Scheduling of Train Driver in Tiwan Rail Companyd

8/11/2019 Scheduling of Train Driver in Tiwan Rail Companyd

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SCHEDULING OF TRAIN DRIVER FOR TAIWAN RAILWAY ADMINISTRATION

Chi-Kang LEE Chao-Hui CHEN

Professor Instructor

Department of Transportation and Department of Traffic and Transportation

Communication Management Engineering and Management

National Cheng Kung University Feng Chia University

1, University Rd., Tainan, 70101 100, Wenhwa Rd., Seatwen Taichung,

Taiwan, R.O.C. 407 Taiwan, R.O.C.

Fax: +886-6-275 3882 Fax: +886-4-24520678

Email: [email protected] Email:[email protected]

ABSTRACT

The paper presents the experience of two approaches in solving driver-scheduling problem for

a depot at Taiwan Railway Administration. The driver-scheduling problem consists of

generating feasible duties, selecting a schedule of duties for the depot, and circulating the

duties in the schedule into a roster for each driver. By using mathematical programming

approach, the set covering problem and the constrained traveling salesman problem are solved

for the pairing and rostering problems respectively. Comparing to TRAs practical solution,

we find a schedule with less duties, but a roster with a little longer cycle length. By using

Genetic algorithm approach, we have a very flexible process to deal with all kinds of rules

and objectives in the pairing and rostering problems. In the study, we consider multiple

objectives for both pairing and rostering problems, and obtain very good solutions.

Key Words: Crew scheduling, Railway transport, Genetic algorithm, Mathematical

Programming.

1. INTRODUCTION

1.1 The Problem

In railway practice, a sequential decision process is widely used for pairing and rostering train

drivers. As illustrated in Figure 1, it includes1.partitioning vehicle blocks into drivers trips

for each depot, 2.pairing trips and generating feasible and potential duties, 3.selecting one

efficient schedule of duties covered the depots daily trips, and 4.circulating of duties and

selecting one efficient roster for a drivers monthly work plan at the depot. An example duty

is shown in Figure 2, where a duty starts and ends at a specific depot, and it total lengthconsists of driving time, general work time, and rest time. The general work time includes

Journal of the Eastern Asia Society for Transportation Studies, Vol.5, October, 2003

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time for engine warm-up, transport time from (to) depot to (from) starting (terminal) station,

time for hand-over or take-over a driving trip, and so on. In TRA, there are compensations

for driving and general works, and the ratio of the compensation for driving time to that for

general work time is 4 to 3. A piece of driving work is a trip or a piece of consecutive

driving trips. A piece of consecutive driving trips is a set of continuous driving trips, where

there is only a small break between any two consecutive trips. A driver may take the train

driving by another diver in a duty, and it is called a deadhead trip for general work. A daily

schedule of duties is a table listed all duties covering all daily trips for the depot. For

example, by using train diagram, Figure 3 shows all trips for the depot; and Figure 4 shows a

set of duties, where each colored line is a feasible duty. There are many criteria for a

feasible duty, such as the total length for a duty, the total length for driving time, and so on.

Each criterion may have different rules for different situations. For example, the definition

of a piece of consecutive trip is different for early trips, daily trips, and night trips. Since alltrips for the depot, illustrated in Figure 3, are covered by the duties shown in Figure 4, the set

of duties is a feasible schedule for the depot. Moreover, Figure 5 shows the schedule of

driving duties for the depot, where five drivers are required for the schedule. Cyclic

assignment is widely used in intercity railway, and a roster for a driver. As illustrated in

Figure 6, it is a cycle of the required duties in the schedule. There are many criteria for

consecutive duties in a roster. For example, the rest time between the two duties should be

longer than the work time of the previous duty. Moreover, each criterion may have different

rules for different situations.

Figure 1: Practical process of driver scheduling

Partitioning vehicle blocks into

driver trips for each depotStep 1

Pairing trips and generating

feasible and potential duties for

the depot

Step 2

Selecting one efficient schedule

of duties for the depotStep 3

Circulating duties and selecting

one efficient roster for the depotStep 4


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trip1

2:17

trip 2

2:54

5:20 6:20 8:37 9:07 11:48 12:28 15:22 16:02

station B station Astation Bststion Adriving time

general work time

rest time

from depot hand over take over to depot

Figure 2: An example duty

time

A

B

Depot

C

t1

t2

t3

t4 t5

t6

t7

t8

t9

t10

t11

t12

t14

t16

t13

Figure 3: Driving trips for a depot

time

A

B

Depot

C

D1

D2

D3

D4 D5

Figure 4: Duties for a depot


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Figure 5: A schedule of duties for a depot

