certain station

b5

b1 b2 b3 b4

Intermediate stationDepot link station

TurnbacksDepot

RouteRoute

s1 s2 s3

s6

s4 s5

rs5s1rs1s5

Figure 1 Rail transit line infrastructure definition

s1 s2 s3 s4 s5

s6 1s1s5tr1

tr4tr3

tr1

tr1

tr1 tr1 tr1

tr2 tr2 tr2 tr2

tr2

tr2

tp

Figure 2 Train route infrastructure definition

denoted as a starting point to an end station with right-handrunning rule

A train route is a group of trains that run bidirectionalbetween two stations on the rail transit line All trains inthe same route have the same size capacity and operatingcharacteristics and additionally they always visit the samesequence of stations We define 119903

119904119894 119904119895as a route linked by the

stations 119904119894and 119904119895 Rail transit line can be characterized by two

main train route styles (1) normal cyclic routes and (2) depotlinking routes The first one comprises the daily operationsof fixed train cyclic running paths with trains stopping andproviding passenger loading services (119903

1199041 1199045 1199031199045 1199041

1199031199041 1199044

and1199031199044 1199041

in Figure 1) The latter refers to the route linking depotwith a main turnback station in which trains sometimes donot stop and cannot provide passenger services (119903

1199046 1199041 1199031199041 1199046

1199031199046 1199042

and 1199031199042 1199046

in Figure 2)A train track path is defined as the detailed train running

path from an original station to a destination station includ-ing the specified tracks in all stations Let 119904

119894(tr119895) describe the

track tr119895of the station 119904

119894 then the train track path from 119904

1(tr4)

to 1199045(tr1) in Figure 2 can be expressed by

tp11199041 1199045

= 1199041(tr4) 1199041(tr2) 1199042(tr2) 1199043(tr2) 1199044(tr2) 1199045(tr1)

(1)

32 Timetable and TUs Circulation In rail transit lines atime-distance diagram has the line (distance) plotted onthe vertical and time on the horizontal axes As shown inFigure 3 the line is divided in sections with uniform speedsThe plot of every run of a train and TUs indicated by anumber shows all scheduled elements (travel time speedetc) of the train on each section and at each terminalThe horizontal axis also shows headways as time distancesbetween subsequent train runs and cycle time (119879

119888) as time

distances between two successive departures related to the


Layover time u2 u2

u2 u2

u2

6 7

1

u1

u1

u1 u1

u1Pull-out train

4 9

Tc

582 3

-depot

s1

s2

s3

s4

s5-

Figure 3 Time-distance diagram for a rail transit line

same TUs from a terminal The whole diagram shows trainarrivalsdepartures at each reference point along the linelayover time as well as locations and times where trainsmeetTime-distance diagram can also show pull-outs and pull-ins of trains from depots for operations on some sectionsdifferent stopping times and so forth

In our model the set of trains considered is given by 119879 =

119879ini

cup119879add where119879ini denotes the set of initial trains that have

a prescribed timetable and 119879add denotes the set of additional

trains that need to be inserted to the original timetableFor each train 119894 isin 119879

ini a timetable is specified consistingof the following

(i) an ordered sequence of trains 119905119894

(ii) an ordered sequence of TUs 119880119895

(iii) an ordered sequence of trains linked byTU 119895 and119906119895=

119905119894 119905

119901

(iv) an ordered sequence of stations 119878119894

= 119891119894 119897119894 isin 119878

that the train 119894 visits where 119891119894is the first (origin)

station and 119897119894is the last (destination) station

(v) the departure time from 119891119894 the arrival time to 119897

119894 and

the arrival and departure times for the intermediatestations in 119878

119894

119891119894 119897119894 of the train 119894

(vi) the exact track path 119896119894that is allocated to the train 119894

on each station(vii) themaximumdeviation for arrival or departure times

of trains(viii) the minimum and the maximum dwell times at each

station in 119878119894

119891119894 119897119894 and the trip time at each section

119887 = 119895 1198951015840

with 119895 1198951015840

isin 119878119894

For each train 119894 isin 119879add a timetable is specified consisting

of the following

(i) a sequence of TUs 119906add119895

(ii) an ordered sequence of new trains 119905add119894

(iii) an ordered sequence of trains linked by TU 119895 and

119906add119895

= 119905add119894

119905add119901


= 119891119894 119897119894 isin 119878



sj+1

bk

sj

haa

rpp

hdd rsp

rss

rps

tde

tac

Figure 4 Illustration of headways and train traveling times

(v) the exact track path 119896119894that is allocated to the train 119894

on each station(vi) the desired departure time window from 119891

119894 the

minimum and the maximum dwell times at eachstation in 119878

119894


119887 = 119895 1198951015840

with 119895 1198951015840

isin 119878119894

33 Headways and Train Traveling Times The minimumheadway on a line is determined by the physical charac-teristics of the system (technology methods of driving andcontrol and required degree of safety) and station operations(rate of boardingalighting departure control etc) In ourmodel we consider the express service strategy so theheadways need to be defined separately for departing andarriving Set ℎ

119889119889to be the minimum headway of two suc-

cessive trains departing from stations and ℎ119886119886

the minimumheadway of two successive trains arriving to stations asshown in Figure 4 Each time when an intermediate station ispassed by a train the spent times in decelerating stoppingand accelerating of the vehicle are saved at the successivestation So this model considers acceleration time (119905ac) anddeceleration time (119905de) as shown in Figure 4 There are fourexecution modes for train traveling at section 119887

119896 namely (1)

bypassing stations 119904119895and 119904119895+1

(119903119901119901) (2) bypassing station 119904

119895

but stopping at station 119904119895+1

(119903119901119904) (3) stopping at station 119904

119895but

bypassing station 119904119895+1

(119903119904119901) and (4) stopping at both stations

119904119895and 119904119895+1

(119903119904119904) So 119903

119901119904 119903119904119901 and 119903

119904119904can be calculated by the

following respectively

119903119901119904

= 119903119901119901

+ 119905de

119903119904119901

= 119903119901119901

+ 119905ac

119903119904119904

= 119903119901119901

+ 119905ac + 119905de

(2)

34 Layover Time and Turnback Operation Layover time isthe time between the scheduled arrival and departure of avehicle at a transit terminal Minimum layover time includesthe dwell time for alighting and boarding of passengersthe time for changing the train operator and conductingany necessary inspections and brake tests and the time formoving and locking the crossover switches and the time forrecovery of the schedule if it is needed Maximum layover


T2

T1

Platform

tbTAS12

u1 u2u3

654321

tRst

Number of crossing points le 1

(a) Turnback operation with crossover located in advance of astation (TAS)

T2

T1

Platform

u1

u2u3

65 43 21

T3

T4

tbTBS12


(b) Turnback operation with crossover located in back of astation (TBS)

Figure 5 Track occupation of turnback operation process at a terminal

time is a function of terminal capacity (number of reversingtracks and platform clearance time) and train arrival rate

There are two typical turnback operations according tothe terminal types turnback operationwith crossover locatedin (1) advance of a station (TAS) and (2) back of a station(TBS) as illustrated in Figure 5 On the condition of TAS ifall trains occupy the same turnback track the second arrivingtrain (V

3in Figure 5(a)) arrival to the station must insure

that the first departing train (V2in Figure 5(a)) which linked

with the first arrival train (V1in Figure 5(a)) has left from

the station Let 119905119877st be the minimum separation time of trainsthat are occupying the same turnback track the occupationtime of each train pair (V

119894 V119895) in which the train V

119894and the

consecutive train V119895share the same TUs at a terminal can be

calculated by

tbTASV119894 V119895 isin [119905119886

V119894 119905119889

V119895 + 119905119877

st] (3)

where 119905119886

V119894 119905119889

V119895 are the arrival time of train V119894and the departure

time of train V119895at the terminal respectively So the capacity

constraint with TAS can be transferred to this problem atany time the number of TUs (same value of the number ofcrossing points as shown in Figure 5(a)) staying in a terminalcannot be more than one

On the other hand on the condition of TBS the arrivingtrain pulls into one platform and then pulls into one of thetail tracks changes direction and then returns to pick uppassengers from the other platform So there is no conflictbetween departing and arriving trains But the maximumnumber of existing TUs at any time in the terminal dependson the number of tail tracks So the capacity constraint withTBS turnback operation can be transferred to this problemat any time the number of TUs staying in a terminal cannot

be more than three (only one tail track can be selected) asshown in Figure 5(b) And the occupation time of each trainpairs (V

119894 V119895) at the terminal can be calculated by

tbTBSV119894 V119895 isin [119905119886

V119894 119905119889

V119895] (4)

4 Model Description

The model of the SATUS problem is developed as a MIPmodel It aims at computing a new timetable accompaniedwith a TUs schedule for a rail transit line and balances severalobjective criterions

41 Sets The sets below contain the basic information for ourmathematical model

119878 set of stations in the rail transit line119861 set of sections between two stations 119887 = (119904

119894 119904119895) in

the rail transit line with 119904119894 119904119895isin 119878

119878119879 set of turnback stations

119879 = 119879ini

cup 119879add set of all trains consisting of

additional trains 119879add and initial trains 119879ini119880 = 119880

inicup119880

add set of all TUs consisting of additionalTUs 119880add and initial TUs 119880ini119877119895 set of all train pairs (119894 119894

1015840

) with 119894 lt 1198941015840 when the

train 119894 and the consecutive train 1198941015840 share the same TUs

at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878119879

119878119894

isin 119878 set of stations that the train 119894 visits119861119894

isin 119861 set of sections that the train 119894 travels along119875 set of time slot in the planning horizon


119891119894 set of first (starting) travelling station of the train

119894 119894 isin 119879119897119894 set of last (ending) travelling station of the train 119894

