Research Article Optimizing the Long-Term Operating Plan...

Research ArticleOptimizing the Long-Term Operating Plan of RailwayMarshalling Station for Capacity Utilization Analysis

Wenliang Zhou1 Xia Yang12 Jin Qin1 and Lianbo Deng1

1 School of Traffic and Transportation Engineering Central South University Changsha 410075 China2Department of Civil and Environmental Engineering Rensselaer Polytechnic Institute Troy AL 12180 USA

Correspondence should be addressed to Lianbo Deng lbdengcsueducn

Received 25 March 2014 Revised 22 August 2014 Accepted 15 September 2014 Published 26 November 2014

Academic Editor Lionel Amodeo

Copyright copy 2014 Wenliang Zhou et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

Not only is the operating plan the basis of organizingmarshalling stationrsquos operation but it is also used to analyze in detail the capac-ity utilization of each facility inmarshalling station In this paper a long-termoperating plan is optimizedmainly for capacity utiliza-tion analysis Firstly amodel is developed tominimize railcarsrsquo average staying timewith the constraints ofminimum time intervalsmarshalling track capacity and so forth Secondly an algorithm is designed to solve thismodel based on genetic algorithm (GA) andsimulation method It divides the plan of whole planning horizon into many subplans and optimizes them with GA one by one inorder to obtain a satisfactory plan with less computing time Finally some numeric examples are constructed to analyze (1) the con-vergence of the algorithm (2) the effect of some algorithm parameters and (3) the influence of arrival train flow on the algorithm

1 Introduction

Railway marshalling station is the main place for disas-sembling and assembling trains in railway freight transportnetworks Generally it can be divided into train arriving yardrailcar marshalling yard and train departure yard consistingof many parallel tracks for different uses separately Thetrain arriving yard connects with the railcar marshallingyard by humps which are used to disassemble trains withgravitational pull while railcar marshalling yard is connectedto train departure yard by some lead tracks which allow forrepeatedly assembling railcars A typical marshalling stationlayout is shown in Figure 1 and the main operations can bedescribed as follows

(1) Inbound trains enter the arriving yard and wait fordisassembling

(2) Disassembling engine pushes inbound train throughthe hump after necessary technical inspections andthen the railcars from dissembling run on differentmarshalling tracks

(3) Assembling engines pull strings of railcars frommarshalling tracks to the departure track to make upoutbound trains

(4) Outbound trains depart from the departure yard afternecessary technical inspections

The above operations are entirely carried out accord-ing to a predetermined operating plan It arranges thearrival track the disassembling starting and ending timethe disassembling engine and track assignments for eachinbound train and the starting time ending time andthe engine of assembling the departure time the compo-nent railcars and storage track for each outbound trainThe improvement of operating plan greatly contributes todecreasing railcarsrsquo staying time in station and enhancingstationrsquos operating performance Besides it has anotherimportant purpose of comprehensively analyzing the capacityutilization of a marshalling station which is very bene-ficial for a railway company as it helps understand thestationrsquos limitations According to a long-term operatingplan the general changing relationship between capacityutilization of each facility and some characteristics of arrivaltrains (eg arrival time distribution) can be obtained byrepeatedly optimizing the long-term operating plan withdifferent arrival train flow which plays a significant rolein the capacity-related decision making for a railroad com-pany

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 251315 13 pageshttpdxdoiorg1011552014251315

2 The Scientific World Journal

Trains arriving yardRailcars marshalling yard

Trains departure yardHump Lead track

Figure 1 The layout of a typical marshalling station

Long-term operational planoptimization problem

Number of inbound trainsand their arrival times

Railcars directioncombinations

Operational constraintsresources and other rules

Number of outbound trains andtheir departure times

Optimization goals such asaverage waiting time of railcars

Figure 2 Long-term operating plan optimization problem

Generally the operating plan of one day is divided intomultiple time periodsrsquo plan called stage operating planwhich arrange the inbound trainsrsquo disassembling outboundtrainsrsquo assembling and shunting locomotive work So farthere are abundant studies on the stage operating plan Liet al [1] comprehensively reviewed the relative research onstage operating plans at marshalling stations Gulbrodsen [2]was one among the first who studied the optimization ofstage operating plan Yagar et al [3] studied the disintegrationsequences of all arrival trains during all stages Assad [4]considered the mutual interaction between different mar-shalling stations on the freight rail transportation networkand presented work on train integration plan Cicerone etal [5] mainly worked on the planning of schedules duringall stages Shafia et al [6] studied the robust of formationmethod for marshalling plans In addition some researcherssuch as Hein [7] Petersen [8 9] Turnquist and Daskin [10]andDimitri [11] also further studied the operations dwellingtimes and delays at marshalling stations

Compared with the abundant studies on stage planoptimization there are much fewer studies on long-termoperating planning They are different in planning scaleand marshalling purpose The stage plan usually uses 3hours as a stage which is relatively small in scale and aimsat providing reference for disassembling assembling andshunting locomotives On the other hand the long-termmarshalling plan is mainly used in analyzing the equipmentutilization conditions of hump arrival yardmarshalling yardand locomotives under various arrival train flows in order todiscover the capacity inefficiency at themarshalling station intime It covers a variety of arrival train flow densities and itsscale is at least ten times even more as big as the stage plan

However achieving a fine operating plan is challenging asit covers too many interrelated decisions It is NP-complete(see [12]) Most of researches struggled to obtain a betterstage operating plan for guiding stationsrsquo operations and the

mainmethods they used include simulation optimization andheuristics search

Simulation optimization is a typical method used tosolve the operating plan problem Gulbrodsen [2] firstlyused it to optimize the stage operating plan Lentink et al[13] established a mathematical model of stage operatingplan with network flow method However the simulationoptimization method has a low efficiency in solving theoperating plan problem due to the large scale

Heuristics search algorithms have been applied in manyfields nowadays as they can obtain a satisfactory solutionwithshorter computing time although they also have difficultyin achieving the best solution Shen et al [14] designed anadaptive colonial selection algorithm out of the immunealgorithm to solve the operating plan of railway marshallingyard Li et al [15] used the hybrid heuristic algorithmbased on the harmony search strategy to optimize the stageoperating plan

Besides Hein [7] and Turnquist and Daskin [10] appliedthe queuing theory to research the operating plan of mar-shalling yard and railcars staying time and their delays inmarshalling yard Ma et al [16] designed a self-learningalgorithm for conflict detection and adjustment to increaseoperating planrsquos fulfillment rates

Compared to the research of stage operating plan veryfew researchers strived to optimize the long-term operatingplan for analyzing the capacity utilization of marshallingstation In fact it can provide time-varying details of capacityutilization while other analysis methods generally supporta single utilization rate and so forth This paper studiesthe optimization problem of long-term operating plan asshown in Figure 2 and its main contributions are as followsAn optimization model of long-term operating plan is builtfirstly and an efficient solving method is designed based onheuristics search (namely GA) and simulation optimizationTo be specific in order to reduce the computing time the

The Scientific World Journal 3

long-term operating plan is divided into many subplansand then optimizes them one by one through combiningsubperiod rolling into GA

This paper is organized as follows The operating planoptimization problem formulation is firstly built in Section 2and then its solution framework is given in Section 3 Underthis framework a simulation method of operating planwith a given assembling sequence of railcars is proposed inSection 4 and the optimization method combining subpe-riod rolling into GA is designed in Section 5 After that thenumerical analysis is provided in Section 6 and at last someconclusions are presented in Section 7

2 Formulation of Operating PlanOptimization Problem

21 Notations and Assumptions All symbols for optimizingoperating plan are denoted as shown in Notations section

All assumptions are given as follows

Assumption 1 Each arrival track (departure track) only storesone inbound train (outbound train) at the same time and itslength is enough to hold all railcars of any train

Assumption 2 All railcarsrsquo size and shape are the sameand they are all allowed to be pushed to the hump fordisassembling

22 Optimization Goal Minimizing railcarsrsquo total stayingtime from their arrival to departure at the station is theprimary goal of operating plan optimization which has beenthe focus in many studies on operating plans for exampleLin and Cheng [17 18] Once railcars arrive at station with aninbound train if they will stay in station until the end of planhorizon railcarsrsquo total staying time is from their arrival timeto the end of plan horizon namely

119911119897 =

119873

sum

119894=1

(119879 minus 119886119894)1003816100381610038161003816119861119894

1003816100381610038161003816 (1)

In factmost railcars will assemble into an outbound train anddepart from station at the end of plan horizon Only a smallpart will stay in station So the time from railcars departingfrom station to the end of plan horizon should be subtractedfrom 119911119897 and railcarsrsquo actual staying time can be calculated asfollows

119911 = 119911119897 minus

119872

sum

119895=1

[(119879 minus 119889119895)

10038161003816100381610038161003816119861119895

10038161003816100381610038161003816] (2)

23 Constraints

231 Used Track Number Constraints At any time trainnumber in the arrival yard cannot exceed the total numberof arrival tracks 119880 the number of trains in the departureyard should not be higher than the total number of departure

tracks 119882 and the maximum number of used marshallingtracks is the capacity 119881

119880

sum

119906=1

L119906 le 119880

119882

sum

119908=1

L119908 le 119882

119881

sum

V=1LV le 119881

(3)

where L119906 L119908 LV are the occupation signs of arrival track119906 marshalling track 119908 and departure track V respectively If119894119906 gt 0 then L119906 = 1 otherwise L119906 = 0 If 119861119908 = 0 thenL119908 = 1 otherwiseL119908 = 0 If 119895V gt 0 thenLV = 1 otherwiseLV = 0

232 Marshalling Track Capacity Constraint The number ofrailcars on each marshalling track shall not exceed its storagecapacity 120585 namely

1003816100381610038161003816119861119908

1003816100381610038161003816le 120585 119908 = 1 2 119882 (4)

And the number of railcars of inbound (outbound) trainmeets the arrival (departure) track capacity requirementaccording to Assumption 1

233 Task Order Constraint of Inbound Train Inboundtrains firstly enter into arrival yard for technical inspectionsand then are pushed up the hump for disassembling byengine So inbound trainsrsquo arrival time starting time andending time of disassembling must meet the following con-straints

119886119894 le 119890119894 119894 = 1 2 119873

119890119894 + 120591 le ℎ119904

119894 119894 = 1 2 119873

ℎ119904

119894+

1003816100381610038161003816119861119894

1003816100381610038161003816

120575

= ℎ119890

119894 119894 = 1 2 119873

(5)

234 Task Order Constraint of Outbound Train All out-bound trains stay in the departure yard for technical inspec-tions after being assembled by combining some strings ofrailcars from marshalling yard and then depart from thereTherefore outbound trainsrsquo starting time and ending timeof assembling and departing time must meet the followingconstraints

max ℎ119890119894119887| 119887 isin 119861119895 le 119903

119904

119895 119895 = 1 2 119872

119903119904

119895+ 120598119891 + (119899119895 minus 1) sdot 120598119886 = 119903

119890

119895 119895 = 1 2 119872

119903119890

119895+ 120591 le 119889119895 119895 = 1 2 119872

(6)

For simplicity railcars stringsrsquo pull time described hereonly distinguishes first track pull and additional track pullsregardless of the number of railcars of each track pull


