Research ArticleLocomotive Schedule Optimization forDa-qin Heavy Haul Railway

Ruiye Su Leishan Zhou and Jinjin Tang

School of Traffic and Transportation Beijing Jiaotong University Beijing 100044 China

Correspondence should be addressed to Jinjin Tang jjtangbjtueducn

Received 18 September 2015 Accepted 6 December 2015

Academic Editor Gisella Tomasini

Copyright copy 2015 Ruiye Su et alThis is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The main difference between locomotive schedule of heavy haul railways and that of regular rail transportation is the numberof locomotives utilized for one train One heavy-loaded train usually has more than one locomotive but a regular train only hasoneThis paper develops an optimizationmodel for themultilocomotive scheduling problem (MLSP) through analyzing the currentlocomotive schedule of Da-qin RailwayThe objective function of our paper is tominimize the total number of utilized locomotivesThe MLSP is nondeterministic polynomial (NP) hard Therefore we convert the multilocomotive traction problem into a single-locomotive traction problemThen the single-locomotive traction problem (SLTP) can be converted into an assignment problemThe Hungarian algorithm is applied to solve the model and obtain the optimal locomotive schedule We use the variance ofdetention time of locomotives at stations to evaluate the stability of locomotive schedule In order to evaluate the effectivenessof the proposed optimization model case studies for 20 kt and 30 kt heavy-loaded combined trains on Da-qin Railway are bothconducted Compared to the current schedules the optimal schedules from the proposed models can save 62 and 47 locomotivesfor 20 kt and 30 kt heavy-loaded combined trains respectively Therefore the effectiveness of the proposed model and its solutionalgorithm are both valid

1 Background

Features of the Heavy Haul Locomotive Schedule Comparedwith regular railways the locomotive schedule of heavy haulrailways has the following features [1]

(1) The heavy-loaded trains transport much heaviergoods than the regular trains So they need high loco-motive traction and braking forces

(2) Heavy-loaded trains generally require multilocomo-tive traction For example one 20 kt heavy-loadedtrain on Da-qin Railway is hauled by two HXD loco-motives

(3) A good coordination and handling system is requiredfor heavy haul locomotives Da-qin Railway Co Ltdimported LOCOTROL technology from the UnitedStates in 2003 in order to improve the coordination oflocomotive control

(4) Heavy-loaded trains travel longer transportation dis-tance As a result they require longer locomotiverouting and more frequent locomotive driver shifts

(5) The gross trainweight of heavy-loaded trains is heavywhich leads to the great power between locomotivesand rolling stocks and tracks

Research Background and Significance As shown in Figure 1the coal demand in China increases every year with the rapiddevelopment of the economy The demand still reaches 359billion tons in 2014 even though a slight decrease is observedcompared to 2013

Da-qin Railway is the earliest double-track electrifiedheavy haul railway in China which carries 10 kt and 20 ktheavy-loaded trains Figure 2 shows the network of Da-qinRailway Based on Figures 1 and 2 it can be found that thefreight volume of Da-qin Railway was more than 200 million

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 607376 14 pageshttpdxdoiorg1011552015607376

2 Mathematical Problems in Engineering

204 254 304 340 330 400 440 426 445 450

2205 2373 2526 2802 2973

3235 3520 3650 3680 3590

000

500

1000

1500

2000

2500

3000

3500

4000

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

Da-qin Railway freight volumeChinarsquos coal demand

Years

100 million tons

Figure 1 Chinarsquos coal demand and Da-qin Railway freight volumefrom 2005 to 2014 [2]

tons in 2005 which already reached the international theoret-ical limit of heavy haul railway transportation Moreover thefreight volume increased to 400 million tons in 2010 and 450million tons in 2014 However the transported cargo volumestill cannot meet the coal demand Freight volume still needsto be improved It is difficult to have higher freight flow den-sity for the current train diagram and increase trainrsquos speedin Da-qin RailwayThe only way to increase freight volume isto increase the transportation capacity of a trainTherefore itneeds to operate 30 kt combined trains in Da-qin Railway

Locomotives are a critical factor for the capacity of arailway The locomotive schedule has a direct impact onefficiency operation cost and benefits of a railwayThereforethe study on the locomotive schedule for 30 kt combinedtrains in Da-qin Railway is necessary

Background of Heavy Haul Locomotive Schedule Research onheavy haul locomotives mainly concentrates on the technicallevel Cheng et al [3] provided technical support for theapplication of Reliability Centered Maintenance (RCM) inrailway locomotive and verified its effectiveness in keepingreliability and reducing maintenance cost Zheng and Wei[4] Chang [5] and Mu et al [6] analyzed the mechanicalresponse of railway track structure under axle loads from25 t to 40 t Raviv and Kaspi [7] and Kumar and Bierlaire [8]proposed a mixed integer linear model that determines theoptimal strategy Studies on optimal locomotive schedule arequite few because of the small density of heavy-loaded trainsshort length of station track and the difficult transportationorganization Meanwhile these researches of heavy haullocomotives are mainly for 10sim20 kt heavy-loaded trainsRelatively speaking researches on the operational level aremainly about regular railways which focus on the followingareas (1) Ji et al [9] used computer technologies to achievedouble-locomotive traction control and determine locomo-tive marshalling schedules (2) Ruppert et al [10] analyzedhow to increase the usage efficiency and reliability of loco-motives (3) Kuo and Nicholls [11] developed a mixed integer

linear program (MILP) to determine the least-cost plan ofallocating locomotives and improve locomotive utilization(4) Aronsson et al [12] built a mixed integer program (MIP)model to solve locomotive routing and scheduling problemand Vaidyanathan et al [13] formulated the locomotiverouting problem (LRP) as an integer programming problemon a suitably constructed space-time network and developedrobust optimization methods to solve it (5) Ghoseiri andGhannadpour [14] Kasalica et al [15] Rouillon et al [16]and Teichmann et al [17] were devoted to the research oflocomotive assignment problem and used different methodsto optimize it and (6) Li et al [18] established amathematicalmodel for generating a locomotiveworking diagramof a traindiagram with paired trains then they used the improvedant colony algorithm to solve it Most of the locomotiveschedule studies are for regular trains and only a few of theminvolve 10sim20 kt heavy-loaded trainsThere are no researchesof locomotive scheduling for 30 kt heavy-loaded trains

2 Analysis of Locomotive Schedule forHeavy-Loaded Combined Trains onDa-qin Railway

21 Organization Forms of Heavy-Loaded Trains on Da-qinRailway There are two organization types for heavy-loadedtrains on Da-qin Railway unit type and combined type In2014 the daily average was 93 in the heavy-load directionon Da-qin Railway which include 65 trains (20 kt heavy-loaded combined) 12 trains (15 kt heavy-loaded combined)13 trains (10 kt heavy-loaded combined) and 3 trains (10 ktheavy-loaded unit) The combined type is adopted for 30 ktheavy-loaded trains

22 Locomotive Utilization Method on Da-qin Railway Theuncertain railway regionmethod is currently used to developthe heavy haul locomotive schedule on Da-qin Railwaywhich aims at reducing locomotive turnaround time [19]Themethod can significantly improve locomotive schedulingefficiency Figure 3 is an illustration of uncertain railroadregions locomotive utilization method

The numbers in Figure 3 mean the running time of thecorresponding section If a locomotive which hauls a traindeparts from station C after 38min it will arrive at station AIn station A there are two traction choices to haul differenttrains The locomotive utilization region of choice (a) is A-B-D-A and the region of choice (b) is A-C-A In station A ifall locomotives haul trains with the same region there will belonger waiting time Therefore the uncertain railway regionmethod can reduce locomotive turnaround time

23 Criteria of Locomotive Schedule The criteria of locomo-tive schedule can reflect thework quality efficiency and bene-fit of locomotives which include quantitative criteria qualitycriteria and equilibrium criteria The critical performancemeasures for these three criteria are the number of availablelocomotives coefficient of locomotive requirement and theequilibrium degree of locomotive schedule respectively

Mathematical Problems in Engineering 3

LuannanDonggang

Caofeidian West

Hud

ong

235

30

Dat

ong

Sout

h 6

0H

anjia

ling

215

23

Dat

ong

coun

ty 3

99

Yang

quan

86

235

Hua

shao

ying

129

020

Zhul

u 18

267

4

Shac

heng

Eas

t 218

263

Beix

inba

o 23

701

4

Yanq

ing

Nor

th 2

625

14

Xiaz

huan

g 30

769

2

Chaw

u 32

72

Chag

ao L

ink

Line

Ping

gu 3

688

44

Jixia

n W

est 4

066

44Cu

ipin

gsha

n 41

796

1

Dad

uan

Link

Lin

eDi

recti

on of

Jixia

nYu

tian

Nor

th 4

408

38

Zunh

ua N

orth

476

767

Qia

nxi 5

083

35

Qia

ncao

Line

Qia

nrsquoan

Nor

th 5

547

27Lu

long

Nor

th 5

688

98

Hou

ying

621

100

Liuc

un S

outh

73

14Li

ucun

Sou

th 2

nd Y

ard

720

7

Qin

huan

gdao

Eas

t 14

417

16725 XihanlingPingwang 10570

KouquanBranch line

Yungang lineDatong

Directionof Gudian

Das

hizh

uang

387

344

Marshalling stationDistrict stationIntermediate station

Figure 2 Network of Da-qin Railway

Station A

Station B

Station C

Station D Locomotive

40 4042

35 37

53Traction choice (a)Traction choice (b)

Figure 3 An illustration of uncertain railroad region locomotiveutilization method

(1) Number of Available Locomotives Number of availablelocomotives is the total number of locomotives which areused in the whole turnaround process [20] The less thenumber is the higher the efficiency will be The graphicmethod and the direct calculation method are generalmethods to determine the number of available locomotivesThe graphic method is based on locomotive schedule todetermine the number of used locomotives The formula ofdirect calculation method is listed below

119872119891 =(119899119904 + 119899119889 + 2 sdot 119899119905) sdot 120579119897

24

= 120583119904 sdot 119870119889(1)

where 119872119891 is the number of available locomotives in eachtraction section 119899119904 is the total number of trains 119899119889 is thenumber of trains with double-locomotive traction 119899119905 is thenumber of trains with three-locomotive traction 120579119897 is thelocomotive complete turnaround time 120583119904 is the number ofsupplied locomotives 119870119889 is the coefficient of locomotiverequirement in traction sections

(2) Coefficient of Locomotive Requirement The coefficient oflocomotive requirement indicates the number of locomotivesrequired per pair of trains in a traction section [20] Onelocomotive completes a turnaround which means it has

completed a traction task The coefficient of locomotiverequirement can be calculated by the formula below

119870119889 =120579119897

24

(2)

(3) Equilibrium Degree of Locomotive Schedule In additionto quantitative and quality criteria the equilibrium criteriaare also very important in evaluating locomotive schedulewhich refer to the difference between detention time of eachlocomotive at stations and running time of each locomotiveAt the same section the running time is the sameThereforein the actual evaluation process the variance of detentiontime of each locomotive at stations 119863 (equilibrium degree)can be used to describe the equilibrium of locomotiveschedule It can be calculated as below

119863 =

1

119899119897

119899119897

sum

119894=1

(119888119894 minus 119864)2 (3)

where 119899119897 is the total number of traction tasks of all locomo-tives and it is numerically equal to (119899119904 + 119899119889 + 2 sdot 119899119905) 119894 is alocomotive that undertakes traction tasks that is 119894 isin 119899119897 119888119894 isthe detention time of a locomotive at station 119864 is the averagedetention time of locomotives at stations

Equation (3) shows that equilibrium degree 119863 refers tothe discrete degree between 119888119894 (119894 = 1 2 3 119899119897) and 119864 Thesmaller the value of119863 is the smaller the difference is and themore balanced the locomotive schedule will be

3 The Optimization Model of Heavy HaulLocomotive Scheduling Problem

31 Description of Locomotive Scheduling Problem The inputsfor the locomotive scheduling problem are railway linelocomotive traction sections and train diagram and so forthBased on these input data the output namely locomotiveschedule can be obtained

4 Mathematical Problems in Engineering

A B C D

Station with depotStation with turnaround depot

Figure 4 The network of hypothetical railway line

A B C D

Figure 5 Schematic diagram of locomotive traction sections

311 Input

(1) Railway Line Railway line A-B-C-D is a double-trackrailway as shown in Figure 4 It is dedicatedly used totransport coal On the railway station B is the station wherethe locomotive depot is located Station A station C andstation D are the stations where locomotive turnarounddepots are located

(2) Locomotive Traction Sections When freight locomotivesundertake traction tasks in section A-D they firstly departfrom station B and haul heavy-loaded trains to arrive atstation A station C or station D Then judge whetherlocomotive preparation is needed or not Finally these loco-motives haul empty-wagon trains and return to the stationwhere the locomotive depot is located while the locomotiveswhich are at turnaround depot can haul the return trainsonly The schematic diagram of locomotive traction sectionsis shown in Figure 5

(3) Train Diagram A complete train diagram determinestrain routes departurearrival time and detention time ateach station In the train diagram we cannot know thelocomotiversquos position andwhich train it will haul As shown inFigure 6 there are 4 pairs of trains on the hypothetical railway(4 heavy-loaded trains and 4 empty-wagon trains) The trainnumbers respectively are 700012 730012 730034 and790012 The train whose train number is 70001 refers to aheavy-loaded train in section A-B and the train whose trainnumber is 70002 refers to an empty-wagon train in section A-B The train whose train number is 73001 or 73003 refers to aheavy-loaded train in section B-C and the train whose trainnumber is 73002 or 73004 refers to an empty-wagon train insection B-CThe train whose train number is 79001 refers to aheavy-loaded train in section D-B and the train whose trainnumber is 79002 refers to an empty-wagon train in sectionD-B

312 Output Based on the input data the potential locomo-tive schedule in section A-D will be obtained (see Figure 7)where a dotted line represents a locomotive Figure 7 showsthat after hauling the train whose train number is 70001 to

station B the locomotive can haul the train whose train num-ber is 70002 79001 or 73001Whenmultilocomotive tractionschemes exist the minimal detention time of locomotives atstations will be adopted to determine the final scheme

32 Model Description Some assumptions are presented inthis model (a) the train diagram is known and invariant in acertain period of time that is the impact of train delay willbe ignored (b) based on the line length of Da-qin Railway(653 km) locomotive routing and locomotive running time(eg a round-trip time more than 17 h) it assumes within24 hours all available locomotives (the total locomotives afterdeducting the locomotives in maintenance and preparation)are without maintenance (c) for the trains with multilo-comotive traction locomotive reconnection between HXDlocomotives and SS4 locomotives is not considered the timeof locomotive reconnection and dismantling was includedin the detention time of locomotives at stations (d) in acertain period of time the locomotive crew working systemin Da-qin Railway is invariant so the effects of locomotivecrew working system and locomotive crew shift mode onlocomotive scheduling are not considered and (e) in thedaily initial time (1800) each station has enough locomotiveswhich at least are equal to the number of locomotives in needthat is the locomotive homing will not be considered

321 Sets In the process of constructing the optimizationmodel of the heavy haul locomotive scheduling problemthere are some notations and sets that will be used

119870 is the set that includes all of stations on the railway

119896 is a station on the railway that is 119896 isin 119870

119873119896119889 is the total number of locomotives arriving atstation 119896

119873119896119891 is the total number of trains departing fromstation 119896

119894 is the locomotive arriving at station 119896

119895 is the train departing from station 119896

119879119896 is the minimal technological time of locomotivesat station 119896

119896119894119889 is the destination station of locomotive 119894 at station119896

119905119894119889119896 is the arrival time of locomotive 119894 at station 119896

119896119895119891 is the departure station of train 119895 from station 119896

119905119895119891119896 is the departure time of train 119895 from station 119896

322 Parameters In the process of developing the opti-mization model the known parameter that will be involvedis 119888119896119894119895 It refers to the waiting time which is equal to thedetention time of locomotive 119894 at station 119896minus theminimal

Mathematical Problems in Engineering 5

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 6 Train diagram in section A-D

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 7 Potential locomotive schedule in section A-D

technological time The waiting time 119888119896119894119895 of locomotives atstations can be calculated by

119888

119896119894119895

=

119905119895119891119896 minus 119905119894119889119896 minus 119879119896 119905119895119891119896 minus 119905119894119889119896 ge 119879119896 119896119894119889 = 119896119895119891

119905119895119891119896 minus 119905119894119889119896 minus 119879119896 + 1440 119905119895119891119896 minus 119905119894119889119896 lt 119879119896 119896119894119889 = 119896119895119891

infin 119896119894119889 = 119896119895119891

(4)

323 Variable Thevariable is assumed to be 119888119896119894119895 which refersto whether locomotive 119894 hauls train 119895 or not at station 119896 Thebinary variable 119888119896119894119895 is defined as follows

119909

119896119894119895 =

1 locomotive 119894 hauls train 119895 at station 119896

0 otherwise(5)

324 Objective Function Completing planned traction tasksis the primary constraint to us Based on that the lessthe number of locomotives is the lower the transportationcost will be Therefore the objective function of heavy haullocomotive scheduling problem is minimizing number oflocomotives that is

min 119885

1015840=

119879119905 + 119879

1440

(6)

where 119879119905 is the total time of trains running from departurestation to destination station and T is the total detention timeof locomotives at stations which can be calculated by

119879 = 119879119870 + 119879119908 (7)

where 119879119870 is the total minimal technological time of locomo-tives at stations and119879119908 is the total waiting time of locomotivesat stations

For the established train diagram 119879119905 is a constant Sothe minimum number of locomotives refers to the minimaldetention time 119879 of locomotives at stations Meanwhile119879119870 is also a constant So the minimal detention time oflocomotives at stations refers to the minimum waiting time119879119908 of locomotives at stations That is

min 119885 = 119879119908 =

119870

sum

119896=1

119873119896119889

sum

119894=1

119873119896119891

sum

119895=1

119888

119896119894119895119909119896119894119895 (8)

Then the equilibrium degree 119863 of locomotive schedulecan be calculated by

119863 =

1

119899119904 + 119899119889 + 2 sdot 119899119905

sdot

119870

sum

119896=1

119873119896119889

sum

119894=1

119873119896119891

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

119899119904 + 119899119889 + 2 sdot 119899119905

)

2

(9)

325 Constraints

(1) Traction Locomotives Constraint Consider the following

119873119896119889

sum

119894=1

119909

119896119894119895 = ℎ forall119895 isin 119873119896119891 forall119896 isin 119870 (10)

This constraint shows that the number of traction loco-motives for each train is ℎ The value of each type of train isshown in Table 1

(2) Traction Tasks Constraint Consider the following

119873119896119891

sum

119895=1

119909

119896119894119895 = 1 forall119894 isin 119873119896119889 forall119896 isin 119870 (11)

6 Mathematical Problems in Engineering

Table 1 The value of each type of trains

Types of heavy-loaded trains Reference value Types of heavy-loaded trains Reference value10 kt heavy-loaded combined trains 2 20 kt heavy-loaded combined trains 210 kt heavy-loaded unit trains 1 30 kt heavy-loaded combined trains 315 kt heavy-loaded combined trains 2 mdash mdash

This constraint shows that each locomotive can only haulone train at the same time

4 Transformation and SolvingProcesses of Heavy Haul LocomotiveScheduling Problem

First of all the locomotive scheduling problem needs to betransformed into an assignment problemThen the optimiza-tion results are obtained by using the Hungarian algorithm

41 Transformation of Heavy Haul Locomotive SchedulingProblem For an unpaired train diagram the number ofarrival trains is not equal to the number of departure trainsat some stations So the unpaired train diagram needs tobe transformed into a paired train diagram before solvingthe locomotive scheduling problem For the paired traindiagram the number of arrival trains is equal to the numberof departure trains at each station Therefore if each trainis hauled by the same type of locomotive the locomotivescheduling problem can be transformed into a standardassignment problem by treating arrival locomotives as ldquoper-sonnelrdquo and departure trains as ldquomissionsrdquo

Therefore the transformation of heavy haul locomotivescheduling problem mainly focuses on the following threeaspects

(1) Transforming Unpaired Train Diagram into Paired TrainDiagram Usually the trainwill be operated smoothly accord-ing to the train diagram Because the number of trains in traindiagram is given on the known railway the number of trainswithin 24 h is determined If the train diagram is unpairedit needs to add some ldquovirtual operation curvesrdquo Then thenumber of arrival trains is equal to the number of departuretrains in each section It assumes that the number of ldquovirtualoperation curvesrdquo is 119884 and the number of actual trains is 119869

(2) Transforming Multilocomotive Traction into Single-Locomotive Traction If the total number of trains is 119884 + 119869the number of trains with double-locomotive traction is Wand the number of trains with three-locomotive traction is119881 (119882 + 119881 le 119869 + 119884) Then the number of ldquovirtual trainsrdquois 119882 + 2119881 when multilocomotive traction is transformedinto single-locomotive traction At the same time it needsto ensure that the information of train diagram is consistentwith before Then trains will be sorted and coded accordingto departure times and the code of trains may be 1 2 3 119869 + 119884 +119882 + 2119881 minus 2 119869 + 119884 +119882 + 2119881 minus 1 119869 + 119884 +119882 + 2119881

(3) Transformation betweenDifferent Locomotive TypesThereare two locomotive types which include SS4 locomotive and

HXD locomotive inDa-qin Railway 10 kt heavy-loaded com-bined trains are hauled by SS4 locomotives while 10 kt heavy-loaded unit trains 15 kt heavy-loaded combined trains and20 kt heavy-loaded combined trains are hauled by HXDlocomotives There is no locomotive reconnection betweenHXD locomotives and SS4 locomotives Therefore locomo-tive scheduling problemwhich involves two locomotive typescan be considered separately for each type in order to obtainthe optimal solutionsThen the optimization objective of thelocomotive schedule on the whole railway is achieved

42 Solution Algorithm of Heavy Haul Locomotive ScheduleIn this paper the optimization model of heavy haul locomo-tive schedule is solved by using Hungarian algorithm TheHungarian algorithm as a classical algorithm can obtain aglobal optimal solution when it is used to solve assignmentproblems With an increasing coefficient matrix dimensionthe algorithm runs a little slower but it can reduce computa-tional efforts muchmore than any enumerationmethod [22]The concrete steps of Hungarian algorithm are as follows

Step 1 (generating coefficient matrix) The coefficient matrixof detention time of locomotives at stations for each station isbuilt firstly and then the coefficient matrix for whole railwaycan be built

Step 2 (transforming coefficient matrix) When performingthe row operations on the coefficient matrix we subtract thelowest value from each element in that row in order to obtainat least one zero element for each row Similarly we also dothis procedure for all column elements

Step 3 (seeking independent zero elements) If the numberof independent zero elements is less than the order ofthe coefficient matrix then go to Step 4 If the numberof independent zero elements is equal to the order of thecoefficient matrix then output the optimization solution andend the algorithm

Step 4 (drawing the least number of straight lines) If thenumber of straight lines is less than the order of the coefficientmatrix then mark the rows and columns and go to Step 5otherwise go to Step 3

Step 5 (transforming coefficient matrix again) Seek mini-mum value of the elements which are not covered by straightlines This value is then subtracted from each element inall marked rows and added to each element in the markedcolumns and then go to Step 3

The flow chart of the Hungarian algorithm is shown inFigure 8

Mathematical Problems in Engineering 7

Begin

equal to thematrix order

Mark the least zero elements and remove the other zero elements

The value of marked zero elements is 1 and

the other is 0 in thecounterpart matrix

Cover all of zero elements by the least straight linesEnd

Generating coefficient matrix

Each row and column element minus the minimum value

Is the number ofstraight lines smaller than

the matrix order

Transform coefficient matrix

again

YesNo

Yes

No

Is the number ofindependent zero elements

Figure 8 Flow chart of the Hungarian algorithm

5 Case Study

51 Evaluating the Feasibility of Heavy Haul LocomotiveSchedule Model The model and its solution algorithm needto be validated before applying them to any real problems

511 Data Source and Processing As shown in Figure 4line A-B-C-D is a double-track railway (section A-D forshort) which is specially used to transport coal Station Bis the station where the locomotive depot is located andthe minimal technological time of locomotives at stations is125min Station A station C and station D are the stationswhere locomotive turnaround depots are located and theminimal technological time of locomotives at stations is50min There are 8 trains (4 pairs) which include 2 trains(1 pair) in section A-B 4 trains (2 pairs) in section B-Cand 2 trains (1 pair) in section B-D on the railway If all oftrains are hauled by single-locomotive Figure 6 is the traindiagram

This basic data needs to be processed as follows

(1) Stations Coding For all stations in section A-D it can codestations from any end of the railway and it starts with stationA

(2) Trains Coding The trains in section A-D are sole How-ever it is necessary for each train to code another sole and

Table 2 Trains coding result in section A-D

Train number Train coding Train number Train coding70001 1 73003 570002 2 73004 673001 3 79001 773002 4 79002 8

simple number due to the long and large train number Thecoding results are listed in Table 2

(3) Time Simplification For the departure time and arrivaltime of each train starting at the initial time (1800) of thetrain diagram time of 24 h can be converted to minutes andfor example 1933 can be converted to the 93rd minute

(4) Generating Coefficient Matrix According to (4) thecoefficient matrix 119888119896119894119895 of the detention time of locomotives atstations will be generated by using C The results are listedbelow

Station A Consider the following

119888

A21 = 50 minus 265 minus 50 + 1440 = 1175

(50 minus 265 = minus215 lt 50)

(12)

8 Mathematical Problems in Engineering

Station B Consider the following

119888

B12 = 220 minus 90 minus 125 = 5 (220 minus 90 = 130 gt 125)

119888

B13 = 280 minus 90 minus 125 = 65 (280 minus 90 = 190 gt 125)

119888

B15 = 516 minus 90 minus 125 = 301

(516 minus 90 = 426 gt 125)

119888

B17 = 150 minus 90 minus 125 + 1440 = 1375

(150 minus 90 = 60 lt 125)

119888

B42 = 220 minus 450 minus 125 + 1440 = 1085

(220 minus 450 = minus230 lt 125)

119888

B43 = 280 minus 450 minus 125 + 1440 = 1145

(280 minus 450 = minus170 lt 125)

119888

B45 = 516 minus 450 minus 125 + 1440 = 1381

(516 minus 450 = 66 lt 125516 minus 450 = 66 lt 125)

119888

B47 = 150 minus 450 minus 125 + 1440 = 1015

(150 minus 450 = minus300 lt 125)

119888

B62 = 220 minus 673 minus 125 + 1440 = 862

(220 minus 673 = minus453 lt 125)

119888

B63 = 280 minus 673 minus 125 + 1440 = 922

(280 minus 673 = minus393 lt 125)

119888

B65 = 516 minus 673 minus 125 + 1440 = 1158

(516 minus 673 = minus157 lt 125)

119888

B67 = 150 minus 673 minus 125 + 1440 = 792

(150 minus 673 = minus523 lt 125)

119888

B82 = 220 minus 363 minus 125 + 1440 = 1172

(220 minus 363 = minus143 lt 125)

119888

B83 = 280 minus 363 minus 125 + 1440 = 1232

(280 minus 363 = minus83 lt 125)

119888

B85 = 516 minus 363 minus 125 = 28

(516 minus 363 = 153 gt 125)

119888

B87 = 150 minus 363 minus 125 + 1440 = 1102

(150 minus 90 = minus213 lt 125)

(13)

Table 3 Coefficient matrix of detention time of locomotives atstations

119894

119895

1 2 3 4 5 6 7 81 infin 5 65 infin 301 infin 1375 infin

2 1175 infin infin infin infin infin infin infin

3 infin infin infin 37 infin 260 infin infin

4 infin 1085 1145 infin 1381 infin 1015 infin

5 infin infin infin 1241 infin 24 infin infin

6 infin 862 922 infin 1158 infin 792 infin

7 infin infin infin infin infin infin infin 148 infin 1172 1232 infin 28 infin 1102 infin

Station C Consider the following

119888

C34 = 407 minus 320 minus 50 = 37 (407 minus 320 = 87 gt 50)

119888

C36 = 630 minus 320 minus 50 = 260

(630 minus 320 = 310 gt 50)

119888

C54 = 407 minus 556 minus 50 + 1440 = 1241

(407 minus 556 = minus149 lt 50)

119888

C56 = 630 minus 556 minus 50 = 24 (630 minus 556 = 74 gt 50)

(14)

Station D Consider the following

119888

D78 = 296 minus 232 minus 50 = 14 (296 minus 232 = 64 gt 50) (15)

The others are equal toinfin the coefficient matrix (119888119896119894119895) isshowed in Table 3

512 Result The Hungarian algorithm is implemented inMATLAB and the optimal locomotive schedule is as follows

The minimum waiting time of locomotives at stations is

119879119908 = 119885 =

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

119888

119896119894119895119909119896119894119895

= 65 + 1175 + 37 + 1085 + 24 + 792 + 14 + 28

= 3220 (min)

(16)

Theminimal technological time of locomotives at stationsis

119879119870 = 125 times 4 + 50 times 4 = 700 (min) (17)