119894 isin 119879

42 Parameters The model uses the following parameterswhich are all assumed to be integer valued

119905min119867

the minimum time of the planning horizon119905max119867

the maximum time of the planning horizon

119909119886ini119894119895

the departure time of the train 119894 from the station119895 119894 isin 119879

ini 119895 isin 119878119894

119909119889ini119894119895

the arrival time of the train 119894 at the station 119895119894 isin 119879

ini 119895 isin 119878119894

ℎ119889119889 the minimum headway time between two con-

secutive departuresℎ119886119886 theminimumheadway time between two consec-

utive arrivals119905ac the acceleration time119905de the deceleration time119903119887 the traveling time of a train without any stops at

stations 119904119894and 119904119895 119887 = (119904

119894 119904119895) isin 119861

dwmin119894119895

the minimum dwell time of the train 119894 if it hasa loading service at the station 119895 119894 isin 119879 119895 isin 119878 = 0otherwisedwmax119894119895

the maximum dwell time of the train 119894 at thestation 119895 119894 isin 119879 119895 isin 119878

119894119862min119895

the minimum layover time at the terminal 119895 119895 isin

119878119879

119862max119895

themaximum layover time at the terminal 119895 119895 isin

119878119879

119872 a sufficiently large positive constant (here giventhe value 3600 times 24 that is the length of the largestconsidered time horizon in seconds)1205821198941198941015840 binary variable = 1 if the train 119894

1015840 shares the sameTUs after the end of the train 119894 119894 1198941015840 isin 119879 119895 isin 119878

119879 (119894 1198941015840) isin

119877119895 = 0 otherwise

119905119888119895 the maximum number of TUs at the same time at

the terminal 119895 119895 isin 119878119879

119905inimax 119878 the maximum deviation of arrival or departuretimes of the initial train 119894 119894 isin 119879

ini

43 Decision Variables The following variables are used inthe model

119909119886

119894119895 the departure time of the train 119894 at the station 119895

119894 isin 119879 119895 isin 119878119894

119909119889

119894119895 the arrival time of the train 119894 at the station 119895 119894 isin 119879

119895 isin 119878119894

120593119894119895 binary variable = 1 if the train 119894 stops at the station

119895 119894 isin 119879 119895 isin 119878119894 = 0 otherwise

120587119889

1198941198941015840119895 binary variable = 1 if the train 119894 departures

before the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878

119894= 0 otherwise120587119886

1198941198941015840119895 binary variable = 1 if the train 119894 arrives before

the train 1198941015840 at the station 119895 119894 1198941015840 isin 119879 119895 isin 119878

119894 = 0otherwise120579119901119903119895

binary variable = 1 if the time slot119901 is within theoccupation time (see (3) and (4)) of the train pairs 119903 atthe terminal 119895 119901 isin 119875 119895 isin 119878

119879 119903 = (119894 1198941015840

) isin 119877119895 119894 1198941015840 isin 119879

= 0 otherwise119899TU119901119895

the number of TUs at the station 119895 in the timeslot 119901 119895 isin 119878

119879 119901 isin 119875

44Objective Functions Weconsider twodifferent objectivesin the view of the following two aspects

(1) high quality for the operation of additional trainswhich can be represented by minimizing the traveltime of the additional trains

min119865119905

119865119905= sum

119894isin119879add

(119909119886

119894119897119894

minus 119909119889

119894119891119894

) (5)

(2) less deviation to existing trains in the originaltimetable this can be represented by minimizing theshift of the initial trains

min119865119904

119865119904= sum

119894isin119879ini119895isin119878119894

[10038161003816100381610038161003816(119909119886

119894119895minus 119909119886ini119894119895

)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889

119894119895minus 119909119889ini119894119895

)10038161003816100381610038161003816]

(6)

45 Constraints In this section we will focus on the con-straints associated with the SATUS problem they are listedas follows

451 Timetable Constraints Consider the following

119909119886

1198941198951015840 = 119909119889

119894119895+ 119903119887+ 119905119886119886

sdot 120593119894119895

+ 119905119886119889

sdot 1205931198941198951015840

119887 = (119895 1198951015840

) isin 119861119894

119894 isin 119879

(7)

119909119889

119894119895minus 119909119886

119894119895ge dwmin119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(8)

119909119889

119894119895minus 119909119886

119894119895le dwmax119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(9)

Constraints (7) define the arrival time to the station 1198951015840

from the departure time at the station 119895 adding the travelingtime at section 119887 which includes the bypassing running time(119903119887) the acceleration time (if a train stops at the station 119895) and

the deceleration time (if a train stops at the station 1198951015840) At each

station the dwell time at the station should not be less thantheminimumdwell time and not bemore than themaximumdwell time if the train needs to stop This fact is depicted inconstraints (8) and (9)


452 Headway Constraints Consider the following

119909119889

119894119895minus 119909119889

1198941015840119895ge ℎ119889119889

sdot 120587119889

1198941198941015840119895minus 119872 sdot (1 minus 120587

119889

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(10)

119909119889

1198941015840119895minus 119909119889

119894119895

ge ℎ119889119889

sdot (1 minus 120587119889

1198941198941015840119895) minus 119872 sdot 120587

119889

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(11)

119909119886

119894119895minus 119909119886

1198941015840119895ge ℎ119886119886

sdot 120587119886

1198941198941015840119895minus 119872 sdot (1 minus 120587

119886

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(12)

119909119886

1198941015840119895minus 119909119886

119894119895

ge ℎ119886119886

sdot (1 minus 120587119886

1198941198941015840119895) minus 119872 sdot 120587

119886

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(13)

120587119886

11989411989410158401198951015840 = 120587119889

1198941198941015840119895 119887 = (119895 119895

1015840

) isin 119861119894

119894 1198941015840

isin 119879 (14)

120587119889

1198941198941015840119895= 120587119886

1198941198941015840119895 119895 isin 119878

119894

119894 1198941015840

isin 119879 (15)

The headway constraints (10)ndash(13) describe theminimumheadway requirements between the departure time and thearrival time of the consecutive trains at the same stationConstraints (14) and (15) enforce the order of the consecutivetrains in all sections meaning that a train is not allowed toovertake another train

453 Time Deviation Constraints Consider the following

119909119886

119894119895minus 119909119886ini119894119895

isin [minus119905inimax 119878 119905

inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

119909119889

119894119895minus 119909119889ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

(16)

Constraints (16) define the deviation for the arrival ordeparture times of a train from its preferred arrival ordeparture times in the initial timetable

454 Layover Time and Turnback Operation ConstraintsConsider the following

119909119889

1198941015840119895minus 119909119886

119894119895isin [119862

min119895

119862max119895

]

(119894 1198941015840

) isin 119877119895

119895 isin 119878119894

119894 1198941015840

isin 119879

(17)

119899TU119901119895

= sum

119903=(1198941198941015840)isin119877119895

120579119901119903119895

119901 isin 119875 119895 isin 119878119879

119894 1198941015840

isin 119879

(18)

119899TU119901119895

le 119905119888119895 119901 isin 119875 119895 isin 119878

119879

(19)

Constraints (17) determine the minimum and the max-imum layover times between two consecutive trains linked

by the same TU at the same station In (18) the total numberof TUs is calculated on the condition that the time slot 119901

is within the occupation time of the train pairs 119903 at theterminal 119895 (see Figure 5) Constraints (19) indicate that thetotal number of TUs in the time slot 119901 at the terminal 119895mustbe equal to or less than the given value based on turnbackoperation style

5 Computational Experiments

51 Simulation Example Our experiments are based on realcases drawn from Shanghai rail transit line 16 This line is5285 km long composed of one main line and one depotlinking line with 11 stations and one depot This line hasdouble tracks on all sections as shown in Figure 6 It is theunique rail transit line in Shanghai that has two differentstopping services (1) slow services in which trains stop at allstations and (2) express services in which trains stop only atLSR XC HN and DSL stations

We implemented the models in Visual Studio 2012 usingIBM ILOG CPLEX 125 as a black-box MIP solver andrunning on a personal computer with an Intel Core i7-3520MCPU at 290GHz and 4GB of RAM This model was rununderWindows 8 64-Bit and default solver values were usedfor all parameters The new time-distance diagram obtainedfrom computation can be displayed by the train plan maker(TPM) software [7 8] In order to reduce the scale of thevariant and the computation time in our model the timestep (eg every 1 sec 5 sec 10 sec 30 sec and 60 sec) can bedefined by the users In this case we define the time step as30 sec and all the time lengths in parameters are the integermultiple of 30 sec

The initial timetable is an actual weekday operationtimetable of the line 16 in March 2014 This timetable whichis named 1601-2 is operated in the interval of 10min by thecyclic trips between LSR and DSL In this case the planninghorizon is defined from 500 to 1000 orsquoclock covering themorning peak hours with 56 trains and 12 TUs Additionallythe possibility of attending 10 different train routes and trackpaths into initial and additional timetables is consideredThese routes and track paths are defined by their originalstation destination station and occupied tracks in everystation as shown in Table 1 The turnback operation mode inEHN and DSL is TAS and on the other hand in DSL is TBS

The computation parameters additional trains with oneTU linking and time windows of the new trains are definedin Tables 2 3 and 4

52 Scenarios In our computation analysis 10 scenarios arestudied and each of them differs from the others mainlyin the points of (1) objective function and (2) maximumdeviation in the arrival or departure times of the initial trainsThe value of maximum deviation should not be too much(better to use less than half of the headways) because theinitial timetable is regularly used by commuter passengersand if there is a big change in it it may cause inconveniencefor the passengers Within these scenarios we also change