235 Minimum Time Interval Constraints Any two sametype tasks including disassembling assembling and trainsrsquodeparture must meet corresponding minimum time intervalrequirements namely

ℎ119904

1198941015840minus ℎ

119890

119894ge 119867119868 forall119894

1015840= 119894

119903119904

1198951015840minus 119903

119890

119895ge 119860119868 forall119895

1015840= 119895

119889119895+1 minus 119889119895 le 119863119868 119895 = 1 2 119872 minus 1

(7)

236 OutboundTrain Size andRailcarDirection CombinationConstraints The outbound railcars direction combinationspecifies which railcars can be put together and their orderon a departure train For example the feasible directioncombination ldquoA1 A2rdquo means that outbound trains can beformed with railcars ldquoA1rdquo or ldquoA2rdquo or ldquoA1 A2rdquo Therefore allrailcars constituting outbound train 119895must belong to a givendirection combination namely

119889119887 isin 119888 isin 119862 forall119887 isin 119861119895 (8)

Meanwhile the number of railcars of each outbound trainmust meet the minimum and maximum requirementsnamely

Γmin le

10038161003816100381610038161003816119861119895

10038161003816100381610038161003816le Γmax 119895 = 1 2 119872 (9)

237 Railcars to Track Assignment Constraints One mar-shalling track can be only assigned to railcars of one directionat any time Railcars of any other direction are allowed to stayin the marshalling track after it is cleared

119889119887 = 1198891198871015840 forall119887 1198871015840isin 119861119908 119908 = 1 2 119882 (10)

3 Solution Framework Based onGA and Simulation

In order tominimize railcarsrsquo staying time in station inboundtrains should be disassembled immediately once they enterthe arrival yard and railcars should be assembled into newoutbound trains once they meet all assembling requirementsIn fact some inbound trains cannot be disassembled in timebecause of the capacity limitation of disassembling engineLikewise some railcars cannot be assembled into outboundtrains timely due to the capacity limitation of assemblingengine So the following two problems have to be solved inthe first place

(1) Which inbound train should be disassembled firstwhich is equal to determining the disassemblingsequence of inbound trains

(2) Which railcars should be first assembled into an out-bound train which means to confirm the assemblingsequence of railcars

If the disassembling sequence of inbound trains is prede-termined it is a priority to assemble railcars whose directionsbelong to the same combination and maximize the number

Decision variable assembling sequence of railcars

Sim

ulat

ion

met

hodG

A

∙ Staying track of inbound train∙ Disassembling plan of inbound train∙ Assembling plan of outbound train∙ Storage track and departure time of outboundtrains

Figure 3 The solution framework of operating plan

2 5 1 4 3 6 5 2 6 4 1 2 3 5 4 5 6 2 1

3 6 5 4 1 6 2 1 3 1 6 2 3 5 3 4 6 1 4

4 2 5 2 1 6 4 2 5 4 5 2 1 2 3 4 2 6 2

1 3 5 2 3 6 5 2 3 2 5 6 1 2 4 3 2 5 4

Fitness 1

Code section 1

Codesection 2

Overlapcode

Fitness 2

Individual 1

Individual 2

Individual 3

Individual 4

middot middot middot




Figure 4 An optimization method combining subperiod rollinginto GA

of railcars to an outbound train Similarly if the assemblingsequence of railcars is pregiven it should firstly disassembleinbound train with the maximum railcars assembled intothe next outbound train Compared with railcarsrsquo assemblingsequence it is more difficult to optimize the disassemblingsequence of inbound trains as it hasmore relative constraintssuch as inbound trainsrsquo arriving sequence and arrival tracknumber Thus GA is chosen to optimize railcarsrsquo assemblingsequence in the paper and based on each acquired assem-bling sequence of railcars a simulation method is used todetermine disassembled trains starting and ending time ofdisassembling and assembling and so forth as shown inFigure 3

With a long scale of planning horizon it is inefficient tosearch the large-scale solution space of railcarsrsquo assemblingsequence within the whole planning horizon Consideringthat a long-term operating plan could be divided intoseveral short period subplans railcarsrsquo assembling sequenceis optimized by combining subperiod rolling into GA inthis paper Firstly a relative short period (eg 1 day) fromthe starting time of planning horizon is chosen and itsassembling sequence is optimized using GA Then roll fortha new same-length period and optimize this periodrsquos codeof each individual while holding previous periodsrsquo codeunchanged The whole operating plan will be obtained bycontinuously optimizing each subperiodrsquos code as shownin Figure 4 Specially there is an overlap code between twoadjacent code sections which contributes to their joiningMoreover the fitness is to evaluate the quality of code sectionswhich starts from individual subperiodrsquos first code and ends


Table 1 Eventsrsquo definition and occurrence prerequisite

Number Event name Definition Occurrence prerequisite

1 Inbound train entering Inbound trains enter arrival yardA Inbound train arrives atstationB Arrival yard has free tracks

2 Disassembling startDisassembling engine pushesinbound train up the hump fordisassembling

A Disassembling engine is freeBThere exist trains in the arrivalyard with technical inspectionscompletedCMarshalling yard has enoughtracks to store disassembledrailcars

3 Disassembling endAll railcars of currentdisassembled train have run onassigned marshalling tracks

A Disassembling train iscompleted

4 Assembling startAssembling engine starts pullingrailcars frommarshalling track toassemble outbound train

A Assembling engine is freeBMarshalling yard has enoughrailcars which can be assembledinto the same outbound trainCThere exist free tracks in thedeparture yard

5 Assembling endAssembling engine stops pullingrailcars and an outbound train isformed

A Assembling train is completed

6 Outbound traindeparture

Outbound train departs fromdeparture yard

A Departure yard has outboundtrains having technicalinspections finishedBMinimum interdepartureinterval is satisfied

at current periodrsquos last code In other words it evaluates theoperating plan till the end of current period

A more detailed explanation of the optimization methodshown in Figure 4 is given as follows The code sections 1and 2 of four individuals in Figure 4 represent the codes ofthe first and second short time periods respectively Eachgene is an integer between 1 and the combination numberof 119862 which represents the index of a direction combinationin 119862 For example code section ldquo25143652641rdquo which meansthat outbound trains will be assembled with direction combi-nations 1198882 1198885 1198881 1198884 1198883 1198886 1198885 1198882 successively represents thecode of individual 1 in the first short time period and codesection ldquo64123545621rdquo expresses the code of individual 1 inthe second short time period It is necessary to point out thatthere is an overlap code ldquo641rdquo between code section 1 andcode section 2 of individual 1 The representing way of otherindividualsrsquo code section is similar to theseWhen optimizingthe operating plan the code section 1 will be optimized firstlyusing GA and then keeping the code section 1 subtractingthe overlap code unchanged GA is used again to optimize thecode section 2Moreover while optimizing the code section 1fitness 1 of each individual is reckoned based on the operatingplan of the first short time period but while optimizing thecode section 2 fitness 2 of each individual is calculated basedon the operating plan from the first short time period to thesecond short time period

4 A Simulation Method of Operating Planwith Given Assembling Sequence of Railcars

41 Definitions of Events Six events related to operating planare defined in Table 1

Each eventrsquos occurrence depends on the satisfaction of itsrelative prerequisites so its occurrence time is the time whenall relative prerequisites are satisfied Each eventrsquos occurrencewould make some facilitiesrsquo state changed According to astationrsquos current state eventsrsquo occurrence time is determinedas follows

411 Inbound Train Entering Inbound trains are allowed toenter the marshalling yard with free tracks when they arriveat station If there are no free tracks at their arriving time theyhave to wait outside of there Denote by 119905119864119880 the earliest timewhen at least one free arrival track exists from now on andby 119906 a free arrival track If there is more than one free arrivaltrack at the same time it represents any one of them So theoccurrence time of inbound train entering event is the largestof 119886119894lowast and 119905119864119880 namely

119905119890 = max 119886119894lowast 119905119864119880 (11)

After this event occurs the time of inbound train 119894lowast entering

arrival yard is 119890119894lowast = 119905119890 and its storage track is 119906119894lowast = 119906Meanwhile the state of track 119906 transfers from being free tobeing occupied that is 119894119906 = 119894

lowast


412 Disassembling Start Denote by 119905119867 the earliest timewhen at least one inbound train satisfies the disassemblingrequirements and by 119901ℎ the earliest time when one disassem-bling engine is free Then the calculation of disassemblingstarting time should consider the following two situationsaccording to the relationship between 119905119867 and 119901ℎ

(1) When 119905119867 ge 119901ℎ In this case disassembling starting time119905119904

ℎis the time when inbound train satisfies the disassembling

requirement namely

119905119904

ℎ= 119905119867 (12)

Suppose train 119894119906 satisfies disassembling requirements on track119906

(2) While 119905119867 lt 119901ℎ In this situation disassembling startingtime 119905119904

ℎis the time of disassembling engine being free namely

119905119904

ℎ= 119901ℎ (13)

As there may be more than one inbound train which satisfiesdisassembling requirements until the time 119901ℎ denote by Ωℎ

the set of these trains In order to assemble more railcarsinto the next outbound train choose train 119894119906 containingmaximum railcars whose directions belong to the nextassembling combination 119888

lowast It satisfies

119872119888lowast

119894119906= max 119872119888

lowast

1198941199061015840| 1198941199061015840 isin Ωℎ (14)

where119872119888lowast

119894119906is the number of railcars whose directions belong

to combination 119888lowast on train 119894119906

When starting disassembling train 119894119906 at ℎ119904119894119906

= 119905119904

ℎ the

disassembling enginersquos state would be transferred from freeto busy and the state of track 119906 would be transferred fromoccupied to free that is

119894ℎ = 119894119906

119901ℎ = ℎ119904

119894119906+

10038161003816100381610038161003816119861119894119906

10038161003816100381610038161003816

120575

+ 119867119868

119894119906 = 0

(15)

413 Disassembling End The event of disassembling endonly occurs after starting disassembling train If all disassem-bling engines do not work now the disassembling endingtime is +infin otherwise it is the time when one disassemblingengine completes humping the current train 119894ℎ That is

119905119890

ℎ=

ℎ119904

119894ℎ+

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816

120575

119894ℎ gt 0

+infin 119894ℎ = 0

(16)

When train 119894ℎ ends disassembling at ℎ119890119894ℎ= 119905

119890

ℎ its railcars stay-

ing tracks are determined as follows If one track has storedrailcars and the car number is less than its maximum storagecapacity then the railcar would be humped into this trackor else any empty track would get this railcar Disassembling

enginersquos state would be transferred from busy to free and thenumber of storage railcars on somemarshalling tracks wouldincrease that is

119894ℎ = 0

119861119908 = 119861119908 cup 119887119896

119894ℎ| 119908

119896

119894ℎ= 119908 119896 = 1 2

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816 forall119908

(17)

414 Assembling Start Denote by 119905119864119881 the time when thedeparture yard has free tracks by 120587 the free assemblingengine and by 119901120587 = min1199011205871015840 its earliest free time Thenumber of railcars whose directions belong to combinations119888lowast reaches the minimum size of outbound train at time 119905119860Then the starting time of assembling is

119905119904

119886= max 119905119864119881 119901120587 119905119860 (18)