The running time of locomotives is

119879119905 = 40 + 45 + 40 + 43 + 40 + 43 + 82 + 67

= 400 (min) (18)

The minimum number of locomotives is

119872 =

119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119879119908

1440

=

400 + 700 + 3220

1440

=

4320

1440

= 3

(19)

Mathematical Problems in Engineering 9

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 9 Locomotive schedule in section A-D

Chawu

299 kmDashizhuang

CuipingshanZunhua North

Qianrsquoan North

Jixian West

HouyingLiucun South

Qinhuangdao EastLuannan

Donggang

Caofeidian West

60 km19 km

11km59 km 78 km 67 km

22 km

24km

71km

66km

44 km

Hudong

Figure 10 The simplified network of Da-qin Railway

The coefficient of locomotive requirement is

119870119889 =119872

119899

=

3

82

= 075 (20)

The locomotive schedule can be expressed as an intuitivediagram which is shown in Figure 9 (dotted lines representlocomotives and solid lines represent trains)

Figure 9 shows that there are 3 traction locomotiveswhich are equal to the calculation result to haul trains

513 Result Analysis

(1) Analysis on the Optimization of Locomotive SchedulingFigures 7 and 9 show that there may be a variety of trainconnections So the number of available locomotives andcoefficient of locomotive requirement are uncertain Theoptimization model of heavy-loaded locomotive schedule isestablished and then it can be solved byHungarian algorithmIt can not only make sure of the uniqueness of locomotivetraction tasks but also complete established traction taskswith a minimum number of locomotives

(2) Analysis on the Equilibrium of Locomotive Schedule Theequilibrium degree of locomotive schedule of Figure 9 can beobtained by using (9) which is

119863 =

1

8

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

8

)

2

=

1

8

[(65 minus

3220

8

)

2

+ (1175 minus

3220

8

)

2

+ (37 minus

3220

8

)

2

+ (1085 minus

3220

8

)

2

+ (24 minus

3220

8

)

2

+ (792 minus

3220

8

)

2

+ (14 minus

3220

8

)

2

+ (28 minus

3220

8

)

2

] = 237027

(21)

It shows that the equilibrium degree of locomotive sched-ule is large The main reason is that there are no trains from600 to 1800 on the railwayTherefore the difference betweenthe detention time of locomotives at stations from 1800 to600 the next day and the detention time of locomotives atstations from 600 to 1800 is relatively large If there are trainsfrom 600 to 1800 the equilibrium degree of locomotiveschedule will be reduced and the locomotive schedule will bemore balanced

52 Analysis of Locomotive Schedule for 20 kt CombinedTrains on Da-qin Railway The previous section shows thatthe model of locomotive schedule is feasible Then theeffectiveness of the proposedmodel also needs to be validatedwhich is conducted in this section

521 Line Simplification Da-qin Railway has 26 stations andthe line length is 653 km Qiancao Railway has 5 stations andthe line length is 137 km Jingtang Port-Donggang RailwayLine has 5 stations and the line length is 45 km Someintermediate stations on the railway do not have the technicaloperation so they are not displayed Only the stationswhere locomotive depot or turnaround depot is located areconsidered The simplified network of Da-qin Railway isshown in Figure 10

Hudong station is the station where the locomotive depotis located and theminimal technological time of locomotives

10 Mathematical Problems in Engineering

at stations is 250min Liucun South station Donggang sta-tion Caofeidian West station and Qinhuangdao East stationare the stations where locomotive turnaround depots arelocated The minimal technological time of locomotives atstations for Liucun South station is 220min The minimaltechnological time of locomotives at stations for Donggangstation and Caofeidian West station is 100min The minimaltechnological time of locomotives at stations for Qinhuang-daoEast station is 125minTheminimal technological time ofthe other stations where the locomotives turn back is 50min

There were 100 heavy-loaded trains in the heavy-loaddirection and 104 empty-wagon trains in the reverse directionon December 10 2014

522 Data Processing In reference to the mentionedmethod complete the data processing For the HXDlocomotives there are 328 trains which are hauled by single-locomotive after conversion so the order of the coefficientmatrix is 328 For the SS4 locomotives there are 56 trainswhich are hauled double-locomotive after conversion sothe order of the coefficient matrix is 56 In this paper C isemployed to deal with such huge data

523 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

119888

119896119894119895119909119896119894119895 = 37116 (min)

119885SS4 =13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10409 (min)

(22)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 164 times 250 + 76 times 220 + 14 times 125 + 44 times 100

+ 30 times 50 = 65370 (min)

119879

SS4119870 = 28 times 250 + 28 times 50 = 8400 (min)

(23)

The running times of locomotives are respectively

119879

HXD119905 = 198474 (min)

119879

SS4119905 = 24391 (min)

(24)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

198474 + 65370 + 37116

1440

=

300960

1440

= 209

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

24391 + 8400 + 10409

1440

times 2 =

43200

1440

times 2

= 60

(25)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

209

1602

= 261

119870

SS4119889 =

119872SS4119899

=

30 times 2

562

= 214

(26)

According to the best connection matrix locomotiveconnection scheme for 20 kt combined trains on Da-qinRailway can be obtainedThe locomotive connection schemeof SS4 locomotives is as follows

H16012 997888rarr 52 (single-locomotive) 997888rarr H16016

997888rarr 54 (single-locomotive) 997888rarr H16004

997888rarr H19609 997888rarr H19504 997888rarr H19605

997888rarr H19512 997888rarr 53 (single-locomotive)

997888rarr H16018 997888rarr 56 (single-locomotive)

997888rarr H16002 997888rarr 48 (single-locomotive)

997888rarr H15506 997888rarr 47 (single-locomotive)

997888rarr H16010 997888rarr 19601 997888rarr H19508

997888rarr 46 (single-locomotive) 997888rarr H16008

997888rarr H15501 997888rarr H15504 997888rarr H19607

997888rarr H19516 997888rarr 19615 997888rarr H19514

997888rarr H28501 997888rarr H28510

997888rarr 51 (single-locomotive) 997888rarr H16014

997888rarr H19603 997888rarr H19502 997888rarr H28507

997888rarr H28512 997888rarr 49 (attached locomotive)

997888rarr H15508 997888rarr H28503 997888rarr H28508

997888rarr 50 (single-locomotive) 997888rarr H16012

997888rarr H19611 997888rarr H19506 997888rarr H28511

997888rarr H28502 997888rarr H15503 997888rarr H15502

997888rarr 55 (attached locomotive) 997888rarr H15510

997888rarr H28505 997888rarr H28506

997888rarr 45 (single-locomotive) 997888rarr H16006

997888rarr H19611

H19613 997888rarr H19510 997888rarr H19613

H28504 997888rarr H28509 997888rarr H28504

(27)

Mathematical Problems in Engineering 11

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

Luannan

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

LuannanH16012 H16014 H16016 H16018H16002 H16004 H16006

H16002 H16004 H16006 H16008 H16010 H160112 H16014H16016 H16018

H15501

H15501

H15503

H15503

H15502 H15504 H15506 H15508 H15510

H28502 H28504 H28506 H28508 H28510 H28512

H15502H28502 H28504 H15504 H128506 H28508 H28510H15506 H15508 H28512

H28501 H28505 H28507 H28509 H28511

H28501 H28503 H28505 H28507 H28509 H28511

H19601 H19603 H19615 H19605 H19607 H19609 H19613 H19611

H19516 H19502 H19504 H19506 H19508 H19510 H19514 H19512

H28503

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

TrainsLocomotivesSingle-locomotives or attached locomotives

H16008 H16010

Figure 11 Locomotive working diagram of SS4 locomotives for 20 kt combined trains

Table 4 The daily number of locomotives on Da-qin Railway onDecember 10 2014 [21]

Types oflocomotive

Daily number oflocomotives

Types oflocomotive

Daily number oflocomotives

HXD1 130 SS4 109HXD2 92 mdash mdash

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 11

524 Analysis of Result

(1) Analyze the Number of Available LocomotivesThe numberof available locomotives (the total number of locomotivesafter deducting locomotives inmaintenance and preparation)onDa-qin Railway onDecember 10 2014 is shown in Table 4

Table 4 indicates that the locomotive schedule obtainedby using themodel can save the number ofHXD locomotives130 + 92 minus 209 = 13 and the number of SS4 locomotives109minus 30times 2 = 49 Therefore if only the minimum number oflocomotives is considered the optimization model of heavyhaul locomotive schedule will be optimized

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

328

13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

328

)

2

= 52379

119863SS4 =1

56

13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

56

)

2

= 74044

(28)

The equilibrium degree of the locomotive schedule forHXD locomotives and SS4 locomotives is smaller than theequilibrium degree of that case which shows that this loco-motive schedule is more balanced The equilibrium degreeof SS4 locomotives is larger than the equilibrium degree ofHXD locomotives which shows that the schedule of HXDlocomotives ismore balanced than SS4 locomotivesThe largeequilibrium degree of SS4 locomotives may be due to thesmall detention time of locomotives at stations

53 Research and Analysis of Locomotive Schedule Optimiza-tion for 30 kt Combined Trains on Da-qin Railway On thebasis of the potential train diagram for 30 kt combined trainsthe practical and feasible locomotive connection scheme canbe determined by using the optimization model

In reference to the mentioned method complete the dataprocessing

531 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

119888

119896119894119895119909119896119894119895 = 44838 (min)

119885SS4 =13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10314 (min)

(29)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 203 times 250 + 64 times 220 + 65 times 100 + 74 times 50

= 75030 (min)

119879

SS4119870 = 26 times 250 + 26 times 50 = 7800 (min)

(30)

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

204 254 304 340 330 400 440 426 445 450

2205 2373 2526 2802 2973

3235 3520 3650 3680 3590

000

500

1000

1500

2000

2500

3000

3500

4000

2005 2006 2007 2008 2009 2010 2011 2012 2013 2014

Da-qin Railway freight volumeChinarsquos coal demand

Years

100 million tons

Figure 1 Chinarsquos coal demand and Da-qin Railway freight volumefrom 2005 to 2014 [2]

tons in 2005 which already reached the international theoret-ical limit of heavy haul railway transportation Moreover thefreight volume increased to 400 million tons in 2010 and 450million tons in 2014 However the transported cargo volumestill cannot meet the coal demand Freight volume still needsto be improved It is difficult to have higher freight flow den-sity for the current train diagram and increase trainrsquos speedin Da-qin RailwayThe only way to increase freight volume isto increase the transportation capacity of a trainTherefore itneeds to operate 30 kt combined trains in Da-qin Railway

Locomotives are a critical factor for the capacity of arailway The locomotive schedule has a direct impact onefficiency operation cost and benefits of a railwayThereforethe study on the locomotive schedule for 30 kt combinedtrains in Da-qin Railway is necessary

Background of Heavy Haul Locomotive Schedule Research onheavy haul locomotives mainly concentrates on the technicallevel Cheng et al [3] provided technical support for theapplication of Reliability Centered Maintenance (RCM) inrailway locomotive and verified its effectiveness in keepingreliability and reducing maintenance cost Zheng and Wei[4] Chang [5] and Mu et al [6] analyzed the mechanicalresponse of railway track structure under axle loads from25 t to 40 t Raviv and Kaspi [7] and Kumar and Bierlaire [8]proposed a mixed integer linear model that determines theoptimal strategy Studies on optimal locomotive schedule arequite few because of the small density of heavy-loaded trainsshort length of station track and the difficult transportationorganization Meanwhile these researches of heavy haullocomotives are mainly for 10sim20 kt heavy-loaded trainsRelatively speaking researches on the operational level aremainly about regular railways which focus on the followingareas (1) Ji et al [9] used computer technologies to achievedouble-locomotive traction control and determine locomo-tive marshalling schedules (2) Ruppert et al [10] analyzedhow to increase the usage efficiency and reliability of loco-motives (3) Kuo and Nicholls [11] developed a mixed integer

linear program (MILP) to determine the least-cost plan ofallocating locomotives and improve locomotive utilization(4) Aronsson et al [12] built a mixed integer program (MIP)model to solve locomotive routing and scheduling problemand Vaidyanathan et al [13] formulated the locomotiverouting problem (LRP) as an integer programming problemon a suitably constructed space-time network and developedrobust optimization methods to solve it (5) Ghoseiri andGhannadpour [14] Kasalica et al [15] Rouillon et al [16]and Teichmann et al [17] were devoted to the research oflocomotive assignment problem and used different methodsto optimize it and (6) Li et al [18] established amathematicalmodel for generating a locomotiveworking diagramof a traindiagram with paired trains then they used the improvedant colony algorithm to solve it Most of the locomotiveschedule studies are for regular trains and only a few of theminvolve 10sim20 kt heavy-loaded trainsThere are no researchesof locomotive scheduling for 30 kt heavy-loaded trains

2 Analysis of Locomotive Schedule forHeavy-Loaded Combined Trains onDa-qin Railway

21 Organization Forms of Heavy-Loaded Trains on Da-qinRailway There are two organization types for heavy-loadedtrains on Da-qin Railway unit type and combined type In2014 the daily average was 93 in the heavy-load directionon Da-qin Railway which include 65 trains (20 kt heavy-loaded combined) 12 trains (15 kt heavy-loaded combined)13 trains (10 kt heavy-loaded combined) and 3 trains (10 ktheavy-loaded unit) The combined type is adopted for 30 ktheavy-loaded trains

22 Locomotive Utilization Method on Da-qin Railway Theuncertain railway regionmethod is currently used to developthe heavy haul locomotive schedule on Da-qin Railwaywhich aims at reducing locomotive turnaround time [19]Themethod can significantly improve locomotive schedulingefficiency Figure 3 is an illustration of uncertain railroadregions locomotive utilization method

The numbers in Figure 3 mean the running time of thecorresponding section If a locomotive which hauls a traindeparts from station C after 38min it will arrive at station AIn station A there are two traction choices to haul differenttrains The locomotive utilization region of choice (a) is A-B-D-A and the region of choice (b) is A-C-A In station A ifall locomotives haul trains with the same region there will belonger waiting time Therefore the uncertain railway regionmethod can reduce locomotive turnaround time

23 Criteria of Locomotive Schedule The criteria of locomo-tive schedule can reflect thework quality efficiency and bene-fit of locomotives which include quantitative criteria qualitycriteria and equilibrium criteria The critical performancemeasures for these three criteria are the number of availablelocomotives coefficient of locomotive requirement and theequilibrium degree of locomotive schedule respectively

Mathematical Problems in Engineering 3

LuannanDonggang

Caofeidian West

Hud

ong

235

30

Dat

ong

Sout

h 6

0H

anjia

ling

215

23

Dat

ong

coun

ty 3

99

Yang

quan

86

235

Hua

shao

ying

129

020

Zhul

u 18

267

4

Shac

heng

Eas

t 218

263

Beix

inba

o 23

701

4

Yanq

ing

Nor

th 2

625

14

Xiaz

huan

g 30

769

2

Chaw

u 32

72

Chag

ao L

ink

Line

Ping

gu 3

688

44

Jixia

n W

est 4

066

44Cu

ipin

gsha

n 41

796

1

Dad

uan

Link

Lin

eDi

recti

on of

Jixia

nYu

tian

Nor

th 4

408

38

Zunh

ua N

orth

476

767

Qia

nxi 5

083

35

Qia

ncao

Line

Qia

nrsquoan

Nor

th 5

547

27Lu

long

Nor

th 5

688

98

Hou

ying

621

100

Liuc

un S

outh

73

14Li

ucun

Sou

th 2

nd Y

ard

720

7

Qin

huan

gdao

Eas

t 14

417

16725 XihanlingPingwang 10570

KouquanBranch line

Yungang lineDatong

Directionof Gudian

Das

hizh

uang

387

344

Marshalling stationDistrict stationIntermediate station

Figure 2 Network of Da-qin Railway

Station A

Station B

Station C

Station D Locomotive

40 4042

35 37

53Traction choice (a)Traction choice (b)

Figure 3 An illustration of uncertain railroad region locomotiveutilization method

(1) Number of Available Locomotives Number of availablelocomotives is the total number of locomotives which areused in the whole turnaround process [20] The less thenumber is the higher the efficiency will be The graphicmethod and the direct calculation method are generalmethods to determine the number of available locomotivesThe graphic method is based on locomotive schedule todetermine the number of used locomotives The formula ofdirect calculation method is listed below

119872119891 =(119899119904 + 119899119889 + 2 sdot 119899119905) sdot 120579119897

24

= 120583119904 sdot 119870119889(1)

where 119872119891 is the number of available locomotives in eachtraction section 119899119904 is the total number of trains 119899119889 is thenumber of trains with double-locomotive traction 119899119905 is thenumber of trains with three-locomotive traction 120579119897 is thelocomotive complete turnaround time 120583119904 is the number ofsupplied locomotives 119870119889 is the coefficient of locomotiverequirement in traction sections

(2) Coefficient of Locomotive Requirement The coefficient oflocomotive requirement indicates the number of locomotivesrequired per pair of trains in a traction section [20] Onelocomotive completes a turnaround which means it has

completed a traction task The coefficient of locomotiverequirement can be calculated by the formula below

119870119889 =120579119897

24

(2)

(3) Equilibrium Degree of Locomotive Schedule In additionto quantitative and quality criteria the equilibrium criteriaare also very important in evaluating locomotive schedulewhich refer to the difference between detention time of eachlocomotive at stations and running time of each locomotiveAt the same section the running time is the sameThereforein the actual evaluation process the variance of detentiontime of each locomotive at stations 119863 (equilibrium degree)can be used to describe the equilibrium of locomotiveschedule It can be calculated as below

119863 =

1

119899119897

119899119897

sum

119894=1

(119888119894 minus 119864)2 (3)

where 119899119897 is the total number of traction tasks of all locomo-tives and it is numerically equal to (119899119904 + 119899119889 + 2 sdot 119899119905) 119894 is alocomotive that undertakes traction tasks that is 119894 isin 119899119897 119888119894 isthe detention time of a locomotive at station 119864 is the averagedetention time of locomotives at stations

Equation (3) shows that equilibrium degree 119863 refers tothe discrete degree between 119888119894 (119894 = 1 2 3 119899119897) and 119864 Thesmaller the value of119863 is the smaller the difference is and themore balanced the locomotive schedule will be

3 The Optimization Model of Heavy HaulLocomotive Scheduling Problem

31 Description of Locomotive Scheduling Problem The inputsfor the locomotive scheduling problem are railway linelocomotive traction sections and train diagram and so forthBased on these input data the output namely locomotiveschedule can be obtained

4 Mathematical Problems in Engineering

A B C D

Station with depotStation with turnaround depot

Figure 4 The network of hypothetical railway line

A B C D

Figure 5 Schematic diagram of locomotive traction sections

311 Input

(1) Railway Line Railway line A-B-C-D is a double-trackrailway as shown in Figure 4 It is dedicatedly used totransport coal On the railway station B is the station wherethe locomotive depot is located Station A station C andstation D are the stations where locomotive turnarounddepots are located

(2) Locomotive Traction Sections When freight locomotivesundertake traction tasks in section A-D they firstly departfrom station B and haul heavy-loaded trains to arrive atstation A station C or station D Then judge whetherlocomotive preparation is needed or not Finally these loco-motives haul empty-wagon trains and return to the stationwhere the locomotive depot is located while the locomotiveswhich are at turnaround depot can haul the return trainsonly The schematic diagram of locomotive traction sectionsis shown in Figure 5

(3) Train Diagram A complete train diagram determinestrain routes departurearrival time and detention time ateach station In the train diagram we cannot know thelocomotiversquos position andwhich train it will haul As shown inFigure 6 there are 4 pairs of trains on the hypothetical railway(4 heavy-loaded trains and 4 empty-wagon trains) The trainnumbers respectively are 700012 730012 730034 and790012 The train whose train number is 70001 refers to aheavy-loaded train in section A-B and the train whose trainnumber is 70002 refers to an empty-wagon train in section A-B The train whose train number is 73001 or 73003 refers to aheavy-loaded train in section B-C and the train whose trainnumber is 73002 or 73004 refers to an empty-wagon train insection B-CThe train whose train number is 79001 refers to aheavy-loaded train in section D-B and the train whose trainnumber is 79002 refers to an empty-wagon train in sectionD-B

312 Output Based on the input data the potential locomo-tive schedule in section A-D will be obtained (see Figure 7)where a dotted line represents a locomotive Figure 7 showsthat after hauling the train whose train number is 70001 to

station B the locomotive can haul the train whose train num-ber is 70002 79001 or 73001Whenmultilocomotive tractionschemes exist the minimal detention time of locomotives atstations will be adopted to determine the final scheme

32 Model Description Some assumptions are presented inthis model (a) the train diagram is known and invariant in acertain period of time that is the impact of train delay willbe ignored (b) based on the line length of Da-qin Railway(653 km) locomotive routing and locomotive running time(eg a round-trip time more than 17 h) it assumes within24 hours all available locomotives (the total locomotives afterdeducting the locomotives in maintenance and preparation)are without maintenance (c) for the trains with multilo-comotive traction locomotive reconnection between HXDlocomotives and SS4 locomotives is not considered the timeof locomotive reconnection and dismantling was includedin the detention time of locomotives at stations (d) in acertain period of time the locomotive crew working systemin Da-qin Railway is invariant so the effects of locomotivecrew working system and locomotive crew shift mode onlocomotive scheduling are not considered and (e) in thedaily initial time (1800) each station has enough locomotiveswhich at least are equal to the number of locomotives in needthat is the locomotive homing will not be considered

321 Sets In the process of constructing the optimizationmodel of the heavy haul locomotive scheduling problemthere are some notations and sets that will be used

119870 is the set that includes all of stations on the railway

119896 is a station on the railway that is 119896 isin 119870

119873119896119889 is the total number of locomotives arriving atstation 119896

119873119896119891 is the total number of trains departing fromstation 119896

119894 is the locomotive arriving at station 119896

119895 is the train departing from station 119896

119879119896 is the minimal technological time of locomotivesat station 119896

119896119894119889 is the destination station of locomotive 119894 at station119896

119905119894119889119896 is the arrival time of locomotive 119894 at station 119896

119896119895119891 is the departure station of train 119895 from station 119896

119905119895119891119896 is the departure time of train 119895 from station 119896

322 Parameters In the process of developing the opti-mization model the known parameter that will be involvedis 119888119896119894119895 It refers to the waiting time which is equal to thedetention time of locomotive 119894 at station 119896minus theminimal

Mathematical Problems in Engineering 5

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 6 Train diagram in section A-D

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 7 Potential locomotive schedule in section A-D

technological time The waiting time 119888119896119894119895 of locomotives atstations can be calculated by

119888

119896119894119895

=

119905119895119891119896 minus 119905119894119889119896 minus 119879119896 119905119895119891119896 minus 119905119894119889119896 ge 119879119896 119896119894119889 = 119896119895119891

119905119895119891119896 minus 119905119894119889119896 minus 119879119896 + 1440 119905119895119891119896 minus 119905119894119889119896 lt 119879119896 119896119894119889 = 119896119895119891

infin 119896119894119889 = 119896119895119891

(4)

323 Variable Thevariable is assumed to be 119888119896119894119895 which refersto whether locomotive 119894 hauls train 119895 or not at station 119896 Thebinary variable 119888119896119894119895 is defined as follows

119909

119896119894119895 =

1 locomotive 119894 hauls train 119895 at station 119896

0 otherwise(5)

324 Objective Function Completing planned traction tasksis the primary constraint to us Based on that the lessthe number of locomotives is the lower the transportationcost will be Therefore the objective function of heavy haullocomotive scheduling problem is minimizing number oflocomotives that is

min 119885

1015840=

119879119905 + 119879

1440

(6)

where 119879119905 is the total time of trains running from departurestation to destination station and T is the total detention timeof locomotives at stations which can be calculated by

119879 = 119879119870 + 119879119908 (7)

where 119879119870 is the total minimal technological time of locomo-tives at stations and119879119908 is the total waiting time of locomotivesat stations

For the established train diagram 119879119905 is a constant Sothe minimum number of locomotives refers to the minimaldetention time 119879 of locomotives at stations Meanwhile119879119870 is also a constant So the minimal detention time oflocomotives at stations refers to the minimum waiting time119879119908 of locomotives at stations That is

min 119885 = 119879119908 =

119870

sum

119896=1

119873119896119889

sum

119894=1

119873119896119891

sum

119895=1

119888

119896119894119895119909119896119894119895 (8)

Then the equilibrium degree 119863 of locomotive schedulecan be calculated by

119863 =

1

119899119904 + 119899119889 + 2 sdot 119899119905

sdot

119870

sum

119896=1

119873119896119889

sum

119894=1

119873119896119891

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

119899119904 + 119899119889 + 2 sdot 119899119905

)

2

(9)

325 Constraints

(1) Traction Locomotives Constraint Consider the following

119873119896119889

sum

119894=1

119909

119896119894119895 = ℎ forall119895 isin 119873119896119891 forall119896 isin 119870 (10)

This constraint shows that the number of traction loco-motives for each train is ℎ The value of each type of train isshown in Table 1

(2) Traction Tasks Constraint Consider the following

119873119896119891

sum

119895=1

119909

119896119894119895 = 1 forall119894 isin 119873119896119889 forall119896 isin 119870 (11)

6 Mathematical Problems in Engineering

Table 1 The value of each type of trains

Types of heavy-loaded trains Reference value Types of heavy-loaded trains Reference value10 kt heavy-loaded combined trains 2 20 kt heavy-loaded combined trains 210 kt heavy-loaded unit trains 1 30 kt heavy-loaded combined trains 315 kt heavy-loaded combined trains 2 mdash mdash

This constraint shows that each locomotive can only haulone train at the same time

4 Transformation and SolvingProcesses of Heavy Haul LocomotiveScheduling Problem

First of all the locomotive scheduling problem needs to betransformed into an assignment problemThen the optimiza-tion results are obtained by using the Hungarian algorithm

41 Transformation of Heavy Haul Locomotive SchedulingProblem For an unpaired train diagram the number ofarrival trains is not equal to the number of departure trainsat some stations So the unpaired train diagram needs tobe transformed into a paired train diagram before solvingthe locomotive scheduling problem For the paired traindiagram the number of arrival trains is equal to the numberof departure trains at each station Therefore if each trainis hauled by the same type of locomotive the locomotivescheduling problem can be transformed into a standardassignment problem by treating arrival locomotives as ldquoper-sonnelrdquo and departure trains as ldquomissionsrdquo

Therefore the transformation of heavy haul locomotivescheduling problem mainly focuses on the following threeaspects

(1) Transforming Unpaired Train Diagram into Paired TrainDiagram Usually the trainwill be operated smoothly accord-ing to the train diagram Because the number of trains in traindiagram is given on the known railway the number of trainswithin 24 h is determined If the train diagram is unpairedit needs to add some ldquovirtual operation curvesrdquo Then thenumber of arrival trains is equal to the number of departuretrains in each section It assumes that the number of ldquovirtualoperation curvesrdquo is 119884 and the number of actual trains is 119869

(2) Transforming Multilocomotive Traction into Single-Locomotive Traction If the total number of trains is 119884 + 119869the number of trains with double-locomotive traction is Wand the number of trains with three-locomotive traction is119881 (119882 + 119881 le 119869 + 119884) Then the number of ldquovirtual trainsrdquois 119882 + 2119881 when multilocomotive traction is transformedinto single-locomotive traction At the same time it needsto ensure that the information of train diagram is consistentwith before Then trains will be sorted and coded accordingto departure times and the code of trains may be 1 2 3 119869 + 119884 +119882 + 2119881 minus 2 119869 + 119884 +119882 + 2119881 minus 1 119869 + 119884 +119882 + 2119881

(3) Transformation betweenDifferent Locomotive TypesThereare two locomotive types which include SS4 locomotive and

HXD locomotive inDa-qin Railway 10 kt heavy-loaded com-bined trains are hauled by SS4 locomotives while 10 kt heavy-loaded unit trains 15 kt heavy-loaded combined trains and20 kt heavy-loaded combined trains are hauled by HXDlocomotives There is no locomotive reconnection betweenHXD locomotives and SS4 locomotives Therefore locomo-tive scheduling problemwhich involves two locomotive typescan be considered separately for each type in order to obtainthe optimal solutionsThen the optimization objective of thelocomotive schedule on the whole railway is achieved

42 Solution Algorithm of Heavy Haul Locomotive ScheduleIn this paper the optimization model of heavy haul locomo-tive schedule is solved by using Hungarian algorithm TheHungarian algorithm as a classical algorithm can obtain aglobal optimal solution when it is used to solve assignmentproblems With an increasing coefficient matrix dimensionthe algorithm runs a little slower but it can reduce computa-tional efforts muchmore than any enumerationmethod [22]The concrete steps of Hungarian algorithm are as follows

Step 1 (generating coefficient matrix) The coefficient matrixof detention time of locomotives at stations for each station isbuilt firstly and then the coefficient matrix for whole railwaycan be built

