research article operational impacts of using restricted

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2013 Article ID 902562 8 pageshttpdxdoiorg1011552013902562

Research ArticleOperational Impacts of Using Restricted Passenger FlowAssignment in High-Speed Train Stop Scheduling Problem

Huiling Fu1 Benjamin R Sperry2 and Lei Nie1

1 School of Traffic and Transportation Beijing Jiaotong University Beijing 100044 China2Ohio University Athens OH 45701 USA

Correspondence should be addressed to Huiling Fu fuhuiling00163com

Received 30 July 2013 Revised 1 October 2013 Accepted 29 October 2013

Academic Editor Wuhong Wang

Copyright copy 2013 Huiling Fu 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

One key decision basis to the train stop scheduling process is the passenger flow assignment that is the estimated passengersrsquo travelpath choices from origins to destinations Many existing assignment approaches are stochastic in nature which causes unbalancedproblems such as low efficiency in train capacity occupancy or an irrational distribution of transfer passengers among stationsThe purpose of this paper is to propose a train stop scheduling approach It combines a passenger flow assignment procedure thatroutes passenger travel paths freely within a train network and is particularly capable of incorporating additional restrictions ongenerating travel paths that better resemble the rail plannerrsquos purpose of utilizing capacity resources by introducing four criteria todefine the feasibility of travel path used by a traveler Our approach also aims at ensuring connectivity and rapidity the two essentialcharacteristics of train service increasingly required by modern high-speed rails The effectiveness of our approach is tested usingthe Chinese high-speed rail network as a real-world example It works well in finding a train stop schedule of good quality whoseoperational indicators dominate those of an existing stochastic approach The paper concludes with a comprehensive operationalimpact analysis further demonstrating the value of our proposed approach

1 Introduction

As high-speed rail (HSR) networks are being constructed andexpandedworldwide rail operators face continuing problemsof efficiently planning their high-speed train services Onesuch problem is the train stop scheduling problem (TSSP)which involves specifying a set or subset of stations withina HSR network where individual trains will stop An effi-cient train stop schedule is crucial to improve train serviceconnectivity particularly for passengers whose origins anddestinations (ODs) are not situated at limited terminals Onthe other hand the goal of scheduling many stops by a trainon its route to ensure good connectivity conflicts with thegoal ofmaintaining the rapidity of that train From a practicalperspective a rail operator will always seek to maintainthe most efficient schedule to provide the optimal balancebetween stop locations and frequencies

Understanding passenger behavior requires a fundamen-tal analysis to be conducted when designing services forany transportation modes [1] For rail a passenger flow

assignment procedure is a key decision basis to develop atrain stop schedule within a train network Most of the workfrom the literature that deals with the TSSP concentrated onselectively allocating train stops at stations along a rail line orin a network where train stop pattern combinations of the so-called non-stop (ie direct express) skip-stop and all-stopare employed Decisions regarding the number of stops madeby a train and at which stations in the network those stopsare to take place should be rationally built on a simulationof passengersrsquo travel paths choice behavior In principlescheduling train stops (or any other train service plans)should be consistent with the specific passenger demandTwo optimization models with the objectives of coveringmore passenger demand with fewer train stops as well assaving more travel time for passengers were developed byHamacher et al A genetic algorithm was introduced andtested on a partial rail network in southern Germany [2]In the context of The Netherlandsrsquo rail network three typesof stations and train lines are usually referred to as regional(R) or stop trains for type 1 interregional (IR) for type 2

2 Mathematical Problems in Engineering

and Intercity (IC) for type 3 Train lines of type 1 halt atall stations they pass Lines of type 2 skip the small stationsof type 1 and so forth Goossens et al established integermodels combined with a multicommodity flow problem formultitype line planning problems to minimize operatorrsquosoperating cost [3] Using a 46 km long six-station transitline in the northeastern US as the background Ulusoy et aloptimized all-stop short-turn and express transit servicesby a cost-efficient operation model and a logit-based modelwas used to estimate the ridership of all the seven pregiventrain stop patterns [4] In the setting of Taiwanrsquos HSR lineChang et al formulated amultiobjectivemodel with the TSSPembedded in that model to yield a train operation planThe objectives included minimizing the operatorrsquos operatingcost and the passengersrsquo travel time loss The model wassolved by a fuzzy mathematical programming approach [5]A bilevel programming model which was combined with anetwork equilibrium analysis of passenger flow assignmenton trains in a lower-level problem was proposed by Leeand Hsieh A numerical case study of the final train stopschedule included seven selectable train stop patterns amongfive stations along the line [6] For the Beijing-ShanghaiChinese HSR line Zhang et al proposed a multiobjective0-1 programming model to solve the TSSP The objectiveswere minimizing train dwell time for travelers improvingthe load of every train andminimizing unsatisfied passengerdemand for seven stations along the line [7] Over a partialrail network mixed of both HSR and conventional rail linesa bilevel programming model was applied by Deng et alwho exploited the user equilibrium theory to generate trainstops together with a process of passenger flow assignment ontrains [8]

From the aforementioned studies we further examinethree aspects of the TSSP including two essential character-istics connectivity and rapidity of a good train stop scheduleas well as the passenger flow assignment as follows And weherein illustrate the motivation for this study

(1) Typically a binary decision variable defining a stopbeing added at a station on a train or not is adopted Indoing this as the number of stations becomes larger itturns out to be hard to ensure a complete connectivitymeaning that travelers cannot always reach theirdestinations using direct or transfer connections Tothe best of our knowledge the first approach in thedirection of guaranteeing connectivity is the use of amixed integer linear programming (MILP) in whicha binary decision variable indicating whether stop(s)is (are) added on a train for a passenger OD isconsidered in the problem formulation working wellin finding satisfactory solutions (see [9])

(2) Frequent train stops result in negative impacts includ-ing reducing trainrsquos rapidity increasing passengersrsquototal travel time as well as train operating costAccording to the literature a common way to avoidthe negative impacts of adding stops on trains is todefine certain classes of trains and then restrict thetotal number of stops trains of each classificationcould make

(3) In the planning phase each rail line generally tendsto be organized towards certain passenger ODsand a few of the major stations in a network aredesigned with good transfer capabilities Accordinglyan ideal train stop schedule would intentionally orga-nize passenger traffic that better resembles plannerrsquospurpose (ie passengers may be appropriately guidedtowards corresponding train services through usingtool eg a seat reservation system) However previ-ous research has not incorporated this idea in theirstochastic passenger flow assignment procedures inthe TSSP

The main goal of the present work is to investigate atrain stop scheduling approach that retains good connectivityand rapidity while also being able to cope with intentionallyorganizing passenger traffic in the passenger flow assignmentprocedure For the purpose of retaining connectivity andrapidity we decided to build on the recent approach of Fuet al [9] using the same way of defining the binary decisionvariable As an improvement a restricted passenger flowassignment will be used in this paper to replace their normalformulation

The paper is organized as follows In Section 2 we for-mally describe the nominal version of our TSSP and illustratethe associated MILP formulation In Section 3 we modifythe MILP formulation by introducing a new passenger flowassignment constraint to deal with rail plannerrsquos purposeas mentioned previously Numerical examples on real-worldinstances from the Chinese HSR network are presented inSection 4 Conclusions and future research are discussed inSection 5

2 The Nominal TSSP

In this section we describe the nominal TSSP that we consi-der in this paper We restrict ourselves to the HSR networkcase used by Fu et al [9] to be able to compare with theirmethod and briefly recall the assumptions prior to illustratingthe MILP formulation Generally in a rail network trainsrunning within a single rail line are known as in-line trainsand trains running across at least two linked up rail linesare known as cross-line trains Both in-line and cross-linetrain OD patterns and potential train operating frequenciesare predetermined as inputs for the model Also the trackcapacity is sufficient to meet the passenger demand byoperating trains with rational load factors

21 Notation In the nominal TSSP the aim is to effectivelyutilize capacity resources and to minimize passengersrsquo gener-alized cost using a combination of train stop patterns Ourunderlying approach partly borrows from Fu et al [9] thenotations of parameters and variables which are outlinedbelow

L set of trains among the given train OD patterns inwhich ℓ119896 represent trains indexed by 119896 (or V as shownin the underlying formulation)

Mathematical Problems in Engineering 3

120576(ℓ119896) the number of train stop patterns that could begenerated for train ℓ119896 in setL

119891(ℓ119896) estimated operating frequency of train ℓ119896

119881 set of stations in network in which V119894 representstations (that can also be indexed by119895 119901 or 119902)

V119894(ℓ119896) stations (may not be stops) on the route of trainℓ119896

119864 set of tracks in network inwhich 119890119897 represent tracksindexed by 119897

119863 set of passenger ODs in which 119889(V119894 V119895) representpassenger demand between station V119894 and V119895

1198631015840(ℓ119896) the collection of possible passenger ODs forwhich stops can be added on train ℓ119896 and 1198631015840(ℓ119896) =

(V119894 V119895) isin 119863 | ℓ119896 isin (V119894 V119895)

L(V119894 V119895) the collection of possible trains on whichstops can be added for a passenger OD (V119894 V119895) andL(V119894 V119895) = ℓ119896 isin 119871 | (V119894 V119895) isin ℓ119896

L(119890119897) the collection of trains in set L with theirroutes covering track 119890119897

119860(V119894 V119895) set of feasible travel sections for travelersof passenger OD (V119894 V119895) within the given train ODpatterns

119886119899(ℓ119896) passenger travel sections in set 119860(V119894 V119895)indexed by 119899 (or 119898) each travel section is uniquelyon one train ℓ119896

120581(ℓ119896) seating capacity of train ℓ119896

ℎ(V119894 V119895) route length between stations V119894 and V119895

119873(ℓ119896) the maximum number of stops that can beadded on train ℓ119896

120578(V119894 ℓ119896) count parameter of whether a stop beingadded at station V119894 on train ℓ119896

119884((V119894 V119894+1) ℓ119896) accumulative passenger flow assignedon train ℓ119896 between two adjacent stations V119894 and V119894+1on the train route

119879(119886119899(ℓ119896)) passengersrsquo generalized cost on travel sec-tions 119886119899(ℓ119896)

120591119894119905(119886119899(ℓ119896)) in-vehicle time on travel sections 119886119899(ℓ119896)

120591V119905(119886119899(ℓ119896)) time converted from the ticket fares bytime value on travel sections 119886119899(ℓ119896)

120591119903119905(119886119899(ℓ119896)) wait time on travel sections 119886119899(ℓ119896)

120575119894119905 120575V119905 120575119903119905 weights of generalized cost components

119909((V119894 V119895) 119886119899(ℓ119896)) binary variable it is 1 only if forpassenger OD (V119894 V119895) stops are added on train ℓ119896 elseit equals 0

119910((V119894 V119895) 119886119899(ℓ119896)) variable of the passenger flowofOD(V119894 V119895) assigned on train ℓ119896 with stops added on it

22 Problem Formulation The nominal MILP formulationfor the TSSP is as follows

min sumℓ119896isinL

sumV119894(ℓ119896)isin119881

(120581 (ℓ119896) sdot 119891 (ℓ119896) minus 119884 ((V119894 V119894+1) ℓ119896))

sdot ℎ (V119894 V119894+1) (1)

min sum(V119894 V119895)isin119863

sum119886119899(ℓ119896)isin119860(V119894 V119895)

119879 (119886119899 (ℓ119896)) sdot 119910 ((V119894 V119895) 119886119899 (ℓ119896))

(2)

st sum

ℓ119896isinL(V119894 V119895)

119910 ((V119894 V119895) 119886119899 (ℓ119896)) = 119889 (V119894 V119895)

forall (V119894 V119895) isin 119863

(3)

sum

ℓ119896isinL(V119894 V119895)

119909 ((V119894 V119895) 119886119899 (ℓ119896)) sdot 120581 (ℓ119896) sdot 119891 (ℓ119896)

ge 119889 (V119894 V119895) forall (V119894 V119895) isin 119863

(4)

sum

(V119894 V119895)isin1198631015840(ℓ119896)119886119899(ℓ119896)supe119890119897

119910 ((V119894 V119895) sdot 119886119899 (ℓ119896))

le 120581 (ℓ119896) sdot 119891 (ℓ119896) forall119890119897 isin 119864 ℓ119896 isin L (119890119897)

(5)

sumV119894isin119881

120578 (V119894 ℓ119896) sdot 119909 ((V119894 V119895) 119886119899 (ℓ119896)) le 119873 (ℓ119896)

forallℓ119896 isin L

(6)

119910 ((V119894 V119895) 119886119899 (ℓ119896)) le 119872 sdot 119909 ((V119894 V119895) 119886119899 (ℓ119896))

forallℓ119896 isin L (V119894 V119895) isin 119863(7)

119909 ((V119894 V119895) 119886119899 (ℓ119896)) isin 0 1 forallℓ119896 isin L (V119894 V119895) isin 119863 (8)

119910 ((V119894 V119895) 119886119899 (ℓ119896)) isin R+ forallℓ119896 isin L (V119894 V119895) isin 119863 (9)

Objective function (1) minimizes total trainsrsquo deadheadkilometers where

119884 ((V119894 V119894+1) ℓ119896)

= sumV119895isin119881

sum119886119899(ℓ119896)isin119860(V119894 V119895)

119910 ((V119894 V119895) 119886119899 (ℓ119896))

forallV119894 (ℓ119896) isin 119881

(10)

Passengersrsquo generalized cost in objective function (2) consistsof three parts in-vehicle time consuming time convertedfrom the ticket fares by time value and wait time Thus119879(119886119899(ℓ119896)) extends as

119879 (119886119899 (ℓ119896)) = 120575119894119905sdot 120591119894119905(119886119899 (ℓ119896)) + 120575

V119905sdot 120591

V119905(119886119899 (ℓ119896))

+ 120575119903119905

sdot 120591119903119905(119886119899 (ℓ119896))

(11)

Because train sets running on the same section do notnecessarily operate at a uniform speed different passengersfor a given OD may have different in-vehicle time Similarlythe ticket fares pricing would adopt a differentiation strategyin terms of either train speed classifications or being basedon in-line trains and cross-line trains Wait time depends ontrain operating frequencies

Constraint (3) imposes passenger flow conservation inthe assignment process Demand-supply constraints are illus-trated in (4) and (5) Constraint (4) ensures the total train


stop frequencies at a given station are adequate to meet thepassenger demand requirements Constraint (5) denotes thatthe flow of different passenger ODs assigned on a giventrain ℓ119896 should not exceed that trainrsquos seating capacity Thecondition of ldquo119886119899(ℓ119896) supe 119890119897rdquo means that only passenger OD(s)using travel sections 119886119899(ℓ119896) which pass through track 119890119897 is(are) taken into account Constraint (6) limits the maximumnumber of stops on train ℓ119896 The count parameter 120578(V119894 ℓ119896)equals 0 if a stop will not be added at station V119894 and equals 1 ifstation V119894 is to be added as a stop for more than one passengerOD Constraint (7) ensures that if stop(s) is (are) not addedon train ℓ119896 for passenger OD (V119894 V119895) its flow assigned on trainℓ119896 equals 0119872 is a very large positive number Finally the twotypes of decision variables are restricted in constraints (8) and(9)

3 Our TSSP with Restricted PassengerFlow Assignment

Passengers may have a large set of travel paths from whichthey choose without any restrictions A main drawbackof the nominal TSSP illustrated in the previous section isthat it assigns passengers onto trains in a stochastic waynot taking into account guiding passengers towards certaintrain services on plannerrsquos capacity resources allocationstrategy Stochastic assignment can strongly affect the qualityof train capacity occupancy causing unbalanced problemsfor example short-distance travelers may preempt seats oflong-distance travelers and be served by a long-distancetrain while long-distance travelers are expelled out of thatservice Additionally the formulation ignores and is not easyto describe transfer behavior of travelers Therefore trainservice connectivity for travelers that mostly obtained fromscheduling additional train stops would instead depend onthe quality of the given train OD patterns

To overcome these problems and stimulated by appli-cations of routing passenger travel paths freely in a publictransit network discussed by for example Goossens et al[3] and Borndorfer et al [10] we convert the nominal TSSPinto a TSSP embedded with a multicommodity flow problem(MCFP) by modifying constraint (3) Subsequently we areable to assign passengers onto trains in a restricted way viaincorporating additional restrictions on generating passengertravel paths Ourmodified version of theMILPmodel (1)ndash(9)includes objective functions (1) and (2) constraints (4)ndash(9)but uses the following new constraint

sumV119902isin119881119886119898(ℓ119896)=(V119901 V119902)

119910 ((V119894 V119895) 119886119898 (ℓ119896))

minus sumV119902isin119881119886119899(ℓ119896)=(V119902 V119901)

119910 ((V119894 V119895) 119886119899 (ℓ119896))

=

119889(V119894 V119895) if V119901 = V1198940 if V119894 = V119901 = V119895minus119889 (V119894 V119895) if V119901 = V119895

forall (V119894 V119895) isin 119863 V119901 isin 119881

(12)

In both versions of the MILP models the passenger flowis assigned on a section (ie arc) basis (the reader interestedin the MCFP applied for passenger flow assignment on apath basis is referred to eg [11]) A travel section in thenominal version equates to a travel path Comparatively sincetransfer can be considered in constraint (12) a travel path nowbecomes splittable That is to say a travel path consists of atleast one travel section two travel sections compose a travelpath if one transfer occurs during one single trip and so forthAccordingly in the modified version we attach transfer timeon travel sections 119886119899(ℓ119896) in (11) denoted by 120591119905119905(119886119899(ℓ119896)) andassign a weight 120575119905119905 to them (11) thus reads