Therefore the next outbound train 119895lowast will be assembled

with component railcars 119861119895 by engine 120587119895lowast = 120587 at 119903119904119895lowast

= 119905119904

119886

Component railcars 119861119895 of train 119895lowast are determined as follows

Firstly choose one direction in combination 119888lowast in a given

sequence Then choose an occupied marshalling track whosestorage railcarsrsquo direction is the same as the selected one Ifits railcars number does not exceed the maximum size ofoutbound train then add them into 119861119895 otherwise choose apart of them just reaching the maximum size

The corresponding state changes include the free assem-bling engine120587 turning into busy and somemarshalling tracksturning into empty or their storage railcars number decreasesthat is

119895120587 = 119895lowast

119861119908 =

119861119908

(119861119895 cap 119861119908)

forall119908

(19)

415 Assembling End The event of assembling end shouldonly occur after starting assembling train If all assemblingengines do not work now the assembling ending time is +infinotherwise it is the time at which one assembling engine 120587

completes assembling the current train 119895120587 namely

119905 = min (119903119904

1198951205871015840+ 120598119891 + (119899119895

1205871015840minus 1) lowast 120598119886) | 1198951205871015840 gt 0 (20)

Therefore the assembling ending time is

119905119890

119886=

+infin if forall119895120587 = 0

119905 else(21)

When train 119895120587 ends assembling at 119903119890119895120587

= 119905119890

119886 assembling engine

120587 turns into free and departure track V storing train 119895120587 turnsinto occupied namely

119895V = 119895120587

119895120587 = 0

119901120587 = 119905119890

119886+ 119867119868

(22)


Choose the earliest event

Yes

No

assemble railcars

Yes

No

Initialize facilities states of the station

Does the occurrencetime of earliest event stay in the

simulation period

Update some related facilities states

Calculate the occurrence time of each event

Update the occurrence of each event

Stop simulation and output theoperating plan

Does the earliest event is theevent of assembling end

Choose the next used combination

Choose the first combination clowast in CQ to

clowast in CQ to assemble railcars

Figure 5 The simulation framework of operating plan

416 Outbound Train Departure The technical inspectionsand the minimum interdeparture interval requirementsshould be satisfied before outbound trains depart Denote by119905119868 the time at which the minimum interdeparture intervalsmeet and by 119905119876 the time at which an outbound train hascompleted the technical inspections

(1) If 119905119868 le 119905119876 then the train departure time is 119905119889 = 119905119876 andsuppose train 119895V completes the technical inspectionfirstly

(2) If 119905119868 gt 119905119876 then the train departure time is 119905119889 =

119905119868 As there may be more than one outbound trainthat completed the technical inspection at that timechoose train 119895V with the maximum number of railcarsso as to make more railcars depart from station

After train 119895V departing from station at 119889119895V = 119905119889 theoccupied departure track V turns into free namely

119895V = 0 (23)

42 The Simulation Framework and Steps As railcars whosedirections belong to the same combination could be assem-bled into the same outbound train railcarsrsquo assemblingsequence is described with a sequence of direction combi-nations For a given combinations sequence 119862119876 determineeach event occurrence time according to facilitiesrsquo usagestates from the starting times of planning horizon Thenchoose the earliest event and update its relevant equipmentrsquosstates This process is repeated to obtain (1) entering timearrival track for staying and disassembling plans of eachinbound train and (2) assembling plan storage departuretracks and departure times of each outbound train until theend time of the planning horizonThe simulation framework

for optimizing the operating plan with a given directioncombination sequence is shown in Figure 5

Denote 119878 = 119894119906 119861119908 119895V 119901120590 119894120590 119901120587 119895120587 todescribe stationrsquos states including the usage states of arrivaltracks marshalling tracks departure tracks disassemblingengines and assembling engines At the start of planninghorizon the usage state of each facility is initialized as follows

(1) All tracks of arrival marshalling and departure yardare empty at first namely

119894119906 = 0 119906 = 1 2 119880

119861119908 = 0 119908 = 1 2 119882

119895V = 0 V = 1 2 119881

(24)

(2) All disassembling engines are free originally namely

119901120590 = 0 forall120590

119894120590 = 0 forall120590

(25)

(3) All assembling engines are free in the beginningnamely

119901120587 = 0 forall120587

119895120587 = 0 forall120587

(26)

Based on the simulation framework shown in Figure 5and events definition in Section 41 the simulation steps foroperating plan with a given combination sequence 119862119876 aredesigned as shown in Algorithm 1


Input combinations sequence 119862119876 planning horizon [119879119904 119879119890] original state 119878(119879119904) at time 119879119904Output operating plan Φ(119879

119904 119879

119890) in period [119879

119904 119879

119890]

StartSet the first inbound train 119894

lowast as the earliest arrival train and 119895lowast= 1

Choose the first combination 119888lowast from 119862119876

Calculate the occurrence time 119905119890 119905119904

ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889 of each event

Determined the earliest event 119902 and its occurrence time 119905119902 = min 119905119890 119905119904

ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889

While 119905119902le 119879

119890

If event 119902 is inbound train entering then119894lowast= 119894

lowast+ 1

If event 119902 is assembling end thenAssemble railcars into outbound train 119895

lowast according to combination 119888lowast

119895lowast= 119895

lowast+ 1

Choose the next combination 119888lowast from 119862119876

Update the state 119878 (119905119902) related to event 119902

Update the occurrence time 119905119890 119905119904

ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889of each event

Update the earliest event 119902 and its occurrence time 119905119902End While

End

Algorithm 1

5 An Optimization Method ofLong-Term Operating Plan CombiningSubperiod Rolling into GA

In the following the optimization steps of this methodare presented after some key technologies of GA are beingexplained

51 Individual Encoding and Decoding Denote by |119862| thecombination number of candidate set 119862 and take theinteger between 1 and |119862| as a gene of individual codeas shown in Figure 6 where 1198881 1198882 119888M are the directioncombinations and only railcars whose directions belongto the same combination can be assembled into the sameoutbound train Thus the indexes of each combination in set119862 are 1 2 119872 respectively which will be used to formindividual code For example the code ldquo24315365 rdquo repre-sents the outbound trains assembled with direction combi-nations 1198882 1198884 1198883 1198881 1198885 1198883 1198886 1198885 successively Different codepositions may have the same gene value Each gene valuerepresents the direction combination of its number Eachcode can be divided into many small sections according tothe subperiods Each sectionrsquos lengthmay differ because of thedifference on assembled trainsrsquo number in each subperiodDenote by 119897119892

119904 1198971015840119892119904the starting and ending position of section 119892

in individual 119904 respectivelyEach cross and variation operation only handles the

code section of current subperiod and the code sections ofprevious subperiods would remain unchanged As shown inFigure 7 the code section 1 of individuals 1199041 1199042 1199043 is attainedafter optimizing the operating plan of subperiod 1 Whenoptimizing the operating plan of subperiod 2 only codesection 2 of individuals 1199041 1199042 1199043 is optimized while codesection 1 of them remains unchanged

For the convenience of individuals cross operation if thecurrent subperiodrsquos code section length of each individual

Directioncombination

2 5Individual code

c1 c2 c3 c4 c5 c6 cM



4 3 1 5 3 6

Figure 6 Encoding method

Code Codesection 1 section 2

1 3 2 4 2 3 1 4 2 1 (3 1)

1 4 3 4 2 3 4 1 2 3 (4)

1 3 1 4 3 2 3 4 1 4 3 2

s1

s2

s3 middot middot middot



Figure 7Themethod of keeping current code sectionrsquos length to bethe same

varies after each genetic iteration at this point some newgenes will be randomly generated and appended to theshorter code sections in order to keep the same length as otherindividual code sections As shown in Figure 7 code sectionldquo31421rdquo is the code section of individual 1199041 after previousgenetic iterations Its length is 5 while the maximum lengthof the code section 2 of individual 1199043 is 7 At the momenttwo genes ldquo31rdquo are randomly generated and appended to thecode section 2 of individual 1199041 which will not affect fitnessrsquosreckon of individual 1199041 and are simply used for keeping thesame code section length as others


Input stationrsquos initial state 1198780 planning horizon [0 119879] sub-period length 119871 overlap lengthΔ119871 population size 120579 maximum generation count119883 for the best individual continuallykeeping unchanged maximum and minimum crossover probability 119901119888

1 119901119888

2 maximum and

minimum mutation probability 1199011198981 119901119898

2

Output operating plan Φ(0 119879) in planning horizon [0 119879]Start

Initialize current sub-period 119892 = 1 and [119879119904

119892 119879

119890

119892] = [0 119871]

Generate new individuals of population 119875 and initialize their code sections of sub-period 119892While 119879119890

119892lt 119879

Set continually unchanged count 119909 = 1 of best individualWhile 119909 le 119883

Compute individualrsquos fitness 119891119894 in population 119875 and determine the optimal 119904lowastChoose 120579 individuals from population 119875 to form a temporary population 119875

1015840 by selection operationTake crossover operation for pairs in population 119875

1015840 and update population 1198751015840

Take mutation operation for individuals of population 1198751015840 and update population 119875

1015840Determine the optimal individual 119904lowast

1198751015840of population 119875

1015840If 119904lowast

1198751015840is better than 119904

lowast then set 119904lowast = 119904lowast

1198751015840 and 119909 = 1 else 119909 = 119909 + 1

Set 119875 = 1198751015840

End whileRoll the sub-period 119892 = 119892 + 1 and 119879

119904

119892= 119879

119890

119892minus Δ119871 119879119890

119892= min119879119904

119892+ 119871 119879

Initialize each individualrsquos code section of sub-period 119892 in population 119875End while

End

Algorithm 2

52 Individual Fitness Calculation Each individualrsquos fitnessis calculated to evaluate the code sections from codersquos firstgene to the last gene of current subperiodrsquos code sectionDenote by 119891119894(119879119904 119879119890) the fitness of individual 119904119894 in subperiod[119879119904 119879119890] Firstly calculate railcarsrsquo staying time119885119894 of individual119904119894 in period [0 119879119890] and transfer the optimization goal fromminimizing 119885119894 to maximizing 119879119890 minus 119885119894 then compute fitness119891119894(119879119904 119879119890) with scale transformation as follows

119891119894 (119879119904 119879119890) =

119890((119879119890minus119885119894)120593timeslog099119909)

sum119904119895isin119878

119890((119879119890minus119885119895)120593timeslog099119909)

(27)

where 119909 is the generation time 120593 is a parameter relating toscale transformation and its value is 150 generally

53 Genetic Operators The basic genetic operators of GA areselection crossover andmutation which are given as followsand more details about the genetic operators can be obtainedin Dimitrirsquos book of Omega A Competent Genetic Algorithmfor Solving Permutation and Scheduling Problems

531 Selection Operator New populationrsquos individuals areselected from the current population by roulette methodbased on their fitness Firstly the selection probability andcumulative probability range of each individual are reckonedaccording to their fitness After that a random numberbetween 0 and 1 is generated and the individual whosecumulative probability range covers this number is selected

532 Crossover Operator The single-point crossover isselected as the crossover operator here Firstly an intersectionof the individual strings is elected randomly Then the