Step 2 (transforming coefficient matrix) When performingthe row operations on the coefficient matrix we subtract thelowest value from each element in that row in order to obtainat least one zero element for each row Similarly we also dothis procedure for all column elements

Step 3 (seeking independent zero elements) If the numberof independent zero elements is less than the order ofthe coefficient matrix then go to Step 4 If the numberof independent zero elements is equal to the order of thecoefficient matrix then output the optimization solution andend the algorithm

Step 4 (drawing the least number of straight lines) If thenumber of straight lines is less than the order of the coefficientmatrix then mark the rows and columns and go to Step 5otherwise go to Step 3

Step 5 (transforming coefficient matrix again) Seek mini-mum value of the elements which are not covered by straightlines This value is then subtracted from each element inall marked rows and added to each element in the markedcolumns and then go to Step 3

The flow chart of the Hungarian algorithm is shown inFigure 8

Mathematical Problems in Engineering 7

Begin

equal to thematrix order

Mark the least zero elements and remove the other zero elements

The value of marked zero elements is 1 and

the other is 0 in thecounterpart matrix

Cover all of zero elements by the least straight linesEnd

Generating coefficient matrix

Each row and column element minus the minimum value

Is the number ofstraight lines smaller than

the matrix order

Transform coefficient matrix

again

YesNo

Yes

No

Is the number ofindependent zero elements

Figure 8 Flow chart of the Hungarian algorithm

5 Case Study

51 Evaluating the Feasibility of Heavy Haul LocomotiveSchedule Model The model and its solution algorithm needto be validated before applying them to any real problems

511 Data Source and Processing As shown in Figure 4line A-B-C-D is a double-track railway (section A-D forshort) which is specially used to transport coal Station Bis the station where the locomotive depot is located andthe minimal technological time of locomotives at stations is125min Station A station C and station D are the stationswhere locomotive turnaround depots are located and theminimal technological time of locomotives at stations is50min There are 8 trains (4 pairs) which include 2 trains(1 pair) in section A-B 4 trains (2 pairs) in section B-Cand 2 trains (1 pair) in section B-D on the railway If all oftrains are hauled by single-locomotive Figure 6 is the traindiagram

This basic data needs to be processed as follows

(1) Stations Coding For all stations in section A-D it can codestations from any end of the railway and it starts with stationA

(2) Trains Coding The trains in section A-D are sole How-ever it is necessary for each train to code another sole and

Table 2 Trains coding result in section A-D

Train number Train coding Train number Train coding70001 1 73003 570002 2 73004 673001 3 79001 773002 4 79002 8

simple number due to the long and large train number Thecoding results are listed in Table 2

(3) Time Simplification For the departure time and arrivaltime of each train starting at the initial time (1800) of thetrain diagram time of 24 h can be converted to minutes andfor example 1933 can be converted to the 93rd minute

(4) Generating Coefficient Matrix According to (4) thecoefficient matrix 119888119896119894119895 of the detention time of locomotives atstations will be generated by using C The results are listedbelow

Station A Consider the following

119888

A21 = 50 minus 265 minus 50 + 1440 = 1175

(50 minus 265 = minus215 lt 50)

(12)

8 Mathematical Problems in Engineering

Station B Consider the following

119888

B12 = 220 minus 90 minus 125 = 5 (220 minus 90 = 130 gt 125)

119888

B13 = 280 minus 90 minus 125 = 65 (280 minus 90 = 190 gt 125)

119888

B15 = 516 minus 90 minus 125 = 301

(516 minus 90 = 426 gt 125)

119888

B17 = 150 minus 90 minus 125 + 1440 = 1375

(150 minus 90 = 60 lt 125)

119888

B42 = 220 minus 450 minus 125 + 1440 = 1085

(220 minus 450 = minus230 lt 125)

119888

B43 = 280 minus 450 minus 125 + 1440 = 1145

(280 minus 450 = minus170 lt 125)

119888

B45 = 516 minus 450 minus 125 + 1440 = 1381

(516 minus 450 = 66 lt 125516 minus 450 = 66 lt 125)

119888

B47 = 150 minus 450 minus 125 + 1440 = 1015

(150 minus 450 = minus300 lt 125)

119888

B62 = 220 minus 673 minus 125 + 1440 = 862

(220 minus 673 = minus453 lt 125)

119888

B63 = 280 minus 673 minus 125 + 1440 = 922

(280 minus 673 = minus393 lt 125)

119888

B65 = 516 minus 673 minus 125 + 1440 = 1158

(516 minus 673 = minus157 lt 125)

119888

B67 = 150 minus 673 minus 125 + 1440 = 792

(150 minus 673 = minus523 lt 125)

119888

B82 = 220 minus 363 minus 125 + 1440 = 1172

(220 minus 363 = minus143 lt 125)

119888

B83 = 280 minus 363 minus 125 + 1440 = 1232

(280 minus 363 = minus83 lt 125)

119888

B85 = 516 minus 363 minus 125 = 28

(516 minus 363 = 153 gt 125)

119888

B87 = 150 minus 363 minus 125 + 1440 = 1102

(150 minus 90 = minus213 lt 125)

(13)

Table 3 Coefficient matrix of detention time of locomotives atstations

119894

119895

1 2 3 4 5 6 7 81 infin 5 65 infin 301 infin 1375 infin

2 1175 infin infin infin infin infin infin infin

3 infin infin infin 37 infin 260 infin infin

4 infin 1085 1145 infin 1381 infin 1015 infin

5 infin infin infin 1241 infin 24 infin infin

6 infin 862 922 infin 1158 infin 792 infin

7 infin infin infin infin infin infin infin 148 infin 1172 1232 infin 28 infin 1102 infin

Station C Consider the following

119888

C34 = 407 minus 320 minus 50 = 37 (407 minus 320 = 87 gt 50)

119888

C36 = 630 minus 320 minus 50 = 260

(630 minus 320 = 310 gt 50)

119888

C54 = 407 minus 556 minus 50 + 1440 = 1241

(407 minus 556 = minus149 lt 50)

119888

C56 = 630 minus 556 minus 50 = 24 (630 minus 556 = 74 gt 50)

(14)

Station D Consider the following

119888

D78 = 296 minus 232 minus 50 = 14 (296 minus 232 = 64 gt 50) (15)

The others are equal toinfin the coefficient matrix (119888119896119894119895) isshowed in Table 3

512 Result The Hungarian algorithm is implemented inMATLAB and the optimal locomotive schedule is as follows

The minimum waiting time of locomotives at stations is

119879119908 = 119885 =

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

119888

119896119894119895119909119896119894119895

= 65 + 1175 + 37 + 1085 + 24 + 792 + 14 + 28

= 3220 (min)

(16)

Theminimal technological time of locomotives at stationsis

119879119870 = 125 times 4 + 50 times 4 = 700 (min) (17)

The running time of locomotives is

119879119905 = 40 + 45 + 40 + 43 + 40 + 43 + 82 + 67

= 400 (min) (18)

The minimum number of locomotives is

119872 =

119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119879119908

1440

=

400 + 700 + 3220

1440

=

4320

1440

= 3

(19)

Mathematical Problems in Engineering 9

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 9 Locomotive schedule in section A-D

Chawu

299 kmDashizhuang

CuipingshanZunhua North

Qianrsquoan North

Jixian West

HouyingLiucun South

Qinhuangdao EastLuannan

Donggang

Caofeidian West

60 km19 km

11km59 km 78 km 67 km

22 km

24km

71km

66km

44 km

Hudong

Figure 10 The simplified network of Da-qin Railway

The coefficient of locomotive requirement is

119870119889 =119872

119899

=

3

82

= 075 (20)

The locomotive schedule can be expressed as an intuitivediagram which is shown in Figure 9 (dotted lines representlocomotives and solid lines represent trains)

Figure 9 shows that there are 3 traction locomotiveswhich are equal to the calculation result to haul trains

513 Result Analysis

(1) Analysis on the Optimization of Locomotive SchedulingFigures 7 and 9 show that there may be a variety of trainconnections So the number of available locomotives andcoefficient of locomotive requirement are uncertain Theoptimization model of heavy-loaded locomotive schedule isestablished and then it can be solved byHungarian algorithmIt can not only make sure of the uniqueness of locomotivetraction tasks but also complete established traction taskswith a minimum number of locomotives

(2) Analysis on the Equilibrium of Locomotive Schedule Theequilibrium degree of locomotive schedule of Figure 9 can beobtained by using (9) which is

119863 =

1

8

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

8

)

2

=

1

8

[(65 minus

3220

8

)

2

+ (1175 minus

3220

8

)

2

+ (37 minus

3220

8

)

2

+ (1085 minus

3220

8

)

2

+ (24 minus

3220

8

)

2

+ (792 minus

3220

8

)

2

+ (14 minus

3220

8

)

2

+ (28 minus

3220

8

)

2

] = 237027

(21)

It shows that the equilibrium degree of locomotive sched-ule is large The main reason is that there are no trains from600 to 1800 on the railwayTherefore the difference betweenthe detention time of locomotives at stations from 1800 to600 the next day and the detention time of locomotives atstations from 600 to 1800 is relatively large If there are trainsfrom 600 to 1800 the equilibrium degree of locomotiveschedule will be reduced and the locomotive schedule will bemore balanced

52 Analysis of Locomotive Schedule for 20 kt CombinedTrains on Da-qin Railway The previous section shows thatthe model of locomotive schedule is feasible Then theeffectiveness of the proposedmodel also needs to be validatedwhich is conducted in this section

521 Line Simplification Da-qin Railway has 26 stations andthe line length is 653 km Qiancao Railway has 5 stations andthe line length is 137 km Jingtang Port-Donggang RailwayLine has 5 stations and the line length is 45 km Someintermediate stations on the railway do not have the technicaloperation so they are not displayed Only the stationswhere locomotive depot or turnaround depot is located areconsidered The simplified network of Da-qin Railway isshown in Figure 10

Hudong station is the station where the locomotive depotis located and theminimal technological time of locomotives

10 Mathematical Problems in Engineering

at stations is 250min Liucun South station Donggang sta-tion Caofeidian West station and Qinhuangdao East stationare the stations where locomotive turnaround depots arelocated The minimal technological time of locomotives atstations for Liucun South station is 220min The minimaltechnological time of locomotives at stations for Donggangstation and Caofeidian West station is 100min The minimaltechnological time of locomotives at stations for Qinhuang-daoEast station is 125minTheminimal technological time ofthe other stations where the locomotives turn back is 50min

There were 100 heavy-loaded trains in the heavy-loaddirection and 104 empty-wagon trains in the reverse directionon December 10 2014

522 Data Processing In reference to the mentionedmethod complete the data processing For the HXDlocomotives there are 328 trains which are hauled by single-locomotive after conversion so the order of the coefficientmatrix is 328 For the SS4 locomotives there are 56 trainswhich are hauled double-locomotive after conversion sothe order of the coefficient matrix is 56 In this paper C isemployed to deal with such huge data

523 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

119888

119896119894119895119909119896119894119895 = 37116 (min)

119885SS4 =13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10409 (min)

(22)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 164 times 250 + 76 times 220 + 14 times 125 + 44 times 100

+ 30 times 50 = 65370 (min)

119879

SS4119870 = 28 times 250 + 28 times 50 = 8400 (min)

(23)

The running times of locomotives are respectively

119879

HXD119905 = 198474 (min)

119879

SS4119905 = 24391 (min)

(24)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

198474 + 65370 + 37116

1440

=

300960

1440

= 209

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

24391 + 8400 + 10409

1440

times 2 =

43200

1440

times 2

= 60

(25)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

209

1602

= 261

119870

SS4119889 =

119872SS4119899

=

30 times 2

562

= 214

(26)

According to the best connection matrix locomotiveconnection scheme for 20 kt combined trains on Da-qinRailway can be obtainedThe locomotive connection schemeof SS4 locomotives is as follows

H16012 997888rarr 52 (single-locomotive) 997888rarr H16016

997888rarr 54 (single-locomotive) 997888rarr H16004

997888rarr H19609 997888rarr H19504 997888rarr H19605

997888rarr H19512 997888rarr 53 (single-locomotive)

997888rarr H16018 997888rarr 56 (single-locomotive)

997888rarr H16002 997888rarr 48 (single-locomotive)

997888rarr H15506 997888rarr 47 (single-locomotive)

997888rarr H16010 997888rarr 19601 997888rarr H19508

997888rarr 46 (single-locomotive) 997888rarr H16008

997888rarr H15501 997888rarr H15504 997888rarr H19607

997888rarr H19516 997888rarr 19615 997888rarr H19514

997888rarr H28501 997888rarr H28510

997888rarr 51 (single-locomotive) 997888rarr H16014

997888rarr H19603 997888rarr H19502 997888rarr H28507

997888rarr H28512 997888rarr 49 (attached locomotive)

997888rarr H15508 997888rarr H28503 997888rarr H28508

997888rarr 50 (single-locomotive) 997888rarr H16012

997888rarr H19611 997888rarr H19506 997888rarr H28511

997888rarr H28502 997888rarr H15503 997888rarr H15502

997888rarr 55 (attached locomotive) 997888rarr H15510

997888rarr H28505 997888rarr H28506

997888rarr 45 (single-locomotive) 997888rarr H16006

997888rarr H19611

H19613 997888rarr H19510 997888rarr H19613

H28504 997888rarr H28509 997888rarr H28504

(27)

Mathematical Problems in Engineering 11

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

Luannan

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

LuannanH16012 H16014 H16016 H16018H16002 H16004 H16006

H16002 H16004 H16006 H16008 H16010 H160112 H16014H16016 H16018

H15501

H15501

H15503

H15503

H15502 H15504 H15506 H15508 H15510

H28502 H28504 H28506 H28508 H28510 H28512

H15502H28502 H28504 H15504 H128506 H28508 H28510H15506 H15508 H28512

H28501 H28505 H28507 H28509 H28511

H28501 H28503 H28505 H28507 H28509 H28511

H19601 H19603 H19615 H19605 H19607 H19609 H19613 H19611

H19516 H19502 H19504 H19506 H19508 H19510 H19514 H19512

H28503

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

TrainsLocomotivesSingle-locomotives or attached locomotives

H16008 H16010

Figure 11 Locomotive working diagram of SS4 locomotives for 20 kt combined trains

Table 4 The daily number of locomotives on Da-qin Railway onDecember 10 2014 [21]

Types oflocomotive

Daily number oflocomotives

Types oflocomotive

Daily number oflocomotives

HXD1 130 SS4 109HXD2 92 mdash mdash

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 11

524 Analysis of Result

(1) Analyze the Number of Available LocomotivesThe numberof available locomotives (the total number of locomotivesafter deducting locomotives inmaintenance and preparation)onDa-qin Railway onDecember 10 2014 is shown in Table 4

Table 4 indicates that the locomotive schedule obtainedby using themodel can save the number ofHXD locomotives130 + 92 minus 209 = 13 and the number of SS4 locomotives109minus 30times 2 = 49 Therefore if only the minimum number oflocomotives is considered the optimization model of heavyhaul locomotive schedule will be optimized

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

328

13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

328

)

2

= 52379

119863SS4 =1

56

13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

56

)

2

= 74044

(28)

The equilibrium degree of the locomotive schedule forHXD locomotives and SS4 locomotives is smaller than theequilibrium degree of that case which shows that this loco-motive schedule is more balanced The equilibrium degreeof SS4 locomotives is larger than the equilibrium degree ofHXD locomotives which shows that the schedule of HXDlocomotives ismore balanced than SS4 locomotivesThe largeequilibrium degree of SS4 locomotives may be due to thesmall detention time of locomotives at stations

53 Research and Analysis of Locomotive Schedule Optimiza-tion for 30 kt Combined Trains on Da-qin Railway On thebasis of the potential train diagram for 30 kt combined trainsthe practical and feasible locomotive connection scheme canbe determined by using the optimization model

In reference to the mentioned method complete the dataprocessing

531 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

119888

119896119894119895119909119896119894119895 = 44838 (min)

119885SS4 =13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10314 (min)

(29)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 203 times 250 + 64 times 220 + 65 times 100 + 74 times 50

= 75030 (min)

119879

SS4119870 = 26 times 250 + 26 times 50 = 7800 (min)

(30)

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

LuannanDonggang

Caofeidian West

Hud

ong

235

30

Dat

ong

Sout

h 6

0H

anjia

ling

215

23

Dat

ong

coun

ty 3

99

Yang

quan

86

235

Hua

shao

ying

129

020

Zhul

u 18

267

4

Shac

heng

Eas

t 218

263

Beix

inba

o 23

701

4

Yanq

ing

Nor

th 2

625

14

Xiaz

huan

g 30

769

2

Chaw

u 32

72

Chag

ao L

ink

Line

Ping

gu 3

688

44

Jixia

n W

est 4

066

44Cu

ipin

gsha

n 41

796

1

Dad

uan

Link

Lin

eDi

recti

on of

Jixia

nYu

tian

Nor

th 4

408

38

Zunh

ua N

orth

476

767

Qia

nxi 5

083

35

Qia

ncao

Line

Qia

nrsquoan

Nor

th 5

547

27Lu

long

Nor

th 5

688

98

Hou

ying

621

100

Liuc

un S

outh

73

14Li

ucun

Sou

th 2

nd Y

ard

720

7

Qin

huan

gdao

Eas

t 14

417

16725 XihanlingPingwang 10570

KouquanBranch line

Yungang lineDatong

Directionof Gudian

Das

hizh

uang

387

344

Marshalling stationDistrict stationIntermediate station

Figure 2 Network of Da-qin Railway

Station A

Station B

Station C

Station D Locomotive

40 4042

35 37

53Traction choice (a)Traction choice (b)

Figure 3 An illustration of uncertain railroad region locomotiveutilization method

(1) Number of Available Locomotives Number of availablelocomotives is the total number of locomotives which areused in the whole turnaround process [20] The less thenumber is the higher the efficiency will be The graphicmethod and the direct calculation method are generalmethods to determine the number of available locomotivesThe graphic method is based on locomotive schedule todetermine the number of used locomotives The formula ofdirect calculation method is listed below

119872119891 =(119899119904 + 119899119889 + 2 sdot 119899119905) sdot 120579119897

24

= 120583119904 sdot 119870119889(1)

where 119872119891 is the number of available locomotives in eachtraction section 119899119904 is the total number of trains 119899119889 is thenumber of trains with double-locomotive traction 119899119905 is thenumber of trains with three-locomotive traction 120579119897 is thelocomotive complete turnaround time 120583119904 is the number ofsupplied locomotives 119870119889 is the coefficient of locomotiverequirement in traction sections

(2) Coefficient of Locomotive Requirement The coefficient oflocomotive requirement indicates the number of locomotivesrequired per pair of trains in a traction section [20] Onelocomotive completes a turnaround which means it has

completed a traction task The coefficient of locomotiverequirement can be calculated by the formula below

119870119889 =120579119897

24

(2)

(3) Equilibrium Degree of Locomotive Schedule In additionto quantitative and quality criteria the equilibrium criteriaare also very important in evaluating locomotive schedulewhich refer to the difference between detention time of eachlocomotive at stations and running time of each locomotiveAt the same section the running time is the sameThereforein the actual evaluation process the variance of detentiontime of each locomotive at stations 119863 (equilibrium degree)can be used to describe the equilibrium of locomotiveschedule It can be calculated as below

119863 =

1

119899119897

119899119897

sum

119894=1

(119888119894 minus 119864)2 (3)

where 119899119897 is the total number of traction tasks of all locomo-tives and it is numerically equal to (119899119904 + 119899119889 + 2 sdot 119899119905) 119894 is alocomotive that undertakes traction tasks that is 119894 isin 119899119897 119888119894 isthe detention time of a locomotive at station 119864 is the averagedetention time of locomotives at stations

Equation (3) shows that equilibrium degree 119863 refers tothe discrete degree between 119888119894 (119894 = 1 2 3 119899119897) and 119864 Thesmaller the value of119863 is the smaller the difference is and themore balanced the locomotive schedule will be

3 The Optimization Model of Heavy HaulLocomotive Scheduling Problem

31 Description of Locomotive Scheduling Problem The inputsfor the locomotive scheduling problem are railway linelocomotive traction sections and train diagram and so forthBased on these input data the output namely locomotiveschedule can be obtained

4 Mathematical Problems in Engineering

A B C D

Station with depotStation with turnaround depot

Figure 4 The network of hypothetical railway line

A B C D

Figure 5 Schematic diagram of locomotive traction sections

311 Input

(1) Railway Line Railway line A-B-C-D is a double-trackrailway as shown in Figure 4 It is dedicatedly used totransport coal On the railway station B is the station wherethe locomotive depot is located Station A station C andstation D are the stations where locomotive turnarounddepots are located

(2) Locomotive Traction Sections When freight locomotivesundertake traction tasks in section A-D they firstly departfrom station B and haul heavy-loaded trains to arrive atstation A station C or station D Then judge whetherlocomotive preparation is needed or not Finally these loco-motives haul empty-wagon trains and return to the stationwhere the locomotive depot is located while the locomotiveswhich are at turnaround depot can haul the return trainsonly The schematic diagram of locomotive traction sectionsis shown in Figure 5

(3) Train Diagram A complete train diagram determinestrain routes departurearrival time and detention time ateach station In the train diagram we cannot know thelocomotiversquos position andwhich train it will haul As shown inFigure 6 there are 4 pairs of trains on the hypothetical railway(4 heavy-loaded trains and 4 empty-wagon trains) The trainnumbers respectively are 700012 730012 730034 and790012 The train whose train number is 70001 refers to aheavy-loaded train in section A-B and the train whose trainnumber is 70002 refers to an empty-wagon train in section A-B The train whose train number is 73001 or 73003 refers to aheavy-loaded train in section B-C and the train whose trainnumber is 73002 or 73004 refers to an empty-wagon train insection B-CThe train whose train number is 79001 refers to aheavy-loaded train in section D-B and the train whose trainnumber is 79002 refers to an empty-wagon train in sectionD-B

312 Output Based on the input data the potential locomo-tive schedule in section A-D will be obtained (see Figure 7)where a dotted line represents a locomotive Figure 7 showsthat after hauling the train whose train number is 70001 to

station B the locomotive can haul the train whose train num-ber is 70002 79001 or 73001Whenmultilocomotive tractionschemes exist the minimal detention time of locomotives atstations will be adopted to determine the final scheme

32 Model Description Some assumptions are presented inthis model (a) the train diagram is known and invariant in acertain period of time that is the impact of train delay willbe ignored (b) based on the line length of Da-qin Railway(653 km) locomotive routing and locomotive running time(eg a round-trip time more than 17 h) it assumes within24 hours all available locomotives (the total locomotives afterdeducting the locomotives in maintenance and preparation)are without maintenance (c) for the trains with multilo-comotive traction locomotive reconnection between HXDlocomotives and SS4 locomotives is not considered the timeof locomotive reconnection and dismantling was includedin the detention time of locomotives at stations (d) in acertain period of time the locomotive crew working systemin Da-qin Railway is invariant so the effects of locomotivecrew working system and locomotive crew shift mode onlocomotive scheduling are not considered and (e) in thedaily initial time (1800) each station has enough locomotiveswhich at least are equal to the number of locomotives in needthat is the locomotive homing will not be considered

321 Sets In the process of constructing the optimizationmodel of the heavy haul locomotive scheduling problemthere are some notations and sets that will be used

119870 is the set that includes all of stations on the railway

119896 is a station on the railway that is 119896 isin 119870

119873119896119889 is the total number of locomotives arriving atstation 119896

119873119896119891 is the total number of trains departing fromstation 119896

119894 is the locomotive arriving at station 119896

119895 is the train departing from station 119896

119879119896 is the minimal technological time of locomotivesat station 119896

119896119894119889 is the destination station of locomotive 119894 at station119896

119905119894119889119896 is the arrival time of locomotive 119894 at station 119896

119896119895119891 is the departure station of train 119895 from station 119896

119905119895119891119896 is the departure time of train 119895 from station 119896

322 Parameters In the process of developing the opti-mization model the known parameter that will be involvedis 119888119896119894119895 It refers to the waiting time which is equal to thedetention time of locomotive 119894 at station 119896minus theminimal

Mathematical Problems in Engineering 5

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 6 Train diagram in section A-D

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 7 Potential locomotive schedule in section A-D

technological time The waiting time 119888119896119894119895 of locomotives atstations can be calculated by

119888

119896119894119895

=

119905119895119891119896 minus 119905119894119889119896 minus 119879119896 119905119895119891119896 minus 119905119894119889119896 ge 119879119896 119896119894119889 = 119896119895119891

119905119895119891119896 minus 119905119894119889119896 minus 119879119896 + 1440 119905119895119891119896 minus 119905119894119889119896 lt 119879119896 119896119894119889 = 119896119895119891

infin 119896119894119889 = 119896119895119891

(4)

323 Variable Thevariable is assumed to be 119888119896119894119895 which refersto whether locomotive 119894 hauls train 119895 or not at station 119896 Thebinary variable 119888119896119894119895 is defined as follows

119909

119896119894119895 =

1 locomotive 119894 hauls train 119895 at station 119896

0 otherwise(5)

324 Objective Function Completing planned traction tasksis the primary constraint to us Based on that the lessthe number of locomotives is the lower the transportationcost will be Therefore the objective function of heavy haullocomotive scheduling problem is minimizing number oflocomotives that is

min 119885

1015840=

119879119905 + 119879

1440

(6)

where 119879119905 is the total time of trains running from departurestation to destination station and T is the total detention timeof locomotives at stations which can be calculated by

119879 = 119879119870 + 119879119908 (7)

where 119879119870 is the total minimal technological time of locomo-tives at stations and119879119908 is the total waiting time of locomotivesat stations

For the established train diagram 119879119905 is a constant Sothe minimum number of locomotives refers to the minimaldetention time 119879 of locomotives at stations Meanwhile119879119870 is also a constant So the minimal detention time oflocomotives at stations refers to the minimum waiting time119879119908 of locomotives at stations That is

min 119885 = 119879119908 =

119870

sum

119896=1

119873119896119889

sum

119894=1

119873119896119891

sum

119895=1

119888

119896119894119895119909119896119894119895 (8)

Then the equilibrium degree 119863 of locomotive schedulecan be calculated by

119863 =

1

119899119904 + 119899119889 + 2 sdot 119899119905

sdot

119870

sum

119896=1

119873119896119889

sum

119894=1

119873119896119891

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

119899119904 + 119899119889 + 2 sdot 119899119905

)

2

(9)

325 Constraints

(1) Traction Locomotives Constraint Consider the following

119873119896119889

sum

119894=1

119909

119896119894119895 = ℎ forall119895 isin 119873119896119891 forall119896 isin 119870 (10)

This constraint shows that the number of traction loco-motives for each train is ℎ The value of each type of train isshown in Table 1

(2) Traction Tasks Constraint Consider the following

119873119896119891

sum

119895=1

119909

119896119894119895 = 1 forall119894 isin 119873119896119889 forall119896 isin 119870 (11)

6 Mathematical Problems in Engineering

Table 1 The value of each type of trains

Types of heavy-loaded trains Reference value Types of heavy-loaded trains Reference value10 kt heavy-loaded combined trains 2 20 kt heavy-loaded combined trains 210 kt heavy-loaded unit trains 1 30 kt heavy-loaded combined trains 315 kt heavy-loaded combined trains 2 mdash mdash

This constraint shows that each locomotive can only haulone train at the same time

4 Transformation and SolvingProcesses of Heavy Haul LocomotiveScheduling Problem

First of all the locomotive scheduling problem needs to betransformed into an assignment problemThen the optimiza-tion results are obtained by using the Hungarian algorithm

41 Transformation of Heavy Haul Locomotive SchedulingProblem For an unpaired train diagram the number ofarrival trains is not equal to the number of departure trainsat some stations So the unpaired train diagram needs tobe transformed into a paired train diagram before solvingthe locomotive scheduling problem For the paired traindiagram the number of arrival trains is equal to the numberof departure trains at each station Therefore if each trainis hauled by the same type of locomotive the locomotivescheduling problem can be transformed into a standardassignment problem by treating arrival locomotives as ldquoper-sonnelrdquo and departure trains as ldquomissionsrdquo

Therefore the transformation of heavy haul locomotivescheduling problem mainly focuses on the following threeaspects

(1) Transforming Unpaired Train Diagram into Paired TrainDiagram Usually the trainwill be operated smoothly accord-ing to the train diagram Because the number of trains in traindiagram is given on the known railway the number of trainswithin 24 h is determined If the train diagram is unpairedit needs to add some ldquovirtual operation curvesrdquo Then thenumber of arrival trains is equal to the number of departuretrains in each section It assumes that the number of ldquovirtualoperation curvesrdquo is 119884 and the number of actual trains is 119869

(2) Transforming Multilocomotive Traction into Single-Locomotive Traction If the total number of trains is 119884 + 119869the number of trains with double-locomotive traction is Wand the number of trains with three-locomotive traction is119881 (119882 + 119881 le 119869 + 119884) Then the number of ldquovirtual trainsrdquois 119882 + 2119881 when multilocomotive traction is transformedinto single-locomotive traction At the same time it needsto ensure that the information of train diagram is consistentwith before Then trains will be sorted and coded accordingto departure times and the code of trains may be 1 2 3 119869 + 119884 +119882 + 2119881 minus 2 119869 + 119884 +119882 + 2119881 minus 1 119869 + 119884 +119882 + 2119881

