research article a branch and bound algorithm for the

Research ArticleA Branch and Bound Algorithm for the Exact Solution ofthe Problem of EMU Circulation Scheduling in Railway Network

Chao Lu Lei-shan Zhou Yi-xiang Yue and Ran Chen

School of Traffic and Transportation Beijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Lei-shan Zhou lshzhoubjtueducn

Received 19 June 2015 Revised 20 November 2015 Accepted 17 December 2015

Academic Editor Francisco Chicano

Copyright copy 2016 Chao Lu et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with the scheduling of Electrical Multiple Units (EMUs) under the condition of their utilization on onesector or within several interacting sectors Based on the introduction of the train connection graph which describes the possibleconnection relationship between trains the integer programming model of EMU circulation planning is constructed In order toanalyzing the resolution of the model a heuristic which shares the characteristics with the existingmethods is introduced firstThismethod consists of two stages one is a greedy strategy to construct a feasible circulation plan fragment and another is to apply astochastic disturbance to it to generate a whole feasible solution or get a new feasible solution Then an exact branch and boundmethod which is based on graph designing is proposed Due to the complexity the lower bound is computed through a polynomialapproximation algorithm which is a modification from the one solving the degree constraint minimum 1-tree problem Then abranching strategy is designed to cope with the maintenance constraints Finally we report extensive computational results on arailway corridor in which the sectors possess the basic feature of railway networks

1 Introduction

The assignment of transportation tools to fulfilling a group oftasks under certain conditions is one of the most importantscheduling activities in many real world applications suchas the Aircraft Routing Problem (ARP) in airline operationVehicle Routing Problem (VRP) in logistics delivery and theLocomotives Scheduling Problem (LSP) in railway systemAs the mobile resources the quality of the utilization of thetransportation tools has a great effect on the efficiency andthe economic benefits of the transportation enterprises Theoperation of these resources also provides a foundation forthe conduction of other works and has an interaction withthem

In high-speed railway network Electrical Multiple Units(EMU) are the main tools used for the passenger transporta-tion They are significantly cost equipment because not onlyis their acquisition or construction expensive but also theyneed power supply and regular maintenance The schedulingof EMU falls in the category of rolling stock circulation prob-lems and the operation plan of them determines the concrete

activities they need to perform when the system is runningIn order not to result in the waste of the precious resourcesthey need to be operated as reasonably as possible

The EMU operation plan in each country differs insmall points In China the EMU operation plan defines thearranging of the train trips and maintenance tasks of eachlevel to each EMU within a certain period of time accordingto the given train timetable the configuration of EMU thecondition of EMU maintenance facilities and the rules ofthe operation of the stations It is a comprehensive planand could be generally divided into three parts namely theEMU circulation plan the EMU allocation plan and EMUmaintenance plan Among these plans the EMU circulationplan mainly determines the connection relationship of thetrain trips and arranges the first level maintenance for EMUsIt is a basis for the other two plans

The optimization of the rolling stock circulation schedul-ing problem has received a lot of attention from researchersSome realistic problems such as the unit coupling or uncou-pling when scheduling the EMU are also considered Cac-chiani et al [1] designed an effective heuristic procedure

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 8537089 14 pageshttpdxdoiorg10115520168537089

2 Mathematical Problems in Engineering

for the train unit assignment problem that were able to findsolutions significantly better than the ldquomanualrdquo solutionsfound by practitioners Cacchiani et al [2] proposed aheuristic which is based on the natural Lagrangian relaxationof a natural integer linear programming of the train unitassignment problem which turns out to be much fasterin practice and still providing solutions of good qualityAbbink et al [3] deal with the tactical problem of findingthe most effective allocation of the train types subtypesand units of rolling stock to the train series such that asmany people as possible can be transported with a seatespecially during the rush hours Lin and Kwan [4] proposeda two-phase approach for the train unit scheduling problemthe first phase assigns and sequences train trips to trainunits considering some real-world scenarios and the secondphase focuses on satisfying the remaining station detailedrequirements Peeters and Kroon [5] focused on the efficientcirculation of train units within a certain scope of railwayline given the timetable and the passengersrsquo seat demandand a branch-and-price algorithm is described Alfieri etal [6] present a solution approach based on an integermulticommodity flow model to determine the appropriatenumbers of train units of different types together with theirefficient circulation on a single line Cacchiani et al [7]present two integer linear programming (ILP) formulationstogether with their relaxations (the linear programming (LP)relaxation and Lagrangian based approach resp) to assignthe train units to the trips with minimum cost When thehigh-speed railway system is running many disturbancescan occur and that would lead to the irregularities of theoperations of the train units In order to avoid such events tosome extent it is reasonable to find a plan that is insensitiveto these disruptions that is the plan is able to cope withrelatively small disruptionswithout structural changes whichis also known as the robustness scheduling or to reactimmediately to those disruptions by applying a recoverystrategy to the plan defined previously which is also knownas the rescheduling process (see eg [8ndash10]) Cadarso andMarın [11] formulate a multicommodity flow model forthe rolling stock problem in rapid transit networks Emptymovements and shunting operations are considered and therobustness is introduced by selectively avoiding empty trainmovements and these operations Cadarso and Marın [12]presents a model to study the robust determining of the bestsequence for each rolling stock in the train network Themethod is based on an approach in which sequences aredesigned once the timetable and the rolling stock assignmenthave been done

As for the particular EMU circulation plan in China it ismainly based on a condition of the utilization of the EMUon one railway sector or within several interacting sectorswhich is generally viewed as a more reasonable mode forthe EMU operation (see eg Zhao et al [13]) Althoughthe scheduling of the EMU shares something in commonwith the scheduling of other kinds of transportation toolsespecially the scheduling of locomotive in ordinary railway ithas its own characteristics Locomotives are often scheduledon a fixed sector and could be viewed as an assignment prob-lem which could be solved by Hungarian algorithm Ahuja

et al [14] formulated the locomotive-scheduling problem asa multicommodity flow problem with side constraints on aweekly space-time network Each locomotive type defines acommodity in the network However the EMUs are sched-uled within several linked sectors and some complicatedconstraints such as the EMU maintenance constraints mustbe taken into account therefore more applicable methodsshould be developed for solving the construction of theEMU circulation plan Zhao and Tomii [15] transformed theoriginal problem into the Traveling Salesperson Problem andintroduced a probability based local search algorithm whosekey points are about the connection of the trains and thegeneration of the maintenance arcs On the basis of thatMiao et al [16] transformed the original problem into amultiple Traveling Salesperson Problem with replenishmentand designed a hierarchical optimization heuristic algorithmShi et al [17] designed a simulated annealing algorithm byintroducing the penalty function and 3-opt neighborhoodstructure which is based on the circular permutation ofall trains Li et al [18] introduces the optimized EMUconnection graph based on which the improved particleswarm optimization algorithm is designed for solving theproblem

Through the analysis of the studies concerning EMUcirculation plan it can be seen that the problem couldbe transformed into some classic optimization problemsand due to the complexity of these problems most of theexisting solution generation methods belong to the rangeof probability based searching heuristics The motivation ofthis paper is to propose an exact algorithm for solving theEMU circulation scheduling problem in high-speed railwaynetwork The algorithm is based on the graph theory andcould be able to deal with the problems of practical sizewithina reasonable time Furthermore we propose a heuristicwhich shares the characteristics with the existing methodsand is based on local search strategy We also make acomparison between these methods

The paper is organized as follows In Section 2 wedescribe the details of the EMU circulation schedulingproblem we study In Section 3 we introduce the conceptof the train connection graph based on which the model ofEMU circulation scheduling in high-speed railway networkis constructed In Section 4 we design the neighbor structureof the solution of the problem and the local search methodis illustrated In Section 5 the exact branch and boundalgorithm for solving the problem is outlined and the detailsof the algorithm such as the calculation of the approximatelower bound and the branching strategy is illustrated In Sec-tion 6 we report the computational results of the comparisonof the proposed methods in a test case

2 Problem Description

In high-speed railway network EMUs are utilized on onerailway sector or within several interacting sectors seeFigure 1There are a set of train trips of up and downdirectionin each sector The EMUs circulation scheduling problemneeds to assign all the tasks of the train trips to a set of EMUs

Mathematical Problems in Engineering 3

A B CSector 1

Sector 2

Sector 3

Figure 1 Utilization of EMU within several interacting sectors

In each sector the train trips of the up direction and the traintrips of the down direction are connected at the endpointstations by certain EMUs At the stations which are linkedwith the EMU maintenance base the EMUs need to havemaintenance if either the accumulative running distance orrunning time reaches the upper bound which is provided inthe first level maintenance document of the EMU

Given a train timetable the circulation plan could beshown as the format in Figure 2 During a period of one daywe use the term circulation plan fragment to indicate a seriesof tasks that an EMU should fulfill which includes severaltrain trips and train connections at stations In Figure 2 acirculation plan fragment is represented by a polygonal linethat links the tasks within one day Each two train trips couldbe connected at the station if the duration between the arrivaltime of one train trip and the departure time of another traintrip is not less than the provided minimum duration time Ifthe EMU needs to have maintenance then the duration ofthe connection should also exceed the minimum durationof the maintenance For example if an EMU conducts thetasks which are provided in fragment A of the circulationplan in the first day it starts from station A and finally arrivesat station B after all the tasks have been conducted it stay atstation B in the night In the second day it starts from stationB and conducts the tasks which are provided in fragmentBof the circulation plan After it finishes serving as train 3102it has maintenance at station A Finally after conducting allthe tasks it returns to station A and stays there in the nightIn the following days it should conduct the tasks which areprovided in another fragment of the circulation plan

The whole circulation plan is formed by connecting allthe fragments through the overnight connection When thefirst EMU conducts the fragment A of the circulation planother EMUs conduct the fragments B C D of thecirculation plan according to the same rules respectively thenthe number of the utilized EMU is known The maintenancein the circulation is of the basic level and the extensivemaintenance which needs much longer time therefore theyare not considered in the circulation plan

It is worth mentioning that one of the objectives pursuedin the scheduling process of the rolling stock circulationwould consider maximization of service to the passengers byminimizing the seat shortages However under the Chineserailway organization mode the types of trains are viewedas the same and the number of travelers of each sector isdealt with when working out the train operation schemewhich provides the train service frequency Therefore theconsideration of the number of travelers when solving theEMU circulation scheduling problem is out of the scope ofthis paper

3 Model of the EMU Circulation Planning

31 Train Connection Graph Given a timetable of severalinteracting railway sectors the corresponding EMU connec-tion graph could be constructed correspondingly which isa useful tool to describe the essentials of the problem ofEMUcirculation schedulingThe notationswith regard to theproblem are listed in Notations section

The parameter 120593119894119895= 1 if and only if there could exist

a connection relationship between trains 119894 and 119895 under thecondition of 119911

119895119889= 119911119894119886and 119888119894119895ge 120576 119888119894119895is calculated as follows

119888 (119894 119895) =

120587119895119889minus 120587119894119886 120587

119895119889minus 120587119894119886ge 120576

120587119895119889minus 120587119894119886+ 119875 120587

119895119889minus 120587119894119886lt 120576

(1)

Parameter 120583119894119895= 1 if and only if 119888

119894119895ge and 119911

119895119889isin119885

Figure 3 shows the connection graph of a certain groupof routes in which we only display the arcs from the up-bound trains to the down-bound trains Each row of nodesrepresents the set of trains that belong to the same routeThe solid arcs are the connection arc whose duration time iswithin one period while the dotted arcs are the connectionarc in which the departing train and the arriving train isdistributed in two periods

32 Model Formulation Weuse the binary decision variables119909119894119895and 119910119894119895to indicate whether the arc (119894 119895) in the connection

graph is selected in the circulation plan and used as themaintenance arc respectively that is if and only if train 119894 isscheduled to be connected by train 119895 in the circulation planthen 119909

119894119895= 1 119910

119894119895= 1 if and only if the arc (119894 119895) is used as the

maintenance arc in the circulation plan Then the model ofEMU circulation planning could be formulated as follows

min 119885 = 1205961sum

119894isin119881

sum

119895isin119881

119888 (119894 119895) 119909119894119895+ 1205962sum

119894isin119881

sum

119895isin119881

119910119894119895 (2)

st sum119894isin119881

119909119894119895= 1 119895 isin 119881 (3)

sum

119895isin119881

119909119894119895= 1 119894 isin 119881 (4)

119909119894119895le 120593119894119895 119894 119895 isin 119881 (5)

sum

119894isin119889(119890)sdotsdotsdot119886(1198901015840)

sum

119895isin119881

119909119894119895119888119894119895+ sum


119905119894le 119871

119890 119890

1015840isin 119882

(6)

sum


sum

119895isin119881

119909119894119895119889119895le 119872 119890 119890

1015840isin 119882 (7)

119910119894119895le 120583119894119895119909119894119895 119894 119895 isin 119881 (8)

119873(119894) + 119873 (119895) = |119881| 119894 119895 isin 119881 (9)

sum

119894isin119881

sum

119895isin119881

119910119894119895ge 1 (10)


Fragment ➀ of the circulation plan

Station A

Station B

Station C

12011301

1303

1305

1307

1309

1311

3102

3104

3106

3108

2102

3112

2104

3116

1203

Sector 1

Sector 2

Overnightconnection

Minimum duration of the connection at stationMinimum duration of the maintenance at station

Time

Space

Connection of two train trips

Fragment ➁ of the circulation plan

Figure 2 Fragments of the circulation plan

Up-bound trains ofroute 1

trains of route n


Down-bound

Down-bound

trains of route n

middot middot middot




Figure 3 The train connection graph

Objective function (2) consists of two terms the firstterm is the total connection time which is closely related tothe number of EMUs needed and this is the main objectiveof this problem the second term is the total times of themaintenance Constraints (3) and (4) impose that each trainshould be used as the starting node and the ending nodeof only one arc in 119866 that is each node of train trip shouldbe visited only once in the circulation plan Constraints (5)impose that the connection between trains should be rea-sonable Constraints (6) and (7) impose that the cumulativerunning time and running distances of an EMU between twomaintenance arcs must not exceed the upper bound of theprescriptive distance and time where 119882 = (119894 119895) 119910

119894119895=

1 is the set of the maintenance arcs The set 119889(119890) sdot sdot sdot 119886(1198901015840)contains all the trains between maintenance arcs 119890 and 1198901015840Constraints (8) impose that an arc could be selected as themaintenance arc if and only if that arc is used as a connectionarc and is able to provide the maintenance Constraints (9)avoid the occurrence of the subloop Given a certain arc setconnecting all the nodes in 119866 (see Figure 4) when arbitrarily

deleting two arcs (represented by the hidden line) then withthe starting or ending node of the deleted arc as the root nodeand searching all the connected nodes using the adjacencymatrix of the existing arcs the number of the nodes that canbe found is denoted as119873(119894) which is linked by the dotted linein Figure 4 Obviously (a) is subjected to constraints (9) while(b) is not subjected to constraints (9) Constraints (10) avoidthe existence of none maintenance loop for the circulationplan