3 1 4 2 4 3 1

2 3 4 1 3 2 4

3 1 4 2 3 2 4

2 3 4 1 4 3 1x

c1

c2

s1

s2

Figure 8 The single-point crossover method

following part of the individual string at the intersectionare exchanged to generate two new individuals A simpleexample is given as shown in Figure 8 Individuals 1199041 and 1199042

are selected for crossover the position 119909 is the intersectionand individuals 1198881 and 1198882 are the new individuals after single-point crossover operator

533MutationOperator Thecombination of uniformmuta-tion and basic bit mutation is adopted as the mutationoperator in order to search freely over the whole search spacein the initial stage and only search in the local scope in thelater algorithm In other words uniform mutation whichmakes each gene value mutated with a larger probabilityis adopted in the early stage while basic bit mutation isemployed in the later stage and each gene value is mutatedwith a smaller probability in this stage

54 Optimization Steps of GA Combined with SubperiodRolling Based on the above key technologies of GA andthe simulation method in Section 4 the optimization stepsfor optimizing the long-term operating plan are designed asshown in Algorithm 2


Table 2 Some parametersrsquo values of marshalling station

Number Parameter name Parameter value1 119880 102 119882 423 119881 74 120585 605 |120590| 16 |120587| 27 119863119868 10min8 120575 3 railcars per min9 119867119868 10min10 119860119868 5min11 120591 45min12 120598119891 10min13 120598119886 15min per pull14 Γmin 50 railcars15 Γmax 140 railcars

6 Numeric Examples

In this section based on the optimization results aboutoperating plan of the marshalling station shown in Figure 1(1) the convergence of the designed algorithm (2) the effectof some algorithmparameters and (3) the influence of arrivaltrain flow on the algorithm will be analyzed

The algorithm is developed with computer language Con the platform of Microsoft Visual Studionet and runs onthe computer with the system of Microsoft Windows XP(Home Edition) RAM configuration of Pentium(R) Dual-Core CPU E5800 320GHZ 319GHz 296GB For GAits population size is 120579 = 100 its maximum and minimumcrossover probability are 119901119888

1= 09 and 119901119888

2= 05 respec-

tively its maximum and minimum mutation probability are119901119898

1= 005 and 119901119898

2= 0005 separately and its maximum

generation count for the best individual continually keepingunchanged is119883 = 50

Some parametersrsquo values of arrival yardmarshalling yardand departure yard are shown in Table 2 In addition railcarsrsquomaximum and minimum number of arrival trains are 50and 100 respectively railcars directions include ldquoADrdquo ldquoAFrdquoldquoAWrdquo ldquoAYrdquo ldquoAHrdquo ldquoAKrdquo ldquoANrdquo ldquoAPrdquo ldquoAJrdquo ldquoBGrdquo ldquoARrdquo ldquoAXrdquo andldquoAVrdquo and the arrival times of inbound trains distribute in thewhole day and their fluctuation is described by their varianceAccording to those parametersrsquo values the railcarsrsquo numberdirection and arrival time of arrival trains are generatedrandomly Moreover railcars with directions belonging tothe same direction combination can be assigned to the sameoutbound train as shown in Table 3

61 Algorithm Convergence The variation relation of railcarsaverage staying time along with the computing time is drawnas shown in Figure 9 when optimizing the 5-day operatingplan with subperiod length of 18 h and overlap length of 2 hIn this example the average number of arrival trains per dayis 30 and the variance of their arrival time is 1

Table 3 Directions combination for assembling outbound train

Number Combination1 AD AF2 AF AW and AY3 AV4 AH AK5 AR AW and AY6 AX7 AN AP AJ and BG

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18220230240250260270280290300310

Railc

ars a

vera

ge st

ayin

g tim

e (m

in)

Computing time (min)

Figure 9 The variation relation of railcars average staying timealong with the computing time

As shown in Figure 9 the 5-day planning horizon isdivided into 8 subperiods namely [0 1080) [960 2040)[1920 3000) [2880 3960) [3840 4920) [4800 5880) [57606840) and [6720 7200] so the 5-day operating plan isaccordingly obtained through rolling optimizing the sub-plans of these 8 subperiods Their computing times are13min 18min 38min 43min 27min 14min 14minand 09min respectively and railcarsrsquo average staying timesare 228min 245min 256min 265min 270min 270min268min and 265min per railcar Although the computingtimes of each subplan are different obviously they areacceptable Railcars average staying times vary a little from250min to 270min and from 228min to 245min for the 1stand 2nd subplans as all facilities are free and can disassembleand assemble trains in time in the beginning Therefore it isnot difficult to draw the conclusion that this algorithm canconverge to a satisfied plan with an acceptable computingtime of 176min

62 Effect of Algorithm Parameter on the Algorithm Whenthe average number of arrival trains per day is 30 and thevariance of their arrival times is 1 railcarsrsquo average stayingtimes and the computing times of 5-day operating plans areoptimized with different subperiod and overlap lengths asshown in Figures 10 and 11 respectively

When the overlap length is 2 h railcars average stayingtime stays in a narrow range of 250sim255min but thecomputing time changes largely from 218min to 127minalong with the increasing of subperiodrsquos length from 6 h to12 hThen with the continuous increase from 12 h to 30 h not


Table 4 The optimization results of different planning horizons and inbound train flows

Planning horizon (d) Inbound trainsnumber

Variance ofarrival time

Railcars average staying time (min) Computing time (min)L = 6 h L = 12 h L = 24 h L = 6 h L = 12 h L = 24 h

5

100 05 268 267 271 95 72 991 271 272 279 97 73 96

150 05 246 247 256 221 134 2271 251 254 273 218 127 221

200 05 317 318 339 472 309 4471 319 321 344 484 316 435

10

200 05 270 273 282 208 192 2121 274 276 283 213 182 205

300 05 248 251 257 682 467 5671 253 258 262 675 472 575

400 05 324 325 332 1102 859 9681 324 326 334 1104 863 972

230235240245250255260265270275280285

6 8 10 12 14 16 18 20 22 24 26 28 30Railc

ars a

vera

ge st

ayin

g tim

e (m

in )

Subperiod (h)p = 2hp = 3h

Figure 10 Railcars average staying time of different subperiod andoverlap lengths

only does railcars average staying time promote faster from253min to 276min but also the computing time increasesquickly from 127min to 276minThe same trend exists whilethe overlap length is 3 h So it has a moderate subperiodlength such as 12 h in this example withwhich a satisfied plancan be obtained with less computing time

In addition two variation curves of railcars averagestaying time are mainly consistent when the overlap lengthsare 2 h and 3 h respectively but the computing time of theformer is slightly less than that of the latter Hence it issuggested to adopt a shorter overlap to optimize the plan

63 Influence of Inbound Train Flow on the AlgorithmThe optimization results of different planning horizons andinbound train flows are shown in Table 4 In these examplesthere are two planning horizons 5 days and 10 days threeinbound trains numbers 20 railcars 30 railcars and 40railcars per day and two variances of their arrival time 05and 1 It is found that the computing time increases obviouslyalong with inbound trainsrsquo number increase from 20 railcars

1012141618202224262830

6 8 10 12 14 16 18 20 22 24 26 28 30

Com

putin

g tim

e (m

in)


Figure 11 Computing times of different subperiod and overlaplengths

to 40 railcars per day and that railcars average staying timeis optimal when inbound trainsrsquo number is 30 Besides thearrival time variances change from 05 to 1 of inbound trainsresulting in a slight increase in railcarsrsquo average staying timeand producing a little effect on the computing time

7 Conclusion

In order to provide more detailed data for analyzing thecapacity utilization ofmarshalling station from the long-termoperating plan amodel of long-termoperating plan is built tominimize railcarsrsquo average staying time under the constraintsof minimum time intervals marshalling track capacitiesand so forth Its solving algorithm is designed based onGA and simulation method in which railcarsrsquo assemblingsequences are optimized by GA and then the operation plansare obtained based on the previously achieved assemblingsequences through the simulationmethod In order to reducethe computing time the whole planning horizon is dividedinto many subperiods which are optimized sequentially As


the optimization of each subplan converges in a short timethe whole algorithm can converge to a satisfied plan with anacceptable computing time Moreover an appropriate lengthof subperiods helps decrease the computing time In additionthe growing of arrival trains number leads to the fast increasein computing time while the change of arrival time varianceshas a slight effect on computing time

This paper specifically optimizes the operating plan ofone type of marshalling station in which trains of up anddowndirections donot interferewith each other but in realitythey may interfere at some other type of marshalling stationsTherefore further study on optimizing the operating plansof those stations can be conducted in the future In additionsome ignored constraints should be taken into account in theoptimization of marshalling stationsrsquo operating plans in thefuture study such as some railcars forbidden to disassemblethrough a hump

Notations

Parameters

119880 Track number of train arrival yard119906 119906 = 1 2 119880 arrival track119882 Track number of railcar marshalling yard119908 119908 = 1 2 119882marshalling track120585 The maximum railcars number that can be held in

each marshalling track119881 Track number of train departure yardV V = 1 2 119881 departure track119873 Inbound train number within planning horizon|120590| Disassembling enginersquos count120590 120590 = 1 2 |120590| disassembling engine|120587| Assembling enginersquos count120587 120587 = 1 2 |120587| assembling engine119888 Railcar direction combination railcars whose

directions belong to the same combination couldbe assembled into the same outbound train

119862 119862 = 119888 set of railcar direction combinations119894 119894 = 1 2 119873 inbound train119886119894 Arrival time of inbound train 119894 at station119861119894 Railcar sequence of train 119894

|119861119894| Railcar number of train 119894

119887119896

119894 119896th railcar of train 119894

119889119896

119894 119896th railcarrsquos direction of train 119894

119894119887 Inbound train of railcar 119887119872 Outbound train number within planning horizon119895 119895 = 1 2 119872 outbound train119863119868 Minimum train interdeparture time119867119868 Minimum disassembling interval119860119868 Minimum assembling interval120575 Humping rate namely the number of

disassembling railcars per minute120598119891 Average time to perform first track pull120598119886 Average time to perform an additional track pullΓmin Minimum railcars number of outbound train

Γmax Maximum railcars number of outbound train120591 Technical inspection time for both inbound and

outbound trains[0 119879] Planning horizon

Variables

119890119894 Arrival time of train 119894 at arrival yard if no freearrival tracks exist when train 119894 arrives at station119890119894 = 119886119894 otherwise 119890119894 = 119886119894

119906119894 Arrival track occupied by train 119894

120590119894 Disassembling engine working for train 119894

ℎ119904

119894 Starting time of disassembling train 119894

ℎ119890

119894 Ending time of disassembling train 119894

119908119896

119894 Assigned marshalling track for 119896th railcar afterhumping train 119894

119888119895 Direction combination for assembling train 119895

120587119895 Assembling engine working for train 119895

119903119904

119895 Starting time of assembling train 119895

119903119890

119895 Ending time of assembling train 119895

119861119895 Railcar sequence of train 119895

V119895 Departure track occupied by train 119895

119889119895 Departure time of train 119895

Intermediate Variables

Φ(119879119904 119879119890) Operating plan in period (119879119904 119879119890)

119894119906 Inbound train stored in arrival track 119906 now if119894119906 = 0 it means no train occupies this track atthis time