Figure 6 Infrastructure of Shanghai rail transit line 16

Table 1 Train routes and track path information in line 16

Route ID Route information Route track path ID Detail track path information

R1 LSR rarr DSL R1-1 LSR (T4 T2) rarr DSL (T1) T4 in EHT WAP and EHNT2 in the other stations

R2 DSL rarr LSR R2-1 DSL (T1) rarr LSR (T1ndashT4) T3 in EHT WAP and EHNT1 in the other stations

R3 DEP rarr EHN R3-1 DEP (T2) rarr EHN (T4)R4 EHN rarr DEP R4-1 EHN (T3) rarr DEP (T1)R5 EHN rarr DSL R5-1 EHN (T4) rarr DSL (T1) T2 in the other stationsR6 DSL rarr EHN R6-1 DSL (T1) rarr EHN (T3) T1 in the other stations

R7 DEP rarr LSR R7-1 DEP (T1) rarr EHN (T3) rarr LSR (T1 T4) T3 in EHTWAP T1 in the other stations

R8 LSR rarr DEP R8-1 LSR (T4 T2) rarr EHN (T4) rarr DEP (T2) T4 in EHTWAP T2 in the other stations

R9 LSR rarr EHN R9-1 LSR (T4 T2) rarr EHN (T3) T4 in EHT WAP T2 in theother stations

R10 EHN rarr LSR R10-1 EHN (T3) rarr LSR (T1 T4) T3 in EHT WAP T1 in theother stations

Table 2 Computation parameters in line 16

Parameter Value119905min119867

500119905max119867

1000ℎ119889119889

180 secℎ119886119886

180 sec119905ac 30 sec119905de 30 secdwmin119894119895

30 secdwmax119894119895

60 sec119862

min119895

DSL (180 sec) LSR (270 sec) EHN (60 sec)119862

max119895

600 sec119905119888119895

2 at DSL 1 at LSR and EHN

the time window of starting time for the new trains Table 5summarizes the studied scenarios

53 Results Table 6 exhibits the computational results of thescenarios carried out on the rail transit line 16 in Shanghaiwith parameters and inputs defined as explained above Thesolution times are less than 1 minute

As summarized in Table 6 the scenarios 1 and 2 have thesame objective value and the computational times are notvery high Inserting the new trains to the initial timetableis mainly restricted by the departure and arrival headwayssince the initial trains are fixed and the express trains cannotovertake all the other trains Figures 7 and 8 show the time-distance diagram obtained by scenarios 1 and 2 respectivelyin which inserting the new trains linked by U2 to the initial


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015 017

007006

021

013 023

027

201

203

001002

003

004

005

006

007

005

U2

U2008

009

007

010011

009

012

011

012

010

014

018

011

008

038

012

040

202 204

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U2008 010 003 005 007 009 011 012

U2001 002 004 006 008 010 003

U2

Figure 7 Time-distance diagram obtained by the scenario 1

Table 3 Parameters of the new trains with one TU linking

New TU ID Train sequence Train number Route ID Track path ID Stopping schemeU1 1 101 R3 R3-1

Original and destination stations (60 sec)HN (45 sec) and other stations (30 sec)

U1 2 102 R5 R5-1U1 3 103 R2 R2-1U1 4 104 R3 R3-1U1 5 105 R6 R6-1U1 6 106 R4 R4-1U2 1 201 R7 R7-1

Original and destination stations (60 sec)XC amp HN (30 sec) and other stations(0 sec)

U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1

Table 4 Time windows of the new trains

Time window scheme ID Time windowTW1 Train ldquo101rdquo 530ndash600 others 500ndash1000TW2 Train ldquo103rdquo 600ndash610 others 500ndash1000TW3 Train ldquo201rdquo 530ndash600 others 500ndash1000TW4 Train ldquo202rdquo 600ndash610 others 500ndash1000

timetables results in the same total traveling times And allthe new trains cause some additional stopping times at somestations For instance let us look at train ldquo201rdquo in scenario 1as seen additional stops at EHN (30 sec) and HSH (30 sec)

have happened and the stopping time at HN is 60 sec whichis longer than the scheduled one (30 sec)

The objective values in scenarios 3 and 4 are differentand it is noticed that scenario 3 needs a higher computationtime due to the wider time window for the pull-out trainldquo201rdquo Figures 9 and 10 illustrate the time-distance diagramobtained by scenarios 3 and 4 respectively the actual effectedtrains and moving time from the initial timetable are quitedifferent because of the fact that the start time windows ofthe new trains are different The restrictions of the headwayand the turnback capacity (at DSL) cause some trains tomoveforward or backward and cause more dwell times at somestations


U2

U2 U2001 003 004002

003

201

005 009

015017

007006

021

013

023

027

203

001002

003

004

005

006

007

005

008

009

007

010011

009

012

011

012

010

014

018

011

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

008

038

202

040

012

204026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

001 002 004 006U2

008 010 003 005 007 009 011 012U2

001 002 004 006 008 010 003


Table 5 Different scenarios

Scenarios ID Objectives (119865119905or 119865119904) Maximum deviation (119889119905

119894) New TU ID Time window scheme ID

1 min 119865119905

0 sec U2 TW32 min 119865

1199050 sec U2 TW4

3 min 119865119904

300 sec U1 TW14 min 119865

119904300 sec U1 TW2

5 min 119865119904

270 sec U1 TW26 min 119865

119904240 sec U1 TW2

7 min 119865119904

210 sec U1 TW28 min 119865

119904180 sec U1 TW2

9 min 119865119904

150 sec U1 TW210 min 119865

119904120 sec U1 TW2

Table 6 Computation results of the scenarios

Scenario ID Objective value Solution time (second)1 265 32 265 23 119 524 139 105 139 126 139 137 139 158 146 99 160 810 No solution

Scenarios 4ndash7 have the same objective values and outputthe same new timetable from computation also the maxi-mumdeviation time of initial timetable is not less than 210 secin the case of adding U1 at the time window TW2 Theobjective value of scenario 9 is 160 sec Scenario 10 has nosolution which means that no new train can be inserted inthe initial timetable since themaximum turnback capacity ofDSL has been reached that is the maximum deviation timeapproaches 120 sec

It implies that 150 sec is the minimum deviation timeon the condition of successfully inserting the trains of U1Figure 11 shows the detailed train line in DSL of scenarios 4and 9 these figures illustrate that train ldquo004rdquo moves to theright for 210 sec and train ldquo019rdquo moves to the left for 60 sec at


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025103

051105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021

013

023

027

101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028022

046

032

016

042

024 048

030

020

044

034

036

001 002 004 006

U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1

Figure 9 Time-distance diagram obtained by the scenario 3 (the dotted lines are the initial trains)

DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

103

051

105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021013

023

027101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1



0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)

Figure 11 Time-distance diagram obtained by the scenarios 4 (a) and 9 (b) (the dotted lines are the initial trains)

DSL as the maximum deviation time is 300 sec (scenario 4)but in scenario 9 (the maximum deviation time is 150 sec)train ldquo004rdquo needs to move right for 120 sec and train ldquo019rdquoneeds to move left for 150 sec at DSL

6 Conclusions and Future Work

In this paper a model and problem formulation for schedul-ing additional TU services have been proposed The maincontribution of the paper is consideration of the timetablescheduling and the TUs scheduling together as an integratedoptimization model with two objectives according to sectionand terminal capacities Additionally a maximum deviationfor arrival or departure times of trains in initial timetablethe strategy of slow services stopping at all stations andexpress services stopping only at some special stations thelinking order and time window of new inserted trains arealso considered in the model The developed model is ageneric one that can be easily modified to adapt any changesin initial timetable or any new scheme of inserting trainslinked by TUs The given example illustrates that rail transitagencies can obtain a reasonable new timetable for differentadministrative goals in amatter of seconds and shows that themodel is well suited to be used in daily operations

However the proposedmodel is not amultiobjective oneinmany real situations creating an appropriate new timetablemeans finding a balance between several objectives such asthe composition ofminimum119865

119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)

where the coefficients of 1205721and 120572

2are hard to evaluate On

the other hand long planning horizon and large number ofnew trains needed to be inserted will make the computationtime longer In order to improve the service level anotherobjective that should be taken into account is how to obtaina regular timetable which has equal intervals between trainsafter adding newonesThese issueswill be addressed in futureresearches

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this article

Acknowledgments

This work was supported by (1) the National Natural Sci-ence Foundation of China (Grant nos 61473210 5100822951208381 and 71071112) (2) the Fundamental Research Fundsfor the Central Universities (Grant no 20123228) (3) iRAGSof Siemens AG in Braunschweig and (4) The Scientific andTechnological Research Council of Turkey (TUBITAK) Theacquisition of the analysis data in the paper is supportedby the Shanghai Shentong Metro Operation ManagementCenter The authors appreciate this support

References

[1] V Vuchic Urban Transit Operations Planning and EconomicsAmerican Society of Civil Engineers Reston Va USA 2005

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

[3] A Caprara L Kroon M Monaci et al ldquoPassenger railwayoptimizationrdquo in Handbooks in Operations Research and Man-agement Science pp 129ndash187 Elsevier San Diego Calif USA2007

[4] V Cacchiani D Huisman M Kidd L Kroon P Toth and LVeelenturf ldquoAn overview of recovery models and algorithmsfor real-time railway reschedulingrdquo Transportation Research BMethodological vol 63 pp 15ndash37 2014

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

[6] A Ceder ldquoPublic-transport automated timetables using evenheadway and even passenger load conceptsrdquo in Proceedings of


the 32nd Australasian Transport Research Forum (ATRF rsquo09)October 2009

[7] Z Jiang J Gao and R Xu ldquoCircle rail transit line timetablescheduling using Rail TPMrdquo in Proceedings of the 12th Interna-tional Conference on Computer System Design and Operation inthe Railways and Other Transit Systems (COMPRAIL rsquo10) pp945ndash952 August-September 2010