Driver 1

Driver 2

Driver 3

Driver 4

Driver 5

Driver 6

Day Day Day3 Day4 Day5 Day6

Holiday

Holiday

D1 D2 D3 D4 D5

D1 D2 D3 D4 D5

D5 Holiday D1 D2 D3 D4

D4 D5 Holiday D1 D2 D3

D3 D4 D5 Holiday D1 D2

D2 D3 D4 D5 Holiday D1

Figure 6: A roster for a depot

1.2 Literature Overview

Many research results have been published on crew pairing problem in operations research

literature, for a review please refer to Barnhart, et al. 1999 and Deasauliners, et al. 1998.The researchers developed efficient methods to solve the problems of step 2 and step 3 in the

Duty

Trip connection

B Depot C

D1

D2

D3

D4

D5

t 7

t 14

t 12

t 1

t 8

t 12t 9

t 6t 3

t 4t 11

t 16

t 5

t 10

t 13

A


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sequential process, e.g. Caprara, et al. 1997 and Yan, et al. 2002. Furthermore, there are

some research results on crew rostering problem in operations research literature, please refer

to Caprara, et al. 1998 as an example. The researchers developed efficient methods to solve

the problem of step 4 in the sequential process. In brief, the crew scheduling problem is a

NP-hard problem to be solved exactly. In addition, most studies are considering the cases

with simple rules for a feasible duty and a feasible roster, and simple objectives for pairing

problem and rostering problem. For example, as illuminated by Kwan 1999, a rule-based

expert system or a heuristic is used to generate feasible and potential duties, if there are

complex rules, which are not easy to be written as simple equations. In recent years, some

studies solve the crew scheduling and related problems by intelligent search methods, e.g.

Beasely, et al. 1996, Chatterjee, et al. 1996, and Levine, et al. 1996. These methods are

suitable to deal with NP hard problem with practical size. Furthermore, these methods are

flexible to deal with various difficult constraints associated with the standard crew schedulingproblem.

In the practice of Taiwan Railway Administration (TRA), there are very complex rules for a

feasible duty and a feasible roster. Moreover, there is a meeting between TRA and her union

every year, discussing possible changes for the rules. Therefore, the study emphasizes the

development of decision method not only for efficiency but also for flexibility. Moreover, a

railway company needs an efficient decision process, and little research has considered the

whole decision process. For example, the generally selected objective for the pairingproblem is the number of duties, and that for the rostering problem is the total cycle length.

The connection of sub-optimal solutions in the sequential process may not result in an optimal

solution as a whole. Therefore, the study emphasizes the development of the method for the

whole decision process, including both the pairing and the rostering problems. Besides, we

will present our experiences in the paper on solving TRAs problem using mathematichal

programming and Genetic algorithms.

2. Mathematical Programming Approach

2.1 Generation of Feasible and Potential Duties

Since the number of feasible duties can be enormous, we need a process generating a certain

number of feasible duties, with high potential to be implemented in the following scheduling

problem. We had an in-depth interview with TRAs crew manager, and examine the

practical duties used in the past and current schedules. Finally, we develop some heuristic

methods to generate two-spell duties, three-spell duties, four-spell duties, and so on. For a

three-spell duty, we can first select the one trip from different stating times as the first, second,


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or third spell in the duty. Moreover, we develop some heuristic methods for over-night

duties.

2.2 Selecting an Efficient Schedule of duties

As illustrated in the following, a bi-objective set covering problem is used to solve the pairing

problem for TRA, where the primary objective is to minimize the total number of duties, and

the secondary objective is to minimize the compensation for the selected duties. As shown

by Ignizio et al. 1984, the bi-level problem can be converted into a single objective and single

level linear program. In this study, we solve the converted single objective LP by

commercial package LINDO. One trouble in solving such a problem is the size of feasible

duty set D . In this study, as shown in Figure 7, we solve the pairing problem with the duty

generation iteratively. We first solve the set covering problem with a small set of feasible

and potential duties, then we try to enrich the rules for generating potential duties, so as tosolve the set covering problem with a big set of feasible and potential duties, and so on. As

the results shown in Figure 8, it does not take long when we reach very good solution at the

3rditeration. After that, the increase of the size of feasible duties has very little effect on the

quality of optimal solution. In the study, the computer time for solving the model with

TRAs practical problem is quite fast. With TRAs practical problem in 2000, the optimal

solution is better than TRAs practical schedule by 3 duties and 389 units of compensation.

