ISSN 0005-1179, Automation and Remote Control, 2018, Vol. 79, No. 3, pp. 506–523.
c
Pleiades Publishing, Ltd., 2018.
Original Russian Text
c
Y. Zinder, A.A. Lazarev, E.G. Musatova, I.A. Tarasov, 2018, published in Avtomatika i Telemekhanika, 2018, No. 3,
pp. 144–166.
OPTIMIZATION, SYSTEM ANALYSIS, AND OPERATIONS RESEARCH
Scheduling the Two-Way Traffic
on a Single-Track Railway with a Siding
Y. Zinder
∗,a
,A.A.Lazarev
∗∗,∗∗∗,∗∗∗∗,∗∗∗∗∗,b
,
E. G. Musatova
∗∗,c
,andI.A.Tarasov
∗∗,∗∗∗,d
∗
Technical University of Sydney, Sydney, Australia
∗∗
Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, Russia
∗∗∗
Lomonosov State University, Moscow, Russia
∗∗∗∗
National Research University Higher School of Economics, Moscow, Russia
∗∗∗∗∗
Moscow Institute of Physics and Technology (State University), Moscow, Russia
e-mail:
a
yakov.zinder@uts.edu.au,
b
jobmath@mail.ru,
c
nekolyap@mail.ru,
d
ia.tarasoff@yandex.ru
Received May 17, 2017
Abstract—The paper is concerned with scheduling the two-way traffic between two stations
connected by a single-track railway with a siding. It is shown that if, for each station, the
order in which trains leave this station is known or can be found, then for various objective
functions an optimal schedule can be constructed in polynomial time using the method of dy-
namic programming. Based on this result, the paper also presents a polynomial-time algorithm
minimising the weighted number of late trains.
Keywords: dynamic programming, polynomial algorithm, railway planning, scheduling theory
DOI: 10.1134/S0005117918030098
1. INTRODUCTION
This paper presents a generalisation of the problem of scheduling the movement of trains on a
single-track railway, previously considered in [1]. For the objective functions, considered in [1], the
proofs below are a new justification of the corresponding algorithms.
The single-track railways are part of many railway networks and often are used for transportation
within factories. The considered problem also arises in situations when one of the tracks of a two-
track railway becomes inaccessible due to maintenance or accidents.
Detailed surveys on models and methods for railway planning are presented in [2–4]. This paper
is a sequel of [1] where the reader can find the related literature review. In particular, [1] analyses
the publications [5–11] which contain interesting results on planning the movement of trains on a
single-track railway.
The considered problem can be stated as follows. There are two sets of trains: N
1
and N
2
.
Trains in the set N
1
are at station 1 and must go to station 2, whereas the trains in the set N
2
are at station 2 and must go to station 1. The station number that is opposite to the station with
number s ∈{1, 2} will be denoted by ¯s. There is a siding between stations, permitting oncoming
trains to pass each other, which can accommodate one train. In the siding, there is the main line
for non-stop movement of trains and the additional line for a train to wait. The train on the main
line goes through the siding without stopping. The speed of trains is the same for all trains and
is constant. The time required for a train to cover the distance between station 1 and the siding,
and between station 2 and the siding, is p
1
and p
2
respectively. Without loss of generality, it will
be assumed that p
1
p
2
. The number of trains in N
1
is n
1
, and the number of trains in N
2
is n
2
.
The trains can depart their stations starting from the point in time t =0.
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