[8] Z Jiang R Xu QWu and J Lv ldquoShared-path routing timetablecomputer designing in rail transit systemrdquo Journal of TongjiUniversity vol 38 no 5 pp 692ndash696 2010

[9] H Niu and X Zhou ldquoOptimizing urban rail timetable undertime-dependent demand and oversaturated conditionsrdquo Trans-portation Research C Emerging Technologies vol 36 pp 212ndash230 2013

[10] M Freyss R Giesen and J C Munoz ldquoContinuous approxi-mation for skip-stop operation in rail transitrdquo TransportationResearch C Emerging Technologies vol 36 pp 419ndash433 2013

[11] A Alfieri R Groot L Kroon and A Schrijver ldquoEfficientcirculation of railway rolling stockrdquo Transportation Science vol40 no 3 pp 378ndash391 2006

[12] P Fioole L Kroon G Maroti and A Schrijver ldquoA rolling stockcirculation model for combining and splitting of passengertrainsrdquo European Journal of Operational Research vol 174 no2 pp 1281ndash1297 2006

[13] L Cadarso A Marın and G Maroti ldquoRecovery of disruptionsin rapid transit networksrdquo Transportation Research E Logisticsand Transportation Review vol 53 no 1 pp 15ndash33 2013

[14] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport 2013

[15] X J Eberlein N H M Wilson C Barnhart and D BernsteinldquoThe real-time deadheading problem in transit operationscontrolrdquo Transportation Research B Methodological vol 32 no2 pp 77ndash100 1997

[16] A Haghani and M Banihashemi ldquoHeuristic approaches forsolving large-scale bus transit vehicle scheduling problem withroute time constraintsrdquo Transportation Research Part A Policyand Practice vol 36 no 4 pp 309ndash333 2002

[17] A Haghani M Banishashemi and K Chiang ldquoA comparativeanalysis of bus transit vehicle scheduling modelsrdquo Transporta-tion Research BMethodological vol 37 no 4 pp 301ndash322 2003

[18] B Yu Z Yang and S Li ldquoReal-time partway deadheadingstrategy based on transit service reliability assessmentrdquo Trans-portationResearchA Policy andPractice vol 46 no 8 pp 1265ndash1279 2012

[19] R L Burdett and E Kozan ldquoTechniques for inserting additionaltrains into existing timetablesrdquo Transportation Research BMethodological vol 43 no 8-9 pp 821ndash836 2009

[20] H Flier T Graffagnino and M Nunkesser ldquoScheduling addi-tional trains on dense corridorsrdquo in Experimental Algorithmsvol 5526 of Lecture Notes in Computer Science pp 149ndash1602009

[21] D Canca E Barrena A Zarzo F Ortega and E Algaba ldquoOpti-mal train reallocation strategies under service disruptionsrdquoProcediamdashSocial and Behavioral Sciences vol 54 pp 402ndash4132012

Submit your manuscripts athttpwwwhindawicom

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


to improve the capacity of some special sections with timewindowsThemost commonway is inserting additional trainservices into the initial timetables

The problem which is called scheduling additional trainunit services (SATUS) problem is a problem in which newtrains connected by a number of TUs start their trip from adepot or reversing tracks after collectively visiting a numberof routes they return to the starting points The SATUSproblem is a complicated one because the efficient circulationof TUs is an important consideration for operators of railtransit trains additionally the large number of trips linksand paths to be considered rapidly increases the numberof variables and constraints in any model developed TheSATUS problem is not a real-time rescheduling problemsince themain difference between it and timetable reschedul-ing in short-term planning or in disruption managementis the absence of uncertainty and the fact that the latter ismuch less time-critical while the first is often thought as atemporary redevelopment strategy of an initial timetable

This paper deals with the problem of SATUS in a doubleparallel rail transit line and a MIP model is formulated forintegration strategies of new trains connected by TUs withthe objective of obtaining higher frequencies in some specialsections and special time periods due to mass passengervolumes

The study contributes a number of new features tocapture the influence of specific elements that have not beenconsidered in studies on the SATUS problem in the relatedliterature First of all the approach decides on timetablesand TUs schedules using an integrated optimization modelaccording to sections and turnback capacities Second amaximum deviation for arrival or departure times of trainsin an initial timetable all-station-stopping policy and expressservice strategy linking orders and time windows of newinserted trains are also considered Finally thismodel has twoobjectives minimizing travel times of additional trains andminimizing shifts of initial trains

The paper is organized as follows In Section 2 a shortreview related to the SATUS problem is provided Afterthat Section 3 introduces a brief summary of the relevantconcepts in the model description A MIP model includingsets parameters decision variables and objective functionsis presented in Section 4 Section 5 illustrates the proposedmodel with an example The conclusions and future studiesare summarized in Section 6

2 Literature Review

The SATUS problem is related to a variety of topics in theliterature The first and foremost is railway transportationTrain scheduling rescheduling and routing problems havehad a great deal of attention in recent years There are twomain timetable variants One of the variants is the periodic(or cyclic) timetable that is repeated every given time periodfor example every hour with only slight differences betweenpeak hours and off-peak hours The other variant is thenonperiodic timetable which allows following the passengerdemands with the frequencies of the trains In both cases

timetables are usually repeated every day although theremaybe differences between weekdays and weekend

Cacchiani et al [3 4] gave a detailed review of theliterature on timetable scheduling The timetable schedulingproblem in a rail transit system in which TUs crews andpassengers are incorporated into a single planning frame-work science is complex various constraints and objectivesshould be considered simultaneously Due to its importanceand complexity which have been acknowledged in variouspublications topics related to this issue have attracted consid-erable attention in the literature A multiphase semiregulartimetable which divides a day into several time periods andeven applies the vehicle-departing interval for each periodmay somehow help to accommodate peak-hour demandwhile maintaining a certain level of service for passengersboarding at nonpeak hours Guihaire and Hao [5] presenteda global review of the crucial strategic and tactical steps oftransit planning and also discussed the scheduling problemwith phase regular for a transit corridor Ceder [6] provided acomprehensive modeling framework for determining vehicledeparture time with either even headways or even averageloads with a special focus on smoothing the transitionsbetween time periods These studies provide useful methodsfor optimizing frequency for a particular time period while aunified framework is critically needed for scheduling meth-ods that can consider uneven headways and time-dependentdemand patterns Jiang et al [7 8] presented a computationaltimetable scheduling method in rail transit line with mul-tiroutes or circle route and a timetable designing softwarenamed Train Plan Maker (TPM) was developed and appliedbymanymetro operation companies in China Niu and Zhou[9] focused on optimizing a passenger train timetable ina heavily congested urban rail corridor A binary integerprogramming model incorporated with passenger loadingand departure events was constructed to provide a theoreticdescription for the problem under consideration Freyss et al[10] focused on the skip-stop operation for rail transit linesusing a single one-way track and the system was modeled byusing a continuous approximation approach

Once the timetable scheduling has been defined therolling stock and TUs assignments must be done An integerprogramming model was considered by Alfieri et al [11] todetermine the rolling stock circulation for multiple rollingstock types on a single line and on a single day and thismodelwas extended by Fioole et al [12] by including combining andsplitting trains as it happens at several locations in the Dutchtimetables Cadarso et al [13] studied the disruptionmanage-ment problem of rapid transit rail networks Besides optimiz-ing timetable and rolling stock schedules they explicitly dealtwith the effects of disruption on the passenger demandsTheyproposed a two-step approach that combines an integratedoptimization model (for the timetable and the rolling stock)with a model for the behaviors of passengers Lin and Kwan[14] proposed a two-phase approach for the TUs schedulingproblemThefirst phase assigned and sequenced train trips toTUs temporarily ignoring some station infrastructure detailswhich was modeled as an integer fixed-charge multicom-modity flow (FCMF) problem The second phase focused onsatisfying the remaining station detailed requirements which









b5

b1 b2 b3 b4


TurnbacksDepot

RouteRoute

s1 s2 s3

s6

s4 s5

rs5s1rs1s5


s1 s2 s3 s4 s5

s6 1s1s5tr1

tr4tr3

tr1

tr1

tr1 tr1 tr1

tr2 tr2 tr2 tr2

tr2

tr2

tp







1199041 1199045 1199031199045 1199041

1199031199041 1199044

and1199031199044 1199041


1199046 1199041 1199031199041 1199046

1199031199046 1199042

and 1199031199042 1199046






1(tr4)


tp11199041 1199045

= 1199041(tr4) 1199041(tr2) 1199042(tr2) 1199043(tr2) 1199044(tr2) 1199045(tr1)

(1)


119888) as time



Layover time u2 u2

u2 u2

u2

6 7

1

u1

u1

u1 u1

u1Pull-out train

4 9

Tc

582 3

-depot

s1

s2

s3

s4

s5-




119879ini








119905119894 119905

119901


= 119891119894 119897119894 isin 119878




119894 and


119894

119891119894 119897119894 of the train 119894




station in 119878119894


119887 = 119895 1198951015840

with 119895 1198951015840

isin 119878119894


of the following




119906add119895

= 119905add119894

119905add119901


= 119891119894 119897119894 isin 119878



sj+1

bk

sj

haa

rpp

hdd rsp

rss

rps

tde

tac




119894 the


119894


119887 = 119895 1198951015840

with 119895 1198951015840

isin 119878119894





119896 namely (1)



119895



119895but



119904119895and 119904119895+1

(119903119904119904) So 119903

119901119904 119903119904119901 and 119903



119903119901119904

= 119903119901119901

+ 119905de

119903119904119901

= 119903119901119901

+ 119905ac

119903119904119904

= 119903119901119901

+ 119905ac + 119905de

(2)