(3) Transformation betweenDifferent Locomotive TypesThereare two locomotive types which include SS4 locomotive and

HXD locomotive inDa-qin Railway 10 kt heavy-loaded com-bined trains are hauled by SS4 locomotives while 10 kt heavy-loaded unit trains 15 kt heavy-loaded combined trains and20 kt heavy-loaded combined trains are hauled by HXDlocomotives There is no locomotive reconnection betweenHXD locomotives and SS4 locomotives Therefore locomo-tive scheduling problemwhich involves two locomotive typescan be considered separately for each type in order to obtainthe optimal solutionsThen the optimization objective of thelocomotive schedule on the whole railway is achieved

42 Solution Algorithm of Heavy Haul Locomotive ScheduleIn this paper the optimization model of heavy haul locomo-tive schedule is solved by using Hungarian algorithm TheHungarian algorithm as a classical algorithm can obtain aglobal optimal solution when it is used to solve assignmentproblems With an increasing coefficient matrix dimensionthe algorithm runs a little slower but it can reduce computa-tional efforts muchmore than any enumerationmethod [22]The concrete steps of Hungarian algorithm are as follows

Step 1 (generating coefficient matrix) The coefficient matrixof detention time of locomotives at stations for each station isbuilt firstly and then the coefficient matrix for whole railwaycan be built

Step 2 (transforming coefficient matrix) When performingthe row operations on the coefficient matrix we subtract thelowest value from each element in that row in order to obtainat least one zero element for each row Similarly we also dothis procedure for all column elements

Step 3 (seeking independent zero elements) If the numberof independent zero elements is less than the order ofthe coefficient matrix then go to Step 4 If the numberof independent zero elements is equal to the order of thecoefficient matrix then output the optimization solution andend the algorithm

Step 4 (drawing the least number of straight lines) If thenumber of straight lines is less than the order of the coefficientmatrix then mark the rows and columns and go to Step 5otherwise go to Step 3

Step 5 (transforming coefficient matrix again) Seek mini-mum value of the elements which are not covered by straightlines This value is then subtracted from each element inall marked rows and added to each element in the markedcolumns and then go to Step 3

The flow chart of the Hungarian algorithm is shown inFigure 8

Mathematical Problems in Engineering 7

Begin

equal to thematrix order

Mark the least zero elements and remove the other zero elements

The value of marked zero elements is 1 and

the other is 0 in thecounterpart matrix

Cover all of zero elements by the least straight linesEnd

Generating coefficient matrix

Each row and column element minus the minimum value

Is the number ofstraight lines smaller than

the matrix order

Transform coefficient matrix

again

YesNo

Yes

No

Is the number ofindependent zero elements

Figure 8 Flow chart of the Hungarian algorithm

5 Case Study

51 Evaluating the Feasibility of Heavy Haul LocomotiveSchedule Model The model and its solution algorithm needto be validated before applying them to any real problems

511 Data Source and Processing As shown in Figure 4line A-B-C-D is a double-track railway (section A-D forshort) which is specially used to transport coal Station Bis the station where the locomotive depot is located andthe minimal technological time of locomotives at stations is125min Station A station C and station D are the stationswhere locomotive turnaround depots are located and theminimal technological time of locomotives at stations is50min There are 8 trains (4 pairs) which include 2 trains(1 pair) in section A-B 4 trains (2 pairs) in section B-Cand 2 trains (1 pair) in section B-D on the railway If all oftrains are hauled by single-locomotive Figure 6 is the traindiagram

This basic data needs to be processed as follows

(1) Stations Coding For all stations in section A-D it can codestations from any end of the railway and it starts with stationA

(2) Trains Coding The trains in section A-D are sole How-ever it is necessary for each train to code another sole and

Table 2 Trains coding result in section A-D

Train number Train coding Train number Train coding70001 1 73003 570002 2 73004 673001 3 79001 773002 4 79002 8

simple number due to the long and large train number Thecoding results are listed in Table 2

(3) Time Simplification For the departure time and arrivaltime of each train starting at the initial time (1800) of thetrain diagram time of 24 h can be converted to minutes andfor example 1933 can be converted to the 93rd minute

(4) Generating Coefficient Matrix According to (4) thecoefficient matrix 119888119896119894119895 of the detention time of locomotives atstations will be generated by using C The results are listedbelow

Station A Consider the following

119888

A21 = 50 minus 265 minus 50 + 1440 = 1175

(50 minus 265 = minus215 lt 50)

(12)

8 Mathematical Problems in Engineering

Station B Consider the following

119888

B12 = 220 minus 90 minus 125 = 5 (220 minus 90 = 130 gt 125)

119888

B13 = 280 minus 90 minus 125 = 65 (280 minus 90 = 190 gt 125)

119888

B15 = 516 minus 90 minus 125 = 301

(516 minus 90 = 426 gt 125)

119888

B17 = 150 minus 90 minus 125 + 1440 = 1375

(150 minus 90 = 60 lt 125)

119888

B42 = 220 minus 450 minus 125 + 1440 = 1085

(220 minus 450 = minus230 lt 125)

119888

B43 = 280 minus 450 minus 125 + 1440 = 1145

(280 minus 450 = minus170 lt 125)

119888

B45 = 516 minus 450 minus 125 + 1440 = 1381

(516 minus 450 = 66 lt 125516 minus 450 = 66 lt 125)

119888

B47 = 150 minus 450 minus 125 + 1440 = 1015

(150 minus 450 = minus300 lt 125)

119888

B62 = 220 minus 673 minus 125 + 1440 = 862

(220 minus 673 = minus453 lt 125)

119888

B63 = 280 minus 673 minus 125 + 1440 = 922

(280 minus 673 = minus393 lt 125)

119888

B65 = 516 minus 673 minus 125 + 1440 = 1158

(516 minus 673 = minus157 lt 125)

119888

B67 = 150 minus 673 minus 125 + 1440 = 792

(150 minus 673 = minus523 lt 125)

119888

B82 = 220 minus 363 minus 125 + 1440 = 1172

(220 minus 363 = minus143 lt 125)

119888

B83 = 280 minus 363 minus 125 + 1440 = 1232

(280 minus 363 = minus83 lt 125)

119888

B85 = 516 minus 363 minus 125 = 28

(516 minus 363 = 153 gt 125)

119888

B87 = 150 minus 363 minus 125 + 1440 = 1102

(150 minus 90 = minus213 lt 125)

(13)

Table 3 Coefficient matrix of detention time of locomotives atstations

119894

119895

1 2 3 4 5 6 7 81 infin 5 65 infin 301 infin 1375 infin

2 1175 infin infin infin infin infin infin infin

3 infin infin infin 37 infin 260 infin infin

4 infin 1085 1145 infin 1381 infin 1015 infin

5 infin infin infin 1241 infin 24 infin infin

6 infin 862 922 infin 1158 infin 792 infin

7 infin infin infin infin infin infin infin 148 infin 1172 1232 infin 28 infin 1102 infin

Station C Consider the following

119888

C34 = 407 minus 320 minus 50 = 37 (407 minus 320 = 87 gt 50)

119888

C36 = 630 minus 320 minus 50 = 260

(630 minus 320 = 310 gt 50)

119888

C54 = 407 minus 556 minus 50 + 1440 = 1241

(407 minus 556 = minus149 lt 50)

119888

C56 = 630 minus 556 minus 50 = 24 (630 minus 556 = 74 gt 50)

(14)

Station D Consider the following

119888

D78 = 296 minus 232 minus 50 = 14 (296 minus 232 = 64 gt 50) (15)

The others are equal toinfin the coefficient matrix (119888119896119894119895) isshowed in Table 3

512 Result The Hungarian algorithm is implemented inMATLAB and the optimal locomotive schedule is as follows

The minimum waiting time of locomotives at stations is

119879119908 = 119885 =

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

119888

119896119894119895119909119896119894119895

= 65 + 1175 + 37 + 1085 + 24 + 792 + 14 + 28

= 3220 (min)

(16)

Theminimal technological time of locomotives at stationsis

119879119870 = 125 times 4 + 50 times 4 = 700 (min) (17)

The running time of locomotives is

119879119905 = 40 + 45 + 40 + 43 + 40 + 43 + 82 + 67

= 400 (min) (18)

The minimum number of locomotives is

119872 =

119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119879119908

1440

=

400 + 700 + 3220

1440

=

4320

1440

= 3

(19)

Mathematical Problems in Engineering 9

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 9 Locomotive schedule in section A-D

Chawu

299 kmDashizhuang

CuipingshanZunhua North

Qianrsquoan North

Jixian West

HouyingLiucun South

Qinhuangdao EastLuannan

Donggang

Caofeidian West

60 km19 km

11km59 km 78 km 67 km

22 km

24km

71km

66km

44 km

Hudong

Figure 10 The simplified network of Da-qin Railway

The coefficient of locomotive requirement is

119870119889 =119872

119899

=

3

82

= 075 (20)

The locomotive schedule can be expressed as an intuitivediagram which is shown in Figure 9 (dotted lines representlocomotives and solid lines represent trains)

Figure 9 shows that there are 3 traction locomotiveswhich are equal to the calculation result to haul trains

513 Result Analysis

(1) Analysis on the Optimization of Locomotive SchedulingFigures 7 and 9 show that there may be a variety of trainconnections So the number of available locomotives andcoefficient of locomotive requirement are uncertain Theoptimization model of heavy-loaded locomotive schedule isestablished and then it can be solved byHungarian algorithmIt can not only make sure of the uniqueness of locomotivetraction tasks but also complete established traction taskswith a minimum number of locomotives

(2) Analysis on the Equilibrium of Locomotive Schedule Theequilibrium degree of locomotive schedule of Figure 9 can beobtained by using (9) which is

119863 =

1

8

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

8

)

2

=

1

8

[(65 minus

3220

8

)

2

+ (1175 minus

3220

8

)

2

+ (37 minus

3220

8

)

2

+ (1085 minus

3220

8

)

2

+ (24 minus

3220

8

)

2

+ (792 minus

3220

8

)

2

+ (14 minus

3220

8

)

2

+ (28 minus

3220

8

)

2

] = 237027

(21)

It shows that the equilibrium degree of locomotive sched-ule is large The main reason is that there are no trains from600 to 1800 on the railwayTherefore the difference betweenthe detention time of locomotives at stations from 1800 to600 the next day and the detention time of locomotives atstations from 600 to 1800 is relatively large If there are trainsfrom 600 to 1800 the equilibrium degree of locomotiveschedule will be reduced and the locomotive schedule will bemore balanced

52 Analysis of Locomotive Schedule for 20 kt CombinedTrains on Da-qin Railway The previous section shows thatthe model of locomotive schedule is feasible Then theeffectiveness of the proposedmodel also needs to be validatedwhich is conducted in this section

521 Line Simplification Da-qin Railway has 26 stations andthe line length is 653 km Qiancao Railway has 5 stations andthe line length is 137 km Jingtang Port-Donggang RailwayLine has 5 stations and the line length is 45 km Someintermediate stations on the railway do not have the technicaloperation so they are not displayed Only the stationswhere locomotive depot or turnaround depot is located areconsidered The simplified network of Da-qin Railway isshown in Figure 10

Hudong station is the station where the locomotive depotis located and theminimal technological time of locomotives

10 Mathematical Problems in Engineering

at stations is 250min Liucun South station Donggang sta-tion Caofeidian West station and Qinhuangdao East stationare the stations where locomotive turnaround depots arelocated The minimal technological time of locomotives atstations for Liucun South station is 220min The minimaltechnological time of locomotives at stations for Donggangstation and Caofeidian West station is 100min The minimaltechnological time of locomotives at stations for Qinhuang-daoEast station is 125minTheminimal technological time ofthe other stations where the locomotives turn back is 50min

There were 100 heavy-loaded trains in the heavy-loaddirection and 104 empty-wagon trains in the reverse directionon December 10 2014

522 Data Processing In reference to the mentionedmethod complete the data processing For the HXDlocomotives there are 328 trains which are hauled by single-locomotive after conversion so the order of the coefficientmatrix is 328 For the SS4 locomotives there are 56 trainswhich are hauled double-locomotive after conversion sothe order of the coefficient matrix is 56 In this paper C isemployed to deal with such huge data

523 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

119888

119896119894119895119909119896119894119895 = 37116 (min)

119885SS4 =13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10409 (min)

(22)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 164 times 250 + 76 times 220 + 14 times 125 + 44 times 100

+ 30 times 50 = 65370 (min)

119879

SS4119870 = 28 times 250 + 28 times 50 = 8400 (min)

(23)

The running times of locomotives are respectively

119879

HXD119905 = 198474 (min)

119879

SS4119905 = 24391 (min)

(24)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

198474 + 65370 + 37116

1440

=

300960

1440

= 209

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

24391 + 8400 + 10409

1440

times 2 =

43200

1440

times 2

= 60

(25)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

209

1602

= 261

119870

SS4119889 =

119872SS4119899

=

30 times 2

562

= 214

(26)

According to the best connection matrix locomotiveconnection scheme for 20 kt combined trains on Da-qinRailway can be obtainedThe locomotive connection schemeof SS4 locomotives is as follows

H16012 997888rarr 52 (single-locomotive) 997888rarr H16016

997888rarr 54 (single-locomotive) 997888rarr H16004

997888rarr H19609 997888rarr H19504 997888rarr H19605

997888rarr H19512 997888rarr 53 (single-locomotive)

997888rarr H16018 997888rarr 56 (single-locomotive)

997888rarr H16002 997888rarr 48 (single-locomotive)

997888rarr H15506 997888rarr 47 (single-locomotive)

997888rarr H16010 997888rarr 19601 997888rarr H19508

997888rarr 46 (single-locomotive) 997888rarr H16008

997888rarr H15501 997888rarr H15504 997888rarr H19607

997888rarr H19516 997888rarr 19615 997888rarr H19514

997888rarr H28501 997888rarr H28510

997888rarr 51 (single-locomotive) 997888rarr H16014

997888rarr H19603 997888rarr H19502 997888rarr H28507

997888rarr H28512 997888rarr 49 (attached locomotive)

997888rarr H15508 997888rarr H28503 997888rarr H28508

997888rarr 50 (single-locomotive) 997888rarr H16012

997888rarr H19611 997888rarr H19506 997888rarr H28511

997888rarr H28502 997888rarr H15503 997888rarr H15502

997888rarr 55 (attached locomotive) 997888rarr H15510

997888rarr H28505 997888rarr H28506

997888rarr 45 (single-locomotive) 997888rarr H16006

997888rarr H19611

H19613 997888rarr H19510 997888rarr H19613

H28504 997888rarr H28509 997888rarr H28504

(27)

Mathematical Problems in Engineering 11

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

Luannan

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

LuannanH16012 H16014 H16016 H16018H16002 H16004 H16006

H16002 H16004 H16006 H16008 H16010 H160112 H16014H16016 H16018

H15501

H15501

H15503

H15503

H15502 H15504 H15506 H15508 H15510

H28502 H28504 H28506 H28508 H28510 H28512

H15502H28502 H28504 H15504 H128506 H28508 H28510H15506 H15508 H28512

H28501 H28505 H28507 H28509 H28511

H28501 H28503 H28505 H28507 H28509 H28511

H19601 H19603 H19615 H19605 H19607 H19609 H19613 H19611

H19516 H19502 H19504 H19506 H19508 H19510 H19514 H19512

H28503

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

TrainsLocomotivesSingle-locomotives or attached locomotives

H16008 H16010

Figure 11 Locomotive working diagram of SS4 locomotives for 20 kt combined trains

Table 4 The daily number of locomotives on Da-qin Railway onDecember 10 2014 [21]

Types oflocomotive

Daily number oflocomotives

Types oflocomotive

Daily number oflocomotives

HXD1 130 SS4 109HXD2 92 mdash mdash

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 11

524 Analysis of Result

(1) Analyze the Number of Available LocomotivesThe numberof available locomotives (the total number of locomotivesafter deducting locomotives inmaintenance and preparation)onDa-qin Railway onDecember 10 2014 is shown in Table 4

Table 4 indicates that the locomotive schedule obtainedby using themodel can save the number ofHXD locomotives130 + 92 minus 209 = 13 and the number of SS4 locomotives109minus 30times 2 = 49 Therefore if only the minimum number oflocomotives is considered the optimization model of heavyhaul locomotive schedule will be optimized

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

328

13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

328

)

2

= 52379

119863SS4 =1

56

13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

56

)

2

= 74044

(28)

The equilibrium degree of the locomotive schedule forHXD locomotives and SS4 locomotives is smaller than theequilibrium degree of that case which shows that this loco-motive schedule is more balanced The equilibrium degreeof SS4 locomotives is larger than the equilibrium degree ofHXD locomotives which shows that the schedule of HXDlocomotives ismore balanced than SS4 locomotivesThe largeequilibrium degree of SS4 locomotives may be due to thesmall detention time of locomotives at stations

53 Research and Analysis of Locomotive Schedule Optimiza-tion for 30 kt Combined Trains on Da-qin Railway On thebasis of the potential train diagram for 30 kt combined trainsthe practical and feasible locomotive connection scheme canbe determined by using the optimization model

In reference to the mentioned method complete the dataprocessing

531 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

119888

119896119894119895119909119896119894119895 = 44838 (min)

119885SS4 =13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10314 (min)

(29)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 203 times 250 + 64 times 220 + 65 times 100 + 74 times 50

= 75030 (min)

119879

SS4119870 = 26 times 250 + 26 times 50 = 7800 (min)

(30)

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

A B C D

Station with depotStation with turnaround depot

Figure 4 The network of hypothetical railway line

A B C D

Figure 5 Schematic diagram of locomotive traction sections

311 Input

(1) Railway Line Railway line A-B-C-D is a double-trackrailway as shown in Figure 4 It is dedicatedly used totransport coal On the railway station B is the station wherethe locomotive depot is located Station A station C andstation D are the stations where locomotive turnarounddepots are located

(2) Locomotive Traction Sections When freight locomotivesundertake traction tasks in section A-D they firstly departfrom station B and haul heavy-loaded trains to arrive atstation A station C or station D Then judge whetherlocomotive preparation is needed or not Finally these loco-motives haul empty-wagon trains and return to the stationwhere the locomotive depot is located while the locomotiveswhich are at turnaround depot can haul the return trainsonly The schematic diagram of locomotive traction sectionsis shown in Figure 5

(3) Train Diagram A complete train diagram determinestrain routes departurearrival time and detention time ateach station In the train diagram we cannot know thelocomotiversquos position andwhich train it will haul As shown inFigure 6 there are 4 pairs of trains on the hypothetical railway(4 heavy-loaded trains and 4 empty-wagon trains) The trainnumbers respectively are 700012 730012 730034 and790012 The train whose train number is 70001 refers to aheavy-loaded train in section A-B and the train whose trainnumber is 70002 refers to an empty-wagon train in section A-B The train whose train number is 73001 or 73003 refers to aheavy-loaded train in section B-C and the train whose trainnumber is 73002 or 73004 refers to an empty-wagon train insection B-CThe train whose train number is 79001 refers to aheavy-loaded train in section D-B and the train whose trainnumber is 79002 refers to an empty-wagon train in sectionD-B

312 Output Based on the input data the potential locomo-tive schedule in section A-D will be obtained (see Figure 7)where a dotted line represents a locomotive Figure 7 showsthat after hauling the train whose train number is 70001 to

station B the locomotive can haul the train whose train num-ber is 70002 79001 or 73001Whenmultilocomotive tractionschemes exist the minimal detention time of locomotives atstations will be adopted to determine the final scheme

32 Model Description Some assumptions are presented inthis model (a) the train diagram is known and invariant in acertain period of time that is the impact of train delay willbe ignored (b) based on the line length of Da-qin Railway(653 km) locomotive routing and locomotive running time(eg a round-trip time more than 17 h) it assumes within24 hours all available locomotives (the total locomotives afterdeducting the locomotives in maintenance and preparation)are without maintenance (c) for the trains with multilo-comotive traction locomotive reconnection between HXDlocomotives and SS4 locomotives is not considered the timeof locomotive reconnection and dismantling was includedin the detention time of locomotives at stations (d) in acertain period of time the locomotive crew working systemin Da-qin Railway is invariant so the effects of locomotivecrew working system and locomotive crew shift mode onlocomotive scheduling are not considered and (e) in thedaily initial time (1800) each station has enough locomotiveswhich at least are equal to the number of locomotives in needthat is the locomotive homing will not be considered

321 Sets In the process of constructing the optimizationmodel of the heavy haul locomotive scheduling problemthere are some notations and sets that will be used

119870 is the set that includes all of stations on the railway

119896 is a station on the railway that is 119896 isin 119870

119873119896119889 is the total number of locomotives arriving atstation 119896

119873119896119891 is the total number of trains departing fromstation 119896

119894 is the locomotive arriving at station 119896

119895 is the train departing from station 119896

119879119896 is the minimal technological time of locomotivesat station 119896

119896119894119889 is the destination station of locomotive 119894 at station119896

119905119894119889119896 is the arrival time of locomotive 119894 at station 119896

119896119895119891 is the departure station of train 119895 from station 119896

119905119895119891119896 is the departure time of train 119895 from station 119896

322 Parameters In the process of developing the opti-mization model the known parameter that will be involvedis 119888119896119894119895 It refers to the waiting time which is equal to thedetention time of locomotive 119894 at station 119896minus theminimal

Mathematical Problems in Engineering 5

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 6 Train diagram in section A-D

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 7 Potential locomotive schedule in section A-D

technological time The waiting time 119888119896119894119895 of locomotives atstations can be calculated by

119888

119896119894119895

=

119905119895119891119896 minus 119905119894119889119896 minus 119879119896 119905119895119891119896 minus 119905119894119889119896 ge 119879119896 119896119894119889 = 119896119895119891

119905119895119891119896 minus 119905119894119889119896 minus 119879119896 + 1440 119905119895119891119896 minus 119905119894119889119896 lt 119879119896 119896119894119889 = 119896119895119891

infin 119896119894119889 = 119896119895119891

(4)

323 Variable Thevariable is assumed to be 119888119896119894119895 which refersto whether locomotive 119894 hauls train 119895 or not at station 119896 Thebinary variable 119888119896119894119895 is defined as follows

119909

119896119894119895 =

1 locomotive 119894 hauls train 119895 at station 119896

0 otherwise(5)

324 Objective Function Completing planned traction tasksis the primary constraint to us Based on that the lessthe number of locomotives is the lower the transportationcost will be Therefore the objective function of heavy haullocomotive scheduling problem is minimizing number oflocomotives that is

min 119885

1015840=

119879119905 + 119879

1440

(6)

where 119879119905 is the total time of trains running from departurestation to destination station and T is the total detention timeof locomotives at stations which can be calculated by

119879 = 119879119870 + 119879119908 (7)

where 119879119870 is the total minimal technological time of locomo-tives at stations and119879119908 is the total waiting time of locomotivesat stations

For the established train diagram 119879119905 is a constant Sothe minimum number of locomotives refers to the minimaldetention time 119879 of locomotives at stations Meanwhile119879119870 is also a constant So the minimal detention time oflocomotives at stations refers to the minimum waiting time119879119908 of locomotives at stations That is

min 119885 = 119879119908 =

119870

sum

119896=1

119873119896119889

sum

119894=1

119873119896119891

sum

119895=1

119888

119896119894119895119909119896119894119895 (8)

Then the equilibrium degree 119863 of locomotive schedulecan be calculated by

119863 =

1

119899119904 + 119899119889 + 2 sdot 119899119905

sdot

119870

sum

119896=1

119873119896119889

sum

119894=1

119873119896119891

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

119899119904 + 119899119889 + 2 sdot 119899119905

)

2

(9)

325 Constraints

(1) Traction Locomotives Constraint Consider the following

119873119896119889

sum

119894=1

119909

119896119894119895 = ℎ forall119895 isin 119873119896119891 forall119896 isin 119870 (10)

This constraint shows that the number of traction loco-motives for each train is ℎ The value of each type of train isshown in Table 1

(2) Traction Tasks Constraint Consider the following

119873119896119891

sum

119895=1

119909

119896119894119895 = 1 forall119894 isin 119873119896119889 forall119896 isin 119870 (11)

6 Mathematical Problems in Engineering

Table 1 The value of each type of trains

Types of heavy-loaded trains Reference value Types of heavy-loaded trains Reference value10 kt heavy-loaded combined trains 2 20 kt heavy-loaded combined trains 210 kt heavy-loaded unit trains 1 30 kt heavy-loaded combined trains 315 kt heavy-loaded combined trains 2 mdash mdash

This constraint shows that each locomotive can only haulone train at the same time

4 Transformation and SolvingProcesses of Heavy Haul LocomotiveScheduling Problem

First of all the locomotive scheduling problem needs to betransformed into an assignment problemThen the optimiza-tion results are obtained by using the Hungarian algorithm

41 Transformation of Heavy Haul Locomotive SchedulingProblem For an unpaired train diagram the number ofarrival trains is not equal to the number of departure trainsat some stations So the unpaired train diagram needs tobe transformed into a paired train diagram before solvingthe locomotive scheduling problem For the paired traindiagram the number of arrival trains is equal to the numberof departure trains at each station Therefore if each trainis hauled by the same type of locomotive the locomotivescheduling problem can be transformed into a standardassignment problem by treating arrival locomotives as ldquoper-sonnelrdquo and departure trains as ldquomissionsrdquo

Therefore the transformation of heavy haul locomotivescheduling problem mainly focuses on the following threeaspects

(1) Transforming Unpaired Train Diagram into Paired TrainDiagram Usually the trainwill be operated smoothly accord-ing to the train diagram Because the number of trains in traindiagram is given on the known railway the number of trainswithin 24 h is determined If the train diagram is unpairedit needs to add some ldquovirtual operation curvesrdquo Then thenumber of arrival trains is equal to the number of departuretrains in each section It assumes that the number of ldquovirtualoperation curvesrdquo is 119884 and the number of actual trains is 119869

(2) Transforming Multilocomotive Traction into Single-Locomotive Traction If the total number of trains is 119884 + 119869the number of trains with double-locomotive traction is Wand the number of trains with three-locomotive traction is119881 (119882 + 119881 le 119869 + 119884) Then the number of ldquovirtual trainsrdquois 119882 + 2119881 when multilocomotive traction is transformedinto single-locomotive traction At the same time it needsto ensure that the information of train diagram is consistentwith before Then trains will be sorted and coded accordingto departure times and the code of trains may be 1 2 3 119869 + 119884 +119882 + 2119881 minus 2 119869 + 119884 +119882 + 2119881 minus 1 119869 + 119884 +119882 + 2119881

(3) Transformation betweenDifferent Locomotive TypesThereare two locomotive types which include SS4 locomotive and

HXD locomotive inDa-qin Railway 10 kt heavy-loaded com-bined trains are hauled by SS4 locomotives while 10 kt heavy-loaded unit trains 15 kt heavy-loaded combined trains and20 kt heavy-loaded combined trains are hauled by HXDlocomotives There is no locomotive reconnection betweenHXD locomotives and SS4 locomotives Therefore locomo-tive scheduling problemwhich involves two locomotive typescan be considered separately for each type in order to obtainthe optimal solutionsThen the optimization objective of thelocomotive schedule on the whole railway is achieved

42 Solution Algorithm of Heavy Haul Locomotive ScheduleIn this paper the optimization model of heavy haul locomo-tive schedule is solved by using Hungarian algorithm TheHungarian algorithm as a classical algorithm can obtain aglobal optimal solution when it is used to solve assignmentproblems With an increasing coefficient matrix dimensionthe algorithm runs a little slower but it can reduce computa-tional efforts muchmore than any enumerationmethod [22]The concrete steps of Hungarian algorithm are as follows

Step 1 (generating coefficient matrix) The coefficient matrixof detention time of locomotives at stations for each station isbuilt firstly and then the coefficient matrix for whole railwaycan be built

Step 2 (transforming coefficient matrix) When performingthe row operations on the coefficient matrix we subtract thelowest value from each element in that row in order to obtainat least one zero element for each row Similarly we also dothis procedure for all column elements

Step 3 (seeking independent zero elements) If the numberof independent zero elements is less than the order ofthe coefficient matrix then go to Step 4 If the numberof independent zero elements is equal to the order of thecoefficient matrix then output the optimization solution andend the algorithm

Step 4 (drawing the least number of straight lines) If thenumber of straight lines is less than the order of the coefficientmatrix then mark the rows and columns and go to Step 5otherwise go to Step 3

Step 5 (transforming coefficient matrix again) Seek mini-mum value of the elements which are not covered by straightlines This value is then subtracted from each element inall marked rows and added to each element in the markedcolumns and then go to Step 3

