METHODOLOGIES AND APPLICATION
Lower and upper bounds for scheduling multiple balancing vehicles in
Ahmed A. Kadri
· Imed Kacem
· Karim Labadi
© Springer-Verlag GmbH Germany, part of Springer Nature 2018
Public bicycle-sharing systems have been implemented in many big cities around the world to face many public transport
problems. The exploitation and the management of such transportation systems imply crucial operational challenges. The
balancing of stations is the most crucial question for their operational efﬁciency and economic viability. In this paper, we
study the balancing problem of stations with multiple vehicles by considering the static case. A mathematical formulation
of the problem is proposed, and two lower bounds based on Eastman’s bound and SPT rule are developed. Moreover, we
proposed four upper bounds based on a genetic algorithm, a greedy search algorithm and two hybrid methods that integrate
a genetic algorithm, a local search method and a branch-and-bound algorithm. The developed lower and upper bounds are
tested and compared on a large set of instances.
Keywords Bicycle-sharing systems · Lower and upper bounds · Greedy search · Hybrid algorithms · Branch-and-bound ·
Vehicle routing problem
The urban population growth, the development and extension
of cities, the environmental and ecological concerns and the
fuel prices have made urban transport systems and its man-
agement a major challenge for many cities around the world.
These factors lead to give more importance to transport sys-
tems that must meet multiple and growing demands for the
mobility of users which become more and more demanding
with the services offered by some public transport compa-
nies. Congestion problems, air pollution, noise nuisance and
road accidents incite public authorities and transport com-
panies to multiply efforts in order to improve and maintain
Communicated by V. Loia.
Ahmed A. Kadri
Université de Lorraine, LCOMS EA 7306, UFR MIM, 3 Rue
Augustin Fresnel, 57000 Metz, France
ECAM-EPMI, 13 Bd de l’Hautil, 95092 Cergy-Pontoise,
the quality of their services in order to make the collective
transport more attractive than the individual one.
At the scientiﬁc and technological levels, the urban trans-
port systems represent a rich area for innovation and research.
In recent years, the sustainable urban mobility modes are
becoming very important due to the growth of the energy
use, the noise and the air pollution. Different new urban
transportation modes are developed such as green vehicles
use for public transport, car-pooling, car-sharing and bicycle-
sharing systems (Laporte et al. 2015). Nowadays, over 1000
bicycle-sharing systems (BSS) are operating in the world as
Vélib’ in Paris, Bicing in Barcelona, Hangzhou Public Bicy-
cle in China, Philly Indego Bikeshare in Philadelphia, etc.
Moreover, about 300 BSS systems are in planning or under
construction (Meddin and DeMaio 2016).
These systems have several advantages from a socioeco-
nomical point of view (Midgley 2009): the optimization of
public spaces, the health beneﬁts, the respect for the environ-
ment and the associated technological progress. However,
despite the large success of this new mode of transporta-
tion and unlike to the traditional transportation systems, there
are still lot of challenging subjects in this area to be studied
(Laporte et al. 2015). Indeed, many questions arise. The main
one is related to an efﬁcient balancing of bicycles distribution
in the various stations that constitute the transport network