119861119908 Railcars set stored in marshalling track 119908 now|119861119908| Number of railcars stored in marshalling track 119908

now119895V Outbound train stored in departure track V now

if 119895V = 0 it shows that no train occupies thistrack at this time

119901120590 The earliest time when the disassembling engine120590 can work from now on

119901120587 The earliest time when the assembling engine 120587can work from now on

119894lowast The next inbound train which will arrive at

station119895lowast The number of trains which has been assembled

or is being processed now119888lowast Direction combination chosen for assembling

next outbound train119894120590 Train for which disassembling engine 120590 works

now if 119894120590 = 0 it shows the engine 120590 is free119895120587 Train for which assembling engine 120587 works now

if 119895120587 = 0 it shows engine 120587 is free

Conflict of Interests

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


Acknowledgments

This research was partially supported by Natural ScienceFoundation of China (Grant no 71401182) and DoctoralScientific Foundation of the Ministry of Education of China(Grant no 20120162120042)

References

[1] H-D Li S-W He B-H Wang and Y-S Shen ldquoSurvey ofstage plan for railway marshalling stationrdquo Journal of the ChinaRailway Society vol 33 no 8 pp 13ndash22 2011

[2] O Gulbrodsen ldquoOptimal planning of marshalling yard byoperation researchrdquo PROC vol 12 no 5 pp 226ndash233 1963

[3] S Yagar F F Saccomanno and Q Shi ldquoAn efficient sequencingmodel for humping in a rail yardrdquo Transportation Research PartA General vol 17 no 4 pp 251ndash262 1983

[4] A A Assad ldquoModelling of rail networks toward a rout-ingmakeup modelrdquo Transportation Research Part B vol 14 no1-2 pp 101ndash114 1980

[5] S Cicerone G Dangelo and G D StefanoWorkshop on Algo-rithmic Approach for Transportation Modeling Optimizationand Systems Internationales Begegnungs Dagstuhl Germany2007

[6] MA Shafia S J Sadjadi andA Jamili ldquoRobust train formationplanningrdquo Proceedings of the Institution of Mechanical EngineersPart F Journal of Rail and Rapid Transit vol 224 no 2 pp 75ndash90 2010

[7] O Hein ldquoA two-stage queue model for a marshalling yardrdquo RailInternational vol 3 no 4 pp 249ndash259 1972

[8] E R Petersen ldquoRailyard modeling part I Prediction of put-through timerdquo Transportation Science vol 11 no 1 pp 37ndash491977

[9] E R Petersen ldquoRailroad modeling part II the effect of yardfacilities on congestionrdquoTransportation Science vol 11 no 1 pp50ndash59 1977

[10] M A Turnquist and M S Daskin ldquoQueueing models ofclassification and connection delay in rail yardsrdquoTransportationScience vol 16 no 2 pp 207ndash230 1982

[11] K DimitriOmeGA A Competent Genetic Algorithm for SolvingPermutation and Scheduling Problems Kluwer Academic Pub-lishers 2002

[12] E Dahlhaus P Horak M Miller and J F Ryan ldquoThe trainmarshalling problemrdquo Discrete Applied Mathematics vol 103no 1ndash3 pp 41ndash54 2000

[13] R M Lentink P Fioole L G Kroon and C Woudt ApplyingOperations Research Techniques to Planning Train ShuntingJohn Wiley amp Sons Hoboken N J USA 2006

[14] Y-S Shen S-W He B-H Wang and M-R Mu ldquoStudy onallocation of wagon-flow in phase plan by using immunealgorithmrdquo Journal of the China Railway Society vol 31 no 4pp 1ndash6 2009

[15] H-D Li S-WHe Y Jing and SWang ldquoWagon-flow allocationoptimization of stage plan at marshaling station in considera-tion of different size limitations of departure trainsrdquo Journal ofthe China Railway Society vol 34 no 7 pp 10ndash17 2012

[16] L Ma J Guo G W Chen and R Guo ldquoResearch on automaticadjustment of the phase plan in railway marshallingrdquo Journal ofTransportation Engineering and Information vol 11 no 3 pp18ndash28 2013

[17] E Lin and C Cheng ldquoYardsim a rail yard simulation frame-work and its implementation in a major railroad in the USrdquo inProceedings of the Winter Simulation Conference (WSC 09) pp2532ndash2541 Austin Tex USA December 2009

[18] E Lin and C Cheng ldquoSimulation and analysis of railroadhump yards in North Americardquo in Proceedings of the WinterSimulationConference (WSC rsquo11) pp 3710ndash3718December 2011

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

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Applied MathematicsJournal of


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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


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Algebra

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


Trains arriving yardRailcars marshalling yard

Trains departure yardHump Lead track

Figure 1 The layout of a typical marshalling station

Long-term operational planoptimization problem

Number of inbound trainsand their arrival times

Railcars directioncombinations

Operational constraintsresources and other rules

Number of outbound trains andtheir departure times

Optimization goals such asaverage waiting time of railcars

Figure 2 Long-term operating plan optimization problem

Generally the operating plan of one day is divided intomultiple time periodsrsquo plan called stage operating planwhich arrange the inbound trainsrsquo disassembling outboundtrainsrsquo assembling and shunting locomotive work So farthere are abundant studies on the stage operating plan Liet al [1] comprehensively reviewed the relative research onstage operating plans at marshalling stations Gulbrodsen [2]was one among the first who studied the optimization ofstage operating plan Yagar et al [3] studied the disintegrationsequences of all arrival trains during all stages Assad [4]considered the mutual interaction between different mar-shalling stations on the freight rail transportation networkand presented work on train integration plan Cicerone etal [5] mainly worked on the planning of schedules duringall stages Shafia et al [6] studied the robust of formationmethod for marshalling plans In addition some researcherssuch as Hein [7] Petersen [8 9] Turnquist and Daskin [10]andDimitri [11] also further studied the operations dwellingtimes and delays at marshalling stations

Compared with the abundant studies on stage planoptimization there are much fewer studies on long-termoperating planning They are different in planning scaleand marshalling purpose The stage plan usually uses 3hours as a stage which is relatively small in scale and aimsat providing reference for disassembling assembling andshunting locomotives On the other hand the long-termmarshalling plan is mainly used in analyzing the equipmentutilization conditions of hump arrival yardmarshalling yardand locomotives under various arrival train flows in order todiscover the capacity inefficiency at themarshalling station intime It covers a variety of arrival train flow densities and itsscale is at least ten times even more as big as the stage plan

However achieving a fine operating plan is challenging asit covers too many interrelated decisions It is NP-complete(see [12]) Most of researches struggled to obtain a betterstage operating plan for guiding stationsrsquo operations and the

mainmethods they used include simulation optimization andheuristics search

Simulation optimization is a typical method used tosolve the operating plan problem Gulbrodsen [2] firstlyused it to optimize the stage operating plan Lentink et al[13] established a mathematical model of stage operatingplan with network flow method However the simulationoptimization method has a low efficiency in solving theoperating plan problem due to the large scale

Heuristics search algorithms have been applied in manyfields nowadays as they can obtain a satisfactory solutionwithshorter computing time although they also have difficultyin achieving the best solution Shen et al [14] designed anadaptive colonial selection algorithm out of the immunealgorithm to solve the operating plan of railway marshallingyard Li et al [15] used the hybrid heuristic algorithmbased on the harmony search strategy to optimize the stageoperating plan

Besides Hein [7] and Turnquist and Daskin [10] appliedthe queuing theory to research the operating plan of mar-shalling yard and railcars staying time and their delays inmarshalling yard Ma et al [16] designed a self-learningalgorithm for conflict detection and adjustment to increaseoperating planrsquos fulfillment rates

Compared to the research of stage operating plan veryfew researchers strived to optimize the long-term operatingplan for analyzing the capacity utilization of marshallingstation In fact it can provide time-varying details of capacityutilization while other analysis methods generally supporta single utilization rate and so forth This paper studiesthe optimization problem of long-term operating plan asshown in Figure 2 and its main contributions are as followsAn optimization model of long-term operating plan is builtfirstly and an efficient solving method is designed based onheuristics search (namely GA) and simulation optimizationTo be specific in order to reduce the computing time the










119911119897 =

119873

sum

119894=1

(119879 minus 119886119894)1003816100381610038161003816119861119894

1003816100381610038161003816 (1)


119911 = 119911119897 minus

119872

sum

119895=1

[(119879 minus 119889119895)

10038161003816100381610038161003816119861119895

10038161003816100381610038161003816] (2)

23 Constraints



119880

sum

119906=1

L119906 le 119880

119882

sum

119908=1

L119908 le 119882

119881

sum

V=1LV le 119881

(3)



1003816100381610038161003816119861119908

1003816100381610038161003816le 120585 119908 = 1 2 119882 (4)



119886119894 le 119890119894 119894 = 1 2 119873

119890119894 + 120591 le ℎ119904

119894 119894 = 1 2 119873

ℎ119904

119894+

1003816100381610038161003816119861119894

1003816100381610038161003816

120575

= ℎ119890

119894 119894 = 1 2 119873

(5)


max ℎ119890119894119887| 119887 isin 119861119895 le 119903

119904

119895 119895 = 1 2 119872

119903119904

119895+ 120598119891 + (119899119895 minus 1) sdot 120598119886 = 119903

119890

119895 119895 = 1 2 119872

119903119890

119895+ 120591 le 119889119895 119895 = 1 2 119872

(6)




ℎ119904

1198941015840minus ℎ

119890

119894ge 119867119868 forall119894

1015840= 119894

119903119904

1198951015840minus 119903

119890

119895ge 119860119868 forall119895

1015840= 119895

119889119895+1 minus 119889119895 le 119863119868 119895 = 1 2 119872 minus 1

(7)




Γmin le

10038161003816100381610038161003816119861119895

10038161003816100381610038161003816le Γmax 119895 = 1 2 119872 (9)


119889119887 = 1198891198871015840 forall119887 1198871015840isin 119861119908 119908 = 1 2 119882 (10)







Sim

ulat

ion

met

hodG

A



2 5 1 4 3 6 5 2 6 4 1 2 3 5 4 5 6 2 1

3 6 5 4 1 6 2 1 3 1 6 2 3 5 3 4 6 1 4

4 2 5 2 1 6 4 2 5 4 5 2 1 2 3 4 2 6 2

1 3 5 2 3 6 5 2 3 2 5 6 1 2 4 3 2 5 4

Fitness 1

Code section 1

Codesection 2

Overlapcode

Fitness 2

Individual 1

Individual 2

Individual 3

Individual 4





























119905119890 = max 119886119894lowast 119905119864119880 (11)



lowast





requirement namely

119905119904

ℎ= 119905119867 (12)




119905119904

ℎ= 119901ℎ (13)



lowast It satisfies

119872119888lowast

119894119906= max 119872119888

lowast

1198941199061015840| 1198941199061015840 isin Ωℎ (14)





= 119905119904

ℎ the


119894ℎ = 119894119906

119901ℎ = ℎ119904

119894119906+

10038161003816100381610038161003816119861119894119906

10038161003816100381610038161003816

120575

+ 119867119868

119894119906 = 0

(15)


119905119890

ℎ=

ℎ119904

119894ℎ+

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816

120575

119894ℎ gt 0

+infin 119894ℎ = 0

(16)