Dj j

xMin (1)

Dj

jjxcMin (2)

tosubject

Dj

jijxa 1 Ti (3)

{ }1,0jx Dj ; (4)

where T is the set of trips, D is the set of feasible and potential duties, jc is the cost or

compensation for dutyj , ija is the indicator parameter for trip i and duty j , and jx is

the model decision variable for the selection of duty j .


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Figure 7: An interactive process for the set covering problem with practical size

Figure 8: Testing results of the pairing problem

2.3 Circulating Duties and Selecting an Efficient Roster

Given the schedule of selected duties, the duty assignment or rostering problem is a

constrained asymmetric traveling salesman problem. As the model shown in the following,

the roster length is the objective, where two duties can be connected without or with a holiday.

Equations 8 to 15 are constrains for a feasible roster. In order to simplify the problem, we do

not formulate the rules accurately. For example, equation 9 represents that there is a holiday

Generate feasible and potential duties

Formulating the set covering problem

LINDO solution

Modifying rules for duty generation

Generating feasible and potential duties

Convergence? End

No

Yes

Master problem

Sub-problem

60

55

50

45

40

Driver

97 194 291 388 485

The size of duty set

56(1st iteration)

46(2nd) 45(3rd) 45(4th)

120000

110000

100000

90000

80000

Compensation

97 194 291 388 485

The size of duty set

117480 (1st iteration)

88032 (2nd) 87339 (3rd) 86306 (4th)


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in a week on average, but the rule is a holiday in every week. However, we are lucky that

testing results give solutions of feasible roster, not only for the constraints in the model but

also for the rules in practice. Moreover, we do not put sub-tour elimination constrains so as

to control the size of the model. The computer time for such a model with TRAs practical

problem is very long. After solving the linear integer programming model by LINDO, we

have to deal with the sub-tour problem by heuristic methods. In the study, we found

multiple optimal solutions for the model, and there are several ways to eliminate the sub-tours

of an optimal solution into one cycle. Given the output of the pairing problem as the input

of the rostering problem, we obtain 19 good rosters with the same value of the objective

function. As shown in Table 1, they have very good characteristics, e.g. low variance of

weekly work-time. An example roster, number 15, is illustrated in Table 2. However, these

rosters are one day longer than TRAs practical solution. That is, testing results in the

pairing and rostering problems give us an example of good schedule but bad roster. Onereason is that TRA take some rules as soft constrains but we put all rules in the model as hard

constrain. In other words, TRAs practical solution is not feasible in our roster model.

+ Ni Nj

ijijijij xdxdMin )(2211 (5)

tosubject

1)( 21 = +Ni

ijij xx Nj (6)

1)( 21 = +Nj

ijij xx Ni (7)

TDxdxd

t

Ni Njijijijij

Nkk

+

)( 2211 (8)

+

Wxdxd

x

Ni Njijijijij

Ni Njij

)( 2211

2

),( ji (9)

ijijij yMHxr 4022 ),( ji (10)

)1(40 22 ijijij yMxrH ),( ji (11)

+

Rxdxd

y

Ni Njijijijij

Ni Njij

)( 2211 ),( ji (12)

0)( 11 ijiij xwr ),( ji (13)

111 ijjk xx = Bkjikji ,,),,( (14)

+

Rxdxd

x

Ni Njijijijij

Bi Bjij

)( 2211

1

),( ji (15)

{ }1,0,, 21 ijijij yxx Nji ),( ; (16)


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where dij1 is the time between duty iand dutyj, and dij

2 is the time between duty i and dutyj

with a holiday; kt is the driving time of duty k; iw is the work time of duty i ;1

ijr is the

rest time between duty i and dutyj ; 2ij

r is the rest time between duties with a holiday; T

is the limit of average daily work time; RWDH ,,, are the minutes of an hour, a day, a week,

and a month; B is the duty set for the duties with 2 hours or more driving-time during 22:00

to 6:00; is the holiday limit for a week; is the holiday limit for a month; is the

limit number of duties with 2 hours or more driving-time during 22:00 to 6:00 in a month;

is a big number; ijy is a 0-1 variable for the rest time between duties great than 40 hours;

1

ijx and2

ijx are 0-1 decision variables for the consecutive connection of duties i and j .