4 Heuristic

41 Generation of the Initial Solution Through the analysis ofthe constraints of the model the format of the solution couldbe written as 119904 = 119890

1 1198902 119890

119896 which satisfies constraints

(3)ndash(10) and the condition of 119886(119890119894) = 119889(119890

119895) for each pair

of adjacent arcs 119890119894 119890119895in 119904 The construction of the solution

starts from the first selected arc which could be any one of 119864and then keeps finding arcs and adds them into the solutionAssuming the current selected arc is 119890

119894 then the basic rules

used to select the next arc 119890119895are

119911119886(119890119894)119886= 119911119889(119890119895)119889

sum

119898isin119881119898 =119886(119890119894)

119909119898119889(119890

119895)= 0

120593119894119895= 1

(11)

The set of nodes that satisfy condition (11) is representedas 119881119889(119890119894) In the process of selecting the connection arcs

the maintenance tasks should be considered simultaneouslySince the existence of constraint (10) then the first arc 119890

1must

be any 119890119898isin 119864 that satisfies 120583

119898= 1 Let 119897[119886(119890

119894)] and 119901[119886(119890

119894)]

be the current accumulative running distance and runningtime when the EMU arrives at the station 119911

119886(119890119894)119886 whose value


i

i1

j

N(j)

N(i)

i2

i3

j1

j2

j3

j4

j5

(a)

i

i1

jN(i)

i2

j1

j2

N(j)

(b)

Figure 4 Illustration of the notation119873(119894)

could be computed using (12) where is the last maintenancearc before 119890

119894 and 1015840 is the next arc after

119897 [119886 (119890119894)] =

119889119886()+ sum

119898isin1015840119890119894

119889119886(119898) 119910119894= 0

119889119886(119890119894) 119910

119894= 1

119901 [119886 (119890119894)] =

119905119886()+ sum

119898isin1015840119890119894

(119888119898+ 119905119886(119898)) 119910

119894= 0

119905119886(119890119894) 119910

119894= 1

(12)

If an EMU arrives at station 119911119886(119890119894)119886

after taking on train119886(119890119894) of current arc 119890

119894in 119904 and the relationship of 119889(119890

119894) = 119886(119890

1)

holds (otherwise the solution is finished) then due to theexistence of the constraints (6) and (7) the next arc to beselected should also to make it be subjected to 119897[119886(119890

119895)] le 119871

119901[119886(119890119895)] le 119872 Therefore 119890

119895is selected as follows

119888 (119886 (119890119894) 119889 (119890

119895)) = min119898isin119881119889(119890119894)

119888119886(119890119894)119898 119897 [119886 (119898)]

le 119871 119901 [119886 (119898)] le 119872 120583119886(119890)119894119898= 0

(13)

We define the set of arcs that satisfies the conditiondefined in (13) as the first arc set If the first arc set doesnot contain any elements then the EMU needs maintenanceand the alternative arc to be selected is a random one that issubjected to the following constraint

min119898isin119881119889(119890119894)

119888119886(119890119894)119898 120583119886(119890119894)119898= 1 le 119888 (119886 (119890

119894) 119889 (119890

119895))

le max119898isin119881119889(119890119894)

119888119886(119890119894)119898 120583119886(119890119894)119898= 1

(14)

We then define the set of arcs that satisfy the conditionmentioned in (14) as the second arc set When such an arcis found then continue to implement the above procedureuntil meeting the node of 119886(119890

1) If the second arc set dose not

contain any elements either then the next arc subjected to allthe constraints does not exist in the process of constructingsolution 119904 As a result it is necessary to improve the structureof the already found part of the solution by means of thestrategy which will be introduced in next section

42 Local Improvement In order to change the structureof a feasible solution or part of an unfinished solution adisturbance should be brought into them In case that asolution needs to continue to be constructed which is a basiccondition in the generation of different feasible solutions let 120585be the already existing feasible part of solution 119904119864

119898120585is the set

of the maintenance arcs of 120585 for a randomly selected element isin 119864119898120585

change the property 119889() as a random node V whichsatisfies the following conditions

sum

119890119898isin120585

119909119889(119890119898)V = 0

119911119889()119889

= 119911V119889

120583119886(119890119894)V = 1

(15)

The elements not belonging to the fragment of 120585 aredeleted from the temporary solution 119904 Then set the newlyfound arc to be the current arc and continue to conduct thesolution construction procedure to finish 119904

In case that a feasible solution 119904 needs to be improvedto get another one the basic idea is to randomly combinethemaintenance arcs with connection arcs and between eachpair of adjacent maintenance arcs the set of connection arcsdo not violate all the constraints

Firstly the component of 120585 derived from 119904 should bedetermined which would be kept temporarily in the nextgenerated solution Since 119904 is a feasible solution then anypart of it is also feasible Therefore in order to ensure therandomness of the new solution 120585 is formed through therandom selection of two maintenance arcs 1198901015840 and from 119904and has the format of 120585 = 1198901015840 isin 119904 which satisfies thecondition of 119886(119890

119894) = 119889(119890

119895) for any adjacent arcs 119890

119894 119890119895in 120585

Note that 1198901015840 could not necessarily be 1198901 Secondly change the

property 119889() as the one satisfying the conditions describedby (15) Then set the newly found arc to be the current arcand continue to conduct the solution construction procedureto finish 119904

When a feasible solution is found function (2) will beused to evaluate the quality of the solutionThewhole processof the heuristic used to find the optimal solution of theproblem is shown in Figure 5


Yes

No

No

Construct the secondedge set

Is the second Yes

max value

Stop

best solutionUpdate the current best

solution

No

Yes

Yes

NoNo

Yes

Start

first edge set

Is the firstedge set Oslash

Compute the value f(s) of the

Is f(s) better than the

Determine ej from the second

determine e from 120585 and changed(e)

Determine e from the120585 of s change

d(e) and update s

Initialization set the current

edge ei = e1 s = Oslash and k = 0

Determine ej from the firstedge set

d(ej) = a(e1) edge set

Update the current edge ei = e

Set k = k + 1 construct 120585 of s

Update the current edge ei ej=

Update s = s ⋃ ei construct the

edge set Oslash

completed s

Does k reach the

Figure 5 The flowchart of the heuristic

5 An Exact Branch and Bound Algorithm

51 Computing the Lower Bound Given a graph with 119899 +1 nodes a 1-tree is defined to be a set of 119899 + 1 arcs thatspan the graph (see eg Fisher [19]) The Degree ConstraintMinimum 1-Tree (DCM1-T) is a 1-tree such that the totalweight of the set of the arcs is minimum and that the degreeof each node does not exceed the limitation of the degreeconstraint

It can be seen from the analysis of the model that underthe condition of the utilization of the EMU within severalinteracting sectors and given the connection graph119866 findingthe optimal circulation plan which satisfies all constraintsis a Degree Constraint Minimum 1-tree Designing ProblemSince DCM1-T is NP-complete (see eg Alexandre andAbilio [20]) there are only methods to find a satisfactoryoptimal solutions In order to solve the problem we drawon the experience of a polynomial approximation algorithm

which is introduced in Subhash and Cesar [21] HoweverSubhash and Cesar [21] focused on the undirected graphwhereas according to Figure 3 the multisector train con-nection graph is a directed incomplete graph besides theconstraints of the degree of a node include both the part ofin-degree constraints and the part of out-degree constraintsObviously the maximum value of the in-degree and the out-degree of each node is 1 All these characteristics about thetrain connection graph are defined asNCDC rulesThereforethe algorithm used in this paper is an improved edition of theoriginal one which could not be applied directly

(1) The Construction of the Basic Connection Tree If none ofthe arcs associated with a node has been selected in 119866 thenwe call the node an idle nodeThe construction process startswith an initial set of nonidle nodes 119862 which could includearbitrary number of nodes For example we could select acertain arc of overnight connection in119866 at the beginning and


then119862 contains two elements namely the departure train tripand the arrival train trip of the arc Let 119880 be the initial set ofidle nodesThe construction process constitutes continuouslyselecting arcs which connect the nodes in 119862 with the nodesin 119880 After an arc is selected the associated node in 119880 istransformed into the set 119862

For each node 119894 isin 119880 its in-degree 119889minus119894and the out-degree

119889

+

119894are initialized to be 0 Let V119903(119894) denote the node in 119862 such

that there is an arc from 119894 to it and that

119888 (119894 V119903 (119894)) = min119895isin119862

119888 (119894 119895) (16)

Let V119905(119894) denote the node in 119862 such that there is an arcfrom it to node 119894 and that

119888 (V119905 (119894) 119894) = min119895isin119862

119888 (119895 119894) (17)

Let 119890lowast119894denote the arc whose cost satisfies the condition

of 119888119890lowast

119894

= min119894isin119880119888(119894 V119903(119894)) 119888(V119905(119894) 119894) If 119890lowast

119894has the form

(119894

lowast V119903(119894lowast)) then the out-degree of node 119894lowast is updated as 119889+

119894lowast =

119889

+

119894lowast + 1 and the in-degree of the node V119903(119894lowast) is updated as119889

minus

V119903(119894lowast) = 119889minus

V119903(119894lowast) + 1 If 119889+

119894lowast gt 1 or 119889minusV119903(119894lowast) gt 1 then the arc 119890lowast

119894

could not be selected and the node V119903(119894) is modified as thenode V119903(119894)1015840 such that

119888 (119894 V119903 (119894)1015840) = min119895isin119862119889

minus

119895lt1

119888 (119894 119895) (18)

Similarly If 119890lowast119894has the form (V119905(119894lowast) 119894lowast) then the out-

degree of the node V119905(119894lowast) is updated as 119889+V119905(119894lowast) = 119889+

V119905(119894lowast) + 1and the in-degree of the node 119894lowast is updated as 119889minus

119894lowast = 119889

minus

119894lowast + 1

If 119889+V119905(119894lowast) gt 1 or 119889minus

119894lowast gt 1 then the arc 119890lowast

119894could not be selected

either and the node V119905(119894) is modified as the node V119905(119894)1015840 suchthat

119888 (V119905 (119894)1015840 119894) = min119895isin119862119889

+

119895lt1

119888 (119895 119894) (19)

The arc 119890lowast119894is redetermined according to the new property

of the 119888(119894 V119903(119894)1015840) and 119888(V119905(119894)1015840 119894) After selecting the arc 119890lowast119894 the

set of119862 is updated as119862 = 119862cup119894lowast and the node 119894lowast is removedfrom the set119880 simultaneously In addition as the componentsof both set 119862 and set 119880 are changed therefore V119903(119894) and V119905(119894)for each 119894 isin 119880 need to update again specifically if 119888(119894 V119903(119894)) lt119888(119894 119894

lowast) set V119903(119894) = 119894lowast and if 119888(V119905(119894) 119894) lt 119888(119894lowast 119894) set V119905(119894) = 119894lowast

If all the nodes are checkedwith the procedurementionedbefore then the basic connection tree (BCT) is gotten How-ever due to the incompleteness of the train connection graphthere are still elements in the set 119880 and the supplementarymeasures need to be applied to continue the construction ofthe DCM1-T

(2) Finishing the Construction of 1-Tree If set 119880 is stillnot empty after the BCT is constructed then it means nonodes could be found in the set of 119880 under the NCDCrules Therefore the remaining nodes in 119880 should first beincorporated into the BCT and the BCT is transformed intoan ordinary tree then some exchanging method should beapplied to the ordinary tree and transform it into a degreeconstraint minimum 1-tree which contains all the nodes in119866

In order to transform as many nodes as possible from set119880 into set 119862 repeat the procedure introduced before and donot consider the in-degree or out-degree constraints of thenodes During this process for forall119894 isin 119880 the correspondingV119903(119894) or V119905(119894) should not belong to the initial119862 which is the setof the nodes before any other node being added into itWhenall the remaining elements in 119880 are transformed to the set 119862the structure of BCT could not be subjected to constraints (3)and (4)

Under the NCDC rules if an arc connecting nodes 119894 and119895 could be selected in the DCM1-T then we denote it as 119894Š 119895Notation [119894 119895] represents the possibility of the arc connectingnodes 119894 and 119895 which could be the form of (119894 119895) or (119895 119894) Wethen introduce two methods for exchanging the structure ofa tree

The dual exchanging method for the ordinary tree (exm-1) let 119878119879(119894) denote the full subtree of BCT rooted at the node119894 119878119881(119894) denote the set of trains which are connected with thenode 119894 of the BCT through the already selected arcsThen thearcs could be exchanged as follows

[119898 119899] 997904rArr [119896 119903 (119896) or 119905 (119896)] forall119889

+

119896isin119862 119889

minus

119896isin119862gt 1 (20)

In (20) [119898 119899] is the arc which is to be added into the BCTand [119896 119903(119896) or 119905(119896)] is the arc which is to be removed from theBCT 119903(119896) or 119905(119896) is some node which is connected with node119896 through an arc and they are determined according to thefollowing conditions

119888 [119896 119903 (119896) or 119905 (119896)] = min119894isin119878119881119896

119888 [119896 119894]

119898 isin 119878119879 (119896)

119899 isin 119878119879 (119903 (119896)) cup 119878119879 (119905 (119896))

119898Š 119899

(21)

The second exchanging method for the 1-tree (exm-2)after applying exm-1 for the BCT and adding the edge [119894 119895]to it where 119894 and 119895 are the nodes whose in-degree and out-degree are not equal then the structure of the modified BCTbecomes 1-tree The exchange is as follows

[119894 119898] + [119894 119895] + [119895 119899] 997904rArr [119898 119899] forall119894 119895 isin 119880 119894Š 119895 (22)

In (22) the arc [119898 119899] is the edge which is to be deletedfrom the BCT and the other three edges are to be addedinto the BCT These edges could be determined through thefollowing condition

119888 [119894 119898] + 119888 [119894 119895] + 119888 [119895 119899] minus 119888 [119898 119899]

= min[ℎ119896]isinBCT

119888 [119894 ℎ] + 119888 [119894 119895] + 119888 [119895 119896] minus 119888 [ℎ 119896]

(23)

Although the exm-1 could be replaced by the exm-2 thelatter is a more conservative method to increase the total costof the BCT when changing its structure to subject it to theconstraints (3) and (4) after a majority of elements in 119880 aretransformed into itTherefore when the initial BCT is gottenthe exm-1 is applied first and if still the119880 = then the exm-2 should be appliedThe whole process of the algorithm used


Start

Yes

Yes

Yes

No

No

Yes

No

No

Yes

No

Yes

No

Initialize the set of C U andr(i) t(i) for each i isin U

Determine the edge elowast

Is exm-1 applied

Apply the exm-1 method Stop

Apply the exm-2 method

Is m gt |V| minus 1

Set arrival node s = t(ilowast)

departure node e = ilowast

Add arc (s e) update C U

Is er(i) lt e(i ilowast) oret(i) lt e(ilowast i)

d+s gt 1 or dminuse gt 1

m = |U|

Set arrival node s = ilowast anddeparture node e = r(ilowast)