The flow chart of the Hungarian algorithm is shown inFigure 8

Mathematical Problems in Engineering 7

Begin

equal to thematrix order

Mark the least zero elements and remove the other zero elements

The value of marked zero elements is 1 and

the other is 0 in thecounterpart matrix

Cover all of zero elements by the least straight linesEnd

Generating coefficient matrix

Each row and column element minus the minimum value

Is the number ofstraight lines smaller than

the matrix order

Transform coefficient matrix

again

YesNo

Yes

No

Is the number ofindependent zero elements

Figure 8 Flow chart of the Hungarian algorithm

5 Case Study

51 Evaluating the Feasibility of Heavy Haul LocomotiveSchedule Model The model and its solution algorithm needto be validated before applying them to any real problems

511 Data Source and Processing As shown in Figure 4line A-B-C-D is a double-track railway (section A-D forshort) which is specially used to transport coal Station Bis the station where the locomotive depot is located andthe minimal technological time of locomotives at stations is125min Station A station C and station D are the stationswhere locomotive turnaround depots are located and theminimal technological time of locomotives at stations is50min There are 8 trains (4 pairs) which include 2 trains(1 pair) in section A-B 4 trains (2 pairs) in section B-Cand 2 trains (1 pair) in section B-D on the railway If all oftrains are hauled by single-locomotive Figure 6 is the traindiagram

This basic data needs to be processed as follows

(1) Stations Coding For all stations in section A-D it can codestations from any end of the railway and it starts with stationA

(2) Trains Coding The trains in section A-D are sole How-ever it is necessary for each train to code another sole and

Table 2 Trains coding result in section A-D

Train number Train coding Train number Train coding70001 1 73003 570002 2 73004 673001 3 79001 773002 4 79002 8

simple number due to the long and large train number Thecoding results are listed in Table 2

(3) Time Simplification For the departure time and arrivaltime of each train starting at the initial time (1800) of thetrain diagram time of 24 h can be converted to minutes andfor example 1933 can be converted to the 93rd minute

(4) Generating Coefficient Matrix According to (4) thecoefficient matrix 119888119896119894119895 of the detention time of locomotives atstations will be generated by using C The results are listedbelow

Station A Consider the following

119888

A21 = 50 minus 265 minus 50 + 1440 = 1175

(50 minus 265 = minus215 lt 50)

(12)

8 Mathematical Problems in Engineering

Station B Consider the following

119888

B12 = 220 minus 90 minus 125 = 5 (220 minus 90 = 130 gt 125)

119888

B13 = 280 minus 90 minus 125 = 65 (280 minus 90 = 190 gt 125)

119888

B15 = 516 minus 90 minus 125 = 301

(516 minus 90 = 426 gt 125)

119888

B17 = 150 minus 90 minus 125 + 1440 = 1375

(150 minus 90 = 60 lt 125)

119888

B42 = 220 minus 450 minus 125 + 1440 = 1085

(220 minus 450 = minus230 lt 125)

119888

B43 = 280 minus 450 minus 125 + 1440 = 1145

(280 minus 450 = minus170 lt 125)

119888

B45 = 516 minus 450 minus 125 + 1440 = 1381

(516 minus 450 = 66 lt 125516 minus 450 = 66 lt 125)

119888

B47 = 150 minus 450 minus 125 + 1440 = 1015

(150 minus 450 = minus300 lt 125)

119888

B62 = 220 minus 673 minus 125 + 1440 = 862

(220 minus 673 = minus453 lt 125)

119888

B63 = 280 minus 673 minus 125 + 1440 = 922

(280 minus 673 = minus393 lt 125)

119888

B65 = 516 minus 673 minus 125 + 1440 = 1158

(516 minus 673 = minus157 lt 125)

119888

B67 = 150 minus 673 minus 125 + 1440 = 792

(150 minus 673 = minus523 lt 125)

119888

B82 = 220 minus 363 minus 125 + 1440 = 1172

(220 minus 363 = minus143 lt 125)

119888

B83 = 280 minus 363 minus 125 + 1440 = 1232

(280 minus 363 = minus83 lt 125)

119888

B85 = 516 minus 363 minus 125 = 28

(516 minus 363 = 153 gt 125)

119888

B87 = 150 minus 363 minus 125 + 1440 = 1102

(150 minus 90 = minus213 lt 125)

(13)

Table 3 Coefficient matrix of detention time of locomotives atstations

119894

119895

1 2 3 4 5 6 7 81 infin 5 65 infin 301 infin 1375 infin

2 1175 infin infin infin infin infin infin infin

3 infin infin infin 37 infin 260 infin infin

4 infin 1085 1145 infin 1381 infin 1015 infin

5 infin infin infin 1241 infin 24 infin infin

6 infin 862 922 infin 1158 infin 792 infin

7 infin infin infin infin infin infin infin 148 infin 1172 1232 infin 28 infin 1102 infin

Station C Consider the following

119888

C34 = 407 minus 320 minus 50 = 37 (407 minus 320 = 87 gt 50)

119888

C36 = 630 minus 320 minus 50 = 260

(630 minus 320 = 310 gt 50)

119888

C54 = 407 minus 556 minus 50 + 1440 = 1241

(407 minus 556 = minus149 lt 50)

119888

C56 = 630 minus 556 minus 50 = 24 (630 minus 556 = 74 gt 50)

(14)

Station D Consider the following

119888

D78 = 296 minus 232 minus 50 = 14 (296 minus 232 = 64 gt 50) (15)

The others are equal toinfin the coefficient matrix (119888119896119894119895) isshowed in Table 3

512 Result The Hungarian algorithm is implemented inMATLAB and the optimal locomotive schedule is as follows

The minimum waiting time of locomotives at stations is

119879119908 = 119885 =

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

119888

119896119894119895119909119896119894119895

= 65 + 1175 + 37 + 1085 + 24 + 792 + 14 + 28

= 3220 (min)

(16)

Theminimal technological time of locomotives at stationsis

119879119870 = 125 times 4 + 50 times 4 = 700 (min) (17)

The running time of locomotives is

119879119905 = 40 + 45 + 40 + 43 + 40 + 43 + 82 + 67

= 400 (min) (18)

The minimum number of locomotives is

119872 =

119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119879119908

1440

=

400 + 700 + 3220

1440

=

4320

1440

= 3

(19)

Mathematical Problems in Engineering 9

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 9 Locomotive schedule in section A-D

Chawu

299 kmDashizhuang

CuipingshanZunhua North

Qianrsquoan North

Jixian West

HouyingLiucun South

Qinhuangdao EastLuannan

Donggang

Caofeidian West

60 km19 km

11km59 km 78 km 67 km

22 km

24km

71km

66km

44 km

Hudong

Figure 10 The simplified network of Da-qin Railway

The coefficient of locomotive requirement is

119870119889 =119872

119899

=

3

82

= 075 (20)

The locomotive schedule can be expressed as an intuitivediagram which is shown in Figure 9 (dotted lines representlocomotives and solid lines represent trains)

Figure 9 shows that there are 3 traction locomotiveswhich are equal to the calculation result to haul trains

513 Result Analysis

(1) Analysis on the Optimization of Locomotive SchedulingFigures 7 and 9 show that there may be a variety of trainconnections So the number of available locomotives andcoefficient of locomotive requirement are uncertain Theoptimization model of heavy-loaded locomotive schedule isestablished and then it can be solved byHungarian algorithmIt can not only make sure of the uniqueness of locomotivetraction tasks but also complete established traction taskswith a minimum number of locomotives

(2) Analysis on the Equilibrium of Locomotive Schedule Theequilibrium degree of locomotive schedule of Figure 9 can beobtained by using (9) which is

119863 =

1

8

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

8

)

2

=

1

8

[(65 minus

3220

8

)

2

+ (1175 minus

3220

8

)

2

+ (37 minus

3220

8

)

2

+ (1085 minus

3220

8

)

2

+ (24 minus

3220

8

)

2

+ (792 minus

3220

8

)

2

+ (14 minus

3220

8

)

2

+ (28 minus

3220

8

)

2

] = 237027

(21)

It shows that the equilibrium degree of locomotive sched-ule is large The main reason is that there are no trains from600 to 1800 on the railwayTherefore the difference betweenthe detention time of locomotives at stations from 1800 to600 the next day and the detention time of locomotives atstations from 600 to 1800 is relatively large If there are trainsfrom 600 to 1800 the equilibrium degree of locomotiveschedule will be reduced and the locomotive schedule will bemore balanced

52 Analysis of Locomotive Schedule for 20 kt CombinedTrains on Da-qin Railway The previous section shows thatthe model of locomotive schedule is feasible Then theeffectiveness of the proposedmodel also needs to be validatedwhich is conducted in this section

521 Line Simplification Da-qin Railway has 26 stations andthe line length is 653 km Qiancao Railway has 5 stations andthe line length is 137 km Jingtang Port-Donggang RailwayLine has 5 stations and the line length is 45 km Someintermediate stations on the railway do not have the technicaloperation so they are not displayed Only the stationswhere locomotive depot or turnaround depot is located areconsidered The simplified network of Da-qin Railway isshown in Figure 10

Hudong station is the station where the locomotive depotis located and theminimal technological time of locomotives

10 Mathematical Problems in Engineering

at stations is 250min Liucun South station Donggang sta-tion Caofeidian West station and Qinhuangdao East stationare the stations where locomotive turnaround depots arelocated The minimal technological time of locomotives atstations for Liucun South station is 220min The minimaltechnological time of locomotives at stations for Donggangstation and Caofeidian West station is 100min The minimaltechnological time of locomotives at stations for Qinhuang-daoEast station is 125minTheminimal technological time ofthe other stations where the locomotives turn back is 50min

There were 100 heavy-loaded trains in the heavy-loaddirection and 104 empty-wagon trains in the reverse directionon December 10 2014

522 Data Processing In reference to the mentionedmethod complete the data processing For the HXDlocomotives there are 328 trains which are hauled by single-locomotive after conversion so the order of the coefficientmatrix is 328 For the SS4 locomotives there are 56 trainswhich are hauled double-locomotive after conversion sothe order of the coefficient matrix is 56 In this paper C isemployed to deal with such huge data

523 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

119888

119896119894119895119909119896119894119895 = 37116 (min)

119885SS4 =13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10409 (min)

(22)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 164 times 250 + 76 times 220 + 14 times 125 + 44 times 100

+ 30 times 50 = 65370 (min)

119879

SS4119870 = 28 times 250 + 28 times 50 = 8400 (min)

(23)

The running times of locomotives are respectively

119879

HXD119905 = 198474 (min)

119879

SS4119905 = 24391 (min)

(24)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

198474 + 65370 + 37116

1440

=

300960

1440

= 209

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

24391 + 8400 + 10409

1440

times 2 =

43200

1440

times 2

= 60

(25)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

209

1602

= 261

119870

SS4119889 =

119872SS4119899

=

30 times 2

562

= 214

(26)

According to the best connection matrix locomotiveconnection scheme for 20 kt combined trains on Da-qinRailway can be obtainedThe locomotive connection schemeof SS4 locomotives is as follows

H16012 997888rarr 52 (single-locomotive) 997888rarr H16016

997888rarr 54 (single-locomotive) 997888rarr H16004

997888rarr H19609 997888rarr H19504 997888rarr H19605

997888rarr H19512 997888rarr 53 (single-locomotive)

997888rarr H16018 997888rarr 56 (single-locomotive)

997888rarr H16002 997888rarr 48 (single-locomotive)

997888rarr H15506 997888rarr 47 (single-locomotive)

997888rarr H16010 997888rarr 19601 997888rarr H19508

997888rarr 46 (single-locomotive) 997888rarr H16008

997888rarr H15501 997888rarr H15504 997888rarr H19607

997888rarr H19516 997888rarr 19615 997888rarr H19514

997888rarr H28501 997888rarr H28510

997888rarr 51 (single-locomotive) 997888rarr H16014

997888rarr H19603 997888rarr H19502 997888rarr H28507

997888rarr H28512 997888rarr 49 (attached locomotive)

997888rarr H15508 997888rarr H28503 997888rarr H28508

997888rarr 50 (single-locomotive) 997888rarr H16012

997888rarr H19611 997888rarr H19506 997888rarr H28511

997888rarr H28502 997888rarr H15503 997888rarr H15502

997888rarr 55 (attached locomotive) 997888rarr H15510

997888rarr H28505 997888rarr H28506

997888rarr 45 (single-locomotive) 997888rarr H16006

997888rarr H19611

H19613 997888rarr H19510 997888rarr H19613

H28504 997888rarr H28509 997888rarr H28504

(27)

Mathematical Problems in Engineering 11

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

Luannan

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

LuannanH16012 H16014 H16016 H16018H16002 H16004 H16006

H16002 H16004 H16006 H16008 H16010 H160112 H16014H16016 H16018

H15501

H15501

H15503

H15503

H15502 H15504 H15506 H15508 H15510

H28502 H28504 H28506 H28508 H28510 H28512

H15502H28502 H28504 H15504 H128506 H28508 H28510H15506 H15508 H28512

H28501 H28505 H28507 H28509 H28511

H28501 H28503 H28505 H28507 H28509 H28511

H19601 H19603 H19615 H19605 H19607 H19609 H19613 H19611

H19516 H19502 H19504 H19506 H19508 H19510 H19514 H19512

H28503

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

TrainsLocomotivesSingle-locomotives or attached locomotives

H16008 H16010

Figure 11 Locomotive working diagram of SS4 locomotives for 20 kt combined trains

Table 4 The daily number of locomotives on Da-qin Railway onDecember 10 2014 [21]

Types oflocomotive

Daily number oflocomotives

Types oflocomotive

Daily number oflocomotives

HXD1 130 SS4 109HXD2 92 mdash mdash

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 11

524 Analysis of Result

(1) Analyze the Number of Available LocomotivesThe numberof available locomotives (the total number of locomotivesafter deducting locomotives inmaintenance and preparation)onDa-qin Railway onDecember 10 2014 is shown in Table 4

Table 4 indicates that the locomotive schedule obtainedby using themodel can save the number ofHXD locomotives130 + 92 minus 209 = 13 and the number of SS4 locomotives109minus 30times 2 = 49 Therefore if only the minimum number oflocomotives is considered the optimization model of heavyhaul locomotive schedule will be optimized

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

328

13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

328

)

2

= 52379

119863SS4 =1

56

13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

56

)

2

= 74044

(28)

The equilibrium degree of the locomotive schedule forHXD locomotives and SS4 locomotives is smaller than theequilibrium degree of that case which shows that this loco-motive schedule is more balanced The equilibrium degreeof SS4 locomotives is larger than the equilibrium degree ofHXD locomotives which shows that the schedule of HXDlocomotives ismore balanced than SS4 locomotivesThe largeequilibrium degree of SS4 locomotives may be due to thesmall detention time of locomotives at stations

53 Research and Analysis of Locomotive Schedule Optimiza-tion for 30 kt Combined Trains on Da-qin Railway On thebasis of the potential train diagram for 30 kt combined trainsthe practical and feasible locomotive connection scheme canbe determined by using the optimization model

In reference to the mentioned method complete the dataprocessing

531 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

119888

119896119894119895119909119896119894119895 = 44838 (min)

119885SS4 =13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10314 (min)

(29)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 203 times 250 + 64 times 220 + 65 times 100 + 74 times 50

= 75030 (min)

119879

SS4119870 = 26 times 250 + 26 times 50 = 7800 (min)

(30)

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 6 Train diagram in section A-D

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 7 Potential locomotive schedule in section A-D

technological time The waiting time 119888119896119894119895 of locomotives atstations can be calculated by

119888

119896119894119895

=

119905119895119891119896 minus 119905119894119889119896 minus 119879119896 119905119895119891119896 minus 119905119894119889119896 ge 119879119896 119896119894119889 = 119896119895119891

119905119895119891119896 minus 119905119894119889119896 minus 119879119896 + 1440 119905119895119891119896 minus 119905119894119889119896 lt 119879119896 119896119894119889 = 119896119895119891

infin 119896119894119889 = 119896119895119891

(4)

323 Variable Thevariable is assumed to be 119888119896119894119895 which refersto whether locomotive 119894 hauls train 119895 or not at station 119896 Thebinary variable 119888119896119894119895 is defined as follows

119909

119896119894119895 =

1 locomotive 119894 hauls train 119895 at station 119896

0 otherwise(5)

324 Objective Function Completing planned traction tasksis the primary constraint to us Based on that the lessthe number of locomotives is the lower the transportationcost will be Therefore the objective function of heavy haullocomotive scheduling problem is minimizing number oflocomotives that is

min 119885

1015840=

119879119905 + 119879

1440

(6)

where 119879119905 is the total time of trains running from departurestation to destination station and T is the total detention timeof locomotives at stations which can be calculated by

119879 = 119879119870 + 119879119908 (7)

where 119879119870 is the total minimal technological time of locomo-tives at stations and119879119908 is the total waiting time of locomotivesat stations

For the established train diagram 119879119905 is a constant Sothe minimum number of locomotives refers to the minimaldetention time 119879 of locomotives at stations Meanwhile119879119870 is also a constant So the minimal detention time oflocomotives at stations refers to the minimum waiting time119879119908 of locomotives at stations That is

min 119885 = 119879119908 =

119870

sum

119896=1

119873119896119889

sum

119894=1

119873119896119891

sum

119895=1

119888

119896119894119895119909119896119894119895 (8)

Then the equilibrium degree 119863 of locomotive schedulecan be calculated by

119863 =

1

119899119904 + 119899119889 + 2 sdot 119899119905

sdot

119870

sum

119896=1

119873119896119889

sum

119894=1

119873119896119891

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

119899119904 + 119899119889 + 2 sdot 119899119905

)

2

(9)

325 Constraints

(1) Traction Locomotives Constraint Consider the following

119873119896119889

sum

119894=1

119909

119896119894119895 = ℎ forall119895 isin 119873119896119891 forall119896 isin 119870 (10)

This constraint shows that the number of traction loco-motives for each train is ℎ The value of each type of train isshown in Table 1

(2) Traction Tasks Constraint Consider the following

119873119896119891

sum

119895=1

119909

119896119894119895 = 1 forall119894 isin 119873119896119889 forall119896 isin 119870 (11)

6 Mathematical Problems in Engineering

Table 1 The value of each type of trains

Types of heavy-loaded trains Reference value Types of heavy-loaded trains Reference value10 kt heavy-loaded combined trains 2 20 kt heavy-loaded combined trains 210 kt heavy-loaded unit trains 1 30 kt heavy-loaded combined trains 315 kt heavy-loaded combined trains 2 mdash mdash

This constraint shows that each locomotive can only haulone train at the same time

4 Transformation and SolvingProcesses of Heavy Haul LocomotiveScheduling Problem

First of all the locomotive scheduling problem needs to betransformed into an assignment problemThen the optimiza-tion results are obtained by using the Hungarian algorithm

41 Transformation of Heavy Haul Locomotive SchedulingProblem For an unpaired train diagram the number ofarrival trains is not equal to the number of departure trainsat some stations So the unpaired train diagram needs tobe transformed into a paired train diagram before solvingthe locomotive scheduling problem For the paired traindiagram the number of arrival trains is equal to the numberof departure trains at each station Therefore if each trainis hauled by the same type of locomotive the locomotivescheduling problem can be transformed into a standardassignment problem by treating arrival locomotives as ldquoper-sonnelrdquo and departure trains as ldquomissionsrdquo

Therefore the transformation of heavy haul locomotivescheduling problem mainly focuses on the following threeaspects

(1) Transforming Unpaired Train Diagram into Paired TrainDiagram Usually the trainwill be operated smoothly accord-ing to the train diagram Because the number of trains in traindiagram is given on the known railway the number of trainswithin 24 h is determined If the train diagram is unpairedit needs to add some ldquovirtual operation curvesrdquo Then thenumber of arrival trains is equal to the number of departuretrains in each section It assumes that the number of ldquovirtualoperation curvesrdquo is 119884 and the number of actual trains is 119869

(2) Transforming Multilocomotive Traction into Single-Locomotive Traction If the total number of trains is 119884 + 119869the number of trains with double-locomotive traction is Wand the number of trains with three-locomotive traction is119881 (119882 + 119881 le 119869 + 119884) Then the number of ldquovirtual trainsrdquois 119882 + 2119881 when multilocomotive traction is transformedinto single-locomotive traction At the same time it needsto ensure that the information of train diagram is consistentwith before Then trains will be sorted and coded accordingto departure times and the code of trains may be 1 2 3 119869 + 119884 +119882 + 2119881 minus 2 119869 + 119884 +119882 + 2119881 minus 1 119869 + 119884 +119882 + 2119881

(3) Transformation betweenDifferent Locomotive TypesThereare two locomotive types which include SS4 locomotive and

HXD locomotive inDa-qin Railway 10 kt heavy-loaded com-bined trains are hauled by SS4 locomotives while 10 kt heavy-loaded unit trains 15 kt heavy-loaded combined trains and20 kt heavy-loaded combined trains are hauled by HXDlocomotives There is no locomotive reconnection betweenHXD locomotives and SS4 locomotives Therefore locomo-tive scheduling problemwhich involves two locomotive typescan be considered separately for each type in order to obtainthe optimal solutionsThen the optimization objective of thelocomotive schedule on the whole railway is achieved

42 Solution Algorithm of Heavy Haul Locomotive ScheduleIn this paper the optimization model of heavy haul locomo-tive schedule is solved by using Hungarian algorithm TheHungarian algorithm as a classical algorithm can obtain aglobal optimal solution when it is used to solve assignmentproblems With an increasing coefficient matrix dimensionthe algorithm runs a little slower but it can reduce computa-tional efforts muchmore than any enumerationmethod [22]The concrete steps of Hungarian algorithm are as follows

Step 1 (generating coefficient matrix) The coefficient matrixof detention time of locomotives at stations for each station isbuilt firstly and then the coefficient matrix for whole railwaycan be built

Step 2 (transforming coefficient matrix) When performingthe row operations on the coefficient matrix we subtract thelowest value from each element in that row in order to obtainat least one zero element for each row Similarly we also dothis procedure for all column elements

Step 3 (seeking independent zero elements) If the numberof independent zero elements is less than the order ofthe coefficient matrix then go to Step 4 If the numberof independent zero elements is equal to the order of thecoefficient matrix then output the optimization solution andend the algorithm

Step 4 (drawing the least number of straight lines) If thenumber of straight lines is less than the order of the coefficientmatrix then mark the rows and columns and go to Step 5otherwise go to Step 3

Step 5 (transforming coefficient matrix again) Seek mini-mum value of the elements which are not covered by straightlines This value is then subtracted from each element inall marked rows and added to each element in the markedcolumns and then go to Step 3

The flow chart of the Hungarian algorithm is shown inFigure 8

Mathematical Problems in Engineering 7

Begin

equal to thematrix order

Mark the least zero elements and remove the other zero elements

The value of marked zero elements is 1 and

the other is 0 in thecounterpart matrix

Cover all of zero elements by the least straight linesEnd

Generating coefficient matrix

Each row and column element minus the minimum value

Is the number ofstraight lines smaller than

the matrix order

Transform coefficient matrix

again

YesNo

Yes

No

Is the number ofindependent zero elements

Figure 8 Flow chart of the Hungarian algorithm

5 Case Study

51 Evaluating the Feasibility of Heavy Haul LocomotiveSchedule Model The model and its solution algorithm needto be validated before applying them to any real problems

511 Data Source and Processing As shown in Figure 4line A-B-C-D is a double-track railway (section A-D forshort) which is specially used to transport coal Station Bis the station where the locomotive depot is located andthe minimal technological time of locomotives at stations is125min Station A station C and station D are the stationswhere locomotive turnaround depots are located and theminimal technological time of locomotives at stations is50min There are 8 trains (4 pairs) which include 2 trains(1 pair) in section A-B 4 trains (2 pairs) in section B-Cand 2 trains (1 pair) in section B-D on the railway If all oftrains are hauled by single-locomotive Figure 6 is the traindiagram

This basic data needs to be processed as follows

(1) Stations Coding For all stations in section A-D it can codestations from any end of the railway and it starts with stationA

(2) Trains Coding The trains in section A-D are sole How-ever it is necessary for each train to code another sole and

Table 2 Trains coding result in section A-D

Train number Train coding Train number Train coding70001 1 73003 570002 2 73004 673001 3 79001 773002 4 79002 8

simple number due to the long and large train number Thecoding results are listed in Table 2

(3) Time Simplification For the departure time and arrivaltime of each train starting at the initial time (1800) of thetrain diagram time of 24 h can be converted to minutes andfor example 1933 can be converted to the 93rd minute

(4) Generating Coefficient Matrix According to (4) thecoefficient matrix 119888119896119894119895 of the detention time of locomotives atstations will be generated by using C The results are listedbelow

Station A Consider the following

119888

A21 = 50 minus 265 minus 50 + 1440 = 1175

(50 minus 265 = minus215 lt 50)

(12)

8 Mathematical Problems in Engineering

Station B Consider the following

119888

B12 = 220 minus 90 minus 125 = 5 (220 minus 90 = 130 gt 125)

119888

B13 = 280 minus 90 minus 125 = 65 (280 minus 90 = 190 gt 125)

119888

B15 = 516 minus 90 minus 125 = 301

(516 minus 90 = 426 gt 125)

119888

B17 = 150 minus 90 minus 125 + 1440 = 1375

(150 minus 90 = 60 lt 125)

119888

B42 = 220 minus 450 minus 125 + 1440 = 1085

(220 minus 450 = minus230 lt 125)

119888

B43 = 280 minus 450 minus 125 + 1440 = 1145

(280 minus 450 = minus170 lt 125)

119888

B45 = 516 minus 450 minus 125 + 1440 = 1381

(516 minus 450 = 66 lt 125516 minus 450 = 66 lt 125)

119888

B47 = 150 minus 450 minus 125 + 1440 = 1015

(150 minus 450 = minus300 lt 125)

119888

B62 = 220 minus 673 minus 125 + 1440 = 862

(220 minus 673 = minus453 lt 125)

119888

B63 = 280 minus 673 minus 125 + 1440 = 922

(280 minus 673 = minus393 lt 125)

119888

B65 = 516 minus 673 minus 125 + 1440 = 1158

(516 minus 673 = minus157 lt 125)

119888

B67 = 150 minus 673 minus 125 + 1440 = 792

(150 minus 673 = minus523 lt 125)

119888

B82 = 220 minus 363 minus 125 + 1440 = 1172

(220 minus 363 = minus143 lt 125)

119888

B83 = 280 minus 363 minus 125 + 1440 = 1232

(280 minus 363 = minus83 lt 125)

119888

B85 = 516 minus 363 minus 125 = 28

(516 minus 363 = 153 gt 125)

119888

B87 = 150 minus 363 minus 125 + 1440 = 1102

(150 minus 90 = minus213 lt 125)

(13)

Table 3 Coefficient matrix of detention time of locomotives atstations

119894

119895

1 2 3 4 5 6 7 81 infin 5 65 infin 301 infin 1375 infin

2 1175 infin infin infin infin infin infin infin

3 infin infin infin 37 infin 260 infin infin

4 infin 1085 1145 infin 1381 infin 1015 infin

5 infin infin infin 1241 infin 24 infin infin

6 infin 862 922 infin 1158 infin 792 infin

7 infin infin infin infin infin infin infin 148 infin 1172 1232 infin 28 infin 1102 infin

Station C Consider the following

119888

C34 = 407 minus 320 minus 50 = 37 (407 minus 320 = 87 gt 50)

119888

C36 = 630 minus 320 minus 50 = 260

(630 minus 320 = 310 gt 50)

119888

C54 = 407 minus 556 minus 50 + 1440 = 1241

(407 minus 556 = minus149 lt 50)

119888

C56 = 630 minus 556 minus 50 = 24 (630 minus 556 = 74 gt 50)

(14)

Station D Consider the following

119888

D78 = 296 minus 232 minus 50 = 14 (296 minus 232 = 64 gt 50) (15)

The others are equal toinfin the coefficient matrix (119888119896119894119895) isshowed in Table 3

512 Result The Hungarian algorithm is implemented inMATLAB and the optimal locomotive schedule is as follows

The minimum waiting time of locomotives at stations is

119879119908 = 119885 =

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

119888

119896119894119895119909119896119894119895

= 65 + 1175 + 37 + 1085 + 24 + 792 + 14 + 28

= 3220 (min)

(16)

Theminimal technological time of locomotives at stationsis

119879119870 = 125 times 4 + 50 times 4 = 700 (min) (17)

The running time of locomotives is

119879119905 = 40 + 45 + 40 + 43 + 40 + 43 + 82 + 67

= 400 (min) (18)

The minimum number of locomotives is

119872 =

119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119879119908

1440

=

400 + 700 + 3220

1440

=

4320

1440

= 3

(19)

Mathematical Problems in Engineering 9

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 9 Locomotive schedule in section A-D

Chawu

299 kmDashizhuang

CuipingshanZunhua North

Qianrsquoan North

Jixian West

HouyingLiucun South

Qinhuangdao EastLuannan

Donggang

Caofeidian West

60 km19 km

11km59 km 78 km 67 km

22 km

24km

71km

66km

44 km

Hudong

Figure 10 The simplified network of Da-qin Railway

The coefficient of locomotive requirement is

119870119889 =119872

119899

=

3

82

= 075 (20)