Table 1: Characteristics of Selected Rosters

Roster 1 2 3 4 5 6 7 8 9 10

Max weekly

work-time

38.00 41.40 39.28 41.15 39.17 39.08 39.5 39.5 40.17 39.50

Min weekly

work-time

28.37 28.37 28.37 28.37 28.87 28.37 28.37 28.37 28.37 28.87

Range 9.63 13.03 10.91 12.78 10.3 10.71 11.13 11.13 11.8 10.63

value 3.5876 4.7803 3.9877 4.7614 3.6348 4.1618 3.7850 4.0139 4.7188 3.6973

Roster 11 12 13 14 15 16 17 18 19 TRAMax weekly

work-time

37.77 37.77 37.77 41.95 37.77 38.38 40.47 44.35 39.95 45.52

Min weekly

work-time

28.37 28.37 28.37 28.37 28.87 28.37 28.37 28.87 28.37 29.72

Range 9.4 9.4 9.4 13.58 8.9 10.01 12.1 15.48 11.58 15.80

Value 4.1930 3.5736 3.8888 4.8329 3.4485 4.1331 4.1620 5.2283 4.2377 6.7749

Table 2: An example roster optimal solution

Roster # 15

Holiday 26 22 29

Holiday 15 4 19 21

Holiday 8 3 25 10

Holiday 9 31 12 30

Holiday 11 27 7 6

Holiday 35 24 5 36 33

Holiday 28 18 20 34

Holiday 32 16 23 17

Holiday 1 14 13 2


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3. GENETIC ALGORITHM APPROACH

3.1 Generating Feasible and Potential Duties

Besides the heuristic methods described in section 2.2, we develop some indicators used to

discard some feasible duties, which have no potential to be selected eventually. They are

efficiency indicator, distribution indicator, and so on. For example, the efficiency indicator

is the ratio of driving time over duty length. As illustrated in Figure 9, there is a negative

relation between the number of generated feasible duties and the efficiency indicator.

Thence, we can choose feasible duties with high values of efficient indicator for solving the

set covering problem.

38 635

28 54

34

5 975

95 6

766

625

0

50000

100000

150000

200000

250000

300000

350000

400000

non 0.15 0.2 0.25 0.3 0.35 0.4

The efficiency indicator

duties

Figure 9: the relation between the number of feasible duties and the efficiency indicator

3.2 Selecting an Efficient Schedule of Duties

The coding method of Genetic algorithm for the set covering problem in the study is binary

coding. As the example illustrated in Table 3, a binary coding is a series of 0-1 numbers,

and each number represent the selection result of a duty. Many parameters have to be

decided so as to have an efficient Genetic algorithm. For example, Figure 10 shows the

effect of population size, and Figure 11 shows the effect of uniform crossover method. After

doing many experiments, we formulate the Genetic algorithm with 200 population size, 2000

generations, roulette wheel method by nonlinear rank, uniform crossover with fusion operator,

100% reproduction rate, 0.05 mutation rate, and Elitism of 10. In addition, several practical

objectives are considered in the algorithm, and they are the number of duties, total length of

duties, total length of work-time, and the variance of work-time. Moreover, there are three

penalty values, and they are the number of uncovered, the number of covered, and the number

of dead-head trips. At last, the fitness function is the sum of objective function and penalty


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function. The weights used in the functions are also tested in many experiments. For

example, the final selected objective function is written as follows.

Obj 0.136 Sj

jx /37 + 0.193 Sj

jjxt /26578 + 0.632 Sj

jjxw /18300 + 0.038

Sj

j

Sj Sj

j

Sj

jjjj

x

xxwxw

1

)(2

/122.46

With TRAs practical dada in 2001, the Genetic algorithm in general produces promising

solutions for the pairing problem. Comparing to TRAs practical solution, all objectives

except the total length of duties can get about 10% improvements by the Genetic algorithm.

Table 3: Coding for the set covering problem

ija Duty INon-binary

Coding

1 1 0 0 0 1

0 0 1 1 1 4

0 1 0 1 0 4

1 0 0 0 1 1

Trip j

0 0 1 1 1 4

Binary

coding1 0 0 1 0

0

10

20

30

40

50

60

70

80

90

100

1 11 21 31 41 51 61 71 81 91generation

objective 100-1000 200-500

Figure 10: The effect of population size


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0

20

40

60

80

100

120

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96generation

objective U-value U-rank U-nlrank U-tournament

Figure 11: The effect of various uniform crossover method

3.3 Circulating Duties and Selecting an Efficient Roster

The coding method of the Genetic algorithm in the study is non-binary coding. As the

example illustrated in Table 4, given a holiday pattern, a gene of roster is represented by a

series of numbers in sequence, while each number is a duty. Note that holiday pattern in

Table 4 is different from the case used in section 2, because holiday regulation is changed in

2001. The objectives of the rostering problem are the cycle length and the variance of

weekly work-time. There are many rules for a feasible roster, and one penalty is formulatedfor each of them. Many parameters have to be decided so as to have an efficient Genetic

algorithm. For example, Figure 12 shows the effect of roulette wheel method. After doing

many experiments, we formulate the Genetic algorithm with 200 population size, 1000

generations, roulette wheel method by nonlinear rank, greedy selection crossover operator,

100% reproduction rate, 0.2 mutation rate, 0.1 rate of inversion operation, and Elitism of 10.

Table 4: The coding of gene for a roster

Holiday Roster

40 hrs 1 25 24 19

24 hrs 16 37 6 7 1324 hrs 27 15 17 34

40 hrs 20 33 23 2

40 hrs 8 5 21 10

40 hrs 3 32 4 9

40 hrs 28 30 11 22

24 hrs 35 18 31 29 36 12

40 hrs 26 14

24 hrs


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62000

64000

66000

68000

70000

72000

74000

76000

78000

80000

82000

1 6 11 16 21 26 31 36 41 46 51 56 61 66 71 76 81 86 91 96

generation

objective By value By rank By nonlinear rank

Figure 12: The effect of roulette wheel method

Table 5 is TRAs practical solution of the pairing and rostering problems. With TRAs

practical dada in 2001, Table 6 is the rostering solution by the Genetic algorithm with the

duties of TRAs schedule; and Table 7 is the rostering solution by the Genetic algorithm, with

the duties of an efficient schedule obtained from the pairing problem by the Genetic algorithm

described in section 3.2. Comparing Table 5 and Table 6, it is evident that the Genetic

algorithm for the rostering problem can decrease the roster length by 1 week. Comparing

Table 5 and Table 7, it is clear that the Genetic algorithms developed in this study canimprove the roster length by 1 week and the schedule by 7 duties. However, with regard to

the cycle length, the combined effect of the Genetic algorithms for the two sub-problems is

similar the Genetic algorithm for the rostering problem. That is, the decrease of the number

of duties in the pairing problem may not have positive impact on the rostering problem.

Further research is necessary to clarify the relationship between the sub-problems.

Table 5: TRAs practical roster in 2001

Week Holiday Roster of duties

Duty and

holidaytime

Worktime

1 40 hrs 836801822833 10318 1931

2 40 hrs 827811803810 9937 2456

3 24 hrs 830808861866832 10067 2141

4 24 hrs 834813804882828 10087 2627

5 40 hrs 812802867826 8593 2055

6 24 hrs 815809829825838 10369 2178

7 24 hrs 814821805847823 9854 2715

8 40 hrs 831881824807837 9975 2197Total 37 duties 79200 18300


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Table 6: One solution of the Genetic algorithm for the rostering problem


Duty and

holiday

time

Work

time

1 24 hrs 836826830815833 8783 2632

2 24 hrs 832828834811821 9224 2569

3 40 hrs 813827829801867 9862 2643

4 24 hrs 807808805802814 9733 2551

5 40 hrs 838831837812881825 9338 2638

6 24 hrs 861803804882823 9230 2599

7 24 hrs 847822824866809810 10070 2668

Total 37duties 66240 18300

Table 7: One roster of the Genetic algorithms for the pairing and rostering problems


Duty and

holiday

time

Work time

1 40 hrs 1216 9 4 10024 2615

2 40 hrs 141122 5 9201 2622

3 24 hrs 242018 3 8565 2625

4 40 hrs 1517 821 9588 2631

5 40 hrs 302326 67 9763 2630

6 40 hrs 11910 2 9484 2631

7 40 hrs 1325292827 9615 2611Total 30duties 66240 18365

4. CONCLUDING REMARKS

In the study, we found the following major points: 1.mathematic programming is not efficient

and flexible for TRAs driver scheduling problem, especially for the rostering problem; 2.the

connection of suboptimal solutions of pairing and rostering problems may not result in the

optimal solution as a whole; 3.Genetic algorithm provides a flexible structure for driver

scheduling problem with multiple objectives and various constrains; and 4.the solution of

Genetic algorithm approach in general give better results the TRAs practical solution.

Moreover, Genetic algorithm may provide a structure to integrate the sub-problems of

generating feasible duties, selecting duties, and rostering duties into an integrated problem.

ACKNOLEDGEMENTS

Authors are grateful to Taiwan National Science Council for the financial support; and the

authors would like to thank their research assistants, Mr. Hsieh, S-H and Mr. Koa, Z-S, for

their computing works.


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Scheduling of Train Driver in Tiwan Rail Companyd

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