Is the edge elowast one

Update the out-degree ofd+s = d+s + 1 dminuse = dminuse + 1

Update nodes r(i) and t(i)and lengths er(i) and et(i)

Does the current Ucontain any elements

et(i) = e(ilowast i) t = i lowast andm = m + 1

Set er(i) = e(i ilowast) r = ilowast or

of er(i r(i))and

Figure 6 Flowchart of the computation of DCM1-T

to find the DCM1-T of the train connection graph is shownin Figure 6

Although the DCM1-T does not consider the main-tenance when connecting the trains it aims at finding acirculation of EMU with the minimum total cost Thereforeit provides an approximate lower bound on the optimal cir-culation plan which considers the maintenance constraintswhile the cost of any feasible circulation plan provides anupper bound on it In order to compute the optimal feasiblecirculation plan we design a branch and bound algorithmwhich could improve the structure of the DCM1-T to make itsubject to the maintenance constraints and increase the totalcost of the DCM1-T as little as possible

52 The Branching Strategy Since the DCM1-T might violateconstraints (6) and (7) then we need first to fix part of theDCM-T in the circulation plan and build some branchesto respectively determine some values of the maintenancevariables then the DCM1-T are calculated for each branchand new parts of the DCM1-T are fixed Repeat this processuntil all the parts of the DCM1-T are feasible with respect tothe constraints (6) and (7) After each branch is calculatedthe best feasible circulation plan could be obtained simulta-neously

Let 119864119873 denote the set of arcs that are included whenrecomputing the DCM1-T Let 119864119883 denote the set of arcsthat are excluded when recomputing the DCM1-T At the


beginning there is only one element in 119864119873 which is thebasic maintenance arc in DCM1-T while the 119864119883 = Ateach level of branch if the objective value of the DCM1-T computed based on the subset (119864119873 119864119883) is smaller thanthe current upper bound then the circulation fragment ℓ =(119890119896 119890119896+1 119890

119896+119899) is fixed which satisfies that 119889(119890

119894) = 119886(119890

119894+1)

for each pair of adjacent arcs 119890119894 119890119894+1isin ℓ in the DCM1-Twhere

119890119896means the adjacent arc of the last arc in 119864119873 119896 = |119864119873| + 1

and 119899 is an integer used to limit the number of the arcs whichwould violate the maintenance constraints Then the set ofmaintenance arcs 119864119872 = 120596

1(119894) 1205962(119894) 120596

120582(119894) is constructed

in which the arrival train of each arc is 119889(119890119896+119899) and 119894 is the

parameter indicating the level of the branching process 120582means the number of the elements of the set 119864119872 and if thenumber of arcs in 119866 that could be selected into the set 119864119872is bigger than 120582 then the actual elements held in 119864119872 are thefirst 120582 lower cost arcs otherwise 119864119872 contains all the arcsBased on 119864119872 and 119864119873 the list of the new subsets could beconstructed as follows

119864119873119895(119894)= 1198641198731198951015840(119894minus1)cup ℓ cup 120596

119895(119894)

119864119883119895(119894)= 1198641198831198951015840(119894minus1)

cup 1205961(119894) 120596

119895minus1(119894) 120596119895+1(119894) 120596

120582(119894)

(24)

For a subset (119864119873119895(119894) 119864119883119895(119894)) which is also the branch 119895

at level 119894 a DCM1-T which includes the arcs in 119864119873119895(119894)

andexcludes the arcs in 119864119883

119895(119894)is calculated and its objective

value is obtained as a lower bound In the whole branchingprocess we apply strategy like the one mentioned in [21]always branch to the left subnode and at each node keepa ldquoleftrdquo bound as the current lower bound of the subset(119864119873119895(119894) 119864119883119895(119894)) and a ldquorightrdquo bound as the lower bound of

the subset (119864119873119895minus1(119894) 119864119883119895minus1(119894)) as shown in Figure 7

At beginning let the time of iteration 119889 = 0 119891(119862119894)

represent the value of the objective function for a given EMUcirculation 119862

119894 119877(119894) and 119871(119894) are the right and the left bounds

of a subset at level 119894 of the branching tree respectively Thenthe algorithm could be summarized as follows

Step 1 Generate an EMU circulation plan 119879119862 which con-siders the maintenance using an arbitrary heuristic set theupper bound 119885 = 119891(119879119862) Initialize set 119864119873

1(0)and set 119864119883

1(0)

Implement the lower bound calculation algorithm to generatethe circulation plan119862

0 which is a DCM1-T Set the left bound

119871(119889) = 119891(1198620) the right bound 119871(119889) = infin and 119889 = 119889 + 1

Step 2 Determine the subset (119864119873120582(119889) 119864119883120582(119889)) based on

which we implement the lower bound calculation algorithmto generate the circulation plan 119862

120582(119889) Set the left bound

119871(119889) = 119891(119862120582(119889)) 119898(119889) = 120582(119889) minus 1 Determine the subset

(119864119873119898(119889) 119864119883119898(119889)) Implement the lower bound calculation

algorithm to generate the circulation plan 119862119898(119889)

set the rightbound 119877(119889) = 119891(119862

119898(119889))

Step 3 If 119871(119889) lt 119885 go to Step 4 otherwise go to Step 5

Step 4 If the circulation plan corresponding to the left bound119871(119889) does not violate the maintenance constraints then a

1(0)

1(2)

1(1)

Lb(2)

Rb(2)

hellip

120582(2)

120582(1)

120582 minus 1(2)

120582 minus 1(1) middot middot middot


1200011 ⋃ 120596120582(1)

1200012 ⋃ 120596120582(2) 1200012 ⋃ 120596120582minus1(2)1200012 ⋃ 1205961(2)

1200011 ⋃ 1205961(1)1200011 ⋃ 120596120582minus1(1)

Figure 7 The branching tree

feasible solution is found Update the upper bound Set 119885 =119871(119889) and go to Step 5 otherwise set 119889 = 119889 + 1 and go toStep 2

Step 5 If119877(119889) lt 119885 then update the left bound and set 119871(119889) =119877(119889) go to Step 6 otherwise go directly to Step 6

Step 6 Set 119898(119889) = 119898(119889) minus 1 if 119898(119889) = 0 go toStep 7 Otherwise if 119898(119889) gt 1 determine the subset(119864119873119898(119889) 119864119883119898(119889)) implement the lower bound calculation


and updatethe right bound 119877(119889) = 119891(119862

119898(119889)) Otherwise set 119877(119889) = infin

When the 119877(119889) is determined if 119877(119889) ge 119885 repeat Step 6Otherwise go to Step 3

Step 7 Go back to the previous level set 119889 = 119889 minus 1 if 119889 gt 1go to Step 5 otherwise stop

It is worth mentioning that the solution made of thecombination of the branch constructed with the lowest costarc in 119864119872 of each level under the condition that thesebranches are able to formalize a feasible solution could beviewed as the result of another greedy strategy namely theusually mentioned Tight Connection which is also a way toquickly generate a feasible EMU circulation plan

53 Improvement of the Heuristic During the generationof a feasible solution introduced in Section 4 due to theincompleteness of the train connection graph there is aninteractive process between the selecting of arcs to constructthe solution and the exchanging of the arcs for the partof the solution which has been already obtained in orderto complete the generation of a feasible solution A lot ofdetection procedures for the arc exchangingmay appear untila whole feasible solution is found in the interactive processtherefore it accounts for the majority of the running time ofthe algorithm

However according to the illustration of Section 42 let120585 = 119890

1015840 isin 119904 be the randomly selected part of either

the unfinished solution or a feasible solution 119904when applyingthe local improvement after property 119889() is changed and thenewly found arc is set to be the current arc if the branchstrategy with a limitation on the levels of the branches is


incorporated into the remaining process of the constructionof the feasible solution then a more reliable way could beobtained to generate the feasible solution faster and with aprobably better objective value

6 Numerical Experiments

Without loss of generality we test our method on the set oftrains on a corridor in which we use the Jing-Hu railwayline between Beijing and Shanghai as reference to settingthe parameters when working out the EMU circulation planFigure 8 shows which segments a sector contains Thisexample has general features in common with other high-speed railway networks that consist of several different trainsectors and these sectors interact with each other

Table 1 shows the data of the train operation scheme usedin the following experiments referring to the corridor119879

1(1198792)

is the set of high-speed trains (medium-speed trains) in eachsector and is twice as much as the trains of each directionof the sector 119889 is the length of each sector The travel timeof each train is different and could be obtained from thetimetable

Stations BJN QFD BBN and SHHQ are able to fulfill thetask of first grade maintenance Other basic parameters areas follows 119871 = 2880min 119872 = 4000 km 120576 = 15min [18]and = 120min [16] Given a train timetable which providesthe condition for EMUs operated within these sectors thenumber of arcs in the train connection graph is 10228The circulation plan computed using the branch and boundalgorithm is as shown in Figure 9

In order to show further computational results we testthe method on other instances of the problem in whichthe timetable of the trains on the corridor is randomlygenerated and other parameters remain the same Figures 10ndash13 compare the convergence curves generated by the heuristicand the branch and bound algorithm on four differentinstances of the problem And for the heuristic we run it twiceand get two curves The modified heuristic is also tested

It can be seen from these figures that the time neededby the branch and bound algorithm to generate an optimalsolution is 2min or so in the test case whose number oftrains corresponds to the scale of amajority of practical casesand the time could be viewed as acceptable At the sametime the convergence time needed by the heuristic reflectsa fluctuation to some extent In addition either the value ofthe total connection cost of the heuristic is bigger than that ofthe branch and bound or the convergence time of the formeralgorithm is longer than that of the latter algorithmHoweveras the heuristic is modified the corresponding solutionsappear to be better to some extent and the convergence timeis shorter compared with the initial heuristic

We also compare among the heuristic branch and boundand tight connectionmethods mentioned in this paper basedon 20 different instances of the problem The results arelisted in Table 2 The second column is the total connectiontime of the initial DCM1-T which is a circulation plan notconsidering the maintenance and columns 2ndash8 are the totalconnection cost (TCC) and the times ofmaintenance (MT) of

BJN

TJN

DZD

TA TZhD

XZhD

BBN

ChZh

ZhJN

ChZh

B

SZhB

ShH

Q

LF

CZhX JN

X

QFD ZZ

h

SZhD D

Y

NJN

DYB

WXD

KShN

Sector 1

Sector 3Sector 2

Figure 8 Train routes on the test railway corridor

Table 1 Train operation scheme of the test corridor

Sector From To 1003816100381610038161003816

1198791

1003816100381610038161003816

1003816100381610038161003816

1198792

1003816100381610038161003816

119889 (km)1 BJN SHHQ 32 times 2 10 times 2 16802 BBN SHHQ 19 times 2 6 times 2 6003 BJN QFD 19 times 2 6 times 2 695

the three algorithms Among them the TTC of the heuristicis the value of the best solutions generated by the heuristic

There are a few observations to be made from this tableFirst there is not a clear proportional relation between thetotal connection cost and the maintenance times that is themaintenance times of a solutionwith a larger total connectioncost might not be more than the maintenance times of asolution with a smaller total connection cost This is becauseunder a given structure of the train timetable the moretimes of EMUs staying at a station at night the larger theaccumulative running time of them after a maintenancewhile the accumulative running distance may not reach thelimitation Second almost all solutions (associated with thenotation ldquolowastrdquo) computed by the branch and bound methodreach the lower bound of the problem which is the totalcost of the initial DCM1-T not considering the maintenanceconstraints except for two solutions but the total connectioncost of these two solutions is very close to their lower boundand smaller than that of the solutions computed by theother two methods too Furthermore there are even somesolutions (associated with the notation ldquolowastlowastrdquo) computed bythe branch and bound method whose total connection costis smaller than the one of the initial DCM1-T This is becausethe outputs obtained by the lower bound algorithm and thecirculation plans obtained by branch and bound all belong todegree constraint 1-tree Since the determining of the DCM1-T is NP-hard and the DCM1-T calculating algorithm couldonly find an approximate solution whose objective valueonly provides an approximate lower bound on the optimalcirculation plan as mentioned before therefore through thecontinuous applying of the DCM1-T calculating algorithmin the process of the branch and bound there would existthe probability of finding a degree constraint 1-tree whoseobjective is better than the one found by directly applyingthe DCM1-T calculating algorithm once Third it can beinferred from the table that for the same case even a slightadjustment of the train timetable may produce completelydifferent optimal EMU circulation plans with different totalconnection costsTherefore the proposed branch and bound


G1001G10

02G10

03G10

04G10

05G10

06G10

07G10

08G10

09G10

10G10

11G10

12G10

13G10

14G10

15G10

16G10

17G10

18G10

19G10

20G10

21G10

22G10

23G10

24G10

25G10

26G10

27G10

28G10

29G10

30G10

31G10

32D1033

D1034D1035

D1036D1037

D1038D1039

D1040D1041

D1042G10

43G10

44G10

45G10

46G10

47G10

48G10

49G10

50G10

51G10

52G10

53G10

54G10

55G10

56G10

57G10

58G10

59G10

60G10

61D1062

D1063D1064

D1065D1066

D1067

G1068

G1069

G1070

G1071

G1072

G1073

G1074

G1075

G1076

G1077

G1078

G1079

G1080

G1081

G1082

G1083

G1084

G1085

G1086

D1087D1088

D1089D1090

D1091D1092

G1093G1094

G1095G1096

G1097G1098

G1099G1100

G1101G1102

G1103G1104

G1105G1106

G1107G1108

G1109G1110

G1111G1112

G1113G1114

G1115G1116

G1117G1118

G1119G1120

G1121G1122

G1123G1124

D1125D1126

D1127D1128

D1129D1130

D1131D1132

D1133D1134

G1135G1136

G1137G1138

G1139G1140

G1141G1142

G1143G1144

G1145G1146

G1147G1148

G1149G1150

G1151G1152

G1153D1154

D1155D1156

D1157D1158

D1159

G1160G1161

G1162G1163

G1164G1165

G1166G1167

G1168G1169

G1170G1171

G1172G1173

G1174G1175

G1176G1177

G1178D1179

D1180D1181

D1182D1183

D1184

G1086

G1015

G1107G1108

D1184

G1122

D1041

G1151

D1091G1030G1031

D1131G1121

G1059

G1153

G1080

D1128

G1060

G1020

G1110

D1067

G1124

G1177

G1061

G1148G1123D1132G1116G1117D1130

D1183

G1081

G1147

D1040

G1149G1115

G1176D1181

G1029

G1178

G1026

D1133

D1037G1082G1032G1019

G1152

G1084G1024

D1158

G1017

G1150

D1066

G1025G1085

D1129

D1038

D1157

G1056

G1028G1027G1023G1083

G1119

D1039D1090

D1127G1120G1118

G1174

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0

BJN

LF

TJN

CZHX

DZD

JNX

TA

QFD

TZHDZZH

XZHD

SZHD

BBN

DY

CHZH

NJN

ZHJNDYB

CHZHB

WXDSZHBKSHN

SHHQ

Figure 9 Train diagram and the circulation plan

0 50 100 150 200 250 300 350 400Time (s)