The locomotive schedule can be expressed as an intuitivediagram which is shown in Figure 9 (dotted lines representlocomotives and solid lines represent trains)

Figure 9 shows that there are 3 traction locomotiveswhich are equal to the calculation result to haul trains

513 Result Analysis

(1) Analysis on the Optimization of Locomotive SchedulingFigures 7 and 9 show that there may be a variety of trainconnections So the number of available locomotives andcoefficient of locomotive requirement are uncertain Theoptimization model of heavy-loaded locomotive schedule isestablished and then it can be solved byHungarian algorithmIt can not only make sure of the uniqueness of locomotivetraction tasks but also complete established traction taskswith a minimum number of locomotives

(2) Analysis on the Equilibrium of Locomotive Schedule Theequilibrium degree of locomotive schedule of Figure 9 can beobtained by using (9) which is

119863 =

1

8

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

8

)

2

=

1

8

[(65 minus

3220

8

)

2

+ (1175 minus

3220

8

)

2

+ (37 minus

3220

8

)

2

+ (1085 minus

3220

8

)

2

+ (24 minus

3220

8

)

2

+ (792 minus

3220

8

)

2

+ (14 minus

3220

8

)

2

+ (28 minus

3220

8

)

2

] = 237027

(21)

It shows that the equilibrium degree of locomotive sched-ule is large The main reason is that there are no trains from600 to 1800 on the railwayTherefore the difference betweenthe detention time of locomotives at stations from 1800 to600 the next day and the detention time of locomotives atstations from 600 to 1800 is relatively large If there are trainsfrom 600 to 1800 the equilibrium degree of locomotiveschedule will be reduced and the locomotive schedule will bemore balanced

52 Analysis of Locomotive Schedule for 20 kt CombinedTrains on Da-qin Railway The previous section shows thatthe model of locomotive schedule is feasible Then theeffectiveness of the proposedmodel also needs to be validatedwhich is conducted in this section

521 Line Simplification Da-qin Railway has 26 stations andthe line length is 653 km Qiancao Railway has 5 stations andthe line length is 137 km Jingtang Port-Donggang RailwayLine has 5 stations and the line length is 45 km Someintermediate stations on the railway do not have the technicaloperation so they are not displayed Only the stationswhere locomotive depot or turnaround depot is located areconsidered The simplified network of Da-qin Railway isshown in Figure 10

Hudong station is the station where the locomotive depotis located and theminimal technological time of locomotives

10 Mathematical Problems in Engineering

at stations is 250min Liucun South station Donggang sta-tion Caofeidian West station and Qinhuangdao East stationare the stations where locomotive turnaround depots arelocated The minimal technological time of locomotives atstations for Liucun South station is 220min The minimaltechnological time of locomotives at stations for Donggangstation and Caofeidian West station is 100min The minimaltechnological time of locomotives at stations for Qinhuang-daoEast station is 125minTheminimal technological time ofthe other stations where the locomotives turn back is 50min

There were 100 heavy-loaded trains in the heavy-loaddirection and 104 empty-wagon trains in the reverse directionon December 10 2014

522 Data Processing In reference to the mentionedmethod complete the data processing For the HXDlocomotives there are 328 trains which are hauled by single-locomotive after conversion so the order of the coefficientmatrix is 328 For the SS4 locomotives there are 56 trainswhich are hauled double-locomotive after conversion sothe order of the coefficient matrix is 56 In this paper C isemployed to deal with such huge data

523 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

119888

119896119894119895119909119896119894119895 = 37116 (min)

119885SS4 =13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10409 (min)

(22)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 164 times 250 + 76 times 220 + 14 times 125 + 44 times 100

+ 30 times 50 = 65370 (min)

119879

SS4119870 = 28 times 250 + 28 times 50 = 8400 (min)

(23)

The running times of locomotives are respectively

119879

HXD119905 = 198474 (min)

119879

SS4119905 = 24391 (min)

(24)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

198474 + 65370 + 37116

1440

=

300960

1440

= 209

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

24391 + 8400 + 10409

1440

times 2 =

43200

1440

times 2

= 60

(25)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

209

1602

= 261

119870

SS4119889 =

119872SS4119899

=

30 times 2

562

= 214

(26)

According to the best connection matrix locomotiveconnection scheme for 20 kt combined trains on Da-qinRailway can be obtainedThe locomotive connection schemeof SS4 locomotives is as follows

H16012 997888rarr 52 (single-locomotive) 997888rarr H16016

997888rarr 54 (single-locomotive) 997888rarr H16004

997888rarr H19609 997888rarr H19504 997888rarr H19605

997888rarr H19512 997888rarr 53 (single-locomotive)

997888rarr H16018 997888rarr 56 (single-locomotive)

997888rarr H16002 997888rarr 48 (single-locomotive)

997888rarr H15506 997888rarr 47 (single-locomotive)

997888rarr H16010 997888rarr 19601 997888rarr H19508

997888rarr 46 (single-locomotive) 997888rarr H16008

997888rarr H15501 997888rarr H15504 997888rarr H19607

997888rarr H19516 997888rarr 19615 997888rarr H19514

997888rarr H28501 997888rarr H28510

997888rarr 51 (single-locomotive) 997888rarr H16014

997888rarr H19603 997888rarr H19502 997888rarr H28507

997888rarr H28512 997888rarr 49 (attached locomotive)

997888rarr H15508 997888rarr H28503 997888rarr H28508

997888rarr 50 (single-locomotive) 997888rarr H16012

997888rarr H19611 997888rarr H19506 997888rarr H28511

997888rarr H28502 997888rarr H15503 997888rarr H15502

997888rarr 55 (attached locomotive) 997888rarr H15510

997888rarr H28505 997888rarr H28506

997888rarr 45 (single-locomotive) 997888rarr H16006

997888rarr H19611

H19613 997888rarr H19510 997888rarr H19613

H28504 997888rarr H28509 997888rarr H28504

(27)

Mathematical Problems in Engineering 11

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

Luannan

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

LuannanH16012 H16014 H16016 H16018H16002 H16004 H16006

H16002 H16004 H16006 H16008 H16010 H160112 H16014H16016 H16018

H15501

H15501

H15503

H15503

H15502 H15504 H15506 H15508 H15510

H28502 H28504 H28506 H28508 H28510 H28512

H15502H28502 H28504 H15504 H128506 H28508 H28510H15506 H15508 H28512

H28501 H28505 H28507 H28509 H28511

H28501 H28503 H28505 H28507 H28509 H28511

H19601 H19603 H19615 H19605 H19607 H19609 H19613 H19611

H19516 H19502 H19504 H19506 H19508 H19510 H19514 H19512

H28503

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

TrainsLocomotivesSingle-locomotives or attached locomotives

H16008 H16010

Figure 11 Locomotive working diagram of SS4 locomotives for 20 kt combined trains

Table 4 The daily number of locomotives on Da-qin Railway onDecember 10 2014 [21]

Types oflocomotive

Daily number oflocomotives

Types oflocomotive

Daily number oflocomotives

HXD1 130 SS4 109HXD2 92 mdash mdash

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 11

524 Analysis of Result

(1) Analyze the Number of Available LocomotivesThe numberof available locomotives (the total number of locomotivesafter deducting locomotives inmaintenance and preparation)onDa-qin Railway onDecember 10 2014 is shown in Table 4

Table 4 indicates that the locomotive schedule obtainedby using themodel can save the number ofHXD locomotives130 + 92 minus 209 = 13 and the number of SS4 locomotives109minus 30times 2 = 49 Therefore if only the minimum number oflocomotives is considered the optimization model of heavyhaul locomotive schedule will be optimized

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

328

13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

328

)

2

= 52379

119863SS4 =1

56

13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

56

)

2

= 74044

(28)

The equilibrium degree of the locomotive schedule forHXD locomotives and SS4 locomotives is smaller than theequilibrium degree of that case which shows that this loco-motive schedule is more balanced The equilibrium degreeof SS4 locomotives is larger than the equilibrium degree ofHXD locomotives which shows that the schedule of HXDlocomotives ismore balanced than SS4 locomotivesThe largeequilibrium degree of SS4 locomotives may be due to thesmall detention time of locomotives at stations

53 Research and Analysis of Locomotive Schedule Optimiza-tion for 30 kt Combined Trains on Da-qin Railway On thebasis of the potential train diagram for 30 kt combined trainsthe practical and feasible locomotive connection scheme canbe determined by using the optimization model

In reference to the mentioned method complete the dataprocessing

531 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

119888

119896119894119895119909119896119894119895 = 44838 (min)

119885SS4 =13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10314 (min)

(29)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 203 times 250 + 64 times 220 + 65 times 100 + 74 times 50

= 75030 (min)

119879

SS4119870 = 26 times 250 + 26 times 50 = 7800 (min)

(30)

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 1 The value of each type of trains

Types of heavy-loaded trains Reference value Types of heavy-loaded trains Reference value10 kt heavy-loaded combined trains 2 20 kt heavy-loaded combined trains 210 kt heavy-loaded unit trains 1 30 kt heavy-loaded combined trains 315 kt heavy-loaded combined trains 2 mdash mdash

This constraint shows that each locomotive can only haulone train at the same time

4 Transformation and SolvingProcesses of Heavy Haul LocomotiveScheduling Problem

First of all the locomotive scheduling problem needs to betransformed into an assignment problemThen the optimiza-tion results are obtained by using the Hungarian algorithm

41 Transformation of Heavy Haul Locomotive SchedulingProblem For an unpaired train diagram the number ofarrival trains is not equal to the number of departure trainsat some stations So the unpaired train diagram needs tobe transformed into a paired train diagram before solvingthe locomotive scheduling problem For the paired traindiagram the number of arrival trains is equal to the numberof departure trains at each station Therefore if each trainis hauled by the same type of locomotive the locomotivescheduling problem can be transformed into a standardassignment problem by treating arrival locomotives as ldquoper-sonnelrdquo and departure trains as ldquomissionsrdquo

Therefore the transformation of heavy haul locomotivescheduling problem mainly focuses on the following threeaspects

(1) Transforming Unpaired Train Diagram into Paired TrainDiagram Usually the trainwill be operated smoothly accord-ing to the train diagram Because the number of trains in traindiagram is given on the known railway the number of trainswithin 24 h is determined If the train diagram is unpairedit needs to add some ldquovirtual operation curvesrdquo Then thenumber of arrival trains is equal to the number of departuretrains in each section It assumes that the number of ldquovirtualoperation curvesrdquo is 119884 and the number of actual trains is 119869

(2) Transforming Multilocomotive Traction into Single-Locomotive Traction If the total number of trains is 119884 + 119869the number of trains with double-locomotive traction is Wand the number of trains with three-locomotive traction is119881 (119882 + 119881 le 119869 + 119884) Then the number of ldquovirtual trainsrdquois 119882 + 2119881 when multilocomotive traction is transformedinto single-locomotive traction At the same time it needsto ensure that the information of train diagram is consistentwith before Then trains will be sorted and coded accordingto departure times and the code of trains may be 1 2 3 119869 + 119884 +119882 + 2119881 minus 2 119869 + 119884 +119882 + 2119881 minus 1 119869 + 119884 +119882 + 2119881

(3) Transformation betweenDifferent Locomotive TypesThereare two locomotive types which include SS4 locomotive and

HXD locomotive inDa-qin Railway 10 kt heavy-loaded com-bined trains are hauled by SS4 locomotives while 10 kt heavy-loaded unit trains 15 kt heavy-loaded combined trains and20 kt heavy-loaded combined trains are hauled by HXDlocomotives There is no locomotive reconnection betweenHXD locomotives and SS4 locomotives Therefore locomo-tive scheduling problemwhich involves two locomotive typescan be considered separately for each type in order to obtainthe optimal solutionsThen the optimization objective of thelocomotive schedule on the whole railway is achieved

42 Solution Algorithm of Heavy Haul Locomotive ScheduleIn this paper the optimization model of heavy haul locomo-tive schedule is solved by using Hungarian algorithm TheHungarian algorithm as a classical algorithm can obtain aglobal optimal solution when it is used to solve assignmentproblems With an increasing coefficient matrix dimensionthe algorithm runs a little slower but it can reduce computa-tional efforts muchmore than any enumerationmethod [22]The concrete steps of Hungarian algorithm are as follows

Step 1 (generating coefficient matrix) The coefficient matrixof detention time of locomotives at stations for each station isbuilt firstly and then the coefficient matrix for whole railwaycan be built

Step 2 (transforming coefficient matrix) When performingthe row operations on the coefficient matrix we subtract thelowest value from each element in that row in order to obtainat least one zero element for each row Similarly we also dothis procedure for all column elements

Step 3 (seeking independent zero elements) If the numberof independent zero elements is less than the order ofthe coefficient matrix then go to Step 4 If the numberof independent zero elements is equal to the order of thecoefficient matrix then output the optimization solution andend the algorithm

Step 4 (drawing the least number of straight lines) If thenumber of straight lines is less than the order of the coefficientmatrix then mark the rows and columns and go to Step 5otherwise go to Step 3

Step 5 (transforming coefficient matrix again) Seek mini-mum value of the elements which are not covered by straightlines This value is then subtracted from each element inall marked rows and added to each element in the markedcolumns and then go to Step 3

The flow chart of the Hungarian algorithm is shown inFigure 8

Mathematical Problems in Engineering 7

Begin

equal to thematrix order

Mark the least zero elements and remove the other zero elements

The value of marked zero elements is 1 and

the other is 0 in thecounterpart matrix

Cover all of zero elements by the least straight linesEnd

Generating coefficient matrix

Each row and column element minus the minimum value

Is the number ofstraight lines smaller than

the matrix order

Transform coefficient matrix

again

YesNo

Yes

No

Is the number ofindependent zero elements

Figure 8 Flow chart of the Hungarian algorithm

5 Case Study

51 Evaluating the Feasibility of Heavy Haul LocomotiveSchedule Model The model and its solution algorithm needto be validated before applying them to any real problems

511 Data Source and Processing As shown in Figure 4line A-B-C-D is a double-track railway (section A-D forshort) which is specially used to transport coal Station Bis the station where the locomotive depot is located andthe minimal technological time of locomotives at stations is125min Station A station C and station D are the stationswhere locomotive turnaround depots are located and theminimal technological time of locomotives at stations is50min There are 8 trains (4 pairs) which include 2 trains(1 pair) in section A-B 4 trains (2 pairs) in section B-Cand 2 trains (1 pair) in section B-D on the railway If all oftrains are hauled by single-locomotive Figure 6 is the traindiagram

This basic data needs to be processed as follows

(1) Stations Coding For all stations in section A-D it can codestations from any end of the railway and it starts with stationA

(2) Trains Coding The trains in section A-D are sole How-ever it is necessary for each train to code another sole and

Table 2 Trains coding result in section A-D

Train number Train coding Train number Train coding70001 1 73003 570002 2 73004 673001 3 79001 773002 4 79002 8

simple number due to the long and large train number Thecoding results are listed in Table 2

(3) Time Simplification For the departure time and arrivaltime of each train starting at the initial time (1800) of thetrain diagram time of 24 h can be converted to minutes andfor example 1933 can be converted to the 93rd minute

(4) Generating Coefficient Matrix According to (4) thecoefficient matrix 119888119896119894119895 of the detention time of locomotives atstations will be generated by using C The results are listedbelow

Station A Consider the following

119888

A21 = 50 minus 265 minus 50 + 1440 = 1175

(50 minus 265 = minus215 lt 50)

(12)

8 Mathematical Problems in Engineering

Station B Consider the following

119888

B12 = 220 minus 90 minus 125 = 5 (220 minus 90 = 130 gt 125)

119888

B13 = 280 minus 90 minus 125 = 65 (280 minus 90 = 190 gt 125)

119888

B15 = 516 minus 90 minus 125 = 301

(516 minus 90 = 426 gt 125)

119888

B17 = 150 minus 90 minus 125 + 1440 = 1375

(150 minus 90 = 60 lt 125)

119888

B42 = 220 minus 450 minus 125 + 1440 = 1085

(220 minus 450 = minus230 lt 125)

119888

B43 = 280 minus 450 minus 125 + 1440 = 1145

(280 minus 450 = minus170 lt 125)

119888

B45 = 516 minus 450 minus 125 + 1440 = 1381

(516 minus 450 = 66 lt 125516 minus 450 = 66 lt 125)

119888

B47 = 150 minus 450 minus 125 + 1440 = 1015

(150 minus 450 = minus300 lt 125)

119888

B62 = 220 minus 673 minus 125 + 1440 = 862

(220 minus 673 = minus453 lt 125)

119888

B63 = 280 minus 673 minus 125 + 1440 = 922

(280 minus 673 = minus393 lt 125)

119888

B65 = 516 minus 673 minus 125 + 1440 = 1158

(516 minus 673 = minus157 lt 125)

119888

B67 = 150 minus 673 minus 125 + 1440 = 792

(150 minus 673 = minus523 lt 125)

119888

B82 = 220 minus 363 minus 125 + 1440 = 1172

(220 minus 363 = minus143 lt 125)

119888

B83 = 280 minus 363 minus 125 + 1440 = 1232

(280 minus 363 = minus83 lt 125)

119888

B85 = 516 minus 363 minus 125 = 28

(516 minus 363 = 153 gt 125)

119888

B87 = 150 minus 363 minus 125 + 1440 = 1102

(150 minus 90 = minus213 lt 125)

(13)

Table 3 Coefficient matrix of detention time of locomotives atstations

119894

119895

1 2 3 4 5 6 7 81 infin 5 65 infin 301 infin 1375 infin

2 1175 infin infin infin infin infin infin infin

3 infin infin infin 37 infin 260 infin infin

4 infin 1085 1145 infin 1381 infin 1015 infin

5 infin infin infin 1241 infin 24 infin infin

6 infin 862 922 infin 1158 infin 792 infin

7 infin infin infin infin infin infin infin 148 infin 1172 1232 infin 28 infin 1102 infin

Station C Consider the following

119888

C34 = 407 minus 320 minus 50 = 37 (407 minus 320 = 87 gt 50)

119888

C36 = 630 minus 320 minus 50 = 260

(630 minus 320 = 310 gt 50)

119888

C54 = 407 minus 556 minus 50 + 1440 = 1241

(407 minus 556 = minus149 lt 50)

119888

C56 = 630 minus 556 minus 50 = 24 (630 minus 556 = 74 gt 50)

(14)

Station D Consider the following

119888

D78 = 296 minus 232 minus 50 = 14 (296 minus 232 = 64 gt 50) (15)

The others are equal toinfin the coefficient matrix (119888119896119894119895) isshowed in Table 3

512 Result The Hungarian algorithm is implemented inMATLAB and the optimal locomotive schedule is as follows

The minimum waiting time of locomotives at stations is

119879119908 = 119885 =

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

119888

119896119894119895119909119896119894119895

= 65 + 1175 + 37 + 1085 + 24 + 792 + 14 + 28

= 3220 (min)

(16)

Theminimal technological time of locomotives at stationsis

119879119870 = 125 times 4 + 50 times 4 = 700 (min) (17)

The running time of locomotives is

119879119905 = 40 + 45 + 40 + 43 + 40 + 43 + 82 + 67

= 400 (min) (18)

The minimum number of locomotives is

119872 =

119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119879119908

1440

=

400 + 700 + 3220

1440

=

4320

1440

= 3

(19)

Mathematical Problems in Engineering 9

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 9 Locomotive schedule in section A-D

Chawu

299 kmDashizhuang

CuipingshanZunhua North

Qianrsquoan North

Jixian West

HouyingLiucun South

Qinhuangdao EastLuannan

Donggang

Caofeidian West

60 km19 km

11km59 km 78 km 67 km

22 km

24km

71km

66km

44 km

Hudong

Figure 10 The simplified network of Da-qin Railway

The coefficient of locomotive requirement is

119870119889 =119872

119899

=

3

82

= 075 (20)

The locomotive schedule can be expressed as an intuitivediagram which is shown in Figure 9 (dotted lines representlocomotives and solid lines represent trains)

Figure 9 shows that there are 3 traction locomotiveswhich are equal to the calculation result to haul trains

513 Result Analysis

(1) Analysis on the Optimization of Locomotive SchedulingFigures 7 and 9 show that there may be a variety of trainconnections So the number of available locomotives andcoefficient of locomotive requirement are uncertain Theoptimization model of heavy-loaded locomotive schedule isestablished and then it can be solved byHungarian algorithmIt can not only make sure of the uniqueness of locomotivetraction tasks but also complete established traction taskswith a minimum number of locomotives

(2) Analysis on the Equilibrium of Locomotive Schedule Theequilibrium degree of locomotive schedule of Figure 9 can beobtained by using (9) which is

119863 =

1

8

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

8

)

2

=

1

8

[(65 minus

3220

8

)

2

+ (1175 minus

3220

8

)

2

+ (37 minus

3220

8

)

2

+ (1085 minus

3220

8

)

2

+ (24 minus

3220

8

)

2

+ (792 minus

3220

8

)

2

+ (14 minus

3220

8

)

2

+ (28 minus

3220

8

)

2

] = 237027

(21)

It shows that the equilibrium degree of locomotive sched-ule is large The main reason is that there are no trains from600 to 1800 on the railwayTherefore the difference betweenthe detention time of locomotives at stations from 1800 to600 the next day and the detention time of locomotives atstations from 600 to 1800 is relatively large If there are trainsfrom 600 to 1800 the equilibrium degree of locomotiveschedule will be reduced and the locomotive schedule will bemore balanced

52 Analysis of Locomotive Schedule for 20 kt CombinedTrains on Da-qin Railway The previous section shows thatthe model of locomotive schedule is feasible Then theeffectiveness of the proposedmodel also needs to be validatedwhich is conducted in this section

521 Line Simplification Da-qin Railway has 26 stations andthe line length is 653 km Qiancao Railway has 5 stations andthe line length is 137 km Jingtang Port-Donggang RailwayLine has 5 stations and the line length is 45 km Someintermediate stations on the railway do not have the technicaloperation so they are not displayed Only the stationswhere locomotive depot or turnaround depot is located areconsidered The simplified network of Da-qin Railway isshown in Figure 10

Hudong station is the station where the locomotive depotis located and theminimal technological time of locomotives

10 Mathematical Problems in Engineering

at stations is 250min Liucun South station Donggang sta-tion Caofeidian West station and Qinhuangdao East stationare the stations where locomotive turnaround depots arelocated The minimal technological time of locomotives atstations for Liucun South station is 220min The minimaltechnological time of locomotives at stations for Donggangstation and Caofeidian West station is 100min The minimaltechnological time of locomotives at stations for Qinhuang-daoEast station is 125minTheminimal technological time ofthe other stations where the locomotives turn back is 50min

There were 100 heavy-loaded trains in the heavy-loaddirection and 104 empty-wagon trains in the reverse directionon December 10 2014

522 Data Processing In reference to the mentionedmethod complete the data processing For the HXDlocomotives there are 328 trains which are hauled by single-locomotive after conversion so the order of the coefficientmatrix is 328 For the SS4 locomotives there are 56 trainswhich are hauled double-locomotive after conversion sothe order of the coefficient matrix is 56 In this paper C isemployed to deal with such huge data

523 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

119888

119896119894119895119909119896119894119895 = 37116 (min)

119885SS4 =13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10409 (min)

(22)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 164 times 250 + 76 times 220 + 14 times 125 + 44 times 100

+ 30 times 50 = 65370 (min)

119879

SS4119870 = 28 times 250 + 28 times 50 = 8400 (min)

(23)

The running times of locomotives are respectively

119879

HXD119905 = 198474 (min)

119879

SS4119905 = 24391 (min)

(24)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

198474 + 65370 + 37116

1440

=

300960

1440

= 209

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

24391 + 8400 + 10409

1440

times 2 =

43200

1440

times 2

= 60

(25)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

209

1602

= 261

119870

SS4119889 =

119872SS4119899

=

30 times 2

562

= 214

(26)

According to the best connection matrix locomotiveconnection scheme for 20 kt combined trains on Da-qinRailway can be obtainedThe locomotive connection schemeof SS4 locomotives is as follows

H16012 997888rarr 52 (single-locomotive) 997888rarr H16016

997888rarr 54 (single-locomotive) 997888rarr H16004

997888rarr H19609 997888rarr H19504 997888rarr H19605

997888rarr H19512 997888rarr 53 (single-locomotive)

997888rarr H16018 997888rarr 56 (single-locomotive)

997888rarr H16002 997888rarr 48 (single-locomotive)

997888rarr H15506 997888rarr 47 (single-locomotive)

997888rarr H16010 997888rarr 19601 997888rarr H19508

997888rarr 46 (single-locomotive) 997888rarr H16008

997888rarr H15501 997888rarr H15504 997888rarr H19607

997888rarr H19516 997888rarr 19615 997888rarr H19514

997888rarr H28501 997888rarr H28510

997888rarr 51 (single-locomotive) 997888rarr H16014

997888rarr H19603 997888rarr H19502 997888rarr H28507

997888rarr H28512 997888rarr 49 (attached locomotive)

997888rarr H15508 997888rarr H28503 997888rarr H28508

997888rarr 50 (single-locomotive) 997888rarr H16012

997888rarr H19611 997888rarr H19506 997888rarr H28511

997888rarr H28502 997888rarr H15503 997888rarr H15502

997888rarr 55 (attached locomotive) 997888rarr H15510

997888rarr H28505 997888rarr H28506

997888rarr 45 (single-locomotive) 997888rarr H16006

997888rarr H19611

H19613 997888rarr H19510 997888rarr H19613

H28504 997888rarr H28509 997888rarr H28504

(27)

Mathematical Problems in Engineering 11

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

Luannan

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

LuannanH16012 H16014 H16016 H16018H16002 H16004 H16006

H16002 H16004 H16006 H16008 H16010 H160112 H16014H16016 H16018

H15501

H15501

H15503

H15503

H15502 H15504 H15506 H15508 H15510

H28502 H28504 H28506 H28508 H28510 H28512

H15502H28502 H28504 H15504 H128506 H28508 H28510H15506 H15508 H28512

H28501 H28505 H28507 H28509 H28511

H28501 H28503 H28505 H28507 H28509 H28511

H19601 H19603 H19615 H19605 H19607 H19609 H19613 H19611

H19516 H19502 H19504 H19506 H19508 H19510 H19514 H19512

H28503

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

TrainsLocomotivesSingle-locomotives or attached locomotives

H16008 H16010

Figure 11 Locomotive working diagram of SS4 locomotives for 20 kt combined trains

Table 4 The daily number of locomotives on Da-qin Railway onDecember 10 2014 [21]

Types oflocomotive

Daily number oflocomotives

Types oflocomotive

Daily number oflocomotives

HXD1 130 SS4 109HXD2 92 mdash mdash

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 11

524 Analysis of Result

(1) Analyze the Number of Available LocomotivesThe numberof available locomotives (the total number of locomotivesafter deducting locomotives inmaintenance and preparation)onDa-qin Railway onDecember 10 2014 is shown in Table 4

Table 4 indicates that the locomotive schedule obtainedby using themodel can save the number ofHXD locomotives130 + 92 minus 209 = 13 and the number of SS4 locomotives109minus 30times 2 = 49 Therefore if only the minimum number oflocomotives is considered the optimization model of heavyhaul locomotive schedule will be optimized

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

328

13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

328

)

2

= 52379

119863SS4 =1

56

13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

56

)

2

= 74044

(28)

The equilibrium degree of the locomotive schedule forHXD locomotives and SS4 locomotives is smaller than theequilibrium degree of that case which shows that this loco-motive schedule is more balanced The equilibrium degreeof SS4 locomotives is larger than the equilibrium degree ofHXD locomotives which shows that the schedule of HXDlocomotives ismore balanced than SS4 locomotivesThe largeequilibrium degree of SS4 locomotives may be due to thesmall detention time of locomotives at stations

53 Research and Analysis of Locomotive Schedule Optimiza-tion for 30 kt Combined Trains on Da-qin Railway On thebasis of the potential train diagram for 30 kt combined trainsthe practical and feasible locomotive connection scheme canbe determined by using the optimization model

In reference to the mentioned method complete the dataprocessing

531 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

119888

119896119894119895119909119896119894119895 = 44838 (min)

119885SS4 =13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10314 (min)

(29)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 203 times 250 + 64 times 220 + 65 times 100 + 74 times 50

= 75030 (min)

119879

SS4119870 = 26 times 250 + 26 times 50 = 7800 (min)

(30)

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

Begin

equal to thematrix order

Mark the least zero elements and remove the other zero elements

The value of marked zero elements is 1 and

the other is 0 in thecounterpart matrix

Cover all of zero elements by the least straight linesEnd

Generating coefficient matrix

Each row and column element minus the minimum value

Is the number ofstraight lines smaller than

the matrix order

Transform coefficient matrix

again

YesNo

Yes

No

Is the number ofindependent zero elements

Figure 8 Flow chart of the Hungarian algorithm

5 Case Study

51 Evaluating the Feasibility of Heavy Haul LocomotiveSchedule Model The model and its solution algorithm needto be validated before applying them to any real problems

511 Data Source and Processing As shown in Figure 4line A-B-C-D is a double-track railway (section A-D forshort) which is specially used to transport coal Station Bis the station where the locomotive depot is located andthe minimal technological time of locomotives at stations is125min Station A station C and station D are the stationswhere locomotive turnaround depots are located and theminimal technological time of locomotives at stations is50min There are 8 trains (4 pairs) which include 2 trains(1 pair) in section A-B 4 trains (2 pairs) in section B-Cand 2 trains (1 pair) in section B-D on the railway If all oftrains are hauled by single-locomotive Figure 6 is the traindiagram

This basic data needs to be processed as follows

(1) Stations Coding For all stations in section A-D it can codestations from any end of the railway and it starts with stationA

(2) Trains Coding The trains in section A-D are sole How-ever it is necessary for each train to code another sole and

Table 2 Trains coding result in section A-D

Train number Train coding Train number Train coding70001 1 73003 570002 2 73004 673001 3 79001 773002 4 79002 8

simple number due to the long and large train number Thecoding results are listed in Table 2

(3) Time Simplification For the departure time and arrivaltime of each train starting at the initial time (1800) of thetrain diagram time of 24 h can be converted to minutes andfor example 1933 can be converted to the 93rd minute

(4) Generating Coefficient Matrix According to (4) thecoefficient matrix 119888119896119894119895 of the detention time of locomotives atstations will be generated by using C The results are listedbelow

Station A Consider the following

119888

A21 = 50 minus 265 minus 50 + 1440 = 1175

(50 minus 265 = minus215 lt 50)

(12)

8 Mathematical Problems in Engineering

Station B Consider the following

119888

B12 = 220 minus 90 minus 125 = 5 (220 minus 90 = 130 gt 125)

119888

B13 = 280 minus 90 minus 125 = 65 (280 minus 90 = 190 gt 125)

119888

B15 = 516 minus 90 minus 125 = 301

(516 minus 90 = 426 gt 125)

119888

B17 = 150 minus 90 minus 125 + 1440 = 1375

(150 minus 90 = 60 lt 125)

119888

B42 = 220 minus 450 minus 125 + 1440 = 1085

(220 minus 450 = minus230 lt 125)

119888

B43 = 280 minus 450 minus 125 + 1440 = 1145

(280 minus 450 = minus170 lt 125)

119888

B45 = 516 minus 450 minus 125 + 1440 = 1381

(516 minus 450 = 66 lt 125516 minus 450 = 66 lt 125)

119888

B47 = 150 minus 450 minus 125 + 1440 = 1015

(150 minus 450 = minus300 lt 125)

119888

B62 = 220 minus 673 minus 125 + 1440 = 862

(220 minus 673 = minus453 lt 125)

119888

B63 = 280 minus 673 minus 125 + 1440 = 922

(280 minus 673 = minus393 lt 125)

119888

B65 = 516 minus 673 minus 125 + 1440 = 1158

(516 minus 673 = minus157 lt 125)

119888

B67 = 150 minus 673 minus 125 + 1440 = 792

(150 minus 673 = minus523 lt 125)

119888

B82 = 220 minus 363 minus 125 + 1440 = 1172

(220 minus 363 = minus143 lt 125)

119888

B83 = 280 minus 363 minus 125 + 1440 = 1232

(280 minus 363 = minus83 lt 125)

119888

B85 = 516 minus 363 minus 125 = 28

(516 minus 363 = 153 gt 125)

119888

B87 = 150 minus 363 minus 125 + 1440 = 1102

(150 minus 90 = minus213 lt 125)

(13)

Table 3 Coefficient matrix of detention time of locomotives atstations

119894

119895

1 2 3 4 5 6 7 81 infin 5 65 infin 301 infin 1375 infin

2 1175 infin infin infin infin infin infin infin

3 infin infin infin 37 infin 260 infin infin

4 infin 1085 1145 infin 1381 infin 1015 infin

5 infin infin infin 1241 infin 24 infin infin

6 infin 862 922 infin 1158 infin 792 infin

7 infin infin infin infin infin infin infin 148 infin 1172 1232 infin 28 infin 1102 infin

Station C Consider the following

119888

C34 = 407 minus 320 minus 50 = 37 (407 minus 320 = 87 gt 50)

119888

C36 = 630 minus 320 minus 50 = 260

(630 minus 320 = 310 gt 50)

119888

C54 = 407 minus 556 minus 50 + 1440 = 1241

(407 minus 556 = minus149 lt 50)

119888

C56 = 630 minus 556 minus 50 = 24 (630 minus 556 = 74 gt 50)

(14)

Station D Consider the following

119888

D78 = 296 minus 232 minus 50 = 14 (296 minus 232 = 64 gt 50) (15)

The others are equal toinfin the coefficient matrix (119888119896119894119895) isshowed in Table 3

512 Result The Hungarian algorithm is implemented inMATLAB and the optimal locomotive schedule is as follows

The minimum waiting time of locomotives at stations is

119879119908 = 119885 =

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

119888

119896119894119895119909119896119894119895

= 65 + 1175 + 37 + 1085 + 24 + 792 + 14 + 28

= 3220 (min)

(16)

Theminimal technological time of locomotives at stationsis

119879119870 = 125 times 4 + 50 times 4 = 700 (min) (17)

The running time of locomotives is

119879119905 = 40 + 45 + 40 + 43 + 40 + 43 + 82 + 67

= 400 (min) (18)

The minimum number of locomotives is

119872 =

119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119879119908

1440

=

400 + 700 + 3220

1440

=

4320

1440

= 3

(19)

Mathematical Problems in Engineering 9

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 9 Locomotive schedule in section A-D

Chawu

299 kmDashizhuang

CuipingshanZunhua North

Qianrsquoan North

Jixian West

HouyingLiucun South

Qinhuangdao EastLuannan

Donggang

Caofeidian West

60 km19 km

11km59 km 78 km 67 km

22 km

24km

71km

66km

44 km

Hudong

Figure 10 The simplified network of Da-qin Railway

The coefficient of locomotive requirement is

119870119889 =119872

119899

=

3

82

= 075 (20)

The locomotive schedule can be expressed as an intuitivediagram which is shown in Figure 9 (dotted lines representlocomotives and solid lines represent trains)

Figure 9 shows that there are 3 traction locomotiveswhich are equal to the calculation result to haul trains

513 Result Analysis

(1) Analysis on the Optimization of Locomotive SchedulingFigures 7 and 9 show that there may be a variety of trainconnections So the number of available locomotives andcoefficient of locomotive requirement are uncertain Theoptimization model of heavy-loaded locomotive schedule isestablished and then it can be solved byHungarian algorithmIt can not only make sure of the uniqueness of locomotivetraction tasks but also complete established traction taskswith a minimum number of locomotives

(2) Analysis on the Equilibrium of Locomotive Schedule Theequilibrium degree of locomotive schedule of Figure 9 can beobtained by using (9) which is

119863 =

1

8

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

8

)

2

=

1

8

[(65 minus

3220

8

)

2

+ (1175 minus

3220

8

)

2

+ (37 minus

3220

8

)

2

+ (1085 minus

3220

8

)

2

+ (24 minus

3220

8

)

2

+ (792 minus

3220

8

)

2

+ (14 minus

3220

8

)

2

+ (28 minus

3220

8

)

2

] = 237027

(21)

It shows that the equilibrium degree of locomotive sched-ule is large The main reason is that there are no trains from600 to 1800 on the railwayTherefore the difference betweenthe detention time of locomotives at stations from 1800 to600 the next day and the detention time of locomotives atstations from 600 to 1800 is relatively large If there are trainsfrom 600 to 1800 the equilibrium degree of locomotiveschedule will be reduced and the locomotive schedule will bemore balanced

52 Analysis of Locomotive Schedule for 20 kt CombinedTrains on Da-qin Railway The previous section shows thatthe model of locomotive schedule is feasible Then theeffectiveness of the proposedmodel also needs to be validatedwhich is conducted in this section

521 Line Simplification Da-qin Railway has 26 stations andthe line length is 653 km Qiancao Railway has 5 stations andthe line length is 137 km Jingtang Port-Donggang RailwayLine has 5 stations and the line length is 45 km Someintermediate stations on the railway do not have the technicaloperation so they are not displayed Only the stationswhere locomotive depot or turnaround depot is located areconsidered The simplified network of Da-qin Railway isshown in Figure 10

Hudong station is the station where the locomotive depotis located and theminimal technological time of locomotives

10 Mathematical Problems in Engineering

at stations is 250min Liucun South station Donggang sta-tion Caofeidian West station and Qinhuangdao East stationare the stations where locomotive turnaround depots arelocated The minimal technological time of locomotives atstations for Liucun South station is 220min The minimaltechnological time of locomotives at stations for Donggangstation and Caofeidian West station is 100min The minimaltechnological time of locomotives at stations for Qinhuang-daoEast station is 125minTheminimal technological time ofthe other stations where the locomotives turn back is 50min

There were 100 heavy-loaded trains in the heavy-loaddirection and 104 empty-wagon trains in the reverse directionon December 10 2014

522 Data Processing In reference to the mentionedmethod complete the data processing For the HXDlocomotives there are 328 trains which are hauled by single-locomotive after conversion so the order of the coefficientmatrix is 328 For the SS4 locomotives there are 56 trainswhich are hauled double-locomotive after conversion sothe order of the coefficient matrix is 56 In this paper C isemployed to deal with such huge data

523 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

119888

119896119894119895119909119896119894119895 = 37116 (min)

119885SS4 =13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10409 (min)

(22)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 164 times 250 + 76 times 220 + 14 times 125 + 44 times 100

+ 30 times 50 = 65370 (min)

119879

SS4119870 = 28 times 250 + 28 times 50 = 8400 (min)

(23)

The running times of locomotives are respectively

119879

HXD119905 = 198474 (min)

119879

SS4119905 = 24391 (min)

(24)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

198474 + 65370 + 37116

1440

=

300960

1440

= 209

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

24391 + 8400 + 10409

1440

times 2 =

43200

1440

times 2

= 60

(25)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

209

1602

= 261

119870

SS4119889 =

119872SS4119899

=

30 times 2

562

= 214

(26)

According to the best connection matrix locomotiveconnection scheme for 20 kt combined trains on Da-qinRailway can be obtainedThe locomotive connection schemeof SS4 locomotives is as follows

H16012 997888rarr 52 (single-locomotive) 997888rarr H16016

997888rarr 54 (single-locomotive) 997888rarr H16004

997888rarr H19609 997888rarr H19504 997888rarr H19605

997888rarr H19512 997888rarr 53 (single-locomotive)

997888rarr H16018 997888rarr 56 (single-locomotive)

997888rarr H16002 997888rarr 48 (single-locomotive)

997888rarr H15506 997888rarr 47 (single-locomotive)

997888rarr H16010 997888rarr 19601 997888rarr H19508

997888rarr 46 (single-locomotive) 997888rarr H16008

997888rarr H15501 997888rarr H15504 997888rarr H19607

997888rarr H19516 997888rarr 19615 997888rarr H19514

997888rarr H28501 997888rarr H28510

997888rarr 51 (single-locomotive) 997888rarr H16014

997888rarr H19603 997888rarr H19502 997888rarr H28507

997888rarr H28512 997888rarr 49 (attached locomotive)

997888rarr H15508 997888rarr H28503 997888rarr H28508

997888rarr 50 (single-locomotive) 997888rarr H16012

997888rarr H19611 997888rarr H19506 997888rarr H28511

997888rarr H28502 997888rarr H15503 997888rarr H15502

997888rarr 55 (attached locomotive) 997888rarr H15510

997888rarr H28505 997888rarr H28506

997888rarr 45 (single-locomotive) 997888rarr H16006

997888rarr H19611

H19613 997888rarr H19510 997888rarr H19613

H28504 997888rarr H28509 997888rarr H28504

(27)

Mathematical Problems in Engineering 11

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

Luannan

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

LuannanH16012 H16014 H16016 H16018H16002 H16004 H16006

H16002 H16004 H16006 H16008 H16010 H160112 H16014H16016 H16018

H15501

H15501

H15503

H15503

H15502 H15504 H15506 H15508 H15510

H28502 H28504 H28506 H28508 H28510 H28512

H15502H28502 H28504 H15504 H128506 H28508 H28510H15506 H15508 H28512

H28501 H28505 H28507 H28509 H28511

H28501 H28503 H28505 H28507 H28509 H28511

H19601 H19603 H19615 H19605 H19607 H19609 H19613 H19611

H19516 H19502 H19504 H19506 H19508 H19510 H19514 H19512

H28503

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

TrainsLocomotivesSingle-locomotives or attached locomotives

H16008 H16010

Figure 11 Locomotive working diagram of SS4 locomotives for 20 kt combined trains

Table 4 The daily number of locomotives on Da-qin Railway onDecember 10 2014 [21]

Types oflocomotive

Daily number oflocomotives

Types oflocomotive

Daily number oflocomotives

HXD1 130 SS4 109HXD2 92 mdash mdash

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 11

524 Analysis of Result

(1) Analyze the Number of Available LocomotivesThe numberof available locomotives (the total number of locomotivesafter deducting locomotives inmaintenance and preparation)onDa-qin Railway onDecember 10 2014 is shown in Table 4

Table 4 indicates that the locomotive schedule obtainedby using themodel can save the number ofHXD locomotives130 + 92 minus 209 = 13 and the number of SS4 locomotives109minus 30times 2 = 49 Therefore if only the minimum number oflocomotives is considered the optimization model of heavyhaul locomotive schedule will be optimized

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

328

13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

328

)

2

= 52379

119863SS4 =1

56

13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

56

)

2

= 74044

(28)

The equilibrium degree of the locomotive schedule forHXD locomotives and SS4 locomotives is smaller than theequilibrium degree of that case which shows that this loco-motive schedule is more balanced The equilibrium degreeof SS4 locomotives is larger than the equilibrium degree ofHXD locomotives which shows that the schedule of HXDlocomotives ismore balanced than SS4 locomotivesThe largeequilibrium degree of SS4 locomotives may be due to thesmall detention time of locomotives at stations

53 Research and Analysis of Locomotive Schedule Optimiza-tion for 30 kt Combined Trains on Da-qin Railway On thebasis of the potential train diagram for 30 kt combined trainsthe practical and feasible locomotive connection scheme canbe determined by using the optimization model

In reference to the mentioned method complete the dataprocessing

531 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

119888

119896119894119895119909119896119894119895 = 44838 (min)

119885SS4 =13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10314 (min)

(29)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 203 times 250 + 64 times 220 + 65 times 100 + 74 times 50

= 75030 (min)

119879

SS4119870 = 26 times 250 + 26 times 50 = 7800 (min)

(30)

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Station B Consider the following

119888

B12 = 220 minus 90 minus 125 = 5 (220 minus 90 = 130 gt 125)

119888

B13 = 280 minus 90 minus 125 = 65 (280 minus 90 = 190 gt 125)

119888

B15 = 516 minus 90 minus 125 = 301

(516 minus 90 = 426 gt 125)

119888

B17 = 150 minus 90 minus 125 + 1440 = 1375

(150 minus 90 = 60 lt 125)

119888

B42 = 220 minus 450 minus 125 + 1440 = 1085

(220 minus 450 = minus230 lt 125)

119888

B43 = 280 minus 450 minus 125 + 1440 = 1145

(280 minus 450 = minus170 lt 125)

119888

B45 = 516 minus 450 minus 125 + 1440 = 1381

(516 minus 450 = 66 lt 125516 minus 450 = 66 lt 125)

119888

B47 = 150 minus 450 minus 125 + 1440 = 1015

(150 minus 450 = minus300 lt 125)

119888

B62 = 220 minus 673 minus 125 + 1440 = 862

(220 minus 673 = minus453 lt 125)

119888

B63 = 280 minus 673 minus 125 + 1440 = 922

(280 minus 673 = minus393 lt 125)

119888

B65 = 516 minus 673 minus 125 + 1440 = 1158

(516 minus 673 = minus157 lt 125)

119888

B67 = 150 minus 673 minus 125 + 1440 = 792

(150 minus 673 = minus523 lt 125)

119888

B82 = 220 minus 363 minus 125 + 1440 = 1172

(220 minus 363 = minus143 lt 125)

119888

B83 = 280 minus 363 minus 125 + 1440 = 1232

(280 minus 363 = minus83 lt 125)

119888

B85 = 516 minus 363 minus 125 = 28

(516 minus 363 = 153 gt 125)

119888

B87 = 150 minus 363 minus 125 + 1440 = 1102

(150 minus 90 = minus213 lt 125)

(13)

Table 3 Coefficient matrix of detention time of locomotives atstations

119894

119895

1 2 3 4 5 6 7 81 infin 5 65 infin 301 infin 1375 infin

2 1175 infin infin infin infin infin infin infin

3 infin infin infin 37 infin 260 infin infin

4 infin 1085 1145 infin 1381 infin 1015 infin

5 infin infin infin 1241 infin 24 infin infin

6 infin 862 922 infin 1158 infin 792 infin

7 infin infin infin infin infin infin infin 148 infin 1172 1232 infin 28 infin 1102 infin

Station C Consider the following

119888

C34 = 407 minus 320 minus 50 = 37 (407 minus 320 = 87 gt 50)

119888

C36 = 630 minus 320 minus 50 = 260

(630 minus 320 = 310 gt 50)

119888

C54 = 407 minus 556 minus 50 + 1440 = 1241

(407 minus 556 = minus149 lt 50)

119888

C56 = 630 minus 556 minus 50 = 24 (630 minus 556 = 74 gt 50)

(14)

Station D Consider the following

119888

D78 = 296 minus 232 minus 50 = 14 (296 minus 232 = 64 gt 50) (15)

The others are equal toinfin the coefficient matrix (119888119896119894119895) isshowed in Table 3

512 Result The Hungarian algorithm is implemented inMATLAB and the optimal locomotive schedule is as follows

The minimum waiting time of locomotives at stations is

119879119908 = 119885 =

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

119888

119896119894119895119909119896119894119895

= 65 + 1175 + 37 + 1085 + 24 + 792 + 14 + 28

= 3220 (min)

(16)

Theminimal technological time of locomotives at stationsis

119879119870 = 125 times 4 + 50 times 4 = 700 (min) (17)

The running time of locomotives is

119879119905 = 40 + 45 + 40 + 43 + 40 + 43 + 82 + 67

= 400 (min) (18)

The minimum number of locomotives is

119872 =

119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119879119908

1440

=

400 + 700 + 3220

1440

=

4320

1440

= 3

(19)

Mathematical Problems in Engineering 9

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 9 Locomotive schedule in section A-D

Chawu

299 kmDashizhuang

CuipingshanZunhua North

Qianrsquoan North

Jixian West

HouyingLiucun South

Qinhuangdao EastLuannan

Donggang

Caofeidian West

60 km19 km

11km59 km 78 km 67 km

22 km

24km

71km

66km

44 km

Hudong

Figure 10 The simplified network of Da-qin Railway

The coefficient of locomotive requirement is

119870119889 =119872

119899

=

3

82

= 075 (20)

The locomotive schedule can be expressed as an intuitivediagram which is shown in Figure 9 (dotted lines representlocomotives and solid lines represent trains)

Figure 9 shows that there are 3 traction locomotiveswhich are equal to the calculation result to haul trains

513 Result Analysis

(1) Analysis on the Optimization of Locomotive SchedulingFigures 7 and 9 show that there may be a variety of trainconnections So the number of available locomotives andcoefficient of locomotive requirement are uncertain Theoptimization model of heavy-loaded locomotive schedule isestablished and then it can be solved byHungarian algorithmIt can not only make sure of the uniqueness of locomotivetraction tasks but also complete established traction taskswith a minimum number of locomotives

(2) Analysis on the Equilibrium of Locomotive Schedule Theequilibrium degree of locomotive schedule of Figure 9 can beobtained by using (9) which is

119863 =

1

8

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

8

)

2

=

1

8

[(65 minus

3220

8

)

2

+ (1175 minus

3220

8

)

2

+ (37 minus

3220

8

)

2

+ (1085 minus

3220

8

)

2

+ (24 minus

3220

8

)

2

+ (792 minus

3220

8

)

2

+ (14 minus

3220

8

)

2

+ (28 minus

3220

8

)

2

] = 237027

(21)

It shows that the equilibrium degree of locomotive sched-ule is large The main reason is that there are no trains from600 to 1800 on the railwayTherefore the difference betweenthe detention time of locomotives at stations from 1800 to600 the next day and the detention time of locomotives atstations from 600 to 1800 is relatively large If there are trainsfrom 600 to 1800 the equilibrium degree of locomotiveschedule will be reduced and the locomotive schedule will bemore balanced

52 Analysis of Locomotive Schedule for 20 kt CombinedTrains on Da-qin Railway The previous section shows thatthe model of locomotive schedule is feasible Then theeffectiveness of the proposedmodel also needs to be validatedwhich is conducted in this section

521 Line Simplification Da-qin Railway has 26 stations andthe line length is 653 km Qiancao Railway has 5 stations andthe line length is 137 km Jingtang Port-Donggang RailwayLine has 5 stations and the line length is 45 km Someintermediate stations on the railway do not have the technicaloperation so they are not displayed Only the stationswhere locomotive depot or turnaround depot is located areconsidered The simplified network of Da-qin Railway isshown in Figure 10

Hudong station is the station where the locomotive depotis located and theminimal technological time of locomotives

10 Mathematical Problems in Engineering

at stations is 250min Liucun South station Donggang sta-tion Caofeidian West station and Qinhuangdao East stationare the stations where locomotive turnaround depots arelocated The minimal technological time of locomotives atstations for Liucun South station is 220min The minimaltechnological time of locomotives at stations for Donggangstation and Caofeidian West station is 100min The minimaltechnological time of locomotives at stations for Qinhuang-daoEast station is 125minTheminimal technological time ofthe other stations where the locomotives turn back is 50min

There were 100 heavy-loaded trains in the heavy-loaddirection and 104 empty-wagon trains in the reverse directionon December 10 2014

522 Data Processing In reference to the mentionedmethod complete the data processing For the HXDlocomotives there are 328 trains which are hauled by single-locomotive after conversion so the order of the coefficientmatrix is 328 For the SS4 locomotives there are 56 trainswhich are hauled double-locomotive after conversion sothe order of the coefficient matrix is 56 In this paper C isemployed to deal with such huge data

523 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

119888

119896119894119895119909119896119894119895 = 37116 (min)

119885SS4 =13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10409 (min)

(22)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 164 times 250 + 76 times 220 + 14 times 125 + 44 times 100

+ 30 times 50 = 65370 (min)

119879

SS4119870 = 28 times 250 + 28 times 50 = 8400 (min)

(23)

The running times of locomotives are respectively

119879

HXD119905 = 198474 (min)

119879

SS4119905 = 24391 (min)

(24)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

198474 + 65370 + 37116

1440

=

300960

1440

= 209

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

24391 + 8400 + 10409

1440

times 2 =

43200

1440

times 2

= 60

(25)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

209

1602

= 261

119870

SS4119889 =

119872SS4119899

=

30 times 2

562

= 214

(26)

According to the best connection matrix locomotiveconnection scheme for 20 kt combined trains on Da-qinRailway can be obtainedThe locomotive connection schemeof SS4 locomotives is as follows

H16012 997888rarr 52 (single-locomotive) 997888rarr H16016

997888rarr 54 (single-locomotive) 997888rarr H16004

997888rarr H19609 997888rarr H19504 997888rarr H19605

997888rarr H19512 997888rarr 53 (single-locomotive)

997888rarr H16018 997888rarr 56 (single-locomotive)

997888rarr H16002 997888rarr 48 (single-locomotive)

997888rarr H15506 997888rarr 47 (single-locomotive)

997888rarr H16010 997888rarr 19601 997888rarr H19508

997888rarr 46 (single-locomotive) 997888rarr H16008

997888rarr H15501 997888rarr H15504 997888rarr H19607

997888rarr H19516 997888rarr 19615 997888rarr H19514

997888rarr H28501 997888rarr H28510

997888rarr 51 (single-locomotive) 997888rarr H16014

997888rarr H19603 997888rarr H19502 997888rarr H28507

997888rarr H28512 997888rarr 49 (attached locomotive)

997888rarr H15508 997888rarr H28503 997888rarr H28508

997888rarr 50 (single-locomotive) 997888rarr H16012

997888rarr H19611 997888rarr H19506 997888rarr H28511

997888rarr H28502 997888rarr H15503 997888rarr H15502

997888rarr 55 (attached locomotive) 997888rarr H15510

997888rarr H28505 997888rarr H28506

997888rarr 45 (single-locomotive) 997888rarr H16006

997888rarr H19611

H19613 997888rarr H19510 997888rarr H19613

H28504 997888rarr H28509 997888rarr H28504

(27)

Mathematical Problems in Engineering 11

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

Luannan

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

LuannanH16012 H16014 H16016 H16018H16002 H16004 H16006

H16002 H16004 H16006 H16008 H16010 H160112 H16014H16016 H16018

H15501

H15501

H15503

H15503

H15502 H15504 H15506 H15508 H15510

H28502 H28504 H28506 H28508 H28510 H28512

H15502H28502 H28504 H15504 H128506 H28508 H28510H15506 H15508 H28512

H28501 H28505 H28507 H28509 H28511

H28501 H28503 H28505 H28507 H28509 H28511

H19601 H19603 H19615 H19605 H19607 H19609 H19613 H19611

H19516 H19502 H19504 H19506 H19508 H19510 H19514 H19512

H28503

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

TrainsLocomotivesSingle-locomotives or attached locomotives

H16008 H16010

Figure 11 Locomotive working diagram of SS4 locomotives for 20 kt combined trains

Table 4 The daily number of locomotives on Da-qin Railway onDecember 10 2014 [21]

Types oflocomotive

Daily number oflocomotives

Types oflocomotive

Daily number oflocomotives

HXD1 130 SS4 109HXD2 92 mdash mdash

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 11

524 Analysis of Result

(1) Analyze the Number of Available LocomotivesThe numberof available locomotives (the total number of locomotivesafter deducting locomotives inmaintenance and preparation)onDa-qin Railway onDecember 10 2014 is shown in Table 4

Table 4 indicates that the locomotive schedule obtainedby using themodel can save the number ofHXD locomotives130 + 92 minus 209 = 13 and the number of SS4 locomotives109minus 30times 2 = 49 Therefore if only the minimum number oflocomotives is considered the optimization model of heavyhaul locomotive schedule will be optimized

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

328

13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

328

)

2

= 52379

119863SS4 =1

56

13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

56

)

2

= 74044

(28)

The equilibrium degree of the locomotive schedule forHXD locomotives and SS4 locomotives is smaller than theequilibrium degree of that case which shows that this loco-motive schedule is more balanced The equilibrium degreeof SS4 locomotives is larger than the equilibrium degree ofHXD locomotives which shows that the schedule of HXDlocomotives ismore balanced than SS4 locomotivesThe largeequilibrium degree of SS4 locomotives may be due to thesmall detention time of locomotives at stations

53 Research and Analysis of Locomotive Schedule Optimiza-tion for 30 kt Combined Trains on Da-qin Railway On thebasis of the potential train diagram for 30 kt combined trainsthe practical and feasible locomotive connection scheme canbe determined by using the optimization model

In reference to the mentioned method complete the dataprocessing

531 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

119888

119896119894119895119909119896119894119895 = 44838 (min)

119885SS4 =13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10314 (min)

(29)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 203 times 250 + 64 times 220 + 65 times 100 + 74 times 50

= 75030 (min)

119879

SS4119870 = 26 times 250 + 26 times 50 = 7800 (min)

(30)

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

18 19 20 21 22 23 0 1 2 3 4 5 6A

B

C

D 2 6

5

7

3 6

6

3

70001

70002

79001 73001 73003

79002

73002 73004

Figure 9 Locomotive schedule in section A-D

Chawu

299 kmDashizhuang

CuipingshanZunhua North

Qianrsquoan North

Jixian West

HouyingLiucun South

Qinhuangdao EastLuannan

Donggang

Caofeidian West

60 km19 km

11km59 km 78 km 67 km

22 km

24km

71km

66km

44 km

Hudong

Figure 10 The simplified network of Da-qin Railway

The coefficient of locomotive requirement is

119870119889 =119872

119899

=

3

82

= 075 (20)

The locomotive schedule can be expressed as an intuitivediagram which is shown in Figure 9 (dotted lines representlocomotives and solid lines represent trains)

Figure 9 shows that there are 3 traction locomotiveswhich are equal to the calculation result to haul trains

513 Result Analysis

(1) Analysis on the Optimization of Locomotive SchedulingFigures 7 and 9 show that there may be a variety of trainconnections So the number of available locomotives andcoefficient of locomotive requirement are uncertain Theoptimization model of heavy-loaded locomotive schedule isestablished and then it can be solved byHungarian algorithmIt can not only make sure of the uniqueness of locomotivetraction tasks but also complete established traction taskswith a minimum number of locomotives