119879 (119886119899 (ℓ119896)) = 120575119894119905sdot 120591119894119905(119886119899 (ℓ119896)) + 120575

V119905sdot 120591

V119905(119886119899 (ℓ119896))

+ 120575119903119905

sdot 120591119903119905(119886119899 (ℓ119896)) + 120575

119905119905sdot 120591119905119905(119886119899 (ℓ119896))

(13)

Apparently 120591119905119905(119886119899(ℓ119896)) is not equal to zero only if the sectionis a part of one travel path with transfer(s)

As the foundation of producing restrictions on generatingpassenger travel paths we first recall the rail plannerrsquospurpose as already mentioned in the introduction sectionand make some further interpretations A passenger rail lineas part of a rail network is built generally tending to beorganized towards certain passenger ODs along the corridorof priority For the sake of efficient train capacity occupancyin-line travelers should be organized onto in-line trains asmuch as possible which is a practical representation of theimportant principle of that passenger travel distance shouldmatch train trip distance Second to adapt to passengerdistributing capacity stations with eligible facilities are rec-ommended as main transfer hubs spread all over the entirerail network From a functional perspective transfer hubshave good performance if one transfer occurs within thesame platform and trains have convenient as well as fastconnections with other high-speed or conventional trains orurban public transport

We next illustrate our adopted restrictions in consider-ation of rail plannerrsquos objectives when generating passengertravel paths in the MCFP A general rule is to limit a feasibletravel paths set for passengers of each OD using two primarycriteria as follows

(1) A travel path or a travel section being part of a pathwith transfer(s) possible for passengers of a certainOD is feasible only if the ratio between the lengthitself and the trip distance of a train onto whichpassengers are probably assigned is greater than agiven baseline value (notated as 120583)

(2) A travel path with transfer(s) is feasible if the transferhub(s) is (are) selected from recommended stations

From the viewpoint of providing better quality trainservices alternative restrictions include the following

(3) a controlled percentage of nonstop trains for travelersare provided

(4) a single passenger is not required to transfer morethan once during a single trip


Lang

fang

Beiji

ng S

outh

Shenyang North

Tian

jin S

outh

Cang

zhou

Wes

t1

Dez

hou

East

Jinan

Wes

t

Taia

n

Zaoz

huan

g

Xuzh

ou E

ast

Suzh

ou E

ast

Beng

bu S

outh

Chuz

hou

Sout

h

Nan

jing

Sout

h

Zhen

jiang

Wes

t

Chan

gzho

u N

orth

Wux

i Eas

t

Suzh

ou N

orth

Shan

ghai

Hon

gqia

o

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Qingdao

Jinan

Shijiazhuang Hefei

Wuhan

Chengdu

19 20

21

22

23

24

25

26

27

28

29Zhengzhou East

Taiyuan East Xian North

Hangzhou East

Stations can serve as terminalsIntermediate stations

Figure 1 Topology of study HSR network

Of course many other restrictions on paths generationare allowed in practice a rail operator could adopt manyother criteria to define the feasibility of a path and choosea desirable set of paths depending on the specific require-ments For example even if criterion (1) ismet a rail operatorcould additionally delete travel paths on a train if that trainprecedes other trains not serving passengers among smallerstations Certain travel paths could be likewise specified topassengers for whom in extreme case no travel paths can befound by following the given criteria

4 Numerical Examples

41 Data Set This section presents a numerical test of theproposed approach for the TSSP using data from a real-worldexample on the Chinese railways We focus our attention onthe example adopted by [9] to allow for comparisons Forthe sake of clarity we stretch their real-world HSR networkto be a topology with each station numbered as shownin Figure 1 Additional information reflecting each line orsectionrsquos operating condition is shown in Table 1 For the 2015forecast year our numerical example includes a total of 303passenger ODs and total 453205 passenger traffic per dayassociated with the Jinghu HSR (the center of the networkwe abbreviate it as JHSR in the following) Due to data sizespassenger flow OD matrix is not shown but is available fromthe authors Table 2 shows the train OD patterns as givenfrom the operational plan which includes 7 in-line trainsand 29 cross-line trains for the JHSR network Other inputparameters of our models are specified in Table 3

42 Discussion of Results Progressing to our appropriatecode based on the original one (used in [9]) both modelsin two versions were applied to the network and solved onan Intel i5 24GHz with 2GB RAM in the environment ofMicrosoft Windows XP using the Lingo 100 optimizer Notethat in the modified version we incorporated all the fourrestrictions into predetermined feasible travel paths set In

addition to the already mentioned criteria we provided thatthe terminal stations in Figure 1 served as transfer hubs and70 percent of travelers of one OD were organized towardsnonstop trains with load factor not less than 085 During thesolving process a layered sequence method was employed toconvert the multiobjective problem to a single-objective oneThe models were first solved under single-objective (1) itsoptimization results were put into constraints as

sumℓ119896isinL

sumV119894(ℓ119896)isin119881

(120581 (ℓ119896) sdot 119891 (ℓ119896) minus 119884 ((V119894 V119894+1) ℓ119896))

sdot ℎ (V119894 V119894+1) = 119872lowast

(14)

with 119872lowast being the optimum solution afterwards the prob-lems were optimized under single-objective (2)

Using a branch-and-bound algorithm our restrictedTSSP took 57minutes of computing time to find an optimumsolution compared to 36 minutes for the nominal TSSP Thefinal train stop schedule obtained for the restricted TSSPis displayed in Table 4 and a comparison of our methodwith the method for the nominal TSSP is shown in Table 5Due to the increase in complexity as expected the efficiencygotten for our solution is lower than that of the nominal onebut the gain in most operational indicators (to be discussedlater) is quite significant and confirms the effectiveness of theproposed approach Because the TSSP has to be solved in theplanning phase the computing time is acceptable

According to Table 5 although it is a straightforwardmodification of an existing approach for the nominal caseour approach turns out to be effective in scheduling trainstops of good quality as indicated by operational metricswhich clearly dominate those of the nominal TSSP In spiteof the differences the two schedules have some similaritiesFirst both methods result in a flexible combination of trainstop patterns that is nonstop skip-stop and all-stop How-ever an ldquoall-stoprdquo train stop pattern is only recommended forshort- or medium-distance trains (eg the train from station13 to 18) Another similarity between the two methods is


Table 1 Line or section information of study HSR network

HSR line or section The year in operation Approximate length (km) Speed classification (kmh)Jinghu (1-18) 2011 1318 350Jingshen (1-19) mdash 676 350Jiaoji (21-20) 2008 362 200sim250Shijiazhuang-Dezhou mdash 180 250Shitai (22-23) 2009 190 250Zhengxu (24-9) mdash 360 350Zhengxi (24-25) 2010 458 350Hening (26-13) 2008 166 200Hewu (26-27) 2008 356 200Wuhan-Chengdu mdash 1260 200Huhang (18-29) 2010 158 350ldquomdashrdquo indicates that the rail line or section is planned or not completely in operation

Table 2 The given train OD patterns

In-line train ODs Cross-line train ODs Train ODs for transfer connections only

(1 13) (1 18) (3 18) (6 13) (6 18) (9 18)(13 18)

(1 20) (1 26) (1 29) (3 20) (3 27) (3 29)(6 19) (6 27) (6 29) (9 20) (9 26) (9 29)(13 20) (13 23) (13 25) (13 29) (18 19)(18 20) (18 23) (18 25) (18 24) (18 28)(18 27) (18 26) (20 19) (20 27) (20 25)

(29 25) (29 24)

(19 1) (20 21) (23 6) (25 9) (24 9) (28 13)(27 13) (26 13) (28 27) (28 26)

Table 3 Input parameters of the models

Parameter Value or descriptive calculationSplit coefficient 120576 (ℓ119896) 11 for in-line trains and 3 for cross-line trainsSeating capacity 120581 (ℓ119896) 1060 seatstrain-setNumber of train stops 119873(ℓ119896) Maximum of 7 times for in-line trains and 8 for cross-line trains

In-vehicle time 120591119894119905 (119886119899(ℓ119896))

Train route length ℎ (V119894 V119895) train travel speed ( accommodates to the rail linersquos speed

classification)

Consuming time 120591V119905(119886119899 (ℓ119896))

Ticket fares (055 RMBkm for 350 kmh train 045 RMBkm for other trains) times ℎ (V119894 V119895) time

value of passenger (45 RMBh on average)

Wait time 120591119903119905(119886119899 (ℓ119896))