55

6

65

7

75

8

9

85

Tota

l con

nect

ing

time

times104

HeuristicHeuristicBampB

Modified heuristicModified heuristic

Figure 10 Convergence curves of the algorithms tested on instance2

method could be used for the problem of the optimization ofnetwork train timetable which is incorporatedwith the EMUcirculation scheduling problem Finally by comparison themean value of the total connection costs of the solutionsgenerated by the tight connection method is (705215 minus655061)655061 = 77 more than the one of the solutionsgenerated by the heuristic and (705215minus590021)590021 =195 more than the one of the solutions generated by thebranch and bound method while mean value of the total

0 50 100 150 200 250 300 350 40055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104




connection costs of the solutions generated by the heuristicis (655061minus590021)590021 = 110more than the one ofthe solutions generated by the branch and bound method

7 Conclusion

Based on the introduction of the train connection graphthe of EMU circulation scheduling model is constructedThrough the analyzing of the features of the structure of the


Table 2 The results of the algorithms

Instance DCM1-T Tight connection Heuristic BampBTCC MT TCC MT TCC MT

1 58667 71627 55 64427 54 58667lowast 542 59843 69923 55 64163 54 58403lowastlowast 533 59807 71327 54 64127 54 58367lowastlowast 574 59855 71375 56 64175 54 59855lowast 565 60100 73060 54 64420 55 58660lowastlowast 556 60144 71664 55 64464 55 60144lowast 547 62756 74272 59 68516 55 62756lowast 548 57503 66143 56 64703 54 57503lowast 569 57133 67213 53 64333 54 58573 5710 58464 72864 55 65664 53 58464lowast 5411 58607 75887 57 64367 54 58607lowast 5412 59784 74184 58 66984 55 59784lowast 5613 61292 68492 56 67052 56 61292lowast 5314 56192 72032 55 63392 55 57632 5315 57347 67427 55 64547 54 57347lowast 5316 57869 67949 56 65069 55 57869lowast 5417 57345 65985 54 64545 54 57345lowast 5518 58618 67250 55 65818 55 58618lowast 5519 60079 70159 57 64399 55 60079lowast 5420 61517 71597 56 67277 55 60077lowastlowast 53

0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104




feasible EMU circulation plan this paper mainly designs anexact branch and bound algorithm for solving the problem

We first put the initial problem down to a graph designingproblem that is the degree constraint minimum 1-treeproblem Due to the complexity of the problem we use apolynomial approximation algorithm to compute the DCM1-T which does not consider the maintenance constraints andprovides an approximately lower bound of the problem Thisalgorithm is modified in order to adjust to the characteristics

0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104




of the train connection graph Then a branch strategy isproposed to cope with the maintenance constraints and togenerate an optimal circulation plan

Through the analysis of the computational results of thenumerical experiments of the comparison of the proposedbranch and bound method with the heuristics which sharesthe features with the existing methods it can be seen that therunning time needed by branch and bound method is morereasonable when dealing with the instance of the problem


with the scale of most practical cases the objective value ofthe solutions generated by the branch and bound methodis better than the one of the solutions generated by otherheuristics and the branch and bound method could beextended to other problems concerning the optimization ofthe train timetable

Notations

119866 The EMU connection graph119881 Set of nodes of the connection graph means all

train trips involved in the planning timehorizon

119894 119895 Indexes of node 119894 119895 isin 119881119911119894119889 119911119894119886 the first station the train 119894 departs from and the

final station the train arrives at along its routerespectively

120587119894119889 120587119894119886 the departure time of the train 119894 at the station119911119894119889

and the arrival time of the train 119894 at thestation 119911

119894119886

119889119894 119905119894 distance between station 119911

119894119889to station 119911

119894119886and

time consumed of the train 119894 running fromstation 119911

119894119889to station 119911

119894119886 respectively

119864 set of arcs of the connection graph means allthe potential connection relationship betweentrain trips

120593119894119895 Binary parameter indicating whether there

exist an arc from train trip 119894 to 119895119890 119890

1015840 (119894 119895) Indexes of the arc 119890 1198901015840 (119894 119895) isin 119864

119888(119894 119895) Connection cost of the arc (119894 119895) which ismeasured as the duration time of the arc

120576 Minimum duration time necessary for EMU toconnect trains at stations

120583119894119895 Binary parameter indicating whether the arc

(119894 119895) could be used as an maintenance arc Minimum duration time of the maintenance

for EMU at stations119885 Set of stations which are able to fulfill the

maintenance task119871 119879 Upper bounds of accumulative running

distance and time for a EMU under the firstlevel maintenance respectively

119886(119890) 119889(119890) Arrival train and departure train the of the arc119890 respectively

1205961 1205962 Weights of the terms in the objective function

119875 the time of the period of one day

Conflict of Interests

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

Acknowledgment

This work was financially supported by The FundamentalResearch Funds for the Central Universities (no2014JBZ008)

References

[1] V Cacchiani A Caprara and P Toth ldquoSolving a real-worldtrain-unit assignment problemrdquo Mathematical Programmingvol 124 no 1-2 pp 207ndash231 2010

[2] V Cacchiani A Caprara and P Toth ldquoA Lagrangian heuristicfor a train-unit assignment problemrdquo Discrete Applied Mathe-matics vol 161 no 12 pp 1707ndash1718 2013

[3] E Abbink B Van den Berg L Kroon andM Salomon ldquoAlloca-tion of railway rolling stock for passenger trainsrdquoTransportationScience vol 38 no 1 pp 33ndash41 2004

[4] Z Lin and R S K Kwan ldquoA two-phase approach for real-worldtrain unit schedulingrdquo Public Transport vol 6 no 1-2 pp 35ndash65 2014

[5] M Peeters and L Kroon ldquoCirculation of railway rolling stocka branch-and-price approachrdquo Erim Report vol 35 no 2 pp538ndash556 2008

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

[7] V Cacchiani A Caprara and P Toth ldquoModels and algorithmsfor the train unit assignment problemrdquo in Combinatorial Opti-mization vol 7422 of Lecture Notes in Computer Science pp 24ndash35 Springer Berlin Germany 2012

[8] V Cacchiani A Caprara L Galli L Kroon G Maroti and PToth ldquoRailway rolling stock planning robustness against largedisruptionsrdquo Transportation Science vol 46 no 2 pp 217ndash2322012

[9] L K Nielsen L Kroon and G Maroti ldquoA rolling horizonapproach for disruption management of railway rolling stockrdquoEuropean Journal of Operational Research vol 220 no 2 pp496ndash509 2012

[10] B Gabriella G Maroti R Dekker D Huisman and L KroonldquoRe-scheduling in railways the rolling stock balancing prob-lemrdquo Research Papers Econometric Institute 2007

[11] L Cadarso and A Marın ldquoRobust rolling stock in rapid transitnetworksrdquo Computers amp Operations Research vol 38 no 8 pp1131ndash1142 2011

[12] L Cadarso and A Marın ldquoRobust routing of rapid transitrolling stockrdquo Public Transport vol 2 no 1 pp 51ndash68 2010

[13] P Zhao H Yang and A Z Hu ldquoResearch on usage of highspeed passenger trains on uncertain railroad regionrdquo Journal ofthe China Railway Society vol 19 no 2 pp 15ndash19 1997

[14] R K Ahuja J Liu J B Orlin D Sharma and L AShughart ldquoSolving real-life locomotive-scheduling problemsrdquoTransportation Science vol 39 no 4 pp 503ndash517 2005

[15] P Zhao and N Tomii ldquoAn algorithm for train-set schedul-ing on weekday based on probabilistic local searchrdquo SystemsEngineeringmdashTheory amp Practice vol 24 no 2 pp 123ndash1282004

[16] J-R Miao Y Wang and Z-X Yang ldquoResearch on the opti-mization of EMU circulation based on optimized connectingnetworkrdquo Journal of the China Railway Society vol 32 no 2 pp1ndash7 2010

[17] F Shi W-L Zhou Y-W Yu and L Qing ldquoOptimized modeland algorithm of motor train-sets scheduling for dedicatedpassenger linesrdquo Journal of the China Railway Society vol 33no 1 pp 8ndash13 2011

[18] H Li B-M Han Q Zhang and R Guo ldquoResearch onoptimization model and algorithm of EMU circulation planrdquoJournal of the China Railway Society vol 35 no 3 pp 1ndash8 2013


[19] M L Fisher ldquoA polynomial algorithm for the degree-constrained minimum K-tree problemrdquo Operations Researchvol 42 no 4 pp 775ndash779 1994

[20] S D C Alexandre and L Abilio ldquoAlgorithms for the degree-constrainedminimum spanning tree problemrdquo Electronic Notesin Discrete Mathematices vol 19 pp 403ndash409 2005

[21] C N Subhash and A H Cesar ldquoDegree-constrained minimumspanning treerdquo Computers amp Operations Research vol 33 pp239ndash249 1980

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


for the train unit assignment problem that were able to findsolutions significantly better than the ldquomanualrdquo solutionsfound by practitioners Cacchiani et al [2] proposed aheuristic which is based on the natural Lagrangian relaxationof a natural integer linear programming of the train unitassignment problem which turns out to be much fasterin practice and still providing solutions of good qualityAbbink et al [3] deal with the tactical problem of findingthe most effective allocation of the train types subtypesand units of rolling stock to the train series such that asmany people as possible can be transported with a seatespecially during the rush hours Lin and Kwan [4] proposeda two-phase approach for the train unit scheduling problemthe first phase assigns and sequences train trips to trainunits considering some real-world scenarios and the secondphase focuses on satisfying the remaining station detailedrequirements Peeters and Kroon [5] focused on the efficientcirculation of train units within a certain scope of railwayline given the timetable and the passengersrsquo seat demandand a branch-and-price algorithm is described Alfieri etal [6] present a solution approach based on an integermulticommodity flow model to determine the appropriatenumbers of train units of different types together with theirefficient circulation on a single line Cacchiani et al [7]present two integer linear programming (ILP) formulationstogether with their relaxations (the linear programming (LP)relaxation and Lagrangian based approach resp) to assignthe train units to the trips with minimum cost When thehigh-speed railway system is running many disturbancescan occur and that would lead to the irregularities of theoperations of the train units In order to avoid such events tosome extent it is reasonable to find a plan that is insensitiveto these disruptions that is the plan is able to cope withrelatively small disruptionswithout structural changes whichis also known as the robustness scheduling or to reactimmediately to those disruptions by applying a recoverystrategy to the plan defined previously which is also knownas the rescheduling process (see eg [8ndash10]) Cadarso andMarın [11] formulate a multicommodity flow model forthe rolling stock problem in rapid transit networks Emptymovements and shunting operations are considered and therobustness is introduced by selectively avoiding empty trainmovements and these operations Cadarso and Marın [12]presents a model to study the robust determining of the bestsequence for each rolling stock in the train network Themethod is based on an approach in which sequences aredesigned once the timetable and the rolling stock assignmenthave been done

As for the particular EMU circulation plan in China it ismainly based on a condition of the utilization of the EMUon one railway sector or within several interacting sectorswhich is generally viewed as a more reasonable mode forthe EMU operation (see eg Zhao et al [13]) Althoughthe scheduling of the EMU shares something in commonwith the scheduling of other kinds of transportation toolsespecially the scheduling of locomotive in ordinary railway ithas its own characteristics Locomotives are often scheduledon a fixed sector and could be viewed as an assignment prob-lem which could be solved by Hungarian algorithm Ahuja

et al [14] formulated the locomotive-scheduling problem asa multicommodity flow problem with side constraints on aweekly space-time network Each locomotive type defines acommodity in the network However the EMUs are sched-uled within several linked sectors and some complicatedconstraints such as the EMU maintenance constraints mustbe taken into account therefore more applicable methodsshould be developed for solving the construction of theEMU circulation plan Zhao and Tomii [15] transformed theoriginal problem into the Traveling Salesperson Problem andintroduced a probability based local search algorithm whosekey points are about the connection of the trains and thegeneration of the maintenance arcs On the basis of thatMiao et al [16] transformed the original problem into amultiple Traveling Salesperson Problem with replenishmentand designed a hierarchical optimization heuristic algorithmShi et al [17] designed a simulated annealing algorithm byintroducing the penalty function and 3-opt neighborhoodstructure which is based on the circular permutation ofall trains Li et al [18] introduces the optimized EMUconnection graph based on which the improved particleswarm optimization algorithm is designed for solving theproblem

Through the analysis of the studies concerning EMUcirculation plan it can be seen that the problem couldbe transformed into some classic optimization problemsand due to the complexity of these problems most of theexisting solution generation methods belong to the rangeof probability based searching heuristics The motivation ofthis paper is to propose an exact algorithm for solving theEMU circulation scheduling problem in high-speed railwaynetwork The algorithm is based on the graph theory andcould be able to deal with the problems of practical sizewithina reasonable time Furthermore we propose a heuristicwhich shares the characteristics with the existing methodsand is based on local search strategy We also make acomparison between these methods

The paper is organized as follows In Section 2 wedescribe the details of the EMU circulation schedulingproblem we study In Section 3 we introduce the conceptof the train connection graph based on which the model ofEMU circulation scheduling in high-speed railway networkis constructed In Section 4 we design the neighbor structureof the solution of the problem and the local search methodis illustrated In Section 5 the exact branch and boundalgorithm for solving the problem is outlined and the detailsof the algorithm such as the calculation of the approximatelower bound and the branching strategy is illustrated In Sec-tion 6 we report the computational results of the comparisonof the proposed methods in a test case

2 Problem Description

In high-speed railway network EMUs are utilized on onerailway sector or within several interacting sectors seeFigure 1There are a set of train trips of up and downdirectionin each sector The EMUs circulation scheduling problemneeds to assign all the tasks of the train trips to a set of EMUs


A B CSector 1

Sector 2

Sector 3











119888 (119894 119895) =

120587119895119889minus 120587119894119886 120587

119895119889minus 120587119894119886ge 120576

120587119895119889minus 120587119894119886+ 119875 120587

119895119889minus 120587119894119886lt 120576

(1)


119894119895ge and 119911

119895119889isin119885




119894119895= 1 119910



min 119885 = 1205961sum

119894isin119881

sum

119895isin119881

119888 (119894 119895) 119909119894119895+ 1205962sum

119894isin119881

sum

119895isin119881

119910119894119895 (2)


119909119894119895= 1 119895 isin 119881 (3)

sum

119895isin119881

119909119894119895= 1 119894 isin 119881 (4)

119909119894119895le 120593119894119895 119894 119895 isin 119881 (5)

sum


sum

119895isin119881

119909119894119895119888119894119895+ sum


119905119894le 119871

119890 119890

1015840isin 119882

(6)

sum


sum

119895isin119881

119909119894119895119889119895le 119872 119890 119890

1015840isin 119882 (7)

119910119894119895le 120583119894119895119909119894119895 119894 119895 isin 119881 (8)

119873(119894) + 119873 (119895) = |119881| 119894 119895 isin 119881 (9)

sum

119894isin119881

sum

119895isin119881

119910119894119895ge 1 (10)



Station A

Station B

Station C

12011301

1303

1305

1307

1309

1311

3102

3104

3106

3108

2102

3112

2104

3116

1203

Sector 1

Sector 2

Overnightconnection


Time

Space





trains of route n


Down-bound

Down-bound

trains of route n







119894119895=



4 Heuristic


1 1198902 119890








119911119886(119890119894)119886= 119911119889(119890119895)119889

sum

119898isin119881119898 =119886(119890119894)

119909119898119889(119890

119895)= 0

120593119894119895= 1

(11)



1must


119898= 1 Let 119897[119886(119890

119894)] and 119901[119886(119890

119894)]


119886(119890119894)119886 whose value


i

i1

j

N(j)

N(i)

i2

i3

j1

j2

j3

j4

j5

(a)

i

i1

jN(i)

i2

j1

j2

N(j)

(b)




119897 [119886 (119890119894)] =

119889119886()+ sum

119898isin1015840119890119894

119889119886(119898) 119910119894= 0

119889119886(119890119894) 119910

119894= 1

119901 [119886 (119890119894)] =

119905119886()+ sum

119898isin1015840119890119894

(119888119898+ 119905119886(119898)) 119910

119894= 0

119905119886(119890119894) 119910

119894= 1

(12)




119894) = 119886(119890

1)


119895)] le 119871

119901[119886(119890119895)] le 119872 Therefore 119890


119888 (119886 (119890119894) 119889 (119890

119895)) = min119898isin119881119889(119890119894)

119888119886(119890119894)119898 119897 [119886 (119898)]

le 119871 119901 [119886 (119898)] le 119872 120583119886(119890)119894119898= 0

(13)


min119898isin119881119889(119890119894)

119888119886(119890119894)119898 120583119886(119890119894)119898= 1 le 119888 (119886 (119890

119894) 119889 (119890

119895))

le max119898isin119881119889(119890119894)

119888119886(119890119894)119898 120583119886(119890119894)119898= 1

(14)





119898120585is the set



sum

119890119898isin120585

119909119889(119890119898)V = 0

119911119889()119889

= 119911V119889

120583119886(119890119894)V = 1

(15)




119894) = 119889(119890


119894 119890119895in 120585





Yes

No

No


Is the second Yes

max value

Stop


solution

No

Yes

Yes

NoNo

Yes

Start

first edge set







d(e) and update s









edge set Oslash

completed s

Does k reach the










119889

+



119888 (119894 V119903 (119894)) = min119895isin119862

119888 (119894 119895) (16)


119888 (V119905 (119894) 119894) = min119895isin119862

119888 (119895 119894) (17)


of 119888119890lowast

119894


119894has the form

(119894


119894lowast =

119889

+


minus

V119903(119894lowast) = 119889minus

V119903(119894lowast) + 1 If 119889+


119894


119888 (119894 V119903 (119894)1015840) = min119895isin119862119889

minus

119895lt1

119888 (119894 119895) (18)




119894lowast = 119889

minus

119894lowast + 1





119888 (V119905 (119894)1015840 119894) = min119895isin119862119889

+

119895lt1

119888 (119895 119894) (19)










[119898 119899] 997904rArr [119896 119903 (119896) or 119905 (119896)] forall119889

+

119896isin119862 119889

minus

119896isin119862gt 1 (20)


119888 [119896 119903 (119896) or 119905 (119896)] = min119894isin119878119881119896

119888 [119896 119894]

119898 isin 119878119879 (119896)

119899 isin 119878119879 (119903 (119896)) cup 119878119879 (119905 (119896))

119898Š 119899

(21)




119888 [119894 119898] + 119888 [119894 119895] + 119888 [119895 119899] minus 119888 [119898 119899]


119888 [119894 ℎ] + 119888 [119894 119895] + 119888 [119895 119896] minus 119888 [ℎ 119896]

(23)



Start

Yes

Yes

Yes

No

No

Yes

No

No

Yes

No

Yes

No



Is exm-1 applied



Is m gt |V| minus 1






m = |U|








of er(i r(i))and









119894) = 119886(119890

119894+1)




1(119894) 1205962(119894) 120596




119864119873119895(119894)= 1198641198731198951015840(119894minus1)cup ℓ cup 120596

119895(119894)

119864119883119895(119894)= 1198641198831198951015840(119894minus1)

cup 1205961(119894) 120596

119895minus1(119894) 120596119895+1(119894) 120596

120582(119894)

(24)












1(0)and set 119864119883

1(0)











119898(119889))



1(0)

1(2)

1(1)

Lb(2)

Rb(2)

hellip

120582(2)

120582(1)

120582 minus 1(2)



1200011 ⋃ 120596120582(1)

1200012 ⋃ 120596120582(2) 1200012 ⋃ 120596120582minus1(2)1200012 ⋃ 1205961(2)

1200011 ⋃ 1205961(1)1200011 ⋃ 120596120582minus1(1)




















1(1198792)






BJN

TJN

DZD

TA TZhD

XZhD

BBN

ChZh

ZhJN

ChZh

B

SZhB

ShH

Q

LF

CZhX JN

X

QFD ZZ

h

SZhD D

Y

NJN

DYB

WXD

KShN

Sector 1

Sector 3Sector 2



Sector From To 1003816100381610038161003816

1198791

1003816100381610038161003816

1003816100381610038161003816

1198792

1003816100381610038161003816





G1001G10

02G10

03G10

04G10

05G10

06G10

07G10

08G10

09G10

10G10

11G10

12G10

13G10

14G10

15G10

16G10

17G10

18G10

19G10

20G10

21G10

22G10

23G10

24G10

25G10

26G10

27G10

28G10

29G10

30G10

31G10

32D1033

D1034D1035

D1036D1037

D1038D1039

D1040D1041

D1042G10

43G10

44G10

45G10

46G10

47G10

48G10

49G10

50G10

51G10

52G10

53G10

54G10

55G10

56G10

57G10

58G10

59G10

60G10

61D1062

D1063D1064

D1065D1066

D1067

G1068

G1069

G1070

G1071

G1072

G1073

G1074

G1075

G1076

G1077

G1078

G1079

G1080

G1081

G1082

G1083

G1084

G1085

G1086

D1087D1088

D1089D1090

D1091D1092

G1093G1094

G1095G1096

G1097G1098

G1099G1100

G1101G1102

G1103G1104

G1105G1106

G1107G1108

G1109G1110

G1111G1112

G1113G1114

G1115G1116

G1117G1118

G1119G1120

G1121G1122

G1123G1124

D1125D1126

D1127D1128

D1129D1130

D1131D1132

D1133D1134

G1135G1136

G1137G1138

G1139G1140

G1141G1142

G1143G1144

G1145G1146

G1147G1148

G1149G1150

G1151G1152

G1153D1154

D1155D1156

D1157D1158

D1159

G1160G1161

G1162G1163

G1164G1165

G1166G1167

G1168G1169

G1170G1171

G1172G1173

G1174G1175

G1176G1177

G1178D1179

D1180D1181

D1182D1183

D1184

G1086

G1015

G1107G1108

D1184

G1122

D1041

G1151

D1091G1030G1031

D1131G1121

G1059

G1153

G1080

D1128

G1060

G1020

G1110

D1067

G1124

G1177

G1061

G1148G1123D1132G1116G1117D1130

D1183

G1081

G1147

D1040

G1149G1115

G1176D1181

G1029

G1178

G1026

D1133

D1037G1082G1032G1019

G1152

G1084G1024

D1158

G1017

G1150

D1066

G1025G1085

D1129

D1038

D1157

G1056

G1028G1027G1023G1083

G1119

D1039D1090

D1127G1120G1118

G1174

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0

BJN

LF

TJN

CZHX

DZD

JNX

TA

QFD

TZHDZZH

XZHD

SZHD

BBN

DY

CHZH

NJN

ZHJNDYB

CHZHB

WXDSZHBKSHN

SHHQ


0 50 100 150 200 250 300 350 400Time (s)

55

6

65

7

75

8

9

85

Tota

l con

nect

ing

time

times104





0 50 100 150 200 250 300 350 40055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104





7 Conclusion






0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104






0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104








Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










A B CSector 1

Sector 2

Sector 3











119888 (119894 119895) =

120587119895119889minus 120587119894119886 120587

119895119889minus 120587119894119886ge 120576

120587119895119889minus 120587119894119886+ 119875 120587

119895119889minus 120587119894119886lt 120576

(1)


119894119895ge and 119911

119895119889isin119885




119894119895= 1 119910



min 119885 = 1205961sum

119894isin119881

sum

119895isin119881

119888 (119894 119895) 119909119894119895+ 1205962sum

119894isin119881

sum

119895isin119881

119910119894119895 (2)


119909119894119895= 1 119895 isin 119881 (3)

sum

119895isin119881

119909119894119895= 1 119894 isin 119881 (4)

119909119894119895le 120593119894119895 119894 119895 isin 119881 (5)

sum


sum

119895isin119881

119909119894119895119888119894119895+ sum


119905119894le 119871

119890 119890

1015840isin 119882

(6)

sum


sum

119895isin119881

119909119894119895119889119895le 119872 119890 119890

1015840isin 119882 (7)

119910119894119895le 120583119894119895119909119894119895 119894 119895 isin 119881 (8)

119873(119894) + 119873 (119895) = |119881| 119894 119895 isin 119881 (9)

sum

119894isin119881

sum

119895isin119881

119910119894119895ge 1 (10)



Station A

Station B

Station C

12011301

1303

1305

1307

1309

1311

3102

3104

3106

3108

2102

3112

2104

3116

1203

Sector 1

Sector 2

Overnightconnection


Time

Space





trains of route n


Down-bound

Down-bound

trains of route n







119894119895=



4 Heuristic


1 1198902 119890








119911119886(119890119894)119886= 119911119889(119890119895)119889

sum

119898isin119881119898 =119886(119890119894)

119909119898119889(119890

119895)= 0

120593119894119895= 1

(11)



1must


119898= 1 Let 119897[119886(119890

119894)] and 119901[119886(119890

119894)]


119886(119890119894)119886 whose value


i

i1

j

N(j)

N(i)

i2

i3

j1

j2

j3

j4

j5

(a)

i

i1

jN(i)

i2

j1

j2

N(j)

(b)




119897 [119886 (119890119894)] =

119889119886()+ sum

119898isin1015840119890119894

119889119886(119898) 119910119894= 0

119889119886(119890119894) 119910

119894= 1

119901 [119886 (119890119894)] =

119905119886()+ sum

119898isin1015840119890119894

(119888119898+ 119905119886(119898)) 119910

119894= 0

119905119886(119890119894) 119910

119894= 1

(12)




119894) = 119886(119890

1)


119895)] le 119871

119901[119886(119890119895)] le 119872 Therefore 119890


119888 (119886 (119890119894) 119889 (119890

119895)) = min119898isin119881119889(119890119894)

119888119886(119890119894)119898 119897 [119886 (119898)]

le 119871 119901 [119886 (119898)] le 119872 120583119886(119890)119894119898= 0

(13)


min119898isin119881119889(119890119894)

119888119886(119890119894)119898 120583119886(119890119894)119898= 1 le 119888 (119886 (119890

119894) 119889 (119890

119895))

le max119898isin119881119889(119890119894)

119888119886(119890119894)119898 120583119886(119890119894)119898= 1

(14)





119898120585is the set



sum

119890119898isin120585

119909119889(119890119898)V = 0

119911119889()119889

= 119911V119889

120583119886(119890119894)V = 1

(15)




119894) = 119889(119890


119894 119890119895in 120585





Yes

No

No


Is the second Yes

max value

Stop


solution

No

Yes

Yes

NoNo

Yes

Start

first edge set







d(e) and update s









edge set Oslash

completed s

Does k reach the










119889

+



119888 (119894 V119903 (119894)) = min119895isin119862

119888 (119894 119895) (16)


119888 (V119905 (119894) 119894) = min119895isin119862

119888 (119895 119894) (17)


of 119888119890lowast

119894


119894has the form

(119894


119894lowast =

119889

+


minus

V119903(119894lowast) = 119889minus

V119903(119894lowast) + 1 If 119889+


119894


119888 (119894 V119903 (119894)1015840) = min119895isin119862119889

minus

119895lt1

119888 (119894 119895) (18)




119894lowast = 119889

minus

119894lowast + 1





119888 (V119905 (119894)1015840 119894) = min119895isin119862119889

+

119895lt1

119888 (119895 119894) (19)










[119898 119899] 997904rArr [119896 119903 (119896) or 119905 (119896)] forall119889

+

119896isin119862 119889

minus

119896isin119862gt 1 (20)


119888 [119896 119903 (119896) or 119905 (119896)] = min119894isin119878119881119896

119888 [119896 119894]

119898 isin 119878119879 (119896)

119899 isin 119878119879 (119903 (119896)) cup 119878119879 (119905 (119896))

119898Š 119899

(21)




119888 [119894 119898] + 119888 [119894 119895] + 119888 [119895 119899] minus 119888 [119898 119899]


119888 [119894 ℎ] + 119888 [119894 119895] + 119888 [119895 119896] minus 119888 [ℎ 119896]

(23)



Start

Yes

Yes

Yes

No

No

Yes

No

No

Yes

No

Yes

No



Is exm-1 applied



Is m gt |V| minus 1






m = |U|








of er(i r(i))and









119894) = 119886(119890

119894+1)




1(119894) 1205962(119894) 120596




119864119873119895(119894)= 1198641198731198951015840(119894minus1)cup ℓ cup 120596

119895(119894)

119864119883119895(119894)= 1198641198831198951015840(119894minus1)

cup 1205961(119894) 120596

119895minus1(119894) 120596119895+1(119894) 120596

120582(119894)

(24)












1(0)and set 119864119883

1(0)











119898(119889))



1(0)

1(2)

1(1)

Lb(2)

Rb(2)

hellip

120582(2)

120582(1)

120582 minus 1(2)



1200011 ⋃ 120596120582(1)

1200012 ⋃ 120596120582(2) 1200012 ⋃ 120596120582minus1(2)1200012 ⋃ 1205961(2)

1200011 ⋃ 1205961(1)1200011 ⋃ 120596120582minus1(1)




















1(1198792)






BJN

TJN

DZD

TA TZhD

XZhD

BBN

ChZh

ZhJN

ChZh

B

SZhB

ShH

Q

LF

CZhX JN

X

QFD ZZ

h

SZhD D

Y

NJN

DYB

WXD

KShN

Sector 1

Sector 3Sector 2



Sector From To 1003816100381610038161003816

1198791

1003816100381610038161003816

1003816100381610038161003816

1198792

1003816100381610038161003816





G1001G10

02G10

03G10

04G10

05G10

06G10

07G10

08G10

09G10

10G10

11G10

12G10

13G10

14G10

15G10

16G10

17G10

18G10

19G10

20G10

21G10

22G10

23G10

24G10

25G10

26G10

27G10

28G10

29G10

30G10

31G10

32D1033

D1034D1035

D1036D1037

D1038D1039

D1040D1041

D1042G10

43G10

44G10

45G10

46G10

47G10

48G10

49G10

50G10

51G10

52G10

53G10

54G10

55G10

56G10

57G10

58G10

59G10

60G10

61D1062

D1063D1064

D1065D1066

D1067

G1068

G1069

G1070

G1071

G1072

G1073

G1074

G1075

G1076

G1077

G1078

G1079

G1080

G1081

G1082

G1083

G1084

G1085

G1086

D1087D1088

D1089D1090

D1091D1092

G1093G1094

G1095G1096

G1097G1098

G1099G1100

G1101G1102

G1103G1104

G1105G1106

G1107G1108

G1109G1110

G1111G1112

G1113G1114

G1115G1116

G1117G1118

G1119G1120

G1121G1122

G1123G1124

D1125D1126

D1127D1128

D1129D1130

D1131D1132

D1133D1134

G1135G1136

G1137G1138

G1139G1140

G1141G1142

G1143G1144

G1145G1146

G1147G1148

G1149G1150

G1151G1152

G1153D1154

D1155D1156

D1157D1158

D1159

G1160G1161

G1162G1163

G1164G1165

G1166G1167

G1168G1169

G1170G1171

G1172G1173

G1174G1175

G1176G1177

G1178D1179

D1180D1181

D1182D1183

D1184

G1086

G1015

G1107G1108

D1184

G1122

D1041

G1151

D1091G1030G1031

D1131G1121

G1059

G1153

G1080

D1128

G1060

G1020

G1110

D1067

G1124

G1177

G1061

G1148G1123D1132G1116G1117D1130

D1183

G1081

G1147

D1040

G1149G1115

G1176D1181

G1029

G1178

G1026

D1133

D1037G1082G1032G1019

G1152

G1084G1024

D1158

G1017

G1150

D1066

G1025G1085

D1129

D1038

D1157

G1056

G1028G1027G1023G1083

G1119

D1039D1090

D1127G1120G1118

G1174

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0

BJN

LF

TJN

CZHX

DZD

JNX

TA

QFD

TZHDZZH

XZHD

SZHD

BBN

DY

CHZH

NJN

ZHJNDYB

CHZHB

WXDSZHBKSHN

SHHQ


0 50 100 150 200 250 300 350 400Time (s)

55

6

65

7

75

8

9

85

Tota

l con

nect

ing

time

times104





0 50 100 150 200 250 300 350 40055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104





7 Conclusion






0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104






0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104








Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Station A

Station B

Station C

12011301

1303

1305

1307

1309

1311

3102

3104

3106

3108

2102

3112

2104

3116

1203

Sector 1

Sector 2

Overnightconnection


Time

Space





trains of route n


Down-bound

Down-bound

trains of route n







119894119895=



4 Heuristic


1 1198902 119890








119911119886(119890119894)119886= 119911119889(119890119895)119889

sum

119898isin119881119898 =119886(119890119894)

119909119898119889(119890

119895)= 0

120593119894119895= 1

(11)



1must


119898= 1 Let 119897[119886(119890

119894)] and 119901[119886(119890

119894)]


119886(119890119894)119886 whose value


i

i1

j

N(j)

N(i)

i2

i3

j1

j2

j3

j4

j5

(a)

i

i1

jN(i)

i2

j1

j2

N(j)

(b)




119897 [119886 (119890119894)] =

119889119886()+ sum

119898isin1015840119890119894

119889119886(119898) 119910119894= 0

119889119886(119890119894) 119910

119894= 1

119901 [119886 (119890119894)] =

119905119886()+ sum

119898isin1015840119890119894

(119888119898+ 119905119886(119898)) 119910

119894= 0

119905119886(119890119894) 119910

119894= 1

(12)




119894) = 119886(119890

1)


119895)] le 119871

119901[119886(119890119895)] le 119872 Therefore 119890


119888 (119886 (119890119894) 119889 (119890

119895)) = min119898isin119881119889(119890119894)

119888119886(119890119894)119898 119897 [119886 (119898)]

le 119871 119901 [119886 (119898)] le 119872 120583119886(119890)119894119898= 0

(13)


min119898isin119881119889(119890119894)

119888119886(119890119894)119898 120583119886(119890119894)119898= 1 le 119888 (119886 (119890

119894) 119889 (119890

119895))

le max119898isin119881119889(119890119894)

119888119886(119890119894)119898 120583119886(119890119894)119898= 1

(14)





119898120585is the set



sum

119890119898isin120585

119909119889(119890119898)V = 0

119911119889()119889

= 119911V119889

120583119886(119890119894)V = 1

(15)




119894) = 119889(119890


119894 119890119895in 120585





Yes

No

No


Is the second Yes

max value

Stop


solution

No

Yes

Yes

NoNo

Yes

Start

first edge set







d(e) and update s









edge set Oslash

completed s

Does k reach the










119889

+



119888 (119894 V119903 (119894)) = min119895isin119862

119888 (119894 119895) (16)


119888 (V119905 (119894) 119894) = min119895isin119862

119888 (119895 119894) (17)


of 119888119890lowast

119894


119894has the form

(119894


119894lowast =

119889

+


minus

V119903(119894lowast) = 119889minus

V119903(119894lowast) + 1 If 119889+


119894


119888 (119894 V119903 (119894)1015840) = min119895isin119862119889

minus

119895lt1

119888 (119894 119895) (18)




119894lowast = 119889

minus

119894lowast + 1





119888 (V119905 (119894)1015840 119894) = min119895isin119862119889

+

119895lt1

119888 (119895 119894) (19)










[119898 119899] 997904rArr [119896 119903 (119896) or 119905 (119896)] forall119889

+

119896isin119862 119889

minus

119896isin119862gt 1 (20)


119888 [119896 119903 (119896) or 119905 (119896)] = min119894isin119878119881119896

119888 [119896 119894]

119898 isin 119878119879 (119896)

119899 isin 119878119879 (119903 (119896)) cup 119878119879 (119905 (119896))

119898Š 119899

(21)




119888 [119894 119898] + 119888 [119894 119895] + 119888 [119895 119899] minus 119888 [119898 119899]


119888 [119894 ℎ] + 119888 [119894 119895] + 119888 [119895 119896] minus 119888 [ℎ 119896]

(23)



Start

Yes

Yes

Yes

No

No

Yes

No

No

Yes

No

Yes

No



Is exm-1 applied



Is m gt |V| minus 1






m = |U|








of er(i r(i))and









119894) = 119886(119890

119894+1)




1(119894) 1205962(119894) 120596




119864119873119895(119894)= 1198641198731198951015840(119894minus1)cup ℓ cup 120596

119895(119894)

119864119883119895(119894)= 1198641198831198951015840(119894minus1)

cup 1205961(119894) 120596

119895minus1(119894) 120596119895+1(119894) 120596

120582(119894)

(24)












1(0)and set 119864119883

1(0)











119898(119889))



1(0)

1(2)

1(1)

Lb(2)

Rb(2)

hellip

120582(2)

120582(1)

120582 minus 1(2)



1200011 ⋃ 120596120582(1)

1200012 ⋃ 120596120582(2) 1200012 ⋃ 120596120582minus1(2)1200012 ⋃ 1205961(2)

1200011 ⋃ 1205961(1)1200011 ⋃ 120596120582minus1(1)




















1(1198792)






BJN

TJN

DZD

TA TZhD

XZhD

BBN

ChZh

ZhJN

ChZh

B

SZhB

ShH

Q

LF

CZhX JN

X

QFD ZZ

h

SZhD D

Y

NJN

DYB

WXD

KShN

Sector 1

Sector 3Sector 2



Sector From To 1003816100381610038161003816

1198791

1003816100381610038161003816

1003816100381610038161003816

1198792

1003816100381610038161003816





G1001G10

02G10

03G10

04G10

05G10

06G10

07G10

08G10

09G10

10G10

11G10

12G10

13G10

14G10

15G10

16G10

17G10

18G10

19G10

20G10

21G10

22G10

23G10

24G10

25G10

26G10

27G10

28G10

29G10

30G10

31G10

32D1033

D1034D1035

D1036D1037

D1038D1039

D1040D1041

D1042G10

43G10

44G10

45G10

46G10

47G10

48G10

49G10

50G10

51G10

52G10

53G10

54G10

55G10

56G10

57G10

58G10

59G10

60G10

61D1062

D1063D1064

D1065D1066

D1067

G1068

G1069

G1070

G1071

G1072

G1073

G1074

G1075

G1076

G1077

G1078

G1079

G1080

G1081

G1082

G1083

G1084

G1085

G1086

D1087D1088

D1089D1090

D1091D1092

G1093G1094

G1095G1096

G1097G1098

G1099G1100

G1101G1102

G1103G1104

G1105G1106

G1107G1108

G1109G1110

G1111G1112

G1113G1114

G1115G1116

G1117G1118

G1119G1120

G1121G1122

G1123G1124

D1125D1126

D1127D1128

D1129D1130

D1131D1132

D1133D1134

G1135G1136

G1137G1138

G1139G1140

G1141G1142

G1143G1144

G1145G1146

G1147G1148

G1149G1150

G1151G1152

G1153D1154

D1155D1156

D1157D1158

D1159

G1160G1161

G1162G1163

G1164G1165

G1166G1167

G1168G1169

G1170G1171

G1172G1173

G1174G1175

G1176G1177

G1178D1179

D1180D1181

D1182D1183

D1184

G1086

G1015

G1107G1108

D1184

G1122

D1041

G1151

D1091G1030G1031

D1131G1121

G1059

G1153

G1080

D1128

G1060

G1020

G1110

D1067

G1124

G1177

G1061

G1148G1123D1132G1116G1117D1130

D1183

G1081

G1147

D1040

G1149G1115

G1176D1181

G1029

G1178

G1026

D1133

D1037G1082G1032G1019

G1152

G1084G1024

D1158

G1017

G1150

D1066

G1025G1085

D1129

D1038

D1157

G1056

G1028G1027G1023G1083

G1119

D1039D1090

D1127G1120G1118

G1174

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0

BJN

LF

TJN

CZHX

DZD

JNX

TA

QFD

TZHDZZH

XZHD

SZHD

BBN

DY

CHZH

NJN

ZHJNDYB

CHZHB

WXDSZHBKSHN

SHHQ


0 50 100 150 200 250 300 350 400Time (s)

55

6

65

7

75

8

9

85

Tota

l con

nect

ing

time

times104





0 50 100 150 200 250 300 350 40055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104





7 Conclusion






0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104






0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104








Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










i

i1

j

N(j)

N(i)

i2

i3

j1

j2

j3

j4

j5

(a)

i

i1

jN(i)

i2

j1

j2

N(j)

(b)




119897 [119886 (119890119894)] =

119889119886()+ sum

119898isin1015840119890119894

119889119886(119898) 119910119894= 0

119889119886(119890119894) 119910

119894= 1

119901 [119886 (119890119894)] =

119905119886()+ sum

119898isin1015840119890119894

(119888119898+ 119905119886(119898)) 119910

119894= 0

119905119886(119890119894) 119910

119894= 1

(12)




119894) = 119886(119890

1)


119895)] le 119871

119901[119886(119890119895)] le 119872 Therefore 119890


119888 (119886 (119890119894) 119889 (119890

119895)) = min119898isin119881119889(119890119894)

119888119886(119890119894)119898 119897 [119886 (119898)]

le 119871 119901 [119886 (119898)] le 119872 120583119886(119890)119894119898= 0

(13)


min119898isin119881119889(119890119894)

119888119886(119890119894)119898 120583119886(119890119894)119898= 1 le 119888 (119886 (119890

119894) 119889 (119890

119895))

le max119898isin119881119889(119890119894)

119888119886(119890119894)119898 120583119886(119890119894)119898= 1

(14)





119898120585is the set



sum

119890119898isin120585

119909119889(119890119898)V = 0

119911119889()119889

= 119911V119889

120583119886(119890119894)V = 1

(15)




119894) = 119889(119890


119894 119890119895in 120585





Yes

No

No


Is the second Yes

max value

Stop


solution

No

Yes

Yes

NoNo

Yes

Start

first edge set







d(e) and update s









edge set Oslash

completed s

Does k reach the










119889

+



119888 (119894 V119903 (119894)) = min119895isin119862

119888 (119894 119895) (16)


119888 (V119905 (119894) 119894) = min119895isin119862

119888 (119895 119894) (17)


of 119888119890lowast

119894


119894has the form

(119894


119894lowast =

119889

+


minus

V119903(119894lowast) = 119889minus

V119903(119894lowast) + 1 If 119889+


119894


119888 (119894 V119903 (119894)1015840) = min119895isin119862119889

minus

119895lt1

119888 (119894 119895) (18)




119894lowast = 119889

minus

119894lowast + 1





119888 (V119905 (119894)1015840 119894) = min119895isin119862119889

+

119895lt1

119888 (119895 119894) (19)










[119898 119899] 997904rArr [119896 119903 (119896) or 119905 (119896)] forall119889

+

119896isin119862 119889

minus

119896isin119862gt 1 (20)


119888 [119896 119903 (119896) or 119905 (119896)] = min119894isin119878119881119896

119888 [119896 119894]

119898 isin 119878119879 (119896)

119899 isin 119878119879 (119903 (119896)) cup 119878119879 (119905 (119896))

119898Š 119899

(21)




119888 [119894 119898] + 119888 [119894 119895] + 119888 [119895 119899] minus 119888 [119898 119899]


119888 [119894 ℎ] + 119888 [119894 119895] + 119888 [119895 119896] minus 119888 [ℎ 119896]

(23)



Start

Yes

Yes

Yes

No

No

Yes

No

No

Yes

No

Yes

No



Is exm-1 applied



Is m gt |V| minus 1






m = |U|








of er(i r(i))and









119894) = 119886(119890

119894+1)




1(119894) 1205962(119894) 120596




119864119873119895(119894)= 1198641198731198951015840(119894minus1)cup ℓ cup 120596

119895(119894)

119864119883119895(119894)= 1198641198831198951015840(119894minus1)

cup 1205961(119894) 120596

119895minus1(119894) 120596119895+1(119894) 120596

120582(119894)

(24)












1(0)and set 119864119883

1(0)











119898(119889))



1(0)

1(2)

1(1)

Lb(2)

Rb(2)

hellip

120582(2)

120582(1)

120582 minus 1(2)



1200011 ⋃ 120596120582(1)

1200012 ⋃ 120596120582(2) 1200012 ⋃ 120596120582minus1(2)1200012 ⋃ 1205961(2)

1200011 ⋃ 1205961(1)1200011 ⋃ 120596120582minus1(1)




















1(1198792)






BJN

TJN

DZD

TA TZhD

XZhD

BBN

ChZh

ZhJN

ChZh

B

SZhB

ShH

Q

LF

CZhX JN

X

QFD ZZ

h

SZhD D

Y

NJN

DYB

WXD

KShN

Sector 1

Sector 3Sector 2



Sector From To 1003816100381610038161003816

1198791

1003816100381610038161003816

1003816100381610038161003816

1198792

1003816100381610038161003816





G1001G10

02G10

03G10

04G10

05G10

06G10

07G10

08G10

09G10

10G10

11G10

12G10

13G10

14G10

15G10

16G10

17G10

18G10

19G10

20G10

21G10

22G10

23G10

24G10

25G10

26G10

27G10

28G10

29G10

30G10

31G10

32D1033

D1034D1035

D1036D1037

D1038D1039

D1040D1041

D1042G10

43G10

44G10

45G10

46G10

47G10

48G10

49G10

50G10

51G10

52G10

53G10

54G10

55G10

56G10

57G10

58G10

59G10

60G10

61D1062

D1063D1064

D1065D1066

D1067

G1068

G1069

G1070

G1071

G1072

G1073

G1074

G1075

G1076

G1077

G1078

G1079

G1080

G1081

G1082

G1083

G1084

G1085

G1086

D1087D1088

D1089D1090

D1091D1092

G1093G1094

G1095G1096

G1097G1098

G1099G1100

G1101G1102

G1103G1104

G1105G1106

G1107G1108

G1109G1110

G1111G1112

G1113G1114

G1115G1116

G1117G1118

G1119G1120

G1121G1122

G1123G1124

D1125D1126

D1127D1128

D1129D1130

D1131D1132

D1133D1134

G1135G1136

G1137G1138

G1139G1140

G1141G1142

G1143G1144

G1145G1146

G1147G1148

G1149G1150

G1151G1152

G1153D1154

D1155D1156

D1157D1158

D1159

G1160G1161

G1162G1163

G1164G1165

G1166G1167

G1168G1169

G1170G1171

G1172G1173

G1174G1175

G1176G1177

G1178D1179

D1180D1181

D1182D1183

D1184

G1086

G1015

G1107G1108

D1184

G1122

D1041

G1151

D1091G1030G1031

D1131G1121

G1059

G1153

G1080

D1128

G1060

G1020

G1110

D1067

G1124

G1177

G1061

G1148G1123D1132G1116G1117D1130

D1183

G1081

G1147

D1040

G1149G1115

G1176D1181

G1029

G1178

G1026

D1133

D1037G1082G1032G1019

G1152

G1084G1024

D1158

G1017

G1150

D1066

G1025G1085

D1129

D1038

D1157

G1056

G1028G1027G1023G1083

G1119

D1039D1090

D1127G1120G1118

G1174

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0

BJN

LF

TJN

CZHX

DZD

JNX

TA

QFD

TZHDZZH

XZHD

SZHD

BBN

DY

CHZH

NJN

ZHJNDYB

CHZHB

WXDSZHBKSHN

SHHQ


0 50 100 150 200 250 300 350 400Time (s)

55

6

65

7

75

8

9

85

Tota

l con

nect

ing

time

times104





0 50 100 150 200 250 300 350 40055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104





7 Conclusion






0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104






0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104








Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Yes

No

No


Is the second Yes

max value

Stop


solution

No

Yes

Yes

NoNo

Yes

Start

first edge set







d(e) and update s









edge set Oslash

completed s

Does k reach the










119889

+



119888 (119894 V119903 (119894)) = min119895isin119862

119888 (119894 119895) (16)


119888 (V119905 (119894) 119894) = min119895isin119862

119888 (119895 119894) (17)


of 119888119890lowast

119894


119894has the form

(119894


119894lowast =

119889

+


minus

V119903(119894lowast) = 119889minus

V119903(119894lowast) + 1 If 119889+


119894


119888 (119894 V119903 (119894)1015840) = min119895isin119862119889

minus

119895lt1

119888 (119894 119895) (18)




119894lowast = 119889

minus

119894lowast + 1





119888 (V119905 (119894)1015840 119894) = min119895isin119862119889

+

119895lt1

119888 (119895 119894) (19)










[119898 119899] 997904rArr [119896 119903 (119896) or 119905 (119896)] forall119889

+

119896isin119862 119889

minus

119896isin119862gt 1 (20)


119888 [119896 119903 (119896) or 119905 (119896)] = min119894isin119878119881119896

119888 [119896 119894]

119898 isin 119878119879 (119896)

119899 isin 119878119879 (119903 (119896)) cup 119878119879 (119905 (119896))

119898Š 119899

(21)




119888 [119894 119898] + 119888 [119894 119895] + 119888 [119895 119899] minus 119888 [119898 119899]


119888 [119894 ℎ] + 119888 [119894 119895] + 119888 [119895 119896] minus 119888 [ℎ 119896]

(23)



Start

Yes

Yes

Yes

No

No

Yes

No

No

Yes

No

Yes

No



Is exm-1 applied



Is m gt |V| minus 1






m = |U|








of er(i r(i))and









119894) = 119886(119890

119894+1)




1(119894) 1205962(119894) 120596




119864119873119895(119894)= 1198641198731198951015840(119894minus1)cup ℓ cup 120596

119895(119894)

119864119883119895(119894)= 1198641198831198951015840(119894minus1)

cup 1205961(119894) 120596

119895minus1(119894) 120596119895+1(119894) 120596

120582(119894)

(24)












1(0)and set 119864119883

1(0)











119898(119889))



1(0)

1(2)

1(1)

Lb(2)

Rb(2)

hellip

120582(2)

120582(1)

120582 minus 1(2)



1200011 ⋃ 120596120582(1)

1200012 ⋃ 120596120582(2) 1200012 ⋃ 120596120582minus1(2)1200012 ⋃ 1205961(2)

1200011 ⋃ 1205961(1)1200011 ⋃ 120596120582minus1(1)




















1(1198792)






BJN

TJN

DZD

TA TZhD

XZhD

BBN

ChZh

ZhJN

ChZh

B

SZhB

ShH

Q

LF

CZhX JN

X

QFD ZZ

h

SZhD D

Y

NJN

DYB

WXD

KShN

Sector 1

Sector 3Sector 2



Sector From To 1003816100381610038161003816

1198791

1003816100381610038161003816

1003816100381610038161003816

1198792

1003816100381610038161003816





G1001G10

02G10

03G10

04G10

05G10

06G10

07G10

08G10

09G10

10G10

11G10

12G10

13G10

14G10

15G10

16G10

17G10

18G10

19G10

20G10

21G10

22G10

23G10

24G10

25G10

26G10

27G10

28G10

29G10

30G10

31G10

32D1033

D1034D1035

D1036D1037

D1038D1039

D1040D1041

D1042G10

43G10

44G10

45G10

46G10

47G10

48G10

49G10

50G10

51G10

52G10

53G10

54G10

55G10

56G10

57G10

58G10

59G10

60G10

61D1062

D1063D1064

D1065D1066

D1067

G1068

G1069

G1070

G1071

G1072

G1073

G1074

G1075

G1076

G1077

G1078

G1079

G1080

G1081

G1082

G1083

G1084

G1085

G1086

D1087D1088

D1089D1090

D1091D1092

G1093G1094

G1095G1096

G1097G1098

G1099G1100

G1101G1102

G1103G1104

G1105G1106

G1107G1108

G1109G1110

G1111G1112

G1113G1114

G1115G1116

G1117G1118

G1119G1120

G1121G1122

G1123G1124

D1125D1126

D1127D1128

D1129D1130

D1131D1132

D1133D1134

G1135G1136

G1137G1138

G1139G1140

G1141G1142

G1143G1144

G1145G1146

G1147G1148

G1149G1150

G1151G1152

G1153D1154

D1155D1156

D1157D1158

D1159

G1160G1161

G1162G1163

G1164G1165

G1166G1167

G1168G1169

G1170G1171

G1172G1173

G1174G1175

G1176G1177

G1178D1179

D1180D1181

D1182D1183

D1184

G1086

G1015

G1107G1108

D1184

G1122

D1041

G1151

D1091G1030G1031

D1131G1121

G1059

G1153

G1080

D1128

G1060

G1020

G1110

D1067

G1124

G1177

G1061

G1148G1123D1132G1116G1117D1130

D1183

G1081

G1147

D1040

G1149G1115

G1176D1181

G1029

G1178

G1026

D1133

D1037G1082G1032G1019

G1152

G1084G1024

D1158

G1017

G1150

D1066

G1025G1085

D1129

D1038

D1157

G1056

G1028G1027G1023G1083

G1119

D1039D1090

D1127G1120G1118

G1174

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0

BJN

LF

TJN

CZHX

DZD

JNX

TA

QFD

TZHDZZH

XZHD

SZHD

BBN

DY

CHZH

NJN

ZHJNDYB

CHZHB

WXDSZHBKSHN

SHHQ


0 50 100 150 200 250 300 350 400Time (s)

55

6

65

7

75

8

9

85

Tota

l con

nect

ing

time

times104





0 50 100 150 200 250 300 350 40055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104





7 Conclusion






0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104






0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104








Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119889

+



119888 (119894 V119903 (119894)) = min119895isin119862

119888 (119894 119895) (16)


119888 (V119905 (119894) 119894) = min119895isin119862

119888 (119895 119894) (17)


of 119888119890lowast

119894


119894has the form

(119894


119894lowast =

119889

+


minus

V119903(119894lowast) = 119889minus

V119903(119894lowast) + 1 If 119889+


119894


119888 (119894 V119903 (119894)1015840) = min119895isin119862119889

minus

119895lt1

119888 (119894 119895) (18)




119894lowast = 119889

minus

119894lowast + 1





119888 (V119905 (119894)1015840 119894) = min119895isin119862119889

+

119895lt1

119888 (119895 119894) (19)










[119898 119899] 997904rArr [119896 119903 (119896) or 119905 (119896)] forall119889

+

119896isin119862 119889

minus

119896isin119862gt 1 (20)


119888 [119896 119903 (119896) or 119905 (119896)] = min119894isin119878119881119896

119888 [119896 119894]

119898 isin 119878119879 (119896)

119899 isin 119878119879 (119903 (119896)) cup 119878119879 (119905 (119896))

119898Š 119899

(21)




119888 [119894 119898] + 119888 [119894 119895] + 119888 [119895 119899] minus 119888 [119898 119899]


119888 [119894 ℎ] + 119888 [119894 119895] + 119888 [119895 119896] minus 119888 [ℎ 119896]

(23)



Start

Yes

Yes

Yes

No

No

Yes

No

No

Yes

No

Yes

No



Is exm-1 applied



Is m gt |V| minus 1






m = |U|








of er(i r(i))and









119894) = 119886(119890

119894+1)




1(119894) 1205962(119894) 120596




119864119873119895(119894)= 1198641198731198951015840(119894minus1)cup ℓ cup 120596

119895(119894)

119864119883119895(119894)= 1198641198831198951015840(119894minus1)

cup 1205961(119894) 120596

119895minus1(119894) 120596119895+1(119894) 120596

120582(119894)

(24)












1(0)and set 119864119883

1(0)











119898(119889))



1(0)

1(2)

1(1)

Lb(2)

Rb(2)

hellip

120582(2)

120582(1)

120582 minus 1(2)



1200011 ⋃ 120596120582(1)

1200012 ⋃ 120596120582(2) 1200012 ⋃ 120596120582minus1(2)1200012 ⋃ 1205961(2)

1200011 ⋃ 1205961(1)1200011 ⋃ 120596120582minus1(1)




















1(1198792)






BJN

TJN

DZD

TA TZhD

XZhD

BBN

ChZh

ZhJN

ChZh

B

SZhB

ShH

Q

LF

CZhX JN

X

QFD ZZ

h

SZhD D

Y

NJN

DYB

WXD

KShN

Sector 1

Sector 3Sector 2



Sector From To 1003816100381610038161003816

1198791

1003816100381610038161003816

1003816100381610038161003816

1198792

1003816100381610038161003816





G1001G10

02G10

03G10

04G10

05G10

06G10

07G10

08G10

09G10

10G10

11G10

12G10

13G10

14G10

15G10

16G10

17G10

18G10

19G10

20G10

21G10

22G10

23G10

24G10

25G10

26G10

27G10

28G10

29G10

30G10

31G10

32D1033

D1034D1035

D1036D1037

D1038D1039

D1040D1041

D1042G10

43G10

44G10

45G10

46G10

47G10

48G10

49G10

50G10

51G10

52G10

53G10

54G10

55G10

56G10

57G10

58G10

59G10

60G10

61D1062

D1063D1064

D1065D1066

D1067

G1068

G1069

G1070

G1071

G1072

G1073

G1074

G1075

G1076

G1077

G1078

G1079

G1080

G1081

G1082

G1083

G1084

G1085

G1086

D1087D1088

D1089D1090

D1091D1092

G1093G1094

G1095G1096

G1097G1098

G1099G1100

G1101G1102

G1103G1104

G1105G1106

G1107G1108

G1109G1110

G1111G1112

G1113G1114

G1115G1116

G1117G1118

G1119G1120

G1121G1122

G1123G1124

D1125D1126

D1127D1128

D1129D1130

D1131D1132

D1133D1134

G1135G1136

G1137G1138

G1139G1140

G1141G1142

G1143G1144

G1145G1146

G1147G1148

G1149G1150

G1151G1152

G1153D1154

D1155D1156

D1157D1158

D1159

G1160G1161

G1162G1163

G1164G1165

G1166G1167

G1168G1169

G1170G1171

G1172G1173

G1174G1175

G1176G1177

G1178D1179

D1180D1181

D1182D1183

D1184

G1086

G1015

G1107G1108

D1184

G1122

D1041

G1151

D1091G1030G1031

D1131G1121

G1059

G1153

G1080

D1128

G1060

G1020

G1110

D1067

G1124

G1177

G1061

G1148G1123D1132G1116G1117D1130

D1183

G1081

G1147

D1040

G1149G1115

G1176D1181

G1029

G1178

G1026

D1133

D1037G1082G1032G1019

G1152

G1084G1024

D1158

G1017

G1150

D1066

G1025G1085

D1129

D1038

D1157

G1056

G1028G1027G1023G1083

G1119

D1039D1090

D1127G1120G1118

G1174

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0

BJN

LF

TJN

CZHX

DZD

JNX

TA

QFD

TZHDZZH

XZHD

SZHD

BBN

DY

CHZH

NJN

ZHJNDYB

CHZHB

WXDSZHBKSHN

SHHQ


0 50 100 150 200 250 300 350 400Time (s)