T2

T1

Platform

tbTAS12

u1 u2u3

654321

tRst



T2

T1

Platform

u1

u2u3

65 43 21

T3

T4

tbTBS12











119894and the


calculated by

tbTASV119894 V119895 isin [119905119886

V119894 119905119889

V119895 + 119905119877

st] (3)

where 119905119886

V119894 119905119889







tbTBSV119894 V119895 isin [119905119886

V119894 119905119889

V119895] (4)

4 Model Description




119894 119904119895) in



119879 = 119879ini



inicup119880


1015840

) with 119894 lt 1198941015840 when the



119878119894






119894 isin 119879


119905min119867



119909119886ini119894119895


ini 119895 isin 119878119894

119909119889ini119894119895


ini 119895 isin 119878119894




stations 119904119894and 119904119895 119887 = (119904

119894 119904119895) isin 119861

dwmin119894119895



119894119862min119895


119878119879

119862max119895


119878119879



119879 (119894 1198941015840) isin

119877119895 = 0 otherwise




ini


119909119886


119894 isin 119879 119895 isin 119878119894

119909119889


119895 isin 119878119894



120587119889



119894= 0 otherwise120587119886



119894 = 0otherwise120579119901119903119895


119879 119903 = (119894 1198941015840

) isin 119877119895 119894 1198941015840 isin 119879

= 0 otherwise119899TU119901119895


119879 119901 isin 119875



min119865119905

119865119905= sum

119894isin119879add

(119909119886

119894119897119894

minus 119909119889

119894119891119894

) (5)


min119865119904

119865119904= sum

119894isin119879ini119895isin119878119894

[10038161003816100381610038161003816(119909119886

119894119895minus 119909119886ini119894119895

)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889

119894119895minus 119909119889ini119894119895

)10038161003816100381610038161003816]

(6)



119909119886

1198941198951015840 = 119909119889

119894119895+ 119903119887+ 119905119886119886

sdot 120593119894119895

+ 119905119886119889

sdot 1205931198941198951015840

119887 = (119895 1198951015840

) isin 119861119894

119894 isin 119879

(7)

119909119889

119894119895minus 119909119886

119894119895ge dwmin119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(8)

119909119889

119894119895minus 119909119886

119894119895le dwmax119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(9)







119909119889

119894119895minus 119909119889

1198941015840119895ge ℎ119889119889

sdot 120587119889

1198941198941015840119895minus 119872 sdot (1 minus 120587

119889

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(10)

119909119889

1198941015840119895minus 119909119889

119894119895

ge ℎ119889119889

sdot (1 minus 120587119889

1198941198941015840119895) minus 119872 sdot 120587

119889

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(11)

119909119886

119894119895minus 119909119886

1198941015840119895ge ℎ119886119886

sdot 120587119886

1198941198941015840119895minus 119872 sdot (1 minus 120587

119886

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(12)

119909119886

1198941015840119895minus 119909119886

119894119895

ge ℎ119886119886

sdot (1 minus 120587119886

1198941198941015840119895) minus 119872 sdot 120587

119886

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(13)

120587119886

11989411989410158401198951015840 = 120587119889

1198941198941015840119895 119887 = (119895 119895

1015840

) isin 119861119894

119894 1198941015840

isin 119879 (14)

120587119889

1198941198941015840119895= 120587119886

1198941198941015840119895 119895 isin 119878

119894

119894 1198941015840

isin 119879 (15)



119909119886

119894119895minus 119909119886ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

119909119889

119894119895minus 119909119889ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

(16)



119909119889

1198941015840119895minus 119909119886

119894119895isin [119862

min119895

119862max119895

]

(119894 1198941015840

) isin 119877119895

119895 isin 119878119894

119894 1198941015840

isin 119879

(17)

119899TU119901119895

= sum

119903=(1198941198941015840)isin119877119895

120579119901119903119895

119901 isin 119875 119895 isin 119878119879

119894 1198941015840

isin 119879

(18)

119899TU119901119895

le 119905119888119895 119901 isin 119875 119895 isin 119878

119879

(19)























500119905max119867

1000ℎ119889119889

180 secℎ119886119886


30 secdwmax119894119895

60 sec119862

min119895


max119895

600 sec119905119888119895






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015 017

007006

021

013 023

027

201

203

001002

003

004

005

006

007

005

U2

U2008

009

007

010011

009

012

011

012

010

014

018

011

008

038

012

040

202 204

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U2008 010 003 005 007 009 011 012

U2001 002 004 006 008 010 003

U2







U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1







U2

U2 U2001 003 004002

003

201

005 009

015017

007006

021

013

023

027

203

001002

003

004

005

006

007

005

008

009

007

010011

009

012

011

012

010

014

018

011

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

008

038

202

040

012

204026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

001 002 004 006U2

008 010 003 005 007 009 011 012U2

001 002 004 006 008 010 003





1 min 119865119905

0 sec U2 TW32 min 119865

1199050 sec U2 TW4

3 min 119865119904

300 sec U1 TW14 min 119865

119904300 sec U1 TW2

5 min 119865119904

270 sec U1 TW26 min 119865

119904240 sec U1 TW2

7 min 119865119904

210 sec U1 TW28 min 119865

119904180 sec U1 TW2

9 min 119865119904

150 sec U1 TW210 min 119865

119904120 sec U1 TW2






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025103

051105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021

013

023

027

101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028022

046

032

016

042

024 048

030

020

044

034

036

001 002 004 006

U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

103

051

105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021013

023

027101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1



0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)






119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















b5

b1 b2 b3 b4


TurnbacksDepot

RouteRoute

s1 s2 s3

s6

s4 s5

rs5s1rs1s5


s1 s2 s3 s4 s5

s6 1s1s5tr1

tr4tr3

tr1

tr1

tr1 tr1 tr1

tr2 tr2 tr2 tr2

tr2

tr2

tp







1199041 1199045 1199031199045 1199041

1199031199041 1199044

and1199031199044 1199041


1199046 1199041 1199031199041 1199046

1199031199046 1199042

and 1199031199042 1199046






1(tr4)


tp11199041 1199045

= 1199041(tr4) 1199041(tr2) 1199042(tr2) 1199043(tr2) 1199044(tr2) 1199045(tr1)

(1)


119888) as time



Layover time u2 u2

u2 u2

u2

6 7

1

u1

u1

u1 u1

u1Pull-out train

4 9

Tc

582 3

-depot

s1

s2

s3

s4

s5-




119879ini








119905119894 119905

119901


= 119891119894 119897119894 isin 119878




119894 and


119894

119891119894 119897119894 of the train 119894




station in 119878119894


119887 = 119895 1198951015840

with 119895 1198951015840

isin 119878119894


of the following




119906add119895

= 119905add119894

119905add119901


= 119891119894 119897119894 isin 119878



sj+1

bk

sj

haa

rpp

hdd rsp

rss

rps

tde

tac




119894 the


119894


119887 = 119895 1198951015840

with 119895 1198951015840

isin 119878119894





119896 namely (1)



119895



119895but



119904119895and 119904119895+1

(119903119904119904) So 119903

119901119904 119903119904119901 and 119903



119903119901119904

= 119903119901119901

+ 119905de

119903119904119901

= 119903119901119901

+ 119905ac

119903119904119904

= 119903119901119901

+ 119905ac + 119905de

(2)



T2

T1

Platform

tbTAS12

u1 u2u3

654321

tRst



T2

T1

Platform

u1

u2u3

65 43 21

T3

T4

tbTBS12











119894and the


calculated by

tbTASV119894 V119895 isin [119905119886

V119894 119905119889

V119895 + 119905119877

st] (3)

where 119905119886

V119894 119905119889







tbTBSV119894 V119895 isin [119905119886

V119894 119905119889

V119895] (4)

4 Model Description




119894 119904119895) in



119879 = 119879ini



inicup119880


1015840

) with 119894 lt 1198941015840 when the



119878119894






119894 isin 119879


119905min119867



119909119886ini119894119895


ini 119895 isin 119878119894

119909119889ini119894119895


ini 119895 isin 119878119894




stations 119904119894and 119904119895 119887 = (119904

119894 119904119895) isin 119861

dwmin119894119895



119894119862min119895


119878119879

119862max119895


119878119879



119879 (119894 1198941015840) isin

119877119895 = 0 otherwise




ini


119909119886


119894 isin 119879 119895 isin 119878119894

119909119889


119895 isin 119878119894



120587119889



119894= 0 otherwise120587119886



119894 = 0otherwise120579119901119903119895


119879 119903 = (119894 1198941015840

) isin 119877119895 119894 1198941015840 isin 119879

= 0 otherwise119899TU119901119895


119879 119901 isin 119875



min119865119905

119865119905= sum

119894isin119879add

(119909119886

119894119897119894

minus 119909119889

119894119891119894

) (5)


min119865119904

119865119904= sum

119894isin119879ini119895isin119878119894

[10038161003816100381610038161003816(119909119886

119894119895minus 119909119886ini119894119895

)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889

119894119895minus 119909119889ini119894119895

)10038161003816100381610038161003816]

(6)



119909119886

1198941198951015840 = 119909119889

119894119895+ 119903119887+ 119905119886119886

sdot 120593119894119895

+ 119905119886119889

sdot 1205931198941198951015840

119887 = (119895 1198951015840

) isin 119861119894

119894 isin 119879

(7)

119909119889

119894119895minus 119909119886

119894119895ge dwmin119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(8)

119909119889

119894119895minus 119909119886

119894119895le dwmax119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(9)







119909119889

119894119895minus 119909119889

1198941015840119895ge ℎ119889119889

sdot 120587119889

1198941198941015840119895minus 119872 sdot (1 minus 120587

119889

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(10)