050119891 (ℓ119896) a fraction (taken as 050 here) of headway which is the inverse of a trainrsquos operating

frequencyTransfer time 120591119905119905 (119886

119899(ℓ119896)) 30 minutes

Weights 120575119894119905 120575V119905 120575119903119905 and 120575119905119905 039 028 012 021Baseline value 120583 080 for in-line travelers and 050 for cross-line (transfer) travelers

that in the interaction with objective functions the obtainedtrain stop schedules perform even better For instance underthe schedule for the restricted TSSP 73 percent of trains areassigned to a number of stops less than restricted as comparedwith 65 percent by the nominal TSSPThis difference is due toa hard rule applied in the restricted TSSP that more nonstoptrains are allowed operations and the consequence arisingfrom this influences the average train travel speeds in thesame way In both methods the entirely assured connectivityfor travelers is not surprising because travelers are alwaystracked on which train(s) they are assigned and stops areaccordingly determined by defining such type of a decisionvariable For example the train ldquo9-11-13-14-17-18rdquo stops at fourintermediate stations for estimated travelers of 10 passenger

ODs assigned on it even though in practice travelers of all15 combinatorial passenger ODs can be absolutely served bythat trainTherefore fromapractical perspective the numberof possible passenger OD combinations served by each trainis necessarily higher than the measures from the computingresults of passenger flow assignment

The indicator of realized passenger traffic represents anotable difference between the two methods The realizedpassenger traffic of our restricted TSSP is significantly largerAs already mentioned in Section 3 this is achieved by allow-ing transfers in travel paths However the gain in reducingpassengersrsquo generalized cost is not that much higher whencompared to the nominal TSSP To make a more accuratecomparison we calculate generalized cost on the scale of


Table 4 Train stop schedule estimated from the method of ourrestricted TSSP

Train OD Train route no Train stop pattern with stationsequence

In-line trains

1 13 1 1-9-13

2 1-2-4-8-9-10-11-12-13

1 18

1 1-182 1-3-183 1-3-6-13-184 1-3-5-6-7-13-185 1-3-6-11-13-16-17-186 1-2-3-5-7-8-9-16-187 1-2-3-4-5-9-15-17-188 1-2-3-6-9-10-11-13-189 1-2-3-4-9-10-12-14-18

10 1-3-6-9-11-13-15-17-18

3 18

1 3-182 3-6-183 3-6-9-184 3-5-6-7-8-15-17-18

5 3-4-5-6-7-8-12-17-18

6 13 1 6-13

2 6-7-9-10-11-12-13

6 18

1 6-9-182 6-13-183 6-7-8-9-14-16-17-184 6-10-11-12-13-16-17-18

9 18 1 9-13-182 9-11-13-14-17-18

13 18 1 13-182 13-14-15-16-17-18

Cross-line trains

1 20 1 1-3-20

2 1-2-3-4-5-20

1 261 1-3-6-9-262 1-3-5-6-8-11-26

3 1-3-4-5-6-7-9-10-11-26

1 29 1 1-3-6-9-13-29

3 20 1 3-5-20

3 27 1 3-9-13-26-27

2 3-4-7-10-12-27

3 29 1 3-5-6-10-11-13-18-29

19 6 1 19-1-5-6

6 27 1 6-8-26-27

Table 4 Continued


6 29 1 6-7-9-29

2 6-8-13-18-29

20 9 1 20-6-9

9 26 1 9-11-26

9 29 1 9-11-13-17-29

20 13 1 20-6-13

23 13 1 23-22-6-7-8-10-11-13

25 13 1 25-9-13

13 29 1 13-18-29

2 13-14-15-16-17-18-29

19 181 19-1-3-6-9-13-182 19-2-7-8-9-10-11-13-17-18

3 19-2-4-5-9-13-14-15-16-18

20 18 1 20-6-14-15-16-17-18

23 18 1 23-22-9-14-15-16-17-18

25 18 1 25-9-12-13-182 25-9-10-11-13-14-15-16-17-18

24 18 1 24-9-13-182 24-9-10-11-12-14-15-16-17-18

28 18 1 28-13-17-18

27 18 1 27-13-15-16-17-18

26 18 1 26-13-182 26-13-15-16-17-18

19 20 1 19-3-4-5-20

20 27 1 20-9-10-11-27

20 25 1 20-6-8-25

25 29 1 25-9-12-13-18-29

24 29 1 24-9-13-18-29

an individual traveler For the case of the restricted TSSP thetotal passengersrsquo generalized cost is 359392times 106 (h)while thecost is 312417 times 106 (h) for the nominal TSSPThis representsa saving of approximately 027 hours per traveler This smallbut meaningful amount of time saving is attributed to that inthe restricted TSSP the decrease of travel time via operatingnonstop trains is greatly offset by the increase of transfer time

The effectiveness of the approach of our restricted TSSP isalso supported by another operational indicator It is apparentfrom Table 5 that the matching degree measured by 120583 for thenominal TSSP is much lower since no limitation is imposedon 120583 when assigning each passenger to his optimal itineraryAn operational outcome of this result is that the average traindeadhead kilometers is nearly 4 percent higher in the nominalTSSP compared to our restricted TSSP


Table 5 Comparison of operational indicators between methods for our restricted TSSP and the nominal TSSP

Operational indicator Restricted TSSP Nominal TSSPUsing combinatorial train stop patterns or not Yes YesPercentage of trains with number of stops less than restricted () 73 65Average train travel speed (kmh) 301 294Realized passenger traffic by train services (people) 453205 380772Connectivity degree for realized passenger traffic () 100 100Percentage of travelers organized onto nonstop trains () 33 21Matching degree measured by 120583 () 100 51Percentage of transfer passengers () 9 0

Transfer hubs in travel paths are distributed at stationsmostly on the JHSR as restricted by the model Stations 13 6 9 and 13 assemble approximately 92 percent of transferpassengers Due to being covered by travel paths of a largeramount of cross-line passenger ODs in this instance stations9 and 13 together are estimated to have a higher probabilityof being transfer hubs selected by approximately 65 percentof transfer passengers Obviously there are clear benefits inassisting rail operator to plan passenger transfer organizationwork as part of the TSSP and as an added benefit theinconvenience of making a transfer connection is reducedrelative to the convenience of direct connection

5 Conclusions

Rail operators develop train stop schedules with the goalsof retaining good connectivity and rapidity to travelerswhile also in the face of requirements for capacity resourcesutilization In this paper we have shown how to incorporaterestricted passenger flow assignment into a TSSP formulationto achieve this purpose To this end two procedures needto be implemented (1) introducing the MCFP constraintintended to route passenger travel paths freely and (2) duringpassenger travel paths generation establishing four criteriato produce restrictions so that the operator can collect adesirable set of travel paths Our approach has been appliedto a real-world HSR network case from the Chinese railwaysalongwith a comparisonwith a nominal train stop schedulingmethod that uses stochastic passenger flow assignment Theresults showed that our approach is very competitive andobtains a train stop schedule solution of good quality inacceptable computing time Future direction of researchinto efficient formulation of the TSSP can be devoted tocollaboratively optimize train operating frequency which istreated as constant value in the present paper

Conflict of Interests

The authors declare that there is no conflict of interests reg-arding the publication of this paper

Acknowledgments

This work was supported by the Fundamental ResearchFunds for the Central Universities (Beijing Jiaotong Univer-sity) under Grant no 2013JBM042 The authors are gratefulto the referees for their valuable comments and suggestions

References

[1] H Guo W Wang W Guo X Jiang and H Bubb ldquoReliabilityanalysis of pedestrian safety crossing in urban traffic environ-mentrdquo Safety Science vol 50 no 4 pp 968ndash973 2012

[2] HWHamacher A Liebers A Schobel DWagner and FWag-ner ldquoLocating new stops in a railway networkrdquo Electronic Notesin Theoretical Computer Science vol 50 no 1 pp 13ndash23 2001

[3] J-W Goossens S van Hoesel and L Kroon ldquoOn solvingmulti-type railway line planning problemsrdquo European Journal of Oper-ational Research vol 168 no 2 pp 403ndash424 2006

[4] Y Y Ulusoy S Chien and C-H Wei ldquoOptimal all-stop short-turn and express transit services under heterogeneous dem-andrdquo Transportation Research Record no 2197 pp 8ndash18 2010

[5] Y-H Chang C-H Yeh and C-C Shen ldquoA multiobjective mo-del for passenger train services planning application to Taiwanrsquoshigh-speed rail linerdquo Transportation Research B vol 34 no 2pp 91ndash106 2000

[6] C Lee andW Hsieh ldquoA demand oriented service planning pro-cessrdquo in Proceedings of the World Congress on Railway ResearchKoln Germany 2001