55

6

65

7

75

8

9

85

Tota

l con

nect

ing

time

times104





0 50 100 150 200 250 300 350 40055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104





7 Conclusion






0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104






0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104








Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Start

Yes

Yes

Yes

No

No

Yes

No

No

Yes

No

Yes

No



Is exm-1 applied



Is m gt |V| minus 1






m = |U|








of er(i r(i))and









119894) = 119886(119890

119894+1)




1(119894) 1205962(119894) 120596




119864119873119895(119894)= 1198641198731198951015840(119894minus1)cup ℓ cup 120596

119895(119894)

119864119883119895(119894)= 1198641198831198951015840(119894minus1)

cup 1205961(119894) 120596

119895minus1(119894) 120596119895+1(119894) 120596

120582(119894)

(24)












1(0)and set 119864119883

1(0)











119898(119889))



1(0)

1(2)

1(1)

Lb(2)

Rb(2)

hellip

120582(2)

120582(1)

120582 minus 1(2)



1200011 ⋃ 120596120582(1)

1200012 ⋃ 120596120582(2) 1200012 ⋃ 120596120582minus1(2)1200012 ⋃ 1205961(2)

1200011 ⋃ 1205961(1)1200011 ⋃ 120596120582minus1(1)




















1(1198792)






BJN

TJN

DZD

TA TZhD

XZhD

BBN

ChZh

ZhJN

ChZh

B

SZhB

ShH

Q

LF

CZhX JN

X

QFD ZZ

h

SZhD D

Y

NJN

DYB

WXD

KShN

Sector 1

Sector 3Sector 2



Sector From To 1003816100381610038161003816

1198791

1003816100381610038161003816

1003816100381610038161003816

1198792

1003816100381610038161003816





G1001G10

02G10

03G10

04G10

05G10

06G10

07G10

08G10

09G10

10G10

11G10

12G10

13G10

14G10

15G10

16G10

17G10

18G10

19G10

20G10

21G10

22G10

23G10

24G10

25G10

26G10

27G10

28G10

29G10

30G10

31G10

32D1033

D1034D1035

D1036D1037

D1038D1039

D1040D1041

D1042G10

43G10

44G10

45G10

46G10

47G10

48G10

49G10

50G10

51G10

52G10

53G10

54G10

55G10

56G10

57G10

58G10

59G10

60G10

61D1062

D1063D1064

D1065D1066

D1067

G1068

G1069

G1070

G1071

G1072

G1073

G1074

G1075

G1076

G1077

G1078

G1079

G1080

G1081

G1082

G1083

G1084

G1085

G1086

D1087D1088

D1089D1090

D1091D1092

G1093G1094

G1095G1096

G1097G1098

G1099G1100

G1101G1102

G1103G1104

G1105G1106

G1107G1108

G1109G1110

G1111G1112

G1113G1114

G1115G1116

G1117G1118

G1119G1120

G1121G1122

G1123G1124

D1125D1126

D1127D1128

D1129D1130

D1131D1132

D1133D1134

G1135G1136

G1137G1138

G1139G1140

G1141G1142

G1143G1144

G1145G1146

G1147G1148

G1149G1150

G1151G1152

G1153D1154

D1155D1156

D1157D1158

D1159

G1160G1161

G1162G1163

G1164G1165

G1166G1167

G1168G1169

G1170G1171

G1172G1173

G1174G1175

G1176G1177

G1178D1179

D1180D1181

D1182D1183

D1184

G1086

G1015

G1107G1108

D1184

G1122

D1041

G1151

D1091G1030G1031

D1131G1121

G1059

G1153

G1080

D1128

G1060

G1020

G1110

D1067

G1124

G1177

G1061

G1148G1123D1132G1116G1117D1130

D1183

G1081

G1147

D1040

G1149G1115

G1176D1181

G1029

G1178

G1026

D1133

D1037G1082G1032G1019

G1152

G1084G1024

D1158

G1017

G1150

D1066

G1025G1085

D1129

D1038

D1157

G1056

G1028G1027G1023G1083

G1119

D1039D1090

D1127G1120G1118

G1174

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0

BJN

LF

TJN

CZHX

DZD

JNX

TA

QFD

TZHDZZH

XZHD

SZHD

BBN

DY

CHZH

NJN

ZHJNDYB

CHZHB

WXDSZHBKSHN

SHHQ


0 50 100 150 200 250 300 350 400Time (s)

55

6

65

7

75

8

9

85

Tota

l con

nect

ing

time

times104





0 50 100 150 200 250 300 350 40055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104





7 Conclusion






0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104






0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104








Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












119894) = 119886(119890

119894+1)




1(119894) 1205962(119894) 120596




119864119873119895(119894)= 1198641198731198951015840(119894minus1)cup ℓ cup 120596

119895(119894)

119864119883119895(119894)= 1198641198831198951015840(119894minus1)

cup 1205961(119894) 120596

119895minus1(119894) 120596119895+1(119894) 120596

120582(119894)

(24)












1(0)and set 119864119883

1(0)











119898(119889))



1(0)

1(2)

1(1)

Lb(2)

Rb(2)

hellip

120582(2)

120582(1)

120582 minus 1(2)



1200011 ⋃ 120596120582(1)

1200012 ⋃ 120596120582(2) 1200012 ⋃ 120596120582minus1(2)1200012 ⋃ 1205961(2)

1200011 ⋃ 1205961(1)1200011 ⋃ 120596120582minus1(1)




















1(1198792)






BJN

TJN

DZD

TA TZhD

XZhD

BBN

ChZh

ZhJN

ChZh

B

SZhB

ShH

Q

LF

CZhX JN

X

QFD ZZ

h

SZhD D

Y

NJN

DYB

WXD

KShN

Sector 1

Sector 3Sector 2



Sector From To 1003816100381610038161003816

1198791

1003816100381610038161003816

1003816100381610038161003816

1198792

1003816100381610038161003816





G1001G10

02G10

03G10

04G10

05G10

06G10

07G10

08G10

09G10

10G10

11G10

12G10

13G10

14G10

15G10

16G10

17G10

18G10

19G10

20G10

21G10

22G10

23G10

24G10

25G10

26G10

27G10

28G10

29G10

30G10

31G10

32D1033

D1034D1035

D1036D1037

D1038D1039

D1040D1041

D1042G10

43G10

44G10

45G10

46G10

47G10

48G10

49G10

50G10

51G10

52G10

53G10

54G10

55G10

56G10

57G10

58G10

59G10

60G10

61D1062

D1063D1064

D1065D1066

D1067

G1068

G1069

G1070

G1071

G1072

G1073

G1074

G1075

G1076

G1077

G1078

G1079

G1080

G1081

G1082

G1083

G1084

G1085

G1086

D1087D1088

D1089D1090

D1091D1092

G1093G1094

G1095G1096

G1097G1098

G1099G1100

G1101G1102

G1103G1104

G1105G1106

G1107G1108

G1109G1110

G1111G1112

G1113G1114

G1115G1116

G1117G1118

G1119G1120

G1121G1122

G1123G1124

D1125D1126

D1127D1128

D1129D1130

D1131D1132

D1133D1134

G1135G1136

G1137G1138

G1139G1140

G1141G1142

G1143G1144

G1145G1146

G1147G1148

G1149G1150

G1151G1152

G1153D1154

D1155D1156

D1157D1158

D1159

G1160G1161

G1162G1163

G1164G1165

G1166G1167

G1168G1169

G1170G1171

G1172G1173

G1174G1175

G1176G1177

G1178D1179

D1180D1181

D1182D1183

D1184

G1086

G1015

G1107G1108

D1184

G1122

D1041

G1151

D1091G1030G1031

D1131G1121

G1059

G1153

G1080

D1128

G1060

G1020

G1110

D1067

G1124

G1177

G1061

G1148G1123D1132G1116G1117D1130

D1183

G1081

G1147

D1040

G1149G1115

G1176D1181

G1029

G1178

G1026

D1133

D1037G1082G1032G1019

G1152

G1084G1024

D1158

G1017

G1150

D1066

G1025G1085

D1129

D1038

D1157

G1056

G1028G1027G1023G1083

G1119

D1039D1090

D1127G1120G1118

G1174

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0

BJN

LF

TJN

CZHX

DZD

JNX

TA

QFD

TZHDZZH

XZHD

SZHD

BBN

DY

CHZH

NJN

ZHJNDYB

CHZHB

WXDSZHBKSHN

SHHQ


0 50 100 150 200 250 300 350 400Time (s)

55

6

65

7

75

8

9

85

Tota

l con

nect

ing

time

times104





0 50 100 150 200 250 300 350 40055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104





7 Conclusion






0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104






0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104








Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














1(1198792)






BJN

TJN

DZD

TA TZhD

XZhD

BBN

ChZh

ZhJN

ChZh

B

SZhB

ShH

Q

LF

CZhX JN

X

QFD ZZ

h

SZhD D

Y

NJN

DYB

WXD

KShN

Sector 1

Sector 3Sector 2



Sector From To 1003816100381610038161003816

1198791

1003816100381610038161003816

1003816100381610038161003816

1198792

1003816100381610038161003816





G1001G10

02G10

03G10

04G10

05G10

06G10

07G10

08G10

09G10

10G10

11G10

12G10

13G10

14G10

15G10

16G10

17G10

18G10

19G10

20G10

21G10

22G10

23G10

24G10

25G10

26G10

27G10

28G10

29G10

30G10

31G10

32D1033

D1034D1035

D1036D1037

D1038D1039

D1040D1041

D1042G10

43G10

44G10

45G10

46G10

47G10

48G10

49G10

50G10

51G10

52G10

53G10

54G10

55G10

56G10

57G10

58G10

59G10

60G10

61D1062

D1063D1064

D1065D1066

D1067

G1068

G1069

G1070

G1071

G1072

G1073

G1074

G1075

G1076

G1077

G1078

G1079

G1080

G1081

G1082

G1083

G1084

G1085

G1086

D1087D1088

D1089D1090

D1091D1092

G1093G1094

G1095G1096

G1097G1098

G1099G1100

G1101G1102

G1103G1104

G1105G1106

G1107G1108

G1109G1110

G1111G1112

G1113G1114

G1115G1116

G1117G1118

G1119G1120

G1121G1122

G1123G1124

D1125D1126

D1127D1128

D1129D1130

D1131D1132

D1133D1134

G1135G1136

G1137G1138

G1139G1140

G1141G1142

G1143G1144

G1145G1146

G1147G1148

G1149G1150

G1151G1152

G1153D1154

D1155D1156

D1157D1158

D1159

G1160G1161

G1162G1163

G1164G1165

G1166G1167

G1168G1169

G1170G1171

G1172G1173

G1174G1175

G1176G1177

G1178D1179

D1180D1181

D1182D1183

D1184

G1086

G1015

G1107G1108

D1184

G1122

D1041

G1151

D1091G1030G1031

D1131G1121

G1059

G1153

G1080

D1128

G1060

G1020

G1110

D1067

G1124

G1177

G1061

G1148G1123D1132G1116G1117D1130

D1183

G1081

G1147

D1040

G1149G1115

G1176D1181

G1029

G1178

G1026

D1133

D1037G1082G1032G1019

G1152

G1084G1024

D1158

G1017

G1150

D1066

G1025G1085

D1129

D1038

D1157

G1056

G1028G1027G1023G1083

G1119

D1039D1090

D1127G1120G1118

G1174

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0

BJN

LF

TJN

CZHX

DZD

JNX

TA

QFD

TZHDZZH

XZHD

SZHD

BBN

DY

CHZH

NJN

ZHJNDYB

CHZHB

WXDSZHBKSHN

SHHQ


0 50 100 150 200 250 300 350 400Time (s)

55

6

65

7

75

8

9

85

Tota

l con

nect

ing

time

times104





0 50 100 150 200 250 300 350 40055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104





7 Conclusion






0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104






0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104








Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










G1001G10

02G10

03G10

04G10

05G10

06G10

07G10

08G10

09G10

10G10

11G10

12G10

13G10

14G10

15G10

16G10

17G10

18G10

19G10

20G10

21G10

22G10

23G10

24G10

25G10

26G10

27G10

28G10

29G10

30G10

31G10

32D1033

D1034D1035

D1036D1037

D1038D1039

D1040D1041

D1042G10

43G10

44G10

45G10

46G10

47G10

48G10

49G10

50G10

51G10

52G10

53G10

54G10

55G10

56G10

57G10

58G10

59G10

60G10

61D1062

D1063D1064

D1065D1066

D1067

G1068

G1069

G1070

G1071

G1072

G1073

G1074

G1075

G1076

G1077

G1078

G1079

G1080

G1081

G1082

G1083

G1084

G1085

G1086

D1087D1088

D1089D1090

D1091D1092

G1093G1094

G1095G1096

G1097G1098

G1099G1100

G1101G1102

G1103G1104

G1105G1106

G1107G1108

G1109G1110

G1111G1112

G1113G1114

G1115G1116

G1117G1118

G1119G1120

G1121G1122

G1123G1124

D1125D1126

D1127D1128

D1129D1130

D1131D1132

D1133D1134

G1135G1136

G1137G1138

G1139G1140

G1141G1142

G1143G1144

G1145G1146

G1147G1148

G1149G1150

G1151G1152

G1153D1154

D1155D1156

D1157D1158

D1159

G1160G1161

G1162G1163

G1164G1165

G1166G1167

G1168G1169

G1170G1171

G1172G1173

G1174G1175

G1176G1177

G1178D1179

D1180D1181

D1182D1183

D1184

G1086

G1015

G1107G1108

D1184

G1122

D1041

G1151

D1091G1030G1031

D1131G1121

G1059

G1153

G1080

D1128

G1060

G1020

G1110

D1067

G1124

G1177

G1061

G1148G1123D1132G1116G1117D1130

D1183

G1081

G1147

D1040

G1149G1115

G1176D1181

G1029

G1178

G1026

D1133

D1037G1082G1032G1019

G1152

G1084G1024

D1158

G1017

G1150

D1066

G1025G1085

D1129

D1038

D1157

G1056

G1028G1027G1023G1083

G1119

D1039D1090

D1127G1120G1118

G1174

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 0

BJN

LF

TJN

CZHX

DZD

JNX

TA

QFD

TZHDZZH

XZHD

SZHD

BBN

DY

CHZH

NJN

ZHJNDYB

CHZHB

WXDSZHBKSHN

SHHQ


0 50 100 150 200 250 300 350 400Time (s)

55

6

65

7

75

8

9

85

Tota

l con

nect

ing

time

times104





0 50 100 150 200 250 300 350 40055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104





7 Conclusion






0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104






0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104








Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













0 50 100 150 200 250 300 350 400 45055

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104






0 50 100 150 200 250 300 350 400 450

6

65

7

75

8

85

Time (s)

Tota

l con

nect

ing

time

times104








Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Notations







119894119886


119894119889to station 119911

119894119886and


119894119889to station 119911


















Acknowledgment


References






























Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article a branch and bound algorithm for the

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