119909119889

1198941015840119895minus 119909119889

119894119895

ge ℎ119889119889

sdot (1 minus 120587119889

1198941198941015840119895) minus 119872 sdot 120587

119889

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(11)

119909119886

119894119895minus 119909119886

1198941015840119895ge ℎ119886119886

sdot 120587119886

1198941198941015840119895minus 119872 sdot (1 minus 120587

119886

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(12)

119909119886

1198941015840119895minus 119909119886

119894119895

ge ℎ119886119886

sdot (1 minus 120587119886

1198941198941015840119895) minus 119872 sdot 120587

119886

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(13)

120587119886

11989411989410158401198951015840 = 120587119889

1198941198941015840119895 119887 = (119895 119895

1015840

) isin 119861119894

119894 1198941015840

isin 119879 (14)

120587119889

1198941198941015840119895= 120587119886

1198941198941015840119895 119895 isin 119878

119894

119894 1198941015840

isin 119879 (15)



119909119886

119894119895minus 119909119886ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

119909119889

119894119895minus 119909119889ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

(16)



119909119889

1198941015840119895minus 119909119886

119894119895isin [119862

min119895

119862max119895

]

(119894 1198941015840

) isin 119877119895

119895 isin 119878119894

119894 1198941015840

isin 119879

(17)

119899TU119901119895

= sum

119903=(1198941198941015840)isin119877119895

120579119901119903119895

119901 isin 119875 119895 isin 119878119879

119894 1198941015840

isin 119879

(18)

119899TU119901119895

le 119905119888119895 119901 isin 119875 119895 isin 119878

119879

(19)























500119905max119867

1000ℎ119889119889

180 secℎ119886119886


30 secdwmax119894119895

60 sec119862

min119895


max119895

600 sec119905119888119895






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015 017

007006

021

013 023

027

201

203

001002

003

004

005

006

007

005

U2

U2008

009

007

010011

009

012

011

012

010

014

018

011

008

038

012

040

202 204

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U2008 010 003 005 007 009 011 012

U2001 002 004 006 008 010 003

U2







U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1







U2

U2 U2001 003 004002

003

201

005 009

015017

007006

021

013

023

027

203

001002

003

004

005

006

007

005

008

009

007

010011

009

012

011

012

010

014

018

011

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

008

038

202

040

012

204026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

001 002 004 006U2

008 010 003 005 007 009 011 012U2

001 002 004 006 008 010 003





1 min 119865119905

0 sec U2 TW32 min 119865

1199050 sec U2 TW4

3 min 119865119904

300 sec U1 TW14 min 119865

119904300 sec U1 TW2

5 min 119865119904

270 sec U1 TW26 min 119865

119904240 sec U1 TW2

7 min 119865119904

210 sec U1 TW28 min 119865

119904180 sec U1 TW2

9 min 119865119904

150 sec U1 TW210 min 119865

119904120 sec U1 TW2






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025103

051105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021

013

023

027

101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028022

046

032

016

042

024 048

030

020

044

034

036

001 002 004 006

U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

103

051

105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021013

023

027101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1



0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)






119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Layover time u2 u2

u2 u2

u2

6 7

1

u1

u1

u1 u1

u1Pull-out train

4 9

Tc

582 3

-depot

s1

s2

s3

s4

s5-




119879ini








119905119894 119905

119901


= 119891119894 119897119894 isin 119878




119894 and


119894

119891119894 119897119894 of the train 119894




station in 119878119894


119887 = 119895 1198951015840

with 119895 1198951015840

isin 119878119894


of the following




119906add119895

= 119905add119894

119905add119901


= 119891119894 119897119894 isin 119878



sj+1

bk

sj

haa

rpp

hdd rsp

rss

rps

tde

tac




119894 the


119894


119887 = 119895 1198951015840

with 119895 1198951015840

isin 119878119894





119896 namely (1)



119895



119895but



119904119895and 119904119895+1

(119903119904119904) So 119903

119901119904 119903119904119901 and 119903



119903119901119904

= 119903119901119901

+ 119905de

119903119904119901

= 119903119901119901

+ 119905ac

119903119904119904

= 119903119901119901

+ 119905ac + 119905de

(2)



T2

T1

Platform

tbTAS12

u1 u2u3

654321

tRst



T2

T1

Platform

u1

u2u3

65 43 21

T3

T4

tbTBS12











119894and the


calculated by

tbTASV119894 V119895 isin [119905119886

V119894 119905119889

V119895 + 119905119877

st] (3)

where 119905119886

V119894 119905119889







tbTBSV119894 V119895 isin [119905119886

V119894 119905119889

V119895] (4)

4 Model Description




119894 119904119895) in



119879 = 119879ini



inicup119880


1015840

) with 119894 lt 1198941015840 when the



119878119894






119894 isin 119879


119905min119867



119909119886ini119894119895


ini 119895 isin 119878119894

119909119889ini119894119895


ini 119895 isin 119878119894




stations 119904119894and 119904119895 119887 = (119904

119894 119904119895) isin 119861

dwmin119894119895



119894119862min119895


119878119879

119862max119895


119878119879



119879 (119894 1198941015840) isin

119877119895 = 0 otherwise




ini


119909119886


119894 isin 119879 119895 isin 119878119894

119909119889


119895 isin 119878119894



120587119889



119894= 0 otherwise120587119886



119894 = 0otherwise120579119901119903119895


119879 119903 = (119894 1198941015840

) isin 119877119895 119894 1198941015840 isin 119879

= 0 otherwise119899TU119901119895


119879 119901 isin 119875



min119865119905

119865119905= sum

119894isin119879add

(119909119886

119894119897119894

minus 119909119889

119894119891119894

) (5)


min119865119904

119865119904= sum

119894isin119879ini119895isin119878119894

[10038161003816100381610038161003816(119909119886

119894119895minus 119909119886ini119894119895

)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889

119894119895minus 119909119889ini119894119895

)10038161003816100381610038161003816]

(6)



119909119886

1198941198951015840 = 119909119889

119894119895+ 119903119887+ 119905119886119886

sdot 120593119894119895

+ 119905119886119889

sdot 1205931198941198951015840

119887 = (119895 1198951015840

) isin 119861119894

119894 isin 119879

(7)

119909119889

119894119895minus 119909119886

119894119895ge dwmin119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(8)

119909119889

119894119895minus 119909119886

119894119895le dwmax119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(9)







119909119889

119894119895minus 119909119889

1198941015840119895ge ℎ119889119889

sdot 120587119889

1198941198941015840119895minus 119872 sdot (1 minus 120587

119889

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(10)

119909119889

1198941015840119895minus 119909119889

119894119895

ge ℎ119889119889

sdot (1 minus 120587119889

1198941198941015840119895) minus 119872 sdot 120587

119889

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(11)

119909119886

119894119895minus 119909119886

1198941015840119895ge ℎ119886119886

sdot 120587119886

1198941198941015840119895minus 119872 sdot (1 minus 120587

119886

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(12)

119909119886

1198941015840119895minus 119909119886

119894119895

ge ℎ119886119886

sdot (1 minus 120587119886

1198941198941015840119895) minus 119872 sdot 120587

119886

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(13)

120587119886

11989411989410158401198951015840 = 120587119889

1198941198941015840119895 119887 = (119895 119895

1015840

) isin 119861119894

119894 1198941015840

isin 119879 (14)

120587119889

1198941198941015840119895= 120587119886

1198941198941015840119895 119895 isin 119878

119894

119894 1198941015840

isin 119879 (15)



119909119886

119894119895minus 119909119886ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

119909119889

119894119895minus 119909119889ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

(16)



119909119889

1198941015840119895minus 119909119886

119894119895isin [119862

min119895

119862max119895

]

(119894 1198941015840

) isin 119877119895

119895 isin 119878119894

119894 1198941015840

isin 119879

(17)

119899TU119901119895

= sum

119903=(1198941198941015840)isin119877119895

120579119901119903119895

119901 isin 119875 119895 isin 119878119879

119894 1198941015840

isin 119879

(18)

119899TU119901119895

le 119905119888119895 119901 isin 119875 119895 isin 119878

119879

(19)























500119905max119867

1000ℎ119889119889

180 secℎ119886119886


30 secdwmax119894119895

60 sec119862

min119895


max119895

600 sec119905119888119895






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015 017

007006

021

013 023

027

201

203

001002

003

004

005

006

007

005

U2

U2008

009

007

010011

009

012

011

012

010

014

018

011

008

038

012

040

202 204

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U2008 010 003 005 007 009 011 012

U2001 002 004 006 008 010 003

U2







U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1







U2

U2 U2001 003 004002

003

201

005 009

015017

007006

021

013

023

027

203

001002

003

004

005

006

007

005

008

009

007

010011

009

012

011

012

010

014

018

011

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

008

038

202

040

012

204026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

001 002 004 006U2

008 010 003 005 007 009 011 012U2

001 002 004 006 008 010 003





1 min 119865119905

0 sec U2 TW32 min 119865

1199050 sec U2 TW4

3 min 119865119904

300 sec U1 TW14 min 119865

119904300 sec U1 TW2

5 min 119865119904

270 sec U1 TW26 min 119865

119904240 sec U1 TW2

7 min 119865119904

210 sec U1 TW28 min 119865

119904180 sec U1 TW2

9 min 119865119904

150 sec U1 TW210 min 119865

119904120 sec U1 TW2






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025103

051105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021

013

023

027

101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028022

046

032

016

042

024 048

030

020

044

034

036

001 002 004 006

U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

103

051

105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021013

023

027101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1



0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)