[7] Y ZhangM Ren andWDu ldquoOptimization of high speed trainoperationrdquo Journal of Southwest Jiaotong University vol 33no 4 pp 400ndash404 1998 (Chinese)

[8] L Deng F Shi and W Zhou ldquoStop schedule plan optimizationfor passenger trainrdquo China Railway Science vol 30 no 4 pp102ndash107 2009

[9] H Fu L Nie B R Sperry andZHe ldquoTrain stop scheduling in ahigh-speed rail network by utilizing a two-stage approachrdquoMathematical Problems in Engineering vol 20102 Article ID579130 11 pages 2012

[10] R BorndorferM Grotschel andM E Pfetsch ldquoA column-gen-eration approach to line planning in public transportrdquo Trans-portation Science vol 41 no 1 pp 123ndash132 2007

[11] R BorndorferM Grotschel andM E Pfetsch ldquoModels for lineplanning in public transportrdquo ZIP-Report 04-10 Konrad-Zuse-Zentrum fur Informationstechnik Berlin Germany 2004

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


and Intercity (IC) for type 3 Train lines of type 1 halt atall stations they pass Lines of type 2 skip the small stationsof type 1 and so forth Goossens et al established integermodels combined with a multicommodity flow problem formultitype line planning problems to minimize operatorrsquosoperating cost [3] Using a 46 km long six-station transitline in the northeastern US as the background Ulusoy et aloptimized all-stop short-turn and express transit servicesby a cost-efficient operation model and a logit-based modelwas used to estimate the ridership of all the seven pregiventrain stop patterns [4] In the setting of Taiwanrsquos HSR lineChang et al formulated amultiobjectivemodel with the TSSPembedded in that model to yield a train operation planThe objectives included minimizing the operatorrsquos operatingcost and the passengersrsquo travel time loss The model wassolved by a fuzzy mathematical programming approach [5]A bilevel programming model which was combined with anetwork equilibrium analysis of passenger flow assignmenton trains in a lower-level problem was proposed by Leeand Hsieh A numerical case study of the final train stopschedule included seven selectable train stop patterns amongfive stations along the line [6] For the Beijing-ShanghaiChinese HSR line Zhang et al proposed a multiobjective0-1 programming model to solve the TSSP The objectiveswere minimizing train dwell time for travelers improvingthe load of every train andminimizing unsatisfied passengerdemand for seven stations along the line [7] Over a partialrail network mixed of both HSR and conventional rail linesa bilevel programming model was applied by Deng et alwho exploited the user equilibrium theory to generate trainstops together with a process of passenger flow assignment ontrains [8]

From the aforementioned studies we further examinethree aspects of the TSSP including two essential character-istics connectivity and rapidity of a good train stop scheduleas well as the passenger flow assignment as follows And weherein illustrate the motivation for this study

(1) Typically a binary decision variable defining a stopbeing added at a station on a train or not is adopted Indoing this as the number of stations becomes larger itturns out to be hard to ensure a complete connectivitymeaning that travelers cannot always reach theirdestinations using direct or transfer connections Tothe best of our knowledge the first approach in thedirection of guaranteeing connectivity is the use of amixed integer linear programming (MILP) in whicha binary decision variable indicating whether stop(s)is (are) added on a train for a passenger OD isconsidered in the problem formulation working wellin finding satisfactory solutions (see [9])

(2) Frequent train stops result in negative impacts includ-ing reducing trainrsquos rapidity increasing passengersrsquototal travel time as well as train operating costAccording to the literature a common way to avoidthe negative impacts of adding stops on trains is todefine certain classes of trains and then restrict thetotal number of stops trains of each classificationcould make

(3) In the planning phase each rail line generally tendsto be organized towards certain passenger ODsand a few of the major stations in a network aredesigned with good transfer capabilities Accordinglyan ideal train stop schedule would intentionally orga-nize passenger traffic that better resembles plannerrsquospurpose (ie passengers may be appropriately guidedtowards corresponding train services through usingtool eg a seat reservation system) However previ-ous research has not incorporated this idea in theirstochastic passenger flow assignment procedures inthe TSSP

The main goal of the present work is to investigate atrain stop scheduling approach that retains good connectivityand rapidity while also being able to cope with intentionallyorganizing passenger traffic in the passenger flow assignmentprocedure For the purpose of retaining connectivity andrapidity we decided to build on the recent approach of Fuet al [9] using the same way of defining the binary decisionvariable As an improvement a restricted passenger flowassignment will be used in this paper to replace their normalformulation

The paper is organized as follows In Section 2 we for-mally describe the nominal version of our TSSP and illustratethe associated MILP formulation In Section 3 we modifythe MILP formulation by introducing a new passenger flowassignment constraint to deal with rail plannerrsquos purposeas mentioned previously Numerical examples on real-worldinstances from the Chinese HSR network are presented inSection 4 Conclusions and future research are discussed inSection 5

2 The Nominal TSSP

In this section we describe the nominal TSSP that we consi-der in this paper We restrict ourselves to the HSR networkcase used by Fu et al [9] to be able to compare with theirmethod and briefly recall the assumptions prior to illustratingthe MILP formulation Generally in a rail network trainsrunning within a single rail line are known as in-line trainsand trains running across at least two linked up rail linesare known as cross-line trains Both in-line and cross-linetrain OD patterns and potential train operating frequenciesare predetermined as inputs for the model Also the trackcapacity is sufficient to meet the passenger demand byoperating trains with rational load factors

21 Notation In the nominal TSSP the aim is to effectivelyutilize capacity resources and to minimize passengersrsquo gener-alized cost using a combination of train stop patterns Ourunderlying approach partly borrows from Fu et al [9] thenotations of parameters and variables which are outlinedbelow

L set of trains among the given train OD patterns inwhich ℓ119896 represent trains indexed by 119896 (or V as shownin the underlying formulation)









(V119894 V119895) isin 119863 | ℓ119896 isin (V119894 V119895)



















sumV119894(ℓ119896)isin119881

(120581 (ℓ119896) sdot 119891 (ℓ119896) minus 119884 ((V119894 V119894+1) ℓ119896))

sdot ℎ (V119894 V119894+1) (1)

min sum(V119894 V119895)isin119863

sum119886119899(ℓ119896)isin119860(V119894 V119895)

119879 (119886119899 (ℓ119896)) sdot 119910 ((V119894 V119895) 119886119899 (ℓ119896))

(2)

st sum

ℓ119896isinL(V119894 V119895)

119910 ((V119894 V119895) 119886119899 (ℓ119896)) = 119889 (V119894 V119895)

forall (V119894 V119895) isin 119863

(3)

sum

ℓ119896isinL(V119894 V119895)

119909 ((V119894 V119895) 119886119899 (ℓ119896)) sdot 120581 (ℓ119896) sdot 119891 (ℓ119896)


(4)

sum

(V119894 V119895)isin1198631015840(ℓ119896)119886119899(ℓ119896)supe119890119897

119910 ((V119894 V119895) sdot 119886119899 (ℓ119896))


(5)


120578 (V119894 ℓ119896) sdot 119909 ((V119894 V119895) 119886119899 (ℓ119896)) le 119873 (ℓ119896)


(6)

119910 ((V119894 V119895) 119886119899 (ℓ119896)) le 119872 sdot 119909 ((V119894 V119895) 119886119899 (ℓ119896))





119884 ((V119894 V119894+1) ℓ119896)

= sumV119895isin119881

sum119886119899(ℓ119896)isin119860(V119894 V119895)

119910 ((V119894 V119895) 119886119899 (ℓ119896))

forallV119894 (ℓ119896) isin 119881

(10)


119879 (119886119899 (ℓ119896)) = 120575119894119905sdot 120591119894119905(119886119899 (ℓ119896)) + 120575

V119905sdot 120591

V119905(119886119899 (ℓ119896))

+ 120575119903119905

sdot 120591119903119905(119886119899 (ℓ119896))

(11)








sumV119902isin119881119886119898(ℓ119896)=(V119901 V119902)

119910 ((V119894 V119895) 119886119898 (ℓ119896))


119910 ((V119894 V119895) 119886119899 (ℓ119896))

=



(12)


119879 (119886119899 (ℓ119896)) = 120575119894119905sdot 120591119894119905(119886119899 (ℓ119896)) + 120575

V119905sdot 120591

V119905(119886119899 (ℓ119896))

+ 120575119903119905

sdot 120591119903119905(119886119899 (ℓ119896)) + 120575

119905119905sdot 120591119905119905(119886119899 (ℓ119896))

(13)










Lang

fang

Beiji

ng S

outh

Shenyang North

Tian

jin S

outh

Cang

zhou

Wes

t1

Dez

hou

East

Jinan

Wes

t

Taia

n

Zaoz

huan

g

Xuzh

ou E

ast

Suzh

ou E

ast

Beng

bu S

outh

Chuz

hou

Sout

h

Nan

jing

Sout

h

Zhen

jiang

Wes

t

Chan

gzho

u N

orth

Wux

i Eas

t

Suzh

ou N

orth

Shan

ghai

Hon

gqia

o

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Qingdao

Jinan

Shijiazhuang Hefei

Wuhan

Chengdu

19 20

21

22

23

24

25

26

27

28

29Zhengzhou East


Hangzhou East








sumℓ119896isinL

sumV119894(ℓ119896)isin119881

(120581 (ℓ119896) sdot 119891 (ℓ119896) minus 119884 ((V119894 V119894+1) ℓ119896))

sdot ℎ (V119894 V119894+1) = 119872lowast

(14)









(1 13) (1 18) (3 18) (6 13) (6 18) (9 18)(13 18)

(1 20) (1 26) (1 29) (3 20) (3 27) (3 29)(6 19) (6 27) (6 29) (9 20) (9 26) (9 29)(13 20) (13 23) (13 25) (13 29) (18 19)(18 20) (18 23) (18 25) (18 24) (18 28)(18 27) (18 26) (20 19) (20 27) (20 25)

(29 25) (29 24)

(19 1) (20 21) (23 6) (25 9) (24 9) (28 13)(27 13) (26 13) (28 27) (28 26)



In-vehicle time 120591119894119905 (119886119899(ℓ119896))


classification)




Wait time 120591119903119905(119886119899 (ℓ119896))



119899(ℓ119896)) 30 minutes








In-line trains

1 13 1 1-9-13

2 1-2-4-8-9-10-11-12-13

1 18

1 1-182 1-3-183 1-3-6-13-184 1-3-5-6-7-13-185 1-3-6-11-13-16-17-186 1-2-3-5-7-8-9-16-187 1-2-3-4-5-9-15-17-188 1-2-3-6-9-10-11-13-189 1-2-3-4-9-10-12-14-18

10 1-3-6-9-11-13-15-17-18

3 18

1 3-182 3-6-183 3-6-9-184 3-5-6-7-8-15-17-18

5 3-4-5-6-7-8-12-17-18

6 13 1 6-13

2 6-7-9-10-11-12-13

6 18

1 6-9-182 6-13-183 6-7-8-9-14-16-17-184 6-10-11-12-13-16-17-18

9 18 1 9-13-182 9-11-13-14-17-18

13 18 1 13-182 13-14-15-16-17-18

Cross-line trains

1 20 1 1-3-20

2 1-2-3-4-5-20

1 261 1-3-6-9-262 1-3-5-6-8-11-26

3 1-3-4-5-6-7-9-10-11-26

1 29 1 1-3-6-9-13-29

3 20 1 3-5-20

3 27 1 3-9-13-26-27

2 3-4-7-10-12-27

3 29 1 3-5-6-10-11-13-18-29

19 6 1 19-1-5-6

6 27 1 6-8-26-27

Table 4 Continued


6 29 1 6-7-9-29

2 6-8-13-18-29

20 9 1 20-6-9

9 26 1 9-11-26

9 29 1 9-11-13-17-29

20 13 1 20-6-13

23 13 1 23-22-6-7-8-10-11-13

25 13 1 25-9-13

13 29 1 13-18-29

2 13-14-15-16-17-18-29

19 181 19-1-3-6-9-13-182 19-2-7-8-9-10-11-13-17-18

3 19-2-4-5-9-13-14-15-16-18

20 18 1 20-6-14-15-16-17-18

23 18 1 23-22-9-14-15-16-17-18

25 18 1 25-9-12-13-182 25-9-10-11-13-14-15-16-17-18

24 18 1 24-9-13-182 24-9-10-11-12-14-15-16-17-18

28 18 1 28-13-17-18

27 18 1 27-13-15-16-17-18

26 18 1 26-13-182 26-13-15-16-17-18

19 20 1 19-3-4-5-20

20 27 1 20-9-10-11-27

20 25 1 20-6-8-25

25 29 1 25-9-12-13-18-29

24 29 1 24-9-13-18-29







5 Conclusions




Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra

















(V119894 V119895) isin 119863 | ℓ119896 isin (V119894 V119895)



















sumV119894(ℓ119896)isin119881

(120581 (ℓ119896) sdot 119891 (ℓ119896) minus 119884 ((V119894 V119894+1) ℓ119896))

sdot ℎ (V119894 V119894+1) (1)

min sum(V119894 V119895)isin119863

sum119886119899(ℓ119896)isin119860(V119894 V119895)

119879 (119886119899 (ℓ119896)) sdot 119910 ((V119894 V119895) 119886119899 (ℓ119896))

(2)

st sum

ℓ119896isinL(V119894 V119895)

119910 ((V119894 V119895) 119886119899 (ℓ119896)) = 119889 (V119894 V119895)

forall (V119894 V119895) isin 119863

(3)

sum

ℓ119896isinL(V119894 V119895)

119909 ((V119894 V119895) 119886119899 (ℓ119896)) sdot 120581 (ℓ119896) sdot 119891 (ℓ119896)


(4)

sum

(V119894 V119895)isin1198631015840(ℓ119896)119886119899(ℓ119896)supe119890119897

119910 ((V119894 V119895) sdot 119886119899 (ℓ119896))


(5)


120578 (V119894 ℓ119896) sdot 119909 ((V119894 V119895) 119886119899 (ℓ119896)) le 119873 (ℓ119896)


(6)

119910 ((V119894 V119895) 119886119899 (ℓ119896)) le 119872 sdot 119909 ((V119894 V119895) 119886119899 (ℓ119896))





119884 ((V119894 V119894+1) ℓ119896)

= sumV119895isin119881

sum119886119899(ℓ119896)isin119860(V119894 V119895)

119910 ((V119894 V119895) 119886119899 (ℓ119896))

forallV119894 (ℓ119896) isin 119881

(10)


119879 (119886119899 (ℓ119896)) = 120575119894119905sdot 120591119894119905(119886119899 (ℓ119896)) + 120575

V119905sdot 120591

V119905(119886119899 (ℓ119896))

+ 120575119903119905

sdot 120591119903119905(119886119899 (ℓ119896))

(11)








sumV119902isin119881119886119898(ℓ119896)=(V119901 V119902)

119910 ((V119894 V119895) 119886119898 (ℓ119896))


119910 ((V119894 V119895) 119886119899 (ℓ119896))

=



(12)


119879 (119886119899 (ℓ119896)) = 120575119894119905sdot 120591119894119905(119886119899 (ℓ119896)) + 120575

V119905sdot 120591

V119905(119886119899 (ℓ119896))

+ 120575119903119905

sdot 120591119903119905(119886119899 (ℓ119896)) + 120575

119905119905sdot 120591119905119905(119886119899 (ℓ119896))

(13)










Lang

fang

Beiji

ng S

outh

Shenyang North

Tian

jin S

outh

Cang

zhou

Wes

t1

Dez

hou

East

Jinan

Wes

t

Taia

n

Zaoz

huan

g

Xuzh

ou E

ast

Suzh

ou E

ast

Beng

bu S

outh

Chuz

hou

Sout

h

Nan

jing

Sout

h

Zhen

jiang

Wes

t

Chan

gzho

u N

orth

Wux

i Eas

t

Suzh

ou N

orth

Shan

ghai

Hon

gqia

o

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Qingdao

Jinan

Shijiazhuang Hefei

Wuhan

Chengdu

19 20

21

22

23

24

25

26

27

28

29Zhengzhou East


Hangzhou East








sumℓ119896isinL

sumV119894(ℓ119896)isin119881

(120581 (ℓ119896) sdot 119891 (ℓ119896) minus 119884 ((V119894 V119894+1) ℓ119896))

sdot ℎ (V119894 V119894+1) = 119872lowast

(14)









(1 13) (1 18) (3 18) (6 13) (6 18) (9 18)(13 18)

(1 20) (1 26) (1 29) (3 20) (3 27) (3 29)(6 19) (6 27) (6 29) (9 20) (9 26) (9 29)(13 20) (13 23) (13 25) (13 29) (18 19)(18 20) (18 23) (18 25) (18 24) (18 28)(18 27) (18 26) (20 19) (20 27) (20 25)

(29 25) (29 24)

(19 1) (20 21) (23 6) (25 9) (24 9) (28 13)(27 13) (26 13) (28 27) (28 26)



In-vehicle time 120591119894119905 (119886119899(ℓ119896))


classification)




Wait time 120591119903119905(119886119899 (ℓ119896))



119899(ℓ119896)) 30 minutes








In-line trains

1 13 1 1-9-13

2 1-2-4-8-9-10-11-12-13

1 18

1 1-182 1-3-183 1-3-6-13-184 1-3-5-6-7-13-185 1-3-6-11-13-16-17-186 1-2-3-5-7-8-9-16-187 1-2-3-4-5-9-15-17-188 1-2-3-6-9-10-11-13-189 1-2-3-4-9-10-12-14-18

10 1-3-6-9-11-13-15-17-18

3 18

1 3-182 3-6-183 3-6-9-184 3-5-6-7-8-15-17-18

5 3-4-5-6-7-8-12-17-18

6 13 1 6-13

2 6-7-9-10-11-12-13

6 18

1 6-9-182 6-13-183 6-7-8-9-14-16-17-184 6-10-11-12-13-16-17-18

9 18 1 9-13-182 9-11-13-14-17-18

13 18 1 13-182 13-14-15-16-17-18

Cross-line trains

1 20 1 1-3-20

2 1-2-3-4-5-20

1 261 1-3-6-9-262 1-3-5-6-8-11-26

3 1-3-4-5-6-7-9-10-11-26

1 29 1 1-3-6-9-13-29

3 20 1 3-5-20

3 27 1 3-9-13-26-27

2 3-4-7-10-12-27

3 29 1 3-5-6-10-11-13-18-29

19 6 1 19-1-5-6

6 27 1 6-8-26-27

Table 4 Continued


6 29 1 6-7-9-29

2 6-8-13-18-29

20 9 1 20-6-9

9 26 1 9-11-26

9 29 1 9-11-13-17-29

20 13 1 20-6-13

23 13 1 23-22-6-7-8-10-11-13

25 13 1 25-9-13

13 29 1 13-18-29

2 13-14-15-16-17-18-29

19 181 19-1-3-6-9-13-182 19-2-7-8-9-10-11-13-17-18

3 19-2-4-5-9-13-14-15-16-18

20 18 1 20-6-14-15-16-17-18

23 18 1 23-22-9-14-15-16-17-18

25 18 1 25-9-12-13-182 25-9-10-11-13-14-15-16-17-18

24 18 1 24-9-13-182 24-9-10-11-12-14-15-16-17-18

28 18 1 28-13-17-18

27 18 1 27-13-15-16-17-18

26 18 1 26-13-182 26-13-15-16-17-18

19 20 1 19-3-4-5-20

20 27 1 20-9-10-11-27

20 25 1 20-6-8-25

25 29 1 25-9-12-13-18-29

24 29 1 24-9-13-18-29







5 Conclusions




Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra














sumV119902isin119881119886119898(ℓ119896)=(V119901 V119902)

119910 ((V119894 V119895) 119886119898 (ℓ119896))


119910 ((V119894 V119895) 119886119899 (ℓ119896))

=



(12)


119879 (119886119899 (ℓ119896)) = 120575119894119905sdot 120591119894119905(119886119899 (ℓ119896)) + 120575

V119905sdot 120591

V119905(119886119899 (ℓ119896))

+ 120575119903119905

sdot 120591119903119905(119886119899 (ℓ119896)) + 120575

119905119905sdot 120591119905119905(119886119899 (ℓ119896))

(13)










Lang

fang

Beiji

ng S

outh

Shenyang North

Tian

jin S

outh

Cang

zhou

Wes

t1

Dez

hou

East

Jinan

Wes

t

Taia

n

Zaoz

huan

g

Xuzh

ou E

ast

Suzh

ou E

ast

Beng

bu S

outh

Chuz

hou

Sout

h

Nan

jing

Sout

h

Zhen

jiang

Wes

t

Chan

gzho

u N

orth

Wux

i Eas

t

Suzh

ou N

orth

Shan

ghai

Hon

gqia

o

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Qingdao

Jinan

Shijiazhuang Hefei

Wuhan

Chengdu

19 20

21

22

23

24

25

26

27

28

29Zhengzhou East


Hangzhou East








sumℓ119896isinL

sumV119894(ℓ119896)isin119881

(120581 (ℓ119896) sdot 119891 (ℓ119896) minus 119884 ((V119894 V119894+1) ℓ119896))

sdot ℎ (V119894 V119894+1) = 119872lowast

(14)









(1 13) (1 18) (3 18) (6 13) (6 18) (9 18)(13 18)

(1 20) (1 26) (1 29) (3 20) (3 27) (3 29)(6 19) (6 27) (6 29) (9 20) (9 26) (9 29)(13 20) (13 23) (13 25) (13 29) (18 19)(18 20) (18 23) (18 25) (18 24) (18 28)(18 27) (18 26) (20 19) (20 27) (20 25)

(29 25) (29 24)

(19 1) (20 21) (23 6) (25 9) (24 9) (28 13)(27 13) (26 13) (28 27) (28 26)



In-vehicle time 120591119894119905 (119886119899(ℓ119896))


classification)




Wait time 120591119903119905(119886119899 (ℓ119896))



119899(ℓ119896)) 30 minutes








In-line trains

1 13 1 1-9-13

2 1-2-4-8-9-10-11-12-13

1 18

1 1-182 1-3-183 1-3-6-13-184 1-3-5-6-7-13-185 1-3-6-11-13-16-17-186 1-2-3-5-7-8-9-16-187 1-2-3-4-5-9-15-17-188 1-2-3-6-9-10-11-13-189 1-2-3-4-9-10-12-14-18

10 1-3-6-9-11-13-15-17-18

3 18

1 3-182 3-6-183 3-6-9-184 3-5-6-7-8-15-17-18

5 3-4-5-6-7-8-12-17-18

6 13 1 6-13

2 6-7-9-10-11-12-13

6 18

1 6-9-182 6-13-183 6-7-8-9-14-16-17-184 6-10-11-12-13-16-17-18

9 18 1 9-13-182 9-11-13-14-17-18

13 18 1 13-182 13-14-15-16-17-18

Cross-line trains

1 20 1 1-3-20

2 1-2-3-4-5-20

1 261 1-3-6-9-262 1-3-5-6-8-11-26

3 1-3-4-5-6-7-9-10-11-26

1 29 1 1-3-6-9-13-29

3 20 1 3-5-20

3 27 1 3-9-13-26-27

2 3-4-7-10-12-27

3 29 1 3-5-6-10-11-13-18-29

19 6 1 19-1-5-6

6 27 1 6-8-26-27

Table 4 Continued


6 29 1 6-7-9-29

2 6-8-13-18-29

20 9 1 20-6-9

9 26 1 9-11-26

9 29 1 9-11-13-17-29

20 13 1 20-6-13

23 13 1 23-22-6-7-8-10-11-13

25 13 1 25-9-13

13 29 1 13-18-29

2 13-14-15-16-17-18-29

19 181 19-1-3-6-9-13-182 19-2-7-8-9-10-11-13-17-18

3 19-2-4-5-9-13-14-15-16-18

20 18 1 20-6-14-15-16-17-18

23 18 1 23-22-9-14-15-16-17-18

25 18 1 25-9-12-13-182 25-9-10-11-13-14-15-16-17-18

24 18 1 24-9-13-182 24-9-10-11-12-14-15-16-17-18

28 18 1 28-13-17-18

27 18 1 27-13-15-16-17-18

26 18 1 26-13-182 26-13-15-16-17-18

19 20 1 19-3-4-5-20

20 27 1 20-9-10-11-27

20 25 1 20-6-8-25

25 29 1 25-9-12-13-18-29

24 29 1 24-9-13-18-29







5 Conclusions




Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Lang

fang

Beiji

ng S

outh

Shenyang North

Tian

jin S

outh

Cang

zhou

Wes

t1

Dez

hou

East

Jinan

Wes

t

Taia

n

Zaoz

huan

g

Xuzh

ou E

ast

Suzh

ou E

ast

Beng

bu S

outh

Chuz

hou

Sout

h

Nan

jing

Sout

h

Zhen

jiang

Wes

t

Chan

gzho

u N

orth

Wux

i Eas

t

Suzh

ou N

orth

Shan

ghai

Hon

gqia

o

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

Qingdao

Jinan

Shijiazhuang Hefei

Wuhan

Chengdu

19 20

21

22

23

24

25

26

27

28

29Zhengzhou East


Hangzhou East








sumℓ119896isinL

sumV119894(ℓ119896)isin119881

(120581 (ℓ119896) sdot 119891 (ℓ119896) minus 119884 ((V119894 V119894+1) ℓ119896))

sdot ℎ (V119894 V119894+1) = 119872lowast

(14)









(1 13) (1 18) (3 18) (6 13) (6 18) (9 18)(13 18)

(1 20) (1 26) (1 29) (3 20) (3 27) (3 29)(6 19) (6 27) (6 29) (9 20) (9 26) (9 29)(13 20) (13 23) (13 25) (13 29) (18 19)(18 20) (18 23) (18 25) (18 24) (18 28)(18 27) (18 26) (20 19) (20 27) (20 25)

(29 25) (29 24)