119890




119894ℎ = 0

119861119908 = 119861119908 cup 119887119896

119894ℎ| 119908

119896

119894ℎ= 119908 119896 = 1 2

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816 forall119908

(17)


119905119904

119886= max 119905119864119881 119901120587 119905119860 (18)



= 119905119904

119886





119895120587 = 119895lowast

119861119908 =

119861119908

(119861119895 cap 119861119908)

forall119908

(19)



119905 = min (119903119904

1198951205871015840+ 120598119891 + (119899119895

1205871015840minus 1) lowast 120598119886) | 1198951205871015840 gt 0 (20)


119905119890

119886=


119905 else(21)


= 119905119890



119895V = 119895120587

119895120587 = 0

119901120587 = 119905119890

119886+ 119867119868

(22)



Yes

No

assemble railcars

Yes

No



simulation period















119895V = 0 (23)





119894119906 = 0 119906 = 1 2 119880

119861119908 = 0 119908 = 1 2 119882

119895V = 0 V = 1 2 119881

(24)


119901120590 = 0 forall120590

119894120590 = 0 forall120590

(25)


119901120587 = 0 forall120587

119895120587 = 0 forall120587

(26)




119904 119879

119890) in period [119879

119904 119879

119890]





ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889 of each event


ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889

While 119905119902le 119879

119890


lowast+ 1



119895lowast= 119895

lowast+ 1




ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889of each event


End

Algorithm 1









2 5Individual code




4 3 1 5 3 6



1 3 2 4 2 3 1 4 2 1 (3 1)

1 4 3 4 2 3 4 1 2 3 (4)

1 3 1 4 3 2 3 4 1 4 3 2

s1

s2








1 119901119888

2 maximum and


2



119892 119879

119890

119892] = [0 119871]


119892lt 119879







1198751015840of population 119875

1015840If 119904lowast

1198751015840is better than 119904


1198751015840 and 119909 = 1 else 119909 = 119909 + 1

Set 119875 = 1198751015840


119904

119892= 119879

119890

119892minus Δ119871 119879119890

119892= min119879119904

119892+ 119871 119879


End

Algorithm 2


119891119894 (119879119904 119879119890) =

119890((119879119890minus119885119894)120593timeslog099119909)

sum119904119895isin119878

119890((119879119890minus119885119895)120593timeslog099119909)

(27)





3 1 4 2 4 3 1

2 3 4 1 3 2 4

3 1 4 2 3 2 4

2 3 4 1 4 3 1x

c1

c2

s1

s2









6 Numeric Examples



1= 09 and 119901119888

2= 05 respec-


1= 005 and 119901119898







0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18220230240250260270280290300310

Railc

ars a

vera

ge st

ayin

g tim

e (m

in)











5

100 05 268 267 271 95 72 991 271 272 279 97 73 96

150 05 246 247 256 221 134 2271 251 254 273 218 127 221

200 05 317 318 339 472 309 4471 319 321 344 484 316 435

10

200 05 270 273 282 208 192 2121 274 276 283 213 182 205

300 05 248 251 257 682 467 5671 253 258 262 675 472 575

400 05 324 325 332 1102 859 9681 324 326 334 1104 863 972

230235240245250255260265270275280285

6 8 10 12 14 16 18 20 22 24 26 28 30Railc

ars a

vera

ge st

ayin

g tim

e (m

in )






1012141618202224262830

6 8 10 12 14 16 18 20 22 24 26 28 30

Com

putin

g tim

e (m

in)




7 Conclusion





Notations

Parameters






119887119896


119889119896






Variables




ℎ119904


ℎ119890


119908119896




119903119904


119903119890






















Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra


















119911119897 =

119873

sum

119894=1

(119879 minus 119886119894)1003816100381610038161003816119861119894

1003816100381610038161003816 (1)


119911 = 119911119897 minus

119872

sum

119895=1

[(119879 minus 119889119895)

10038161003816100381610038161003816119861119895

10038161003816100381610038161003816] (2)

23 Constraints



119880

sum

119906=1

L119906 le 119880

119882

sum

119908=1

L119908 le 119882

119881

sum

V=1LV le 119881

(3)



1003816100381610038161003816119861119908

1003816100381610038161003816le 120585 119908 = 1 2 119882 (4)



119886119894 le 119890119894 119894 = 1 2 119873

119890119894 + 120591 le ℎ119904

119894 119894 = 1 2 119873

ℎ119904

119894+

1003816100381610038161003816119861119894

1003816100381610038161003816

120575

= ℎ119890

119894 119894 = 1 2 119873

(5)


max ℎ119890119894119887| 119887 isin 119861119895 le 119903

119904

119895 119895 = 1 2 119872

119903119904

119895+ 120598119891 + (119899119895 minus 1) sdot 120598119886 = 119903

119890

119895 119895 = 1 2 119872

119903119890

119895+ 120591 le 119889119895 119895 = 1 2 119872

(6)




ℎ119904

1198941015840minus ℎ

119890

119894ge 119867119868 forall119894

1015840= 119894

119903119904

1198951015840minus 119903

119890

119895ge 119860119868 forall119895

1015840= 119895

119889119895+1 minus 119889119895 le 119863119868 119895 = 1 2 119872 minus 1

(7)




Γmin le

10038161003816100381610038161003816119861119895

10038161003816100381610038161003816le Γmax 119895 = 1 2 119872 (9)


119889119887 = 1198891198871015840 forall119887 1198871015840isin 119861119908 119908 = 1 2 119882 (10)







Sim

ulat

ion

met

hodG

A



2 5 1 4 3 6 5 2 6 4 1 2 3 5 4 5 6 2 1

3 6 5 4 1 6 2 1 3 1 6 2 3 5 3 4 6 1 4

4 2 5 2 1 6 4 2 5 4 5 2 1 2 3 4 2 6 2

1 3 5 2 3 6 5 2 3 2 5 6 1 2 4 3 2 5 4

Fitness 1

Code section 1

Codesection 2

Overlapcode

Fitness 2

Individual 1

Individual 2

Individual 3

Individual 4





























119905119890 = max 119886119894lowast 119905119864119880 (11)



lowast





requirement namely

119905119904

ℎ= 119905119867 (12)




119905119904

ℎ= 119901ℎ (13)



lowast It satisfies

119872119888lowast

119894119906= max 119872119888

lowast

1198941199061015840| 1198941199061015840 isin Ωℎ (14)





= 119905119904

ℎ the


119894ℎ = 119894119906

119901ℎ = ℎ119904

119894119906+

10038161003816100381610038161003816119861119894119906

10038161003816100381610038161003816

120575

+ 119867119868

119894119906 = 0

(15)


119905119890

ℎ=

ℎ119904

119894ℎ+

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816

120575

119894ℎ gt 0

+infin 119894ℎ = 0

(16)


119890




119894ℎ = 0

119861119908 = 119861119908 cup 119887119896

119894ℎ| 119908

119896

119894ℎ= 119908 119896 = 1 2

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816 forall119908

(17)


119905119904

119886= max 119905119864119881 119901120587 119905119860 (18)



= 119905119904

119886





119895120587 = 119895lowast

119861119908 =

119861119908

(119861119895 cap 119861119908)

forall119908

(19)



119905 = min (119903119904

1198951205871015840+ 120598119891 + (119899119895

1205871015840minus 1) lowast 120598119886) | 1198951205871015840 gt 0 (20)


119905119890

119886=


119905 else(21)


= 119905119890



119895V = 119895120587

119895120587 = 0

119901120587 = 119905119890

119886+ 119867119868

(22)



Yes

No

assemble railcars

Yes

No



simulation period















119895V = 0 (23)





119894119906 = 0 119906 = 1 2 119880

119861119908 = 0 119908 = 1 2 119882

119895V = 0 V = 1 2 119881

(24)


119901120590 = 0 forall120590

119894120590 = 0 forall120590

(25)


119901120587 = 0 forall120587

119895120587 = 0 forall120587

(26)




119904 119879

119890) in period [119879

119904 119879

119890]





ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889 of each event


ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889

While 119905119902le 119879

119890


lowast+ 1



119895lowast= 119895

lowast+ 1




ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889of each event


End

Algorithm 1









2 5Individual code




4 3 1 5 3 6



1 3 2 4 2 3 1 4 2 1 (3 1)

1 4 3 4 2 3 4 1 2 3 (4)

1 3 1 4 3 2 3 4 1 4 3 2

s1

s2








1 119901119888

2 maximum and


2



119892 119879

119890

119892] = [0 119871]


119892lt 119879







1198751015840of population 119875

1015840If 119904lowast

1198751015840is better than 119904


1198751015840 and 119909 = 1 else 119909 = 119909 + 1

Set 119875 = 1198751015840


119904

119892= 119879

119890

119892minus Δ119871 119879119890

119892= min119879119904

119892+ 119871 119879


End

Algorithm 2


119891119894 (119879119904 119879119890) =

119890((119879119890minus119885119894)120593timeslog099119909)

sum119904119895isin119878

119890((119879119890minus119885119895)120593timeslog099119909)

(27)





3 1 4 2 4 3 1

2 3 4 1 3 2 4

3 1 4 2 3 2 4

2 3 4 1 4 3 1x

c1

c2

s1

s2









6 Numeric Examples



1= 09 and 119901119888

2= 05 respec-


1= 005 and 119901119898







0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18220230240250260270280290300310

Railc

ars a

vera

ge st

ayin

g tim

e (m

in)











5

100 05 268 267 271 95 72 991 271 272 279 97 73 96

150 05 246 247 256 221 134 2271 251 254 273 218 127 221

200 05 317 318 339 472 309 4471 319 321 344 484 316 435

10

200 05 270 273 282 208 192 2121 274 276 283 213 182 205

300 05 248 251 257 682 467 5671 253 258 262 675 472 575

400 05 324 325 332 1102 859 9681 324 326 334 1104 863 972

230235240245250255260265270275280285

6 8 10 12 14 16 18 20 22 24 26 28 30Railc

ars a

vera

ge st

ayin

g tim

e (m

in )






1012141618202224262830

6 8 10 12 14 16 18 20 22 24 26 28 30

Com

putin

g tim

e (m

in)




7 Conclusion





Notations

Parameters






119887119896


119889119896






Variables




ℎ119904


ℎ119890


119908119896




119903119904


119903119890






















Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











ℎ119904

1198941015840minus ℎ

119890

119894ge 119867119868 forall119894

1015840= 119894

119903119904

1198951015840minus 119903

119890

119895ge 119860119868 forall119895

1015840= 119895

119889119895+1 minus 119889119895 le 119863119868 119895 = 1 2 119872 minus 1

(7)




Γmin le

10038161003816100381610038161003816119861119895

10038161003816100381610038161003816le Γmax 119895 = 1 2 119872 (9)


119889119887 = 1198891198871015840 forall119887 1198871015840isin 119861119908 119908 = 1 2 119882 (10)







Sim

ulat

ion

met

hodG

A



2 5 1 4 3 6 5 2 6 4 1 2 3 5 4 5 6 2 1

3 6 5 4 1 6 2 1 3 1 6 2 3 5 3 4 6 1 4

4 2 5 2 1 6 4 2 5 4 5 2 1 2 3 4 2 6 2

1 3 5 2 3 6 5 2 3 2 5 6 1 2 4 3 2 5 4

Fitness 1

Code section 1

Codesection 2

Overlapcode

Fitness 2

Individual 1

Individual 2

Individual 3

Individual 4





























119905119890 = max 119886119894lowast 119905119864119880 (11)



lowast





requirement namely

119905119904

ℎ= 119905119867 (12)




119905119904

ℎ= 119901ℎ (13)



lowast It satisfies

119872119888lowast

119894119906= max 119872119888

lowast

1198941199061015840| 1198941199061015840 isin Ωℎ (14)





= 119905119904

ℎ the


119894ℎ = 119894119906

119901ℎ = ℎ119904

119894119906+

10038161003816100381610038161003816119861119894119906

10038161003816100381610038161003816

120575

+ 119867119868

119894119906 = 0

(15)


119905119890

ℎ=

ℎ119904

119894ℎ+

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816

120575

119894ℎ gt 0

+infin 119894ℎ = 0

(16)


119890




119894ℎ = 0

119861119908 = 119861119908 cup 119887119896

119894ℎ| 119908

119896

119894ℎ= 119908 119896 = 1 2

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816 forall119908

(17)


119905119904

119886= max 119905119864119881 119901120587 119905119860 (18)



= 119905119904

119886





119895120587 = 119895lowast

119861119908 =

119861119908

(119861119895 cap 119861119908)

forall119908

(19)



119905 = min (119903119904

1198951205871015840+ 120598119891 + (119899119895

1205871015840minus 1) lowast 120598119886) | 1198951205871015840 gt 0 (20)


119905119890

119886=


119905 else(21)


= 119905119890



119895V = 119895120587

119895120587 = 0

119901120587 = 119905119890

119886+ 119867119868

(22)



Yes

No

assemble railcars

Yes

No



simulation period















119895V = 0 (23)





119894119906 = 0 119906 = 1 2 119880

119861119908 = 0 119908 = 1 2 119882

119895V = 0 V = 1 2 119881

(24)


119901120590 = 0 forall120590

119894120590 = 0 forall120590

(25)


119901120587 = 0 forall120587

119895120587 = 0 forall120587

(26)




119904 119879

119890) in period [119879

119904 119879

119890]





ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889 of each event


ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889

While 119905119902le 119879

119890


lowast+ 1



119895lowast= 119895

lowast+ 1




ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889of each event


End

Algorithm 1









2 5Individual code




4 3 1 5 3 6



1 3 2 4 2 3 1 4 2 1 (3 1)

1 4 3 4 2 3 4 1 2 3 (4)

1 3 1 4 3 2 3 4 1 4 3 2

s1

s2








1 119901119888

2 maximum and


2



119892 119879

119890

119892] = [0 119871]


119892lt 119879







1198751015840of population 119875

1015840If 119904lowast

1198751015840is better than 119904


1198751015840 and 119909 = 1 else 119909 = 119909 + 1

Set 119875 = 1198751015840


119904

119892= 119879

119890

119892minus Δ119871 119879119890

119892= min119879119904

119892+ 119871 119879


End

Algorithm 2


119891119894 (119879119904 119879119890) =

119890((119879119890minus119885119894)120593timeslog099119909)

sum119904119895isin119878

119890((119879119890minus119885119895)120593timeslog099119909)

(27)





3 1 4 2 4 3 1

2 3 4 1 3 2 4

3 1 4 2 3 2 4

2 3 4 1 4 3 1x

c1

c2

s1

s2









6 Numeric Examples



1= 09 and 119901119888

2= 05 respec-


1= 005 and 119901119898







0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18220230240250260270280290300310

Railc

ars a

vera

ge st

ayin

g tim

e (m

in)











5

100 05 268 267 271 95 72 991 271 272 279 97 73 96

150 05 246 247 256 221 134 2271 251 254 273 218 127 221

200 05 317 318 339 472 309 4471 319 321 344 484 316 435

10

200 05 270 273 282 208 192 2121 274 276 283 213 182 205

300 05 248 251 257 682 467 5671 253 258 262 675 472 575

400 05 324 325 332 1102 859 9681 324 326 334 1104 863 972

230235240245250255260265270275280285

6 8 10 12 14 16 18 20 22 24 26 28 30Railc

ars a

vera

ge st

ayin

g tim

e (m

in )






1012141618202224262830

6 8 10 12 14 16 18 20 22 24 26 28 30

Com

putin

g tim

e (m

in)




7 Conclusion





Notations

Parameters






119887119896


119889119896






Variables




ℎ119904


ℎ119890


119908119896




119903119904


119903119890






















Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra






























119905119890 = max 119886119894lowast 119905119864119880 (11)



lowast





requirement namely

119905119904

ℎ= 119905119867 (12)




119905119904

ℎ= 119901ℎ (13)



lowast It satisfies

119872119888lowast

119894119906= max 119872119888

lowast

1198941199061015840| 1198941199061015840 isin Ωℎ (14)





= 119905119904

ℎ the


119894ℎ = 119894119906

119901ℎ = ℎ119904

119894119906+

10038161003816100381610038161003816119861119894119906

10038161003816100381610038161003816

120575

+ 119867119868

119894119906 = 0

(15)


119905119890

ℎ=

ℎ119904

119894ℎ+

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816

120575

119894ℎ gt 0

+infin 119894ℎ = 0

(16)


119890




119894ℎ = 0

119861119908 = 119861119908 cup 119887119896

119894ℎ| 119908

119896

119894ℎ= 119908 119896 = 1 2

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816 forall119908

(17)


119905119904

119886= max 119905119864119881 119901120587 119905119860 (18)



= 119905119904

119886





119895120587 = 119895lowast

119861119908 =

119861119908

(119861119895 cap 119861119908)

forall119908

(19)



119905 = min (119903119904

1198951205871015840+ 120598119891 + (119899119895

1205871015840minus 1) lowast 120598119886) | 1198951205871015840 gt 0 (20)


119905119890

119886=


119905 else(21)


= 119905119890



119895V = 119895120587

119895120587 = 0

119901120587 = 119905119890

119886+ 119867119868

(22)



Yes

No

assemble railcars

Yes

No



simulation period















119895V = 0 (23)





119894119906 = 0 119906 = 1 2 119880

119861119908 = 0 119908 = 1 2 119882

119895V = 0 V = 1 2 119881

(24)


119901120590 = 0 forall120590

119894120590 = 0 forall120590

(25)


119901120587 = 0 forall120587

119895120587 = 0 forall120587

(26)




119904 119879

119890) in period [119879

119904 119879

119890]





ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889 of each event


ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889

While 119905119902le 119879

119890


lowast+ 1



119895lowast= 119895

lowast+ 1




ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889of each event


End

Algorithm 1









2 5Individual code




4 3 1 5 3 6



1 3 2 4 2 3 1 4 2 1 (3 1)

1 4 3 4 2 3 4 1 2 3 (4)

1 3 1 4 3 2 3 4 1 4 3 2

s1

s2








1 119901119888

2 maximum and


2



119892 119879

119890

119892] = [0 119871]


119892lt 119879







1198751015840of population 119875

1015840If 119904lowast

1198751015840is better than 119904


1198751015840 and 119909 = 1 else 119909 = 119909 + 1

Set 119875 = 1198751015840


119904

119892= 119879

119890

119892minus Δ119871 119879119890

119892= min119879119904

119892+ 119871 119879


End

Algorithm 2


119891119894 (119879119904 119879119890) =

119890((119879119890minus119885119894)120593timeslog099119909)

sum119904119895isin119878

119890((119879119890minus119885119895)120593timeslog099119909)

(27)





3 1 4 2 4 3 1

2 3 4 1 3 2 4

3 1 4 2 3 2 4

2 3 4 1 4 3 1x

c1

c2

s1

s2









6 Numeric Examples



1= 09 and 119901119888

2= 05 respec-


1= 005 and 119901119898







0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18220230240250260270280290300310

Railc

ars a

vera

ge st

ayin

g tim

e (m

in)











5

100 05 268 267 271 95 72 991 271 272 279 97 73 96

150 05 246 247 256 221 134 2271 251 254 273 218 127 221

200 05 317 318 339 472 309 4471 319 321 344 484 316 435

10

200 05 270 273 282 208 192 2121 274 276 283 213 182 205

300 05 248 251 257 682 467 5671 253 258 262 675 472 575

400 05 324 325 332 1102 859 9681 324 326 334 1104 863 972

230235240245250255260265270275280285

6 8 10 12 14 16 18 20 22 24 26 28 30Railc

ars a

vera

ge st

ayin

g tim

e (m

in )






1012141618202224262830

6 8 10 12 14 16 18 20 22 24 26 28 30

Com

putin

g tim

e (m

in)




7 Conclusion





Notations

Parameters






119887119896


119889119896






Variables




ℎ119904


ℎ119890


119908119896




119903119904


119903119890






















Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra













requirement namely

119905119904

ℎ= 119905119867 (12)




119905119904

ℎ= 119901ℎ (13)



lowast It satisfies

119872119888lowast

119894119906= max 119872119888

lowast

1198941199061015840| 1198941199061015840 isin Ωℎ (14)





= 119905119904

ℎ the


119894ℎ = 119894119906

119901ℎ = ℎ119904

119894119906+

10038161003816100381610038161003816119861119894119906

10038161003816100381610038161003816

120575

+ 119867119868

119894119906 = 0

(15)


119905119890

ℎ=

ℎ119904

119894ℎ+

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816

120575

119894ℎ gt 0

+infin 119894ℎ = 0

(16)


119890




119894ℎ = 0

119861119908 = 119861119908 cup 119887119896

119894ℎ| 119908

119896

119894ℎ= 119908 119896 = 1 2

10038161003816100381610038161003816119861119894ℎ

10038161003816100381610038161003816 forall119908

(17)


119905119904

119886= max 119905119864119881 119901120587 119905119860 (18)



= 119905119904

119886





119895120587 = 119895lowast

119861119908 =

119861119908

(119861119895 cap 119861119908)

forall119908

(19)



119905 = min (119903119904

1198951205871015840+ 120598119891 + (119899119895

1205871015840minus 1) lowast 120598119886) | 1198951205871015840 gt 0 (20)


119905119890

119886=


119905 else(21)


= 119905119890



119895V = 119895120587

119895120587 = 0

119901120587 = 119905119890

119886+ 119867119868

(22)



Yes

No

assemble railcars

Yes

No



simulation period















119895V = 0 (23)





119894119906 = 0 119906 = 1 2 119880

119861119908 = 0 119908 = 1 2 119882

119895V = 0 V = 1 2 119881

(24)


119901120590 = 0 forall120590

119894120590 = 0 forall120590

(25)


119901120587 = 0 forall120587

119895120587 = 0 forall120587

(26)




119904 119879

119890) in period [119879

119904 119879

119890]





ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889 of each event


ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889

While 119905119902le 119879

119890


lowast+ 1



119895lowast= 119895

lowast+ 1




ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889of each event


End

Algorithm 1









2 5Individual code




4 3 1 5 3 6



1 3 2 4 2 3 1 4 2 1 (3 1)

1 4 3 4 2 3 4 1 2 3 (4)

1 3 1 4 3 2 3 4 1 4 3 2

s1

s2








1 119901119888

2 maximum and


2



119892 119879

119890

119892] = [0 119871]


119892lt 119879







1198751015840of population 119875

1015840If 119904lowast

1198751015840is better than 119904


1198751015840 and 119909 = 1 else 119909 = 119909 + 1

Set 119875 = 1198751015840


119904

119892= 119879

119890

119892minus Δ119871 119879119890

119892= min119879119904

119892+ 119871 119879


End

Algorithm 2


119891119894 (119879119904 119879119890) =

119890((119879119890minus119885119894)120593timeslog099119909)

sum119904119895isin119878

119890((119879119890minus119885119895)120593timeslog099119909)

(27)





3 1 4 2 4 3 1

2 3 4 1 3 2 4

3 1 4 2 3 2 4

2 3 4 1 4 3 1x

c1

c2

s1

s2









6 Numeric Examples



1= 09 and 119901119888

2= 05 respec-


1= 005 and 119901119898







0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18220230240250260270280290300310

Railc

ars a

vera

ge st

ayin

g tim

e (m

in)











5

100 05 268 267 271 95 72 991 271 272 279 97 73 96

150 05 246 247 256 221 134 2271 251 254 273 218 127 221

200 05 317 318 339 472 309 4471 319 321 344 484 316 435

10

200 05 270 273 282 208 192 2121 274 276 283 213 182 205

300 05 248 251 257 682 467 5671 253 258 262 675 472 575

400 05 324 325 332 1102 859 9681 324 326 334 1104 863 972

230235240245250255260265270275280285

6 8 10 12 14 16 18 20 22 24 26 28 30Railc

ars a

vera

ge st

ayin

g tim

e (m

in )






1012141618202224262830

6 8 10 12 14 16 18 20 22 24 26 28 30

Com

putin

g tim

e (m

in)




7 Conclusion





Notations

Parameters






119887119896


119889119896






Variables




ℎ119904


ℎ119890


119908119896




119903119904


119903119890






















Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











Yes

No

assemble railcars

Yes

No



simulation period















119895V = 0 (23)





119894119906 = 0 119906 = 1 2 119880

119861119908 = 0 119908 = 1 2 119882

119895V = 0 V = 1 2 119881

(24)


119901120590 = 0 forall120590

119894120590 = 0 forall120590

(25)


119901120587 = 0 forall120587

119895120587 = 0 forall120587

(26)




119904 119879

119890) in period [119879

119904 119879

119890]





ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889 of each event


ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889

While 119905119902le 119879

119890


lowast+ 1



119895lowast= 119895

lowast+ 1




ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889of each event


End

Algorithm 1









2 5Individual code




4 3 1 5 3 6



1 3 2 4 2 3 1 4 2 1 (3 1)

1 4 3 4 2 3 4 1 2 3 (4)

1 3 1 4 3 2 3 4 1 4 3 2

s1

s2








1 119901119888

2 maximum and


2



119892 119879

119890

119892] = [0 119871]


119892lt 119879







1198751015840of population 119875

1015840If 119904lowast

1198751015840is better than 119904


1198751015840 and 119909 = 1 else 119909 = 119909 + 1

Set 119875 = 1198751015840


119904

119892= 119879

119890

119892minus Δ119871 119879119890

119892= min119879119904

119892+ 119871 119879


End

Algorithm 2


119891119894 (119879119904 119879119890) =

119890((119879119890minus119885119894)120593timeslog099119909)

sum119904119895isin119878

119890((119879119890minus119885119895)120593timeslog099119909)

(27)





3 1 4 2 4 3 1

2 3 4 1 3 2 4

3 1 4 2 3 2 4

2 3 4 1 4 3 1x

c1

c2

s1

s2









6 Numeric Examples



1= 09 and 119901119888

2= 05 respec-


1= 005 and 119901119898







0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18220230240250260270280290300310

Railc

ars a

vera

ge st

ayin

g tim

e (m

in)











5

100 05 268 267 271 95 72 991 271 272 279 97 73 96

150 05 246 247 256 221 134 2271 251 254 273 218 127 221

200 05 317 318 339 472 309 4471 319 321 344 484 316 435

10

200 05 270 273 282 208 192 2121 274 276 283 213 182 205

300 05 248 251 257 682 467 5671 253 258 262 675 472 575

400 05 324 325 332 1102 859 9681 324 326 334 1104 863 972

230235240245250255260265270275280285

6 8 10 12 14 16 18 20 22 24 26 28 30Railc

ars a

vera

ge st

ayin

g tim

e (m

in )






1012141618202224262830

6 8 10 12 14 16 18 20 22 24 26 28 30

Com

putin

g tim

e (m

in)




7 Conclusion





Notations

Parameters






119887119896


119889119896






Variables




ℎ119904


ℎ119890


119908119896




119903119904


119903119890






















Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119904 119879

119890) in period [119879

119904 119879

119890]





ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889 of each event


ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889

While 119905119902le 119879

119890


lowast+ 1



119895lowast= 119895

lowast+ 1




ℎ 119905119890

ℎ 119905119904

119886 119905119890

119886 119905119889of each event


End

Algorithm 1









2 5Individual code




4 3 1 5 3 6



1 3 2 4 2 3 1 4 2 1 (3 1)

1 4 3 4 2 3 4 1 2 3 (4)

1 3 1 4 3 2 3 4 1 4 3 2

s1

s2








1 119901119888

2 maximum and


2



119892 119879

119890

119892] = [0 119871]


119892lt 119879







1198751015840of population 119875

1015840If 119904lowast

1198751015840is better than 119904


1198751015840 and 119909 = 1 else 119909 = 119909 + 1

Set 119875 = 1198751015840


119904

119892= 119879

119890

119892minus Δ119871 119879119890

119892= min119879119904

119892+ 119871 119879


End

Algorithm 2


119891119894 (119879119904 119879119890) =

119890((119879119890minus119885119894)120593timeslog099119909)

sum119904119895isin119878

119890((119879119890minus119885119895)120593timeslog099119909)

(27)





3 1 4 2 4 3 1

2 3 4 1 3 2 4

3 1 4 2 3 2 4

2 3 4 1 4 3 1x

c1

c2

s1

s2









6 Numeric Examples



1= 09 and 119901119888

2= 05 respec-


1= 005 and 119901119898







0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18220230240250260270280290300310

Railc

ars a

vera

ge st

ayin

g tim

e (m

in)











5

100 05 268 267 271 95 72 991 271 272 279 97 73 96

150 05 246 247 256 221 134 2271 251 254 273 218 127 221

200 05 317 318 339 472 309 4471 319 321 344 484 316 435

10

200 05 270 273 282 208 192 2121 274 276 283 213 182 205

300 05 248 251 257 682 467 5671 253 258 262 675 472 575

400 05 324 325 332 1102 859 9681 324 326 334 1104 863 972

230235240245250255260265270275280285

6 8 10 12 14 16 18 20 22 24 26 28 30Railc

ars a

vera

ge st

ayin

g tim

e (m

in )






1012141618202224262830

6 8 10 12 14 16 18 20 22 24 26 28 30

Com

putin

g tim

e (m

in)




7 Conclusion





Notations

Parameters






119887119896


119889119896






Variables




ℎ119904


ℎ119890


119908119896




119903119904


119903119890






















Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1 119901119888

2 maximum and


2



119892 119879

119890

119892] = [0 119871]


119892lt 119879







1198751015840of population 119875

1015840If 119904lowast

1198751015840is better than 119904


1198751015840 and 119909 = 1 else 119909 = 119909 + 1

Set 119875 = 1198751015840


119904

119892= 119879

119890

119892minus Δ119871 119879119890

119892= min119879119904

119892+ 119871 119879


End

Algorithm 2


119891119894 (119879119904 119879119890) =

119890((119879119890minus119885119894)120593timeslog099119909)

sum119904119895isin119878

119890((119879119890minus119885119895)120593timeslog099119909)

(27)





3 1 4 2 4 3 1

2 3 4 1 3 2 4

3 1 4 2 3 2 4

2 3 4 1 4 3 1x

c1

c2

s1

s2









6 Numeric Examples



1= 09 and 119901119888

2= 05 respec-


1= 005 and 119901119898







0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18220230240250260270280290300310

Railc

ars a

vera

ge st

ayin

g tim

e (m

in)











5

100 05 268 267 271 95 72 991 271 272 279 97 73 96

150 05 246 247 256 221 134 2271 251 254 273 218 127 221

200 05 317 318 339 472 309 4471 319 321 344 484 316 435

10

200 05 270 273 282 208 192 2121 274 276 283 213 182 205

300 05 248 251 257 682 467 5671 253 258 262 675 472 575

400 05 324 325 332 1102 859 9681 324 326 334 1104 863 972

230235240245250255260265270275280285

6 8 10 12 14 16 18 20 22 24 26 28 30Railc

ars a

vera

ge st

ayin

g tim

e (m

in )






1012141618202224262830

6 8 10 12 14 16 18 20 22 24 26 28 30

Com

putin

g tim

e (m

in)




7 Conclusion





Notations

Parameters






119887119896


119889119896






Variables




ℎ119904


ℎ119890


119908119896




119903119904


119903119890






















Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












6 Numeric Examples



1= 09 and 119901119888

2= 05 respec-


1= 005 and 119901119898







0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18220230240250260270280290300310

Railc

ars a

vera

ge st

ayin

g tim

e (m

in)











5

100 05 268 267 271 95 72 991 271 272 279 97 73 96

150 05 246 247 256 221 134 2271 251 254 273 218 127 221

200 05 317 318 339 472 309 4471 319 321 344 484 316 435

10

200 05 270 273 282 208 192 2121 274 276 283 213 182 205

300 05 248 251 257 682 467 5671 253 258 262 675 472 575

400 05 324 325 332 1102 859 9681 324 326 334 1104 863 972

230235240245250255260265270275280285

6 8 10 12 14 16 18 20 22 24 26 28 30Railc

ars a

vera

ge st

ayin

g tim

e (m

in )






1012141618202224262830

6 8 10 12 14 16 18 20 22 24 26 28 30

Com

putin

g tim

e (m

in)




7 Conclusion





Notations

Parameters






119887119896


119889119896






Variables




ℎ119904


ℎ119890


119908119896




119903119904


119903119890






















Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra














5

100 05 268 267 271 95 72 991 271 272 279 97 73 96

150 05 246 247 256 221 134 2271 251 254 273 218 127 221

200 05 317 318 339 472 309 4471 319 321 344 484 316 435

10

200 05 270 273 282 208 192 2121 274 276 283 213 182 205

300 05 248 251 257 682 467 5671 253 258 262 675 472 575

400 05 324 325 332 1102 859 9681 324 326 334 1104 863 972

230235240245250255260265270275280285

6 8 10 12 14 16 18 20 22 24 26 28 30Railc

ars a

vera

ge st

ayin

g tim

e (m

in )






1012141618202224262830

6 8 10 12 14 16 18 20 22 24 26 28 30

Com

putin

g tim

e (m

in)




7 Conclusion





Notations

Parameters






119887119896


119889119896






Variables




ℎ119904


ℎ119890


119908119896




119903119904


119903119890






















Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra












Notations

Parameters






119887119896


119889119896






Variables




ℎ119904


ℎ119890


119908119896




119903119904


119903119890






















Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Acknowledgments


References


























Volume 2014




Journal of











Journal of


Function Spaces






Algebra
















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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