(2) Analysis on the Equilibrium of Locomotive Schedule Theequilibrium degree of locomotive schedule of Figure 9 can beobtained by using (9) which is

119863 =

1

8

4

sum

119896=1

8

sum

119894=1

8

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

8

)

2

=

1

8

[(65 minus

3220

8

)

2

+ (1175 minus

3220

8

)

2

+ (37 minus

3220

8

)

2

+ (1085 minus

3220

8

)

2

+ (24 minus

3220

8

)

2

+ (792 minus

3220

8

)

2

+ (14 minus

3220

8

)

2

+ (28 minus

3220

8

)

2

] = 237027

(21)

It shows that the equilibrium degree of locomotive sched-ule is large The main reason is that there are no trains from600 to 1800 on the railwayTherefore the difference betweenthe detention time of locomotives at stations from 1800 to600 the next day and the detention time of locomotives atstations from 600 to 1800 is relatively large If there are trainsfrom 600 to 1800 the equilibrium degree of locomotiveschedule will be reduced and the locomotive schedule will bemore balanced

52 Analysis of Locomotive Schedule for 20 kt CombinedTrains on Da-qin Railway The previous section shows thatthe model of locomotive schedule is feasible Then theeffectiveness of the proposedmodel also needs to be validatedwhich is conducted in this section

521 Line Simplification Da-qin Railway has 26 stations andthe line length is 653 km Qiancao Railway has 5 stations andthe line length is 137 km Jingtang Port-Donggang RailwayLine has 5 stations and the line length is 45 km Someintermediate stations on the railway do not have the technicaloperation so they are not displayed Only the stationswhere locomotive depot or turnaround depot is located areconsidered The simplified network of Da-qin Railway isshown in Figure 10

Hudong station is the station where the locomotive depotis located and theminimal technological time of locomotives

10 Mathematical Problems in Engineering

at stations is 250min Liucun South station Donggang sta-tion Caofeidian West station and Qinhuangdao East stationare the stations where locomotive turnaround depots arelocated The minimal technological time of locomotives atstations for Liucun South station is 220min The minimaltechnological time of locomotives at stations for Donggangstation and Caofeidian West station is 100min The minimaltechnological time of locomotives at stations for Qinhuang-daoEast station is 125minTheminimal technological time ofthe other stations where the locomotives turn back is 50min

There were 100 heavy-loaded trains in the heavy-loaddirection and 104 empty-wagon trains in the reverse directionon December 10 2014

522 Data Processing In reference to the mentionedmethod complete the data processing For the HXDlocomotives there are 328 trains which are hauled by single-locomotive after conversion so the order of the coefficientmatrix is 328 For the SS4 locomotives there are 56 trainswhich are hauled double-locomotive after conversion sothe order of the coefficient matrix is 56 In this paper C isemployed to deal with such huge data

523 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

119888

119896119894119895119909119896119894119895 = 37116 (min)

119885SS4 =13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10409 (min)

(22)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 164 times 250 + 76 times 220 + 14 times 125 + 44 times 100

+ 30 times 50 = 65370 (min)

119879

SS4119870 = 28 times 250 + 28 times 50 = 8400 (min)

(23)

The running times of locomotives are respectively

119879

HXD119905 = 198474 (min)

119879

SS4119905 = 24391 (min)

(24)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

198474 + 65370 + 37116

1440

=

300960

1440

= 209

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

24391 + 8400 + 10409

1440

times 2 =

43200

1440

times 2

= 60

(25)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

209

1602

= 261

119870

SS4119889 =

119872SS4119899

=

30 times 2

562

= 214

(26)

According to the best connection matrix locomotiveconnection scheme for 20 kt combined trains on Da-qinRailway can be obtainedThe locomotive connection schemeof SS4 locomotives is as follows

H16012 997888rarr 52 (single-locomotive) 997888rarr H16016

997888rarr 54 (single-locomotive) 997888rarr H16004

997888rarr H19609 997888rarr H19504 997888rarr H19605

997888rarr H19512 997888rarr 53 (single-locomotive)

997888rarr H16018 997888rarr 56 (single-locomotive)

997888rarr H16002 997888rarr 48 (single-locomotive)

997888rarr H15506 997888rarr 47 (single-locomotive)

997888rarr H16010 997888rarr 19601 997888rarr H19508

997888rarr 46 (single-locomotive) 997888rarr H16008

997888rarr H15501 997888rarr H15504 997888rarr H19607

997888rarr H19516 997888rarr 19615 997888rarr H19514

997888rarr H28501 997888rarr H28510

997888rarr 51 (single-locomotive) 997888rarr H16014

997888rarr H19603 997888rarr H19502 997888rarr H28507

997888rarr H28512 997888rarr 49 (attached locomotive)

997888rarr H15508 997888rarr H28503 997888rarr H28508

997888rarr 50 (single-locomotive) 997888rarr H16012

997888rarr H19611 997888rarr H19506 997888rarr H28511

997888rarr H28502 997888rarr H15503 997888rarr H15502

997888rarr 55 (attached locomotive) 997888rarr H15510

997888rarr H28505 997888rarr H28506

997888rarr 45 (single-locomotive) 997888rarr H16006

997888rarr H19611

H19613 997888rarr H19510 997888rarr H19613

H28504 997888rarr H28509 997888rarr H28504

(27)

Mathematical Problems in Engineering 11

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

Luannan

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

LuannanH16012 H16014 H16016 H16018H16002 H16004 H16006

H16002 H16004 H16006 H16008 H16010 H160112 H16014H16016 H16018

H15501

H15501

H15503

H15503

H15502 H15504 H15506 H15508 H15510

H28502 H28504 H28506 H28508 H28510 H28512

H15502H28502 H28504 H15504 H128506 H28508 H28510H15506 H15508 H28512

H28501 H28505 H28507 H28509 H28511

H28501 H28503 H28505 H28507 H28509 H28511

H19601 H19603 H19615 H19605 H19607 H19609 H19613 H19611

H19516 H19502 H19504 H19506 H19508 H19510 H19514 H19512

H28503

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

TrainsLocomotivesSingle-locomotives or attached locomotives

H16008 H16010

Figure 11 Locomotive working diagram of SS4 locomotives for 20 kt combined trains

Table 4 The daily number of locomotives on Da-qin Railway onDecember 10 2014 [21]

Types oflocomotive

Daily number oflocomotives

Types oflocomotive

Daily number oflocomotives

HXD1 130 SS4 109HXD2 92 mdash mdash

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 11

524 Analysis of Result

(1) Analyze the Number of Available LocomotivesThe numberof available locomotives (the total number of locomotivesafter deducting locomotives inmaintenance and preparation)onDa-qin Railway onDecember 10 2014 is shown in Table 4

Table 4 indicates that the locomotive schedule obtainedby using themodel can save the number ofHXD locomotives130 + 92 minus 209 = 13 and the number of SS4 locomotives109minus 30times 2 = 49 Therefore if only the minimum number oflocomotives is considered the optimization model of heavyhaul locomotive schedule will be optimized

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

328

13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

328

)

2

= 52379

119863SS4 =1

56

13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

56

)

2

= 74044

(28)

The equilibrium degree of the locomotive schedule forHXD locomotives and SS4 locomotives is smaller than theequilibrium degree of that case which shows that this loco-motive schedule is more balanced The equilibrium degreeof SS4 locomotives is larger than the equilibrium degree ofHXD locomotives which shows that the schedule of HXDlocomotives ismore balanced than SS4 locomotivesThe largeequilibrium degree of SS4 locomotives may be due to thesmall detention time of locomotives at stations

53 Research and Analysis of Locomotive Schedule Optimiza-tion for 30 kt Combined Trains on Da-qin Railway On thebasis of the potential train diagram for 30 kt combined trainsthe practical and feasible locomotive connection scheme canbe determined by using the optimization model

In reference to the mentioned method complete the dataprocessing

531 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

119888

119896119894119895119909119896119894119895 = 44838 (min)

119885SS4 =13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10314 (min)

(29)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 203 times 250 + 64 times 220 + 65 times 100 + 74 times 50

= 75030 (min)

119879

SS4119870 = 26 times 250 + 26 times 50 = 7800 (min)

(30)

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

at stations is 250min Liucun South station Donggang sta-tion Caofeidian West station and Qinhuangdao East stationare the stations where locomotive turnaround depots arelocated The minimal technological time of locomotives atstations for Liucun South station is 220min The minimaltechnological time of locomotives at stations for Donggangstation and Caofeidian West station is 100min The minimaltechnological time of locomotives at stations for Qinhuang-daoEast station is 125minTheminimal technological time ofthe other stations where the locomotives turn back is 50min

There were 100 heavy-loaded trains in the heavy-loaddirection and 104 empty-wagon trains in the reverse directionon December 10 2014

522 Data Processing In reference to the mentionedmethod complete the data processing For the HXDlocomotives there are 328 trains which are hauled by single-locomotive after conversion so the order of the coefficientmatrix is 328 For the SS4 locomotives there are 56 trainswhich are hauled double-locomotive after conversion sothe order of the coefficient matrix is 56 In this paper C isemployed to deal with such huge data

523 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

119888

119896119894119895119909119896119894119895 = 37116 (min)

119885SS4 =13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10409 (min)

(22)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 164 times 250 + 76 times 220 + 14 times 125 + 44 times 100

+ 30 times 50 = 65370 (min)

119879

SS4119870 = 28 times 250 + 28 times 50 = 8400 (min)

(23)

The running times of locomotives are respectively

119879

HXD119905 = 198474 (min)

119879

SS4119905 = 24391 (min)

(24)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

198474 + 65370 + 37116

1440

=

300960

1440

= 209

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

24391 + 8400 + 10409

1440

times 2 =

43200

1440

times 2

= 60

(25)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

209

1602

= 261

119870

SS4119889 =

119872SS4119899

=

30 times 2

562

= 214

(26)

According to the best connection matrix locomotiveconnection scheme for 20 kt combined trains on Da-qinRailway can be obtainedThe locomotive connection schemeof SS4 locomotives is as follows

H16012 997888rarr 52 (single-locomotive) 997888rarr H16016

997888rarr 54 (single-locomotive) 997888rarr H16004

997888rarr H19609 997888rarr H19504 997888rarr H19605

997888rarr H19512 997888rarr 53 (single-locomotive)

997888rarr H16018 997888rarr 56 (single-locomotive)

997888rarr H16002 997888rarr 48 (single-locomotive)

997888rarr H15506 997888rarr 47 (single-locomotive)

997888rarr H16010 997888rarr 19601 997888rarr H19508

997888rarr 46 (single-locomotive) 997888rarr H16008

997888rarr H15501 997888rarr H15504 997888rarr H19607

997888rarr H19516 997888rarr 19615 997888rarr H19514

997888rarr H28501 997888rarr H28510

997888rarr 51 (single-locomotive) 997888rarr H16014

997888rarr H19603 997888rarr H19502 997888rarr H28507

997888rarr H28512 997888rarr 49 (attached locomotive)

997888rarr H15508 997888rarr H28503 997888rarr H28508

997888rarr 50 (single-locomotive) 997888rarr H16012

997888rarr H19611 997888rarr H19506 997888rarr H28511

997888rarr H28502 997888rarr H15503 997888rarr H15502

997888rarr 55 (attached locomotive) 997888rarr H15510

997888rarr H28505 997888rarr H28506

997888rarr 45 (single-locomotive) 997888rarr H16006

997888rarr H19611

H19613 997888rarr H19510 997888rarr H19613

H28504 997888rarr H28509 997888rarr H28504

(27)

Mathematical Problems in Engineering 11

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

Luannan

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

LuannanH16012 H16014 H16016 H16018H16002 H16004 H16006

H16002 H16004 H16006 H16008 H16010 H160112 H16014H16016 H16018

H15501

H15501

H15503

H15503

H15502 H15504 H15506 H15508 H15510

H28502 H28504 H28506 H28508 H28510 H28512

H15502H28502 H28504 H15504 H128506 H28508 H28510H15506 H15508 H28512

H28501 H28505 H28507 H28509 H28511

H28501 H28503 H28505 H28507 H28509 H28511

H19601 H19603 H19615 H19605 H19607 H19609 H19613 H19611

H19516 H19502 H19504 H19506 H19508 H19510 H19514 H19512

H28503

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

TrainsLocomotivesSingle-locomotives or attached locomotives

H16008 H16010

Figure 11 Locomotive working diagram of SS4 locomotives for 20 kt combined trains

Table 4 The daily number of locomotives on Da-qin Railway onDecember 10 2014 [21]

Types oflocomotive

Daily number oflocomotives

Types oflocomotive

Daily number oflocomotives

HXD1 130 SS4 109HXD2 92 mdash mdash

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 11

524 Analysis of Result

(1) Analyze the Number of Available LocomotivesThe numberof available locomotives (the total number of locomotivesafter deducting locomotives inmaintenance and preparation)onDa-qin Railway onDecember 10 2014 is shown in Table 4

Table 4 indicates that the locomotive schedule obtainedby using themodel can save the number ofHXD locomotives130 + 92 minus 209 = 13 and the number of SS4 locomotives109minus 30times 2 = 49 Therefore if only the minimum number oflocomotives is considered the optimization model of heavyhaul locomotive schedule will be optimized

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

328

13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

328

)

2

= 52379

119863SS4 =1

56

13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

56

)

2

= 74044

(28)

The equilibrium degree of the locomotive schedule forHXD locomotives and SS4 locomotives is smaller than theequilibrium degree of that case which shows that this loco-motive schedule is more balanced The equilibrium degreeof SS4 locomotives is larger than the equilibrium degree ofHXD locomotives which shows that the schedule of HXDlocomotives ismore balanced than SS4 locomotivesThe largeequilibrium degree of SS4 locomotives may be due to thesmall detention time of locomotives at stations

53 Research and Analysis of Locomotive Schedule Optimiza-tion for 30 kt Combined Trains on Da-qin Railway On thebasis of the potential train diagram for 30 kt combined trainsthe practical and feasible locomotive connection scheme canbe determined by using the optimization model

In reference to the mentioned method complete the dataprocessing

531 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

119888

119896119894119895119909119896119894119895 = 44838 (min)

119885SS4 =13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10314 (min)

(29)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 203 times 250 + 64 times 220 + 65 times 100 + 74 times 50

= 75030 (min)

119879

SS4119870 = 26 times 250 + 26 times 50 = 7800 (min)

(30)

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

Luannan

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

LuannanH16012 H16014 H16016 H16018H16002 H16004 H16006

H16002 H16004 H16006 H16008 H16010 H160112 H16014H16016 H16018

H15501

H15501

H15503

H15503

H15502 H15504 H15506 H15508 H15510

H28502 H28504 H28506 H28508 H28510 H28512

H15502H28502 H28504 H15504 H128506 H28508 H28510H15506 H15508 H28512

H28501 H28505 H28507 H28509 H28511

H28501 H28503 H28505 H28507 H28509 H28511

H19601 H19603 H19615 H19605 H19607 H19609 H19613 H19611

H19516 H19502 H19504 H19506 H19508 H19510 H19514 H19512

H28503

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

TrainsLocomotivesSingle-locomotives or attached locomotives

H16008 H16010

Figure 11 Locomotive working diagram of SS4 locomotives for 20 kt combined trains

Table 4 The daily number of locomotives on Da-qin Railway onDecember 10 2014 [21]

Types oflocomotive

Daily number oflocomotives

Types oflocomotive

Daily number oflocomotives

HXD1 130 SS4 109HXD2 92 mdash mdash

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 11

524 Analysis of Result

(1) Analyze the Number of Available LocomotivesThe numberof available locomotives (the total number of locomotivesafter deducting locomotives inmaintenance and preparation)onDa-qin Railway onDecember 10 2014 is shown in Table 4

Table 4 indicates that the locomotive schedule obtainedby using themodel can save the number ofHXD locomotives130 + 92 minus 209 = 13 and the number of SS4 locomotives109minus 30times 2 = 49 Therefore if only the minimum number oflocomotives is considered the optimization model of heavyhaul locomotive schedule will be optimized

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

328

13

sum

119896=1

328

sum

119894=1

328

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

328

)

2

= 52379

119863SS4 =1

56

13

sum

119896=1

56

sum

119894=1

56

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

56

)

2

= 74044

(28)

The equilibrium degree of the locomotive schedule forHXD locomotives and SS4 locomotives is smaller than theequilibrium degree of that case which shows that this loco-motive schedule is more balanced The equilibrium degreeof SS4 locomotives is larger than the equilibrium degree ofHXD locomotives which shows that the schedule of HXDlocomotives ismore balanced than SS4 locomotivesThe largeequilibrium degree of SS4 locomotives may be due to thesmall detention time of locomotives at stations

53 Research and Analysis of Locomotive Schedule Optimiza-tion for 30 kt Combined Trains on Da-qin Railway On thebasis of the potential train diagram for 30 kt combined trainsthe practical and feasible locomotive connection scheme canbe determined by using the optimization model

In reference to the mentioned method complete the dataprocessing

531 Result The results are illustrated as followsThe minimum waiting times of locomotives at stations

are respectively

119885HXD =13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

119888

119896119894119895119909119896119894119895 = 44838 (min)

119885SS4 =13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

119888

119896119894119895119909119896119894119895 = 10314 (min)

(29)

The minimal technological times of locomotives at sta-tions are respectively

119879

HXD119870 = 203 times 250 + 64 times 220 + 65 times 100 + 74 times 50

= 75030 (min)

119879

SS4119870 = 26 times 250 + 26 times 50 = 7800 (min)

(30)

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

Table 5 The average number of locomotives in Da-qin Railway in 2014

Types of locomotive The possessive quantity oflocomotives

The locomotive number of20 kt combined trains

The locomotive number of30 kt combined trains

HXD 251 209 251SS4 105 60 58

The running times of locomotives are respectively

119879

HXD119905 = 241572 (min)

119879

SS4119905 = 23646 (min)

(31)

According to that data the minimum numbers of loco-motives are respectively

119872HXD =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

241572 + 75030 + 44838

1440

=

361440

1440

= 251

119872SS4 =119879119905 + 119879

1440

=

119879119905 + 119879119870 + 119885

1440

=

23646 + 7800 + 10314

1440

=

41760

1440

= 29

(32)

So the coefficients of locomotive requirement are respec-tively

119870

HXD119889 =

119872HXD119899

=

251

59 + 3 + 20

= 306

119870

SS4119889 =

119872SS4119899

=

29 times 2

522

= 223

(33)

532 The Locomotive Connection Scheme and the LocomotiveSchedule According to the best connection matrix obtainedlocomotive connection scheme for 30 kt combined train onDa-qin Railway can be obtained The locomotive connectionscheme of SS4 locomotives is as follows

H16004 997888rarr H28503 997888rarr H28508 997888rarr H17215

997888rarr H17208 997888rarr H17203 997888rarr H17214

997888rarr H28507 997888rarr H28512 997888rarr H28501

997888rarr H28506 997888rarr H28515 997888rarr H28504

997888rarr H15505 997888rarr H15502 997888rarr H19605

997888rarr H19502 997888rarr H17209 997888rarr H17202

997888rarr H17211 997888rarr H17206 997888rarr H17213

997888rarr H17210 997888rarr H15503 997888rarr H15506

997888rarr H28505 997888rarr H28510 997888rarr H17201

997888rarr H17212 997888rarr H17205 997888rarr H17216

997888rarr H28509 997888rarr H28514 997888rarr H16003

997888rarr H16008 997888rarr H28511 997888rarr H28516

997888rarr H17207 997888rarr H17204 997888rarr H28513

997888rarr H28502 997888rarr H16005 997888rarr H16002

997888rarr H16007 997888rarr H16004

H19601 997888rarr H19504 997888rarr H15501 997888rarr H15504

997888rarr H16001 997888rarr H16006 997888rarr H19603

997888rarr H19506 997888rarr H19601(34)

The locomotive connection scheme can be represented asa locomotive working diagram which is shown in Figure 12

533 Analysis of Result

(1) Analyze the Number of Available Locomotives The dailyaverage number of available locomotives in 2014 can betreated as the daily number of locomotives which is shownin Table 5

Analyzing the data in Table 5 it is obvious that the loco-motive schedule just meets the needs of HXD locomotivesand saves the number of SS4 locomotives 105 minus 29 times 2 = 47Therefore for the given train diagram for 30 kt combinedtrains all trains will be operated smoothly

(2) Analyze the Equilibrium of Locomotive ScheduleThe equi-librium degree of locomotive schedule of HXD locomotivesand SS4 locomotives is respectively

119863HXD =1

406

13

sum

119896=1

406

sum

119894=1

406

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

406

)

2

= 40812

119863SS4 =1

52

13

sum

119896=1

52

sum

119894=1

52

sum

119895=1

(119888

119896119894119895119909119896119894119895 minus

119885

52

)

2

= 47201

(35)

Compared to the optimal locomotive schedule of thecurrent diagram the equilibrium degree of the potentiallocomotive schedule is smaller It shows that the potentiallocomotive schedule is more balanced The equilibriumdegree of SS4 locomotives is larger than the equilibriumdegree of HXD locomotives which shows that the scheduleof HXD locomotives is more balanced than that of SS4locomotives

6 Conclusions

The rationality of locomotive schedule has a direct impacton the transportation efficiency and benefit Therefore it is

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 13

Hudong

Chawu

DashizhuangJixian West

Cuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun South

Qianrsquoan North

Luannan

Donggang

18 19 20 2221 43224 123 5 6 7 1098 11 12 1413 171615 18

18 19 20 2221 224 123 43 5 6 7 1098 11 12 1413 171615 18

Hudong

Chawu

DashizhuangJixian WestCuipingshan

Zunhua North

Qianrsquoan North

HouyingLiucun SouthQianrsquoan North

Luannan

Donggang

H19601 H19603 H19605H28501 H28503 H28505 H28507 H28509 H28511 H28513 H28515H15501 H15503 H15505

H28502 H28504 H28506 H28508 H28510 H28512 H28514 H28516

H15502 H15504

H17201 H17203 H17205 H17207 H17209 H17211 H17213 H17215

H17202 H17204 H17206 H17208 H17210

H16001 H16003 H16005 H16007

H16004 H16006

H19502 H19504 H19506

H15506

H17212 H17214 H17216

H16002 H16008

Trains

Locomotives

Figure 12 Locomotive working diagram of SS4 locomotives for 30 kt combined trains

necessary to study how to improve the locomotive schedulingefficiency

The following conclusions can be drawn on the compar-isons of the locomotive schedule for 30 kt combined trainsof the potential train diagram and the optimized locomotiveschedule for 20 kt combined trains of the current traindiagram

(1) For the given feasible train diagram for 30 kt com-bined trains on Da-qin Railway all trains will beoperated smoothly The HXD locomotives just meetthe needs while the remaining number of SS4 loco-motives is 44 Therefore in the actual transportationoperations it needs to strengthen monitoring andmaintenance of HXD locomotives and for SS4 loco-motives It is an obvious effective way that increasesthe quantities of locomotives for operating trains toreduce the equilibrium degree of locomotive sched-ule

(2) Increasing the minimal technological time of loco-motives at stations can reduce the equilibrium degreeof the locomotive schedule and make locomotiveschedule more balanced

(3) Strengthen reconnection between different locomo-tive types At present 15 kt combined trains are hauledby two HXD locomotives The hauling weight ofone HXD locomotive is 10 kilotons so there will besome waste of hauling weight The hauling weight ofone SS4 locomotive is 5 kilotons Therefore if thereconnection operation and synchronization controlbetween HXD locomotives and SS4 locomotives canbe realized by LOCOTROL technology one 15 kt

combined train will be successfully hauled by oneHXD locomotive and one SS4 locomotive which willrelease the hauling weight and make the transporta-tion tasks complete more smoothly

Conflict of Interests

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

Acknowledgments

The material in this paper is based on research supported byldquoThe Fundamental Research Funds for the Central Universi-tiesrdquo under Grant no 2015JBM057 ldquoResearch on Assessmentand Improvement Train Schedule for High Speed RailwayBased on Multi-Resource Constraintrdquo and China RailwayCorporation under Grant no 2014X001-B ldquoStudy on Trans-portation Scheduling for 30 kt Combined Train in Da-qinRailwayrdquo The real data in this paper is based on researchsupported byTaiyuanRailwayAdministrationWewould liketo extendmy sincere gratitude to ZhitongHuang and PeihengLi for their careful review

References

[1] XWu ldquoStudy on traffic organization systemofDa-qin RailwayrdquoChinese Railway vol 6 pp 11ndash15 2009

[2] National Bureau of Statistics of the Peoplersquos Republic of ChinaChina Statistical Yearbook China Statistics Press 2006ndash2015

[3] Z H Cheng F X Li S Wu and P Gao ldquoApplication ofreliability centered maintenance in railway locomotivemdasha case

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 Mathematical Problems in Engineering

studyrdquo in Proceedings of the 1st International Conference onMaintenance Engineering pp 269ndash275 July 2006

[4] F Y Zheng and Q C Wei ldquoHeavy-axle load mechanical effectson railway tracksrdquo in Proceedings of the 2nd InternationalConference on Railway Engineering NewTechnologies of RailwayEngineering (ICRE rsquo12) pp 233ndash236 Beijing China July 2012

[5] W H Chang Research on the Mechanical Properties of 30t Axle Heavy Haul Railway Track Structure Beijing JiaotongUniversity 2009

[6] XMu C L Yang andM Li ldquoEnlightenment of foreign railwayheavy haul transport to the development of Chinarsquos railwayfreight transportrdquo Railway Economics Research vol 1 2013

[7] T Raviv and M Kaspi ldquoThe locomotive fleet fueling problemrdquoOperations Research Letters vol 40 no 1 pp 39ndash45 2012

[8] V P Kumar and M Bierlaire ldquoOptimizing fueling decisions forlocomotives in railroad networksrdquo Transportation Science vol49 no 1 pp 149ndash159 2015

[9] B Ji X M Diao and S X Li ldquoStudy on intercommunicationbetween locomotives for heavy haul train on Da-qin linerdquoRailway Locomotive amp Car vol 30 pp 12ndash14 2010

[10] D Ruppert G Hull B Green R Golden G Binns and MLatino ldquoRoot cause analysis increases locomotive reliabilityat AMTRAKrdquo in Proceedings of the ASMEIEEE Joint RailConference and the ASME Internal Combustion Engine DivisionSpring Technical Conference pp 305ndash310 Pueblo Colo USAMarch 2007

[11] C-C Kuo and G M Nicholls ldquoA mathematical modelingapproach to improving locomotive utilization at a freightrailroadrdquo Omega vol 35 no 5 pp 472ndash485 2007

[12] M Aronsson P Kreuger and J Gjerdrum ldquoAn efficient MIPmodel for locomotive routing and schedulingrdquo in Proceedingsof the 12th International Conference on Computer System Designand Operation in Railways and Other Transit Systems vol 114pp 963ndash973 Beijing China September 2010

[13] B Vaidyanathan R K Ahuja and J B Orlin ldquoThe locomotiverouting problemrdquoTransportation Science vol 42 no 4 pp 492ndash507 2008

[14] K Ghoseiri and S F Ghannadpour ldquoA hybrid genetic algorithmfor multi-depot homogenous locomotive assignment with timewindowsrdquo Applied Soft Computing Journal vol 10 no 1 pp 53ndash65 2010

[15] S Kasalica D Mandic and V Vukadinovic ldquoLocomotiveassignment optimization including train delaysrdquo PROMETmdashTrafficampTransportation vol 25 no 5 pp 421ndash429 2013

[16] S Rouillon G Desaulniers and F Soumis ldquoAn extendedbranch-and-bound method for locomotive assignmentrdquo Trans-portation Research Part B Methodological vol 40 no 5 pp404ndash423 2006

[17] D Teichmann M Dorda K Golc and H Bınova ldquoLocomotiveassignment problemwith heterogeneous vehicle fleet and hiringexternal locomotivesrdquo Mathematical Problems in Engineeringvol 2015 Article ID 583909 7 pages 2015

[18] C B Li C Ning Q Q Guo and J Cheng ldquoAn improved antcolony algorithm for making locomotive working diagramrdquo inProceedings of the 2nd International Conference ofModelling andSimulation pp 63ndash68 Manchester UK 2009

[19] F S Zhu C M Gao J P Zhang and B Ji ldquoOptimizingplan discussion on the Da-qin railway heavy-loaded trainslocomotive routingrdquo Railway Locomotive amp Car vol 29 pp 4ndash10 2001

[20] H Yang Railway Transport Organization China Railway Pub-lishing House Beijing China 3rd edition 2006

[21] TaiYuan RailWay Administration Technical Information ofTrain Schedule TaiYuan RailWay Administration 2012

[22] C J Guo andDY Lei ldquoModel ofwagonsrsquo placing-in and taking-out problem in a railway station and its Heuristic algorithmrdquoMathematical Problems in Engineering vol 2014 Article ID493809 8 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of