Hostname: page-component-848d4c4894-nmvwc Total loading time: 0 Render date: 2024-06-17T04:36:43.420Z Has data issue: false hasContentIssue false

Balancing the stations of a self service “bike hire” system

Published online by Cambridge University Press:  10 May 2011

Mike Benchimol
Affiliation:
Université Paris Est, ENPC, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France. mike.benchimol@eleves.enpc.fr; benoit.chappert@eleves.enpc.fr; de-la-ta@eleves.enpc.fr; frederic.meunier@enpc.fr; ludovic.robinet@eleves.enpc.fr
Pascal Benchimol
Affiliation:
École Polytechnique, 91128 Palaiseau Cedex, France. pascal.benchimol@polytechnique.edu; fabien.laroche@polytechnique.edu
Benoît Chappert
Affiliation:
Université Paris Est, ENPC, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France. mike.benchimol@eleves.enpc.fr; benoit.chappert@eleves.enpc.fr; de-la-ta@eleves.enpc.fr; frederic.meunier@enpc.fr; ludovic.robinet@eleves.enpc.fr
Arnaud de la Taille
Affiliation:
Université Paris Est, ENPC, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France. mike.benchimol@eleves.enpc.fr; benoit.chappert@eleves.enpc.fr; de-la-ta@eleves.enpc.fr; frederic.meunier@enpc.fr; ludovic.robinet@eleves.enpc.fr
Fabien Laroche
Affiliation:
École Polytechnique, 91128 Palaiseau Cedex, France. pascal.benchimol@polytechnique.edu; fabien.laroche@polytechnique.edu
Frédéric Meunier
Affiliation:
Université Paris Est, ENPC, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France. mike.benchimol@eleves.enpc.fr; benