Job-shop Problems with Objectives Appropriate for Train Scheduling in

a Single-track Railway

Omid Gholami

, Yuri N. Sotskov

and Frank Werner

Islamic Azad University, Mahmudabad Branch, Moalem, Mahmudabad, Iran

United Institute of Informatics Problems, National Academy of Sciences of Belarus, Minsk, Belarus

Faculty of Mathematics, Otto-von-Guericke-University, Magdeburg, Germany

Keywords:

Job-shop, Train Scheduling, Regular Criteria.

Abstract:

A train scheduling problem in a single-track railway is studied using a mixed graph model for a job-shop with

appropriate criteria. There are several performance evaluations for a train schedule. Optimizing a train sched-

ule subtends minimizing total tardiness of the trains, minimizing the sum of train transit times, minimizing the

makespan for a train schedule, etc. Since the corresponding job-shop problems with the above three criteria

are NP-hard, several heuristic algorithms have been developed using different priorities based on the release

times of the jobs, the job due-dates and the job completion times. Experiments on a computer were used for

evaluating the quality and efﬁciency of the heuristic algorithms developed for appropriate job-shop problems.

The release times, due-dates and completion times of the jobs have been used as input parameters (priorities)

in the computer simulation to see the effect of them on the quality of the schedules with different objective

functions. The efﬁciency of the developed heuristics was demonstrated via a simulation on a set of randomly

generated instances of small and medium sizes. The computational results showed that one heuristic algorithm

outperformed the other algorithms tested for two of the three objective functions under consideration.

1 INTRODUCTION

This paper addresses the problem of generating an ef-

ﬁcient schedule of passenger and freight trains in a

single-track railway. We use the terminology from

(Lusby et al., 2011) for train scheduling and that from

(Tanaev et al., 1994) for machine scheduling.

In the world, the railway trafﬁc is increasing from

year to year. The employment of railroads grows both

for passenger and freight transportation. When the

density of train moving is increasing, the train sched-

ule becomes more difﬁcult both as generation and

control are concerned. During the last decades, a lot

of new algorithms and software have been developed

and published in the OR literature and in special liter-

ature in order to produce a better tool for generating

an accurate and reliable train schedule.

In this paper, it is shown how one can ﬁnd a train

schedule which is close to an optimal one with three

objective functions for a single-track railway. A pos-

sible way to achieve a proper train schedule uses job-

shop scheduling (Burdett and Kozan, 2010; Mascis

and Pacciarelli, 2002; Szpigel, 1973), although job-

shop problems are fairly complicated since they be-

long to the class of NP-hard problems (Brucker et al.,

1997; Brucker et al., 2007; Tanaev et al., 1994). In

order to achieve a practical size of a job-shop prob-

lem, which can be solved within a reasonable time,

we propose and test several heuristic algorithms for

three objective functions which are appropriate for

train scheduling. In Sections 3–6, we consider a rail-

way network provided that a pair of sequential sta-

tions can be connected by at most one single-track

(a railroad section). In particular, this is the case for

most railway systems in countries of the Middle East.

2 LITERATURE REVIEW

In (Zhou and Zhong, 2007), a resource-constrained

project scheduling was used for a single-track

timetabling problem. Both the track segments and sta-

tions were modelled as limited resources. A branch-

and-bound algorithm has been developed in order to

obtain a feasible train timetable with a guaranteed

level of optimality. A lower bound based on La-

grangian relaxation was used to relax the segment and

425

Gholami O., N. Sotskov Y. and Werner F..

Job-shop Problems with Objectives Appropriate for Train Scheduling in a Single-track Railway.

DOI: 10.5220/0004054404250430

In Proceedings of the 2nd International Conference on Simulation and Modeling Methodologies, Technologies and Applications (SIMULTECH-2012),

pages 425-430

ISBN: 978-989-8565-20-4

 2012 SCITEPRESS (Science and Technology Publications, Lda.)

station capacity constraints. A lower bound was used

to estimate the least train delay. An upper bound was

constructed via a beam search heuristic. In (Cai and

Goh, 1994), a heuristic algorithm was developed for

train scheduling in a single-track railway under the as-

sumption that all trains moving in the same direction

must have the same speed. A greedy heuristic was

proposed based on a local optimality criterion in the

event of a potential crossing conﬂict.

The paper (Liu and Kozan, 2011) was devoted to

train scheduling problems when prioritized trains and

non-prioritized trains are simultaneously traversed in

a single-track railway. No-wait conditions arise be-

cause the prioritized trains (e.g., an express passen-

ger train has a higher priority) should traverse contin-

uously without interruptions. Non-prioritized trains

(e.g., a freight train) are allowed to either enter the

next section immediately (if it is free) or to remain in

a section until the next section on the route becomes

available. A generic algorithm has been developed

to construct a feasible train timetable in terms of the

given train order. The proposed algorithm comprises

several recursively used procedures to guarantee the

feasibility by satisfying the no-waiting, a deadlock-

free condition, and a conﬂict-free constraint.

B. Szpigel (Szpigel, 1973) was the ﬁrst who iden-

tiﬁed the similarities between a job-shop problem and

train scheduling in a single-track railway. The former

was solved in (Szpigel, 1973) using a branch-and-

bound algorithm, where the initial linear program-

ming problem excludes the order constraints. Branch-

ing is required if the current solution contains trains

which are in a conﬂict (i.e., when trains turn out to

be located on the same railroad section at the same

time). The objective was to minimize the weighted

sum of the train transit times. Computational results

for 5 single-track sections and 10 trains have been re-

ported. The same problem was considered in (Carey

and Lockwood, 1995) via binary mixed integer pro-

gramming similarly to that considered in (Jovanovic

and Harker, 1991). The temporal constraints were

identical to those used in (Szpigel, 1973). The objec-

tive was to minimize the deviation from the ideal ar-

rival times and the departure times for the trains to be

scheduled. In (Mladenovic and Cangalovic, 2007), a

job-shop problem was used to solve the train schedul-

ing problem, where a route was interpreted as follows:

The route is a sequence of the facilities the train must

cross from the origin to the destination. Assuming

that the train trips are jobs to be scheduled, which re-

quire elements of the infrastructure as restricted re-

sources, it was done by mapping the initial problem

into a special case of a job-shop problem. In order

to solve the job-shop problem, a constraint program-

ming approach has been developed. A support for

ﬁnding quickly a good schedule was offered by an

original separation and a bound-and-search heuristic.

To improve the time performance, a surrogate objec-

tive function was used which had a smaller domain

than the actual objective function.

In (Dorfman and Medanic, 2004), a discrete-event

model was used to schedule the trafﬁc on a railway

network. This model was computationally efﬁcient

and generated near optimal schedules with respect to

a number of time-of-travel-related criteria. In (Bur-

dett and Kozan, 2010), train scheduling was inter-

preted in terms of a job-shop problem with parallel

machines. A disjunctive graph model was used in sev-

eral algorithms with the makespan objective. It was

demonstrated that solutions with a good quality may

be obtained within a reasonable CPU-time.

3 PROBLEM SETTINGS AND

TESTING

One of the main problems in the management of a

railway network is the train scheduling (timetabling)

problem, in which it is necessary to determine a

schedule (timetable) for a set of given trains that does

not violate the railway constraints. This problem has

to be solved at the tactical level of the railway plan-

ning process (Lusby et al., 2011). For the case of a

single-track railway, train scheduling may be inter-

preted as the following job-shop problems.

There are n jobs J = {J

, J

, . . . , J

} to be pro-

cessed on m different machines M = {M

, M

, . . . ,

}. The time p

i j

> 0 needed for processing an

operation O

i j

of a job J

∈ J on the correspond-

ing machine M

∈ M is known. Operation preemp-

tions are not allowed, and the machine routes O

, O

, . . . , O

) for the jobs J

∈ J may be differ-

ent. A job J

∈ J is available for processing from time-

point r

≥ 0. The time-point d

deﬁnes a due-date for

completing the job J

. A machine M

∈ M can pro-

cess a job J

∈ J at most once. So, any two operations

i j

and O

, j 6= k, of the same job J

∈ J have to

be processed by different machines of the set M , i.e.,

inequality n

≤ m holds (such a problem is called a

classical job-shop).

One objective is to ﬁnd a schedule minimiz-

ing the sum γ =

∑

i=1

of the tardiness times T

max{0,C

− d

} of the jobs J

∈ J . Hereafter, C

de-

notes the completion time of a job J

∈ J . Accord-

ing to the three-ﬁeld notation α|β|γ used for machine

scheduling problems, the above job-shop problem is

denoted as J|r

∑

. If γ =

∑

i=1

, then this prob-

lem is denoted as J|r

∑

. If γ = max

i=1

, then it is

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Applications

426

denoted as J|r

max

The problems J|r

∑

, J|r

∑

and J|r

max

arise in train scheduling for a single-track railway: To

determine the best train schedule among those which

do not violate the single-track capacities. For passen-

ger trains, the criteria

∑

and

∑

are more impor-

tant than C

max

while for freight trains, the criteria C

max

and

∑

are more important than

∑

In a job-shop approach to train scheduling, the

trains and railroad sections are synonymous with the

jobs J

∈ J and the machines M

∈ M , respectively.

An operation O

i j

is regarded as a movement of a train

∈ J across a railroad section M

∈ M (in the route

, machine M

processes operation O

i j

). The pos-

itive number p

i j

denotes the time required for train

∈ J to travel through the railroad section M

∈ M .

The non-negative number r

denotes the departure

time of train J

∈ J , which is given in the ofﬁcial train

timetable. The positive number d

denotes the ofﬁcial

arrival time of train J

∈ J (a due-date for the desired

completion time C

of a job J

∈ J ) at the terminal sta-

tion in the route O

. Note that for train scheduling the

inequality m ≥ n usually holds.

The problems J|r

∑

, J|r

∑

and J|r

max

are

complicated in the computational sense since their

special cases belong to the class of NP-hard prob-

lems (Tanaev et al., 1994). In order to achieve a

practical size of a classical job-shop problem, which

can be solved heuristically within a reasonable time,

we coded a shifting bottleneck algorithm, which was

originated in (Adams et al., 1988) for the job-shop

problem J||C

max

. However, testing the program real-

izing a shifting bottleneck algorithm for the problem

J|r

∑

showed an unsatisfactorily large CPU-time

when the number n of trains was large and the num-

ber m of railroad sections was no less than n (Sotskov

and Gholami, 2012). A simulation showed that this

algorithm can handle 125 operations (e.g., 25 trains

on 5 railroad sections or 5 trains on 25 railroad sec-

tions) within half an hour of CPU-time. For larger

job-shops with m ≥ n, the CPU-time grows quickly.

In Sections 4 and 5, we develop heuristic algo-

rithms, which run faster than the shifting bottleneck

algorithm providing a quality of the objective function

values which is close to the quality of the schedules

constructed by the shifting bottleneck algorithm.

4 MIXED GRAPH MODEL

The problems J|r

∑

, J|r

∑

and J|r

max

de-

scribed in Section 3 can be formulated using a mixed

graph model G = (Q , C , D) (Tanaev et al., 1994) or a

disjunctive graph model (Sussmann, 1972).

Let Q denote the set of operations O

i j

, J

∈ J ,

j ∈ {1, 2, . . . , n

}, to be executed by the machines M

and a dummy operation O

associated with the be-

ginning of a schedule and n dummy operations O

i,n

associated with the completion of the jobs J

∈ J .

Two operations O

i j

and O

, which have to be

executed by the same machine M

∈ M , cannot be

simultaneously processed by this machine. This re-

striction is presented by an edge [O

i j

, O

] ∈ D. Two

consecutive operations O

i j

and O

i, j+1

of the same

job J

∈ J are connected by an arc (O

i j

, O

i, j+1

) ∈ C ,

where 1 ≤ j ≤ n

− 1. The arc (O

i j

, O

i, j+1

) means

that operation O

i, j+1

has to be started after the com-

pletion of operation O

i j

. The processing time p

i j

prescribed to the arc (O

i j

, O

i, j+1

) ∈ C , and the two

processing times p

i j

and p

are prescribed to the edge

i j

, O

] ∈ D. For the dummy operation O

∈ Q , the

arc (O

, O

) with the weight r

is included into the

set C for each job J

∈ J . For the dummy operation

i,n

∈ Q , the arc (O

, O

i,n

) with the weight p

is included into the set C .

The problems J|r

∑

, J|r

∑

and J|r

max

are

modelled by a mixed graph G = (Q , C , D). The due-

dates d

are used when calculating the objective func-

tion γ =

∑

i=1

for a schedule constructed.

Since operation preemptions are not allowed, a

schedule on a mixed graph G = (O, C , D) may be de-

ﬁned as a sequence of the starting times s

= (s

= 0,

, s

, . . . , s

, s

, . . . , s

, s

, . . . , s

, s

) of all the

operations Q such that the conjunctive constraint

− s

≥ p

i j

(1)

has to be satisﬁed for each arc (O

i j

, O

) ∈ C , and the

disjunctive constraint

either s

− s

≥ p

i j

or s

− s

≥ p

(2)

has to be satisﬁed for each edge [O

i j

, O

] ∈ D.

Using the above weighted mixed graph G =

(Q , C , D), to deﬁne a feasible sequence s

of the start-

ing times, one has to replace each edge [O

i j

, O

] ∈ D

by either the arc (O

i j

, O

) with the weight p

i j

or the

arc (O

, O

i j

) with the weight p

respecting the dis-

junctive constraint (2) in such a way that no circuit

arises in the obtained digraph. As a result, the set of

edges D will be substituted by a chosen set D

, the

mixed graph G = (Q , C , D) will be transformed into

a circuit-free digraph G

= (Q , C

, ∅), and an op-

eration sequence for each machine of the set M will

be determined. Since the cardinality of the set Q is

equal to |Q | = 1 +

∑

i=1

+1), using the critical path

method in O(n

) time, one can build a unique semi-

active schedule deﬁned by the weighted digraph G

A schedule is called semiactive if no operation O

i j

∈ J , j ∈ {1, 2, . . . , n

}, can start earlier without de-

laying the processing of some operation from the set

Job-shopProblemswithObjectivesAppropriateforTrainSchedulinginaSingle-trackRailway

427

Q or (and) without altering the processing sequence

of the operations on any of the machines M .

The main complexity of the problem J|r

|γ with

a regular criterion γ is to ﬁnd an optimal circuit-free

digraph G

= (Q , C

, ∅) generated by the mixed

graph G = (Q , C , D). In other words, it is neces-

sary to ﬁnd such a set of arcs D

for substituting the

set of edges D in the mixed graph G that the objec-

tive function γ has the minimal value among all other

circuit-free digraphs generated by the mixed graph G

via replacing each edge [O

i j

, O

] ∈ D either by the

arc (O

i j

, O

) ∈ D

or by the arc (O

, O

i j

) ∈ D

5 HEURISTIC ALGORITHMS

We developed three types of algorithms denoted as

Ordinal, Max-PT and Min-PT. The release times,

completion times and due-dates are used as priorities

in ordering the jobs J

∈ J for processing on the same

machine from the set M . Nine heuristic algorithms

of three types with three priority rules are developed.

5.1 Ordinal-algorithm

The Ordinal-algorithm generates a sequence of the

operations O

i j

on different machines of the set M in

the order as they are requested for processing the jobs

∈ J . In the ﬁrst iteration, the Ordinal-algorithm

ﬁnds the ﬁrst request (i.e., operation O

) of a job

∈ J for the machine M

∈ M processing opera-

tion O

. Then, depending on which priority rule is

used, the Ordinal-algorithm computes either the re-

lease time or the completion time or the due-date as

the priority of operation O

. For example, let the

Ordinal-algorithm use the release time of operation

as its priority. Then the algorithm compares the

release time r

of operation O

with the release times

of all operations O

of the other jobs J

∈ J , i 6= j, on

the same machine M

∈ M processing operation O

If the release time r

is smaller than the release time

of the operations of the other jobs on the same ma-

chine M

∈ M , then an arc starting from operation

and ending in operation O

has to be added to

the digraph (Q , C ,

0). Otherwise, the symmetric arc

, O

) has to be added to the digraph (Q , C ,

0).

The release time r

denotes the earliest start time

of operation O

which can be computed due to the re-

cursion r

= max{r

i j

+ p

i j

}, where the maximum is

taken over all operations O

i j

∈ Q preceding operation

in the digraph already constructed. The release

time of the source operation O

is equal to zero.

The above procedure is repeated for the second

job request (iteration 2), then for the third job request

(iteration 3) and so on until all machine requests have

been satisﬁed. We called this version of the algorithm

as Ordinal-SRT (from Shortest Release Time).

The other two versions of the Ordinal-algorithm

based on either the completion time priority or the

due-date priority are called Ordinal-SCT (Shortest

Completion Time) and Ordinal-SDD (Shortest Due-

Date), respectively.

5.2 MaxPT-algorithm

The MaxPT-algorithm (Maximum Processing Time)

tends to schedule ﬁrst the jobs that need more pro-

cessing time on all machines M

∈ M .

In the ﬁrst step, the MaxPT-algorithm calculates

the sum of the processing times (total processing

time) of all operations O

i j

, j ∈ {1, 2, . . . , n

}, for each

job J

∈ J . Before scheduling, the maximum sum of

the processing times of a job J

∈ J is equal to the

length of a critical path in the digraph (Q , C ,

0).

The MaxPT-algorithm sorts the jobs J in non-

increasing order of their total processing times and

selects a job with the maximum total processing time

to be processed next. The MaxPT-algorithm starts to

process the ﬁrst request (operation O

) of the job J

with maximum total processing time, then the second

request of the same job and so on until the last request

of job J

. At each operation, the MaxPT-algorithm

computes one of the three priorities (either the release

time, the completion time or the due-date) depending

on the version of the algorithm. The chosen priority

is compared with those of all operations of the other

jobs on the same machine M

. Then either the arc

, O

) or the arc (O

, O

) is added to the digraph

(Q , C ,

0)) depending on the larger priority of job J

job J

. The added arc deﬁnes the order of processing

the jobs J

and J

on machine M

Then the MaxPT-algorithm repeats the same pro-

cedure for the other jobs that are sorted by non-

increasing sums of their processing times. We call

this version of the algorithm which uses the release

time as priority as MaxPTRT-algorithm (Maximum

Processing Time, Release Time).

The MaxPTCT-algorithm (Maximum Processing

Time and Completion Time) is another version that

compares the job completion times as priorities,

and the MaxPTDD-algorithm (Maximum Processing

Time, Due-Date) compares the due-dates as priorities.

5.3 MinPT-algorithm

The MinPT-algorithm (Minimum Processing Time)

is basically similar to its counterpart, the MaxPT-

algorithm but in contrast to the latter one, the MinPT-

SIMULTECH2012-2ndInternationalConferenceonSimulationandModelingMethodologies,Technologiesand

Applications

428

Figure 1: Objective function values of the obtained sched-

ules for job-shop problems with the criterion C

max

algorithm schedules ﬁrst the job J

∈ J that needs the

smallest total processing time on all machines M .

The MinPT-algorithm sorts the jobs in non-

decreasing order of their total processing times and

then schedules the jobs on each machine M

∈ M in

non-decreasing order of the corresponding priorities.

The three versions of the MinPT-algorithm

are the MinPTRT-algorithm (Minimum Processing

Time, Release Time), the MinPTCT-algorithm (Min-

imum Processing Time, Completion Time) and the

MinPTDD-algorithm (Minimum Processing Time,

Due-Date).

6 COMPUTATIONAL RESULTS

Three versions of the three types of algorithms (Or-

dinal, Max-PT and Min-PT) have been coded in Bor-

land Delphi. For the simulation, a laptop computer

with the following speciﬁcation has been used: Intel,

coreTM 2 Duo, CPU T6400, 2.00 GHz and 2GB In-

ternal Memory, Windows 7, Ultimate 32 bit.

We were interested in investigating experimen-

tally the effect of choosing different types of algo-

rithms and different priorities for the three objective

functions. We randomly generated job-shop problems

of size n × m, where n = m ∈ {5, 6, . . . , 12} to see the

effect of these algorithms on different objective func-

tions for job-shop problems.

First, we compared the makespan objective func-

tion for randomly generated problems J|r

max

different size n × m, where n = m. We compared

the makespan values obtained by the nine algo-

rithms developed for instances with the same input

data. Each series includes 10 randomly generated in-

stances. The results are presented in Fig. 1, which

shows that the quality of a schedule obtained by the

algorithms generally depends on the input data, but

both the OrdinalSCT-algorithm and the OrdinalSRT-

algorithm slightly outperform the other algorithms for

Figure 2: Objective function values of the obtained sched-

ules for job-shop problems with the criterion

∑

Figure 3: Objective function values of the obtained sched-

ules for job-shop problems with the criterion

∑

the makespan criterion.

In the next experiments, the nine algorithms were

used to solve heuristically the job-shop problems with

the objective function

∑

. Figure 2 shows that the

OrdinalSCT-algorithm provides schedules with the

best quality among the algorithms tested.

In the last experiments, we evaluated the objective

function

∑

. Figure 3 shows that the OrdinalSCT-

algorithm again provides schedules with the best

quality among the algorithms tested.

Table 1: CPU-time in seconds for randomly generated in-

stances with n jobs and m = n machines

n = m = 7 8 9 10 11 12

OrdinalSDD 2 4 9 31 145 2801

OrdinalSCT 3 5 9 66 208 6510

OrdinalSRT 2 4 5 34 212 3244

MaxPTDD 2 9 17 73 639 11452

MaxPTCT 3 9 19 58 449 4500

MaxPTRT 2 8 11 47 376 11710

MinPTDD 3 9 18 188 325 3986

MinPTCT 2 7 13 63 324 4594

MinPTRT 1 4 11 57 299 8340

In Table 1, the CPU-time taken by the nine algo-

rithms to solve heuristically different job-shop prob-

Job-shopProblemswithObjectivesAppropriateforTrainSchedulinginaSingle-trackRailway

429

lems are given. For all randomly generated problems

of size n = 5 = m and n = 6 = m tested, the CPU-time

was less than 1 s.

7 CONCLUSIONS

The problem of ﬁnding an optimal train schedule in-

cludes several criteria. We considered three of them

and developed nine heuristic algorithms to solve the

corresponding job-shop problems. We coded dif-

ferent variants of heuristic algorithms and compared

three parameters for job-shop scheduling to ﬁnd suit-

able heuristic algorithms for three objective functions.

The simulation showed that the OrdinalSCT-

algorithm generates a good schedule to minimize the

makespan, the sum of job completion times and the

sum of job tardiness. It was shown that the use of

a more complicated algorithm (like the shifting bot-

tleneck one) for solving train scheduling problems

needed more CPU-time for the case when m ≥ n with

only a slight improvement of the objective function

values.

For future research, we recommend to compare

more parameters for those objective functions appro-

priate for train scheduling. Note that the OrdinalSCT-

algorithm may be generalized to the weighted objec-

tive functions

∑

and

∑

allowing a scheduler

to take into account different priorities of the trains.

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