(19 1) (20 21) (23 6) (25 9) (24 9) (28 13)(27 13) (26 13) (28 27) (28 26)



In-vehicle time 120591119894119905 (119886119899(ℓ119896))


classification)




Wait time 120591119903119905(119886119899 (ℓ119896))



119899(ℓ119896)) 30 minutes








In-line trains

1 13 1 1-9-13

2 1-2-4-8-9-10-11-12-13

1 18

1 1-182 1-3-183 1-3-6-13-184 1-3-5-6-7-13-185 1-3-6-11-13-16-17-186 1-2-3-5-7-8-9-16-187 1-2-3-4-5-9-15-17-188 1-2-3-6-9-10-11-13-189 1-2-3-4-9-10-12-14-18

10 1-3-6-9-11-13-15-17-18

3 18

1 3-182 3-6-183 3-6-9-184 3-5-6-7-8-15-17-18

5 3-4-5-6-7-8-12-17-18

6 13 1 6-13

2 6-7-9-10-11-12-13

6 18

1 6-9-182 6-13-183 6-7-8-9-14-16-17-184 6-10-11-12-13-16-17-18

9 18 1 9-13-182 9-11-13-14-17-18

13 18 1 13-182 13-14-15-16-17-18

Cross-line trains

1 20 1 1-3-20

2 1-2-3-4-5-20

1 261 1-3-6-9-262 1-3-5-6-8-11-26

3 1-3-4-5-6-7-9-10-11-26

1 29 1 1-3-6-9-13-29

3 20 1 3-5-20

3 27 1 3-9-13-26-27

2 3-4-7-10-12-27

3 29 1 3-5-6-10-11-13-18-29

19 6 1 19-1-5-6

6 27 1 6-8-26-27

Table 4 Continued


6 29 1 6-7-9-29

2 6-8-13-18-29

20 9 1 20-6-9

9 26 1 9-11-26

9 29 1 9-11-13-17-29

20 13 1 20-6-13

23 13 1 23-22-6-7-8-10-11-13

25 13 1 25-9-13

13 29 1 13-18-29

2 13-14-15-16-17-18-29

19 181 19-1-3-6-9-13-182 19-2-7-8-9-10-11-13-17-18

3 19-2-4-5-9-13-14-15-16-18

20 18 1 20-6-14-15-16-17-18

23 18 1 23-22-9-14-15-16-17-18

25 18 1 25-9-12-13-182 25-9-10-11-13-14-15-16-17-18

24 18 1 24-9-13-182 24-9-10-11-12-14-15-16-17-18

28 18 1 28-13-17-18

27 18 1 27-13-15-16-17-18

26 18 1 26-13-182 26-13-15-16-17-18

19 20 1 19-3-4-5-20

20 27 1 20-9-10-11-27

20 25 1 20-6-8-25

25 29 1 25-9-12-13-18-29

24 29 1 24-9-13-18-29







5 Conclusions




Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra














(1 13) (1 18) (3 18) (6 13) (6 18) (9 18)(13 18)

(1 20) (1 26) (1 29) (3 20) (3 27) (3 29)(6 19) (6 27) (6 29) (9 20) (9 26) (9 29)(13 20) (13 23) (13 25) (13 29) (18 19)(18 20) (18 23) (18 25) (18 24) (18 28)(18 27) (18 26) (20 19) (20 27) (20 25)

(29 25) (29 24)

(19 1) (20 21) (23 6) (25 9) (24 9) (28 13)(27 13) (26 13) (28 27) (28 26)



In-vehicle time 120591119894119905 (119886119899(ℓ119896))


classification)




Wait time 120591119903119905(119886119899 (ℓ119896))



119899(ℓ119896)) 30 minutes








In-line trains

1 13 1 1-9-13

2 1-2-4-8-9-10-11-12-13

1 18

1 1-182 1-3-183 1-3-6-13-184 1-3-5-6-7-13-185 1-3-6-11-13-16-17-186 1-2-3-5-7-8-9-16-187 1-2-3-4-5-9-15-17-188 1-2-3-6-9-10-11-13-189 1-2-3-4-9-10-12-14-18

10 1-3-6-9-11-13-15-17-18

3 18

1 3-182 3-6-183 3-6-9-184 3-5-6-7-8-15-17-18

5 3-4-5-6-7-8-12-17-18

6 13 1 6-13

2 6-7-9-10-11-12-13

6 18

1 6-9-182 6-13-183 6-7-8-9-14-16-17-184 6-10-11-12-13-16-17-18

9 18 1 9-13-182 9-11-13-14-17-18

13 18 1 13-182 13-14-15-16-17-18

Cross-line trains

1 20 1 1-3-20

2 1-2-3-4-5-20

1 261 1-3-6-9-262 1-3-5-6-8-11-26

3 1-3-4-5-6-7-9-10-11-26

1 29 1 1-3-6-9-13-29

3 20 1 3-5-20

3 27 1 3-9-13-26-27

2 3-4-7-10-12-27

3 29 1 3-5-6-10-11-13-18-29

19 6 1 19-1-5-6

6 27 1 6-8-26-27

Table 4 Continued


6 29 1 6-7-9-29

2 6-8-13-18-29

20 9 1 20-6-9

9 26 1 9-11-26

9 29 1 9-11-13-17-29

20 13 1 20-6-13

23 13 1 23-22-6-7-8-10-11-13

25 13 1 25-9-13

13 29 1 13-18-29

2 13-14-15-16-17-18-29

19 181 19-1-3-6-9-13-182 19-2-7-8-9-10-11-13-17-18

3 19-2-4-5-9-13-14-15-16-18

20 18 1 20-6-14-15-16-17-18

23 18 1 23-22-9-14-15-16-17-18

25 18 1 25-9-12-13-182 25-9-10-11-13-14-15-16-17-18

24 18 1 24-9-13-182 24-9-10-11-12-14-15-16-17-18

28 18 1 28-13-17-18

27 18 1 27-13-15-16-17-18

26 18 1 26-13-182 26-13-15-16-17-18

19 20 1 19-3-4-5-20

20 27 1 20-9-10-11-27

20 25 1 20-6-8-25

25 29 1 25-9-12-13-18-29

24 29 1 24-9-13-18-29







5 Conclusions




Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra












In-line trains

1 13 1 1-9-13

2 1-2-4-8-9-10-11-12-13

1 18

1 1-182 1-3-183 1-3-6-13-184 1-3-5-6-7-13-185 1-3-6-11-13-16-17-186 1-2-3-5-7-8-9-16-187 1-2-3-4-5-9-15-17-188 1-2-3-6-9-10-11-13-189 1-2-3-4-9-10-12-14-18

10 1-3-6-9-11-13-15-17-18

3 18

1 3-182 3-6-183 3-6-9-184 3-5-6-7-8-15-17-18

5 3-4-5-6-7-8-12-17-18

6 13 1 6-13

2 6-7-9-10-11-12-13

6 18

1 6-9-182 6-13-183 6-7-8-9-14-16-17-184 6-10-11-12-13-16-17-18

9 18 1 9-13-182 9-11-13-14-17-18

13 18 1 13-182 13-14-15-16-17-18

Cross-line trains

1 20 1 1-3-20

2 1-2-3-4-5-20

1 261 1-3-6-9-262 1-3-5-6-8-11-26

3 1-3-4-5-6-7-9-10-11-26

1 29 1 1-3-6-9-13-29

3 20 1 3-5-20

3 27 1 3-9-13-26-27

2 3-4-7-10-12-27

3 29 1 3-5-6-10-11-13-18-29

19 6 1 19-1-5-6

6 27 1 6-8-26-27

Table 4 Continued


6 29 1 6-7-9-29

2 6-8-13-18-29

20 9 1 20-6-9

9 26 1 9-11-26

9 29 1 9-11-13-17-29

20 13 1 20-6-13

23 13 1 23-22-6-7-8-10-11-13

25 13 1 25-9-13

13 29 1 13-18-29

2 13-14-15-16-17-18-29

19 181 19-1-3-6-9-13-182 19-2-7-8-9-10-11-13-17-18

3 19-2-4-5-9-13-14-15-16-18

20 18 1 20-6-14-15-16-17-18

23 18 1 23-22-9-14-15-16-17-18

25 18 1 25-9-12-13-182 25-9-10-11-13-14-15-16-17-18

24 18 1 24-9-13-182 24-9-10-11-12-14-15-16-17-18

28 18 1 28-13-17-18

27 18 1 27-13-15-16-17-18

26 18 1 26-13-182 26-13-15-16-17-18

19 20 1 19-3-4-5-20

20 27 1 20-9-10-11-27

20 25 1 20-6-8-25

25 29 1 25-9-12-13-18-29

24 29 1 24-9-13-18-29







5 Conclusions




Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra













5 Conclusions




Acknowledgments


References



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article operational impacts of using restricted

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