119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










T2

T1

Platform

tbTAS12

u1 u2u3

654321

tRst



T2

T1

Platform

u1

u2u3

65 43 21

T3

T4

tbTBS12











119894and the


calculated by

tbTASV119894 V119895 isin [119905119886

V119894 119905119889

V119895 + 119905119877

st] (3)

where 119905119886

V119894 119905119889







tbTBSV119894 V119895 isin [119905119886

V119894 119905119889

V119895] (4)

4 Model Description




119894 119904119895) in



119879 = 119879ini



inicup119880


1015840

) with 119894 lt 1198941015840 when the



119878119894






119894 isin 119879


119905min119867



119909119886ini119894119895


ini 119895 isin 119878119894

119909119889ini119894119895


ini 119895 isin 119878119894




stations 119904119894and 119904119895 119887 = (119904

119894 119904119895) isin 119861

dwmin119894119895



119894119862min119895


119878119879

119862max119895


119878119879



119879 (119894 1198941015840) isin

119877119895 = 0 otherwise




ini


119909119886


119894 isin 119879 119895 isin 119878119894

119909119889


119895 isin 119878119894



120587119889



119894= 0 otherwise120587119886



119894 = 0otherwise120579119901119903119895


119879 119903 = (119894 1198941015840

) isin 119877119895 119894 1198941015840 isin 119879

= 0 otherwise119899TU119901119895


119879 119901 isin 119875



min119865119905

119865119905= sum

119894isin119879add

(119909119886

119894119897119894

minus 119909119889

119894119891119894

) (5)


min119865119904

119865119904= sum

119894isin119879ini119895isin119878119894

[10038161003816100381610038161003816(119909119886

119894119895minus 119909119886ini119894119895

)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889

119894119895minus 119909119889ini119894119895

)10038161003816100381610038161003816]

(6)



119909119886

1198941198951015840 = 119909119889

119894119895+ 119903119887+ 119905119886119886

sdot 120593119894119895

+ 119905119886119889

sdot 1205931198941198951015840

119887 = (119895 1198951015840

) isin 119861119894

119894 isin 119879

(7)

119909119889

119894119895minus 119909119886

119894119895ge dwmin119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(8)

119909119889

119894119895minus 119909119886

119894119895le dwmax119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(9)







119909119889

119894119895minus 119909119889

1198941015840119895ge ℎ119889119889

sdot 120587119889

1198941198941015840119895minus 119872 sdot (1 minus 120587

119889

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(10)

119909119889

1198941015840119895minus 119909119889

119894119895

ge ℎ119889119889

sdot (1 minus 120587119889

1198941198941015840119895) minus 119872 sdot 120587

119889

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(11)

119909119886

119894119895minus 119909119886

1198941015840119895ge ℎ119886119886

sdot 120587119886

1198941198941015840119895minus 119872 sdot (1 minus 120587

119886

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(12)

119909119886

1198941015840119895minus 119909119886

119894119895

ge ℎ119886119886

sdot (1 minus 120587119886

1198941198941015840119895) minus 119872 sdot 120587

119886

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(13)

120587119886

11989411989410158401198951015840 = 120587119889

1198941198941015840119895 119887 = (119895 119895

1015840

) isin 119861119894

119894 1198941015840

isin 119879 (14)

120587119889

1198941198941015840119895= 120587119886

1198941198941015840119895 119895 isin 119878

119894

119894 1198941015840

isin 119879 (15)



119909119886

119894119895minus 119909119886ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

119909119889

119894119895minus 119909119889ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

(16)



119909119889

1198941015840119895minus 119909119886

119894119895isin [119862

min119895

119862max119895

]

(119894 1198941015840

) isin 119877119895

119895 isin 119878119894

119894 1198941015840

isin 119879

(17)

119899TU119901119895

= sum

119903=(1198941198941015840)isin119877119895

120579119901119903119895

119901 isin 119875 119895 isin 119878119879

119894 1198941015840

isin 119879

(18)

119899TU119901119895

le 119905119888119895 119901 isin 119875 119895 isin 119878

119879

(19)























500119905max119867

1000ℎ119889119889

180 secℎ119886119886


30 secdwmax119894119895

60 sec119862

min119895


max119895

600 sec119905119888119895






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015 017

007006

021

013 023

027

201

203

001002

003

004

005

006

007

005

U2

U2008

009

007

010011

009

012

011

012

010

014

018

011

008

038

012

040

202 204

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U2008 010 003 005 007 009 011 012

U2001 002 004 006 008 010 003

U2







U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1







U2

U2 U2001 003 004002

003

201

005 009

015017

007006

021

013

023

027

203

001002

003

004

005

006

007

005

008

009

007

010011

009

012

011

012

010

014

018

011

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

008

038

202

040

012

204026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

001 002 004 006U2

008 010 003 005 007 009 011 012U2

001 002 004 006 008 010 003





1 min 119865119905

0 sec U2 TW32 min 119865

1199050 sec U2 TW4

3 min 119865119904

300 sec U1 TW14 min 119865

119904300 sec U1 TW2

5 min 119865119904

270 sec U1 TW26 min 119865

119904240 sec U1 TW2

7 min 119865119904

210 sec U1 TW28 min 119865

119904180 sec U1 TW2

9 min 119865119904

150 sec U1 TW210 min 119865

119904120 sec U1 TW2






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025103

051105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021

013

023

027

101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028022

046

032

016

042

024 048

030

020

044

034

036

001 002 004 006

U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

103

051

105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021013

023

027101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1



0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)






119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894 isin 119879


119905min119867



119909119886ini119894119895


ini 119895 isin 119878119894

119909119889ini119894119895


ini 119895 isin 119878119894




stations 119904119894and 119904119895 119887 = (119904

119894 119904119895) isin 119861

dwmin119894119895



119894119862min119895


119878119879

119862max119895


119878119879



119879 (119894 1198941015840) isin

119877119895 = 0 otherwise




ini


119909119886


119894 isin 119879 119895 isin 119878119894

119909119889


119895 isin 119878119894



120587119889



119894= 0 otherwise120587119886



119894 = 0otherwise120579119901119903119895


119879 119903 = (119894 1198941015840

) isin 119877119895 119894 1198941015840 isin 119879

= 0 otherwise119899TU119901119895


119879 119901 isin 119875



min119865119905

119865119905= sum

119894isin119879add

(119909119886

119894119897119894

minus 119909119889

119894119891119894

) (5)


min119865119904

119865119904= sum

119894isin119879ini119895isin119878119894

[10038161003816100381610038161003816(119909119886

119894119895minus 119909119886ini119894119895

)10038161003816100381610038161003816+10038161003816100381610038161003816(119909119889

119894119895minus 119909119889ini119894119895

)10038161003816100381610038161003816]

(6)



119909119886

1198941198951015840 = 119909119889

119894119895+ 119903119887+ 119905119886119886

sdot 120593119894119895

+ 119905119886119889

sdot 1205931198941198951015840

119887 = (119895 1198951015840

) isin 119861119894

119894 isin 119879

(7)

119909119889

119894119895minus 119909119886

119894119895ge dwmin119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(8)

119909119889

119894119895minus 119909119886

119894119895le dwmax119894119895

sdot 120593119894119895 119894 isin 119879 119895 isin 119878

119894

(9)







119909119889

119894119895minus 119909119889

1198941015840119895ge ℎ119889119889

sdot 120587119889

1198941198941015840119895minus 119872 sdot (1 minus 120587

119889

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(10)

119909119889

1198941015840119895minus 119909119889

119894119895

ge ℎ119889119889

sdot (1 minus 120587119889

1198941198941015840119895) minus 119872 sdot 120587

119889

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(11)

119909119886

119894119895minus 119909119886

1198941015840119895ge ℎ119886119886

sdot 120587119886

1198941198941015840119895minus 119872 sdot (1 minus 120587

119886

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(12)

119909119886

1198941015840119895minus 119909119886

119894119895

ge ℎ119886119886

sdot (1 minus 120587119886

1198941198941015840119895) minus 119872 sdot 120587

119886

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(13)

120587119886

11989411989410158401198951015840 = 120587119889

1198941198941015840119895 119887 = (119895 119895

1015840

) isin 119861119894

119894 1198941015840

isin 119879 (14)

120587119889

1198941198941015840119895= 120587119886

1198941198941015840119895 119895 isin 119878

119894

119894 1198941015840

isin 119879 (15)



119909119886

119894119895minus 119909119886ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

119909119889

119894119895minus 119909119889ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

(16)



119909119889

1198941015840119895minus 119909119886

119894119895isin [119862

min119895

119862max119895

]

(119894 1198941015840

) isin 119877119895

119895 isin 119878119894

119894 1198941015840

isin 119879

(17)

119899TU119901119895

= sum

119903=(1198941198941015840)isin119877119895

120579119901119903119895

119901 isin 119875 119895 isin 119878119879

119894 1198941015840

isin 119879

(18)

119899TU119901119895

le 119905119888119895 119901 isin 119875 119895 isin 119878

119879

(19)























500119905max119867

1000ℎ119889119889

180 secℎ119886119886


30 secdwmax119894119895

60 sec119862

min119895


max119895

600 sec119905119888119895






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015 017

007006

021

013 023

027

201

203

001002

003

004

005

006

007

005

U2

U2008

009

007

010011

009

012

011

012

010

014

018

011

008

038

012

040

202 204

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U2008 010 003 005 007 009 011 012

U2001 002 004 006 008 010 003

U2







U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1







U2

U2 U2001 003 004002

003

201

005 009

015017

007006

021

013

023

027

203

001002

003

004

005

006

007

005

008

009

007

010011

009

012

011

012

010

014

018

011

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

008

038

202

040

012

204026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

001 002 004 006U2

008 010 003 005 007 009 011 012U2

001 002 004 006 008 010 003





1 min 119865119905

0 sec U2 TW32 min 119865

1199050 sec U2 TW4

3 min 119865119904

300 sec U1 TW14 min 119865

119904300 sec U1 TW2

5 min 119865119904

270 sec U1 TW26 min 119865

119904240 sec U1 TW2

7 min 119865119904

210 sec U1 TW28 min 119865

119904180 sec U1 TW2

9 min 119865119904

150 sec U1 TW210 min 119865

119904120 sec U1 TW2






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025103

051105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021

013

023

027

101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028022

046

032

016

042

024 048

030

020

044

034

036

001 002 004 006

U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

103

051

105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021013

023

027101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1



0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)






119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119909119889

119894119895minus 119909119889

1198941015840119895ge ℎ119889119889

sdot 120587119889

1198941198941015840119895minus 119872 sdot (1 minus 120587

119889

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(10)

119909119889

1198941015840119895minus 119909119889

119894119895

ge ℎ119889119889

sdot (1 minus 120587119889

1198941198941015840119895) minus 119872 sdot 120587

119889

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(11)

119909119886

119894119895minus 119909119886

1198941015840119895ge ℎ119886119886

sdot 120587119886

1198941198941015840119895minus 119872 sdot (1 minus 120587

119886

1198941198941015840119895)

119894 1198941015840

isin 119879 119895 isin 119878119894

(12)

119909119886

1198941015840119895minus 119909119886

119894119895

ge ℎ119886119886

sdot (1 minus 120587119886

1198941198941015840119895) minus 119872 sdot 120587

119886

1198941198941015840119895

119894 1198941015840

isin 119879 119895 isin 119878119894

(13)

120587119886

11989411989410158401198951015840 = 120587119889

1198941198941015840119895 119887 = (119895 119895

1015840

) isin 119861119894

119894 1198941015840

isin 119879 (14)

120587119889

1198941198941015840119895= 120587119886

1198941198941015840119895 119895 isin 119878

119894

119894 1198941015840

isin 119879 (15)



119909119886

119894119895minus 119909119886ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

119909119889

119894119895minus 119909119889ini119894119895


inimax 119878]

119887 = (119895 1198951015840

) isin 119861119894

119895 isin 119878119894

119894 isin 119879

(16)



119909119889

1198941015840119895minus 119909119886

119894119895isin [119862

min119895

119862max119895

]

(119894 1198941015840

) isin 119877119895

119895 isin 119878119894

119894 1198941015840

isin 119879

(17)

119899TU119901119895

= sum

119903=(1198941198941015840)isin119877119895

120579119901119903119895

119901 isin 119875 119895 isin 119878119879

119894 1198941015840

isin 119879

(18)

119899TU119901119895

le 119905119888119895 119901 isin 119875 119895 isin 119878

119879

(19)























500119905max119867

1000ℎ119889119889

180 secℎ119886119886


30 secdwmax119894119895

60 sec119862

min119895


max119895

600 sec119905119888119895






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015 017

007006

021

013 023

027

201

203

001002

003

004

005

006

007

005

U2

U2008

009

007

010011

009

012

011

012

010

014

018

011

008

038

012

040

202 204

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U2008 010 003 005 007 009 011 012

U2001 002 004 006 008 010 003

U2







U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1







U2

U2 U2001 003 004002

003

201

005 009

015017

007006

021

013

023

027

203

001002

003

004

005

006

007

005

008

009

007

010011

009

012

011

012

010

014

018

011

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

008

038

202

040

012

204026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

001 002 004 006U2

008 010 003 005 007 009 011 012U2

001 002 004 006 008 010 003





1 min 119865119905

0 sec U2 TW32 min 119865

1199050 sec U2 TW4

3 min 119865119904

300 sec U1 TW14 min 119865

119904300 sec U1 TW2

5 min 119865119904

270 sec U1 TW26 min 119865

119904240 sec U1 TW2

7 min 119865119904

210 sec U1 TW28 min 119865

119904180 sec U1 TW2

9 min 119865119904

150 sec U1 TW210 min 119865

119904120 sec U1 TW2






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025103

051105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021

013

023

027

101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028022

046

032

016

042

024 048

030

020

044

034

036

001 002 004 006

U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

103

051

105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021013

023

027101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1



0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)






119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra






















500119905max119867

1000ℎ119889119889

180 secℎ119886119886


30 secdwmax119894119895

60 sec119862

min119895


max119895

600 sec119905119888119895






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015 017

007006

021

013 023

027

201

203

001002

003

004

005

006

007

005

U2

U2008

009

007

010011

009

012

011

012

010

014

018

011

008

038

012

040

202 204

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U2008 010 003 005 007 009 011 012

U2001 002 004 006 008 010 003

U2







U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1







U2

U2 U2001 003 004002

003

201

005 009

015017

007006

021

013

023

027

203

001002

003

004

005

006

007

005

008

009

007

010011

009

012

011

012

010

014

018

011

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

008

038

202

040

012

204026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

001 002 004 006U2

008 010 003 005 007 009 011 012U2

001 002 004 006 008 010 003





1 min 119865119905

0 sec U2 TW32 min 119865

1199050 sec U2 TW4

3 min 119865119904

300 sec U1 TW14 min 119865

119904300 sec U1 TW2

5 min 119865119904

270 sec U1 TW26 min 119865

119904240 sec U1 TW2

7 min 119865119904

210 sec U1 TW28 min 119865

119904180 sec U1 TW2

9 min 119865119904

150 sec U1 TW210 min 119865

119904120 sec U1 TW2






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025103

051105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021

013

023

027

101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028022

046

032

016

042

024 048

030

020

044

034

036

001 002 004 006

U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

103

051

105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021013

023

027101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1



0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)






119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015 017

007006

021

013 023

027

201

203

001002

003

004

005

006

007

005

U2

U2008

009

007

010011

009

012

011

012

010

014

018

011

008

038

012

040

202 204

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U2008 010 003 005 007 009 011 012

U2001 002 004 006 008 010 003

U2







U2 2 202 R9 R9-1U2 3 203 R10 R10-1U2 4 204 R8 R8-1







U2

U2 U2001 003 004002

003

201

005 009

015017

007006

021

013

023

027

203

001002

003

004

005

006

007

005

008

009

007

010011

009

012

011

012

010

014

018

011

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

008

038

202

040

012

204026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

001 002 004 006U2

008 010 003 005 007 009 011 012U2

001 002 004 006 008 010 003





1 min 119865119905

0 sec U2 TW32 min 119865

1199050 sec U2 TW4

3 min 119865119904

300 sec U1 TW14 min 119865

119904300 sec U1 TW2

5 min 119865119904

270 sec U1 TW26 min 119865

119904240 sec U1 TW2

7 min 119865119904

210 sec U1 TW28 min 119865

119904180 sec U1 TW2

9 min 119865119904

150 sec U1 TW210 min 119865

119904120 sec U1 TW2






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025103

051105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021

013

023

027

101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028022

046

032

016

042

024 048

030

020

044

034

036

001 002 004 006

U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

103

051

105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021013

023

027101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1



0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)






119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










U2

U2 U2001 003 004002

003

201

005 009

015017

007006

021

013

023

027

203

001002

003

004

005

006

007

005

008

009

007

010011

009

012

011

012

010

014

018

011

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

051

045

031

055

041

047

029

053

043

033

057

035

059DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

008

038

202

040

012

204026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

001 002 004 006U2

008 010 003 005 007 009 011 012U2

001 002 004 006 008 010 003





1 min 119865119905

0 sec U2 TW32 min 119865

1199050 sec U2 TW4

3 min 119865119904

300 sec U1 TW14 min 119865

119904300 sec U1 TW2

5 min 119865119904

270 sec U1 TW26 min 119865

119904240 sec U1 TW2

7 min 119865119904

210 sec U1 TW28 min 119865

119904180 sec U1 TW2

9 min 119865119904

150 sec U1 TW210 min 119865

119904120 sec U1 TW2






DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025103

051105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021

013

023

027

101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028022

046

032

016

042

024 048

030

020

044

034

036

001 002 004 006

U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

103

051

105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021013

023

027101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1



0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)






119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025103

051105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021

013

023

027

101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028022

046

032

016

042

024 048

030

020

044

034

036

001 002 004 006

U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1


DSL

LGA

SY

EHN

HN

WAP

XC

EHTHSH

EZP

LSR

DEP

500 510 520 530 540 550 600 610 620 630 640 650 700 710 720 730 740 750 800 810 820 830 840 850 900

500 510 520 540530 550 600 610 620 630 640 650 700 810 820 830 840 850 900710 720 730 740 750 800

003 005 007 009 011 012 001 002 004 006 008 010 003 005 007 009 011 012 001

037

061

039

019

049

025

103

051

105

045

031

055

041

047

029

053

043

033

057

035

059

001003

004002

003

005009

015017

007006

021013

023

027101

106

001

002

003

004

005

006

007

005

U1

U1

008

009

007

010011

009

012

011

012

010

014

018

102

011

008

038

012

040

104

026

050

028

022

046

032

016

042

024

048

030

020

044

034

036

001 002 004 006U1

008 010 003 005 007 009 011 012 001 002 004 006 008 010 003

U1

U1

U1 U1



0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)






119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










0540 550 600 610 620

DSL

LGA

SY

003 005 007U1

019

002

004102

103

025

029

(a)

DSL0

540 550 600 610 620

LGA

SY

019

003 005 007U1

002

004

102

103

025

029

(b)






119905and119865119904(119865119905+119904

= 119865119905sdot1205721+119865119904sdot1205722)






Acknowledgments


References































Volume 2014




Journal of











Journal of


Function Spaces






Algebra

































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article scheduling additional train unit services on rail … · 2019. 7. 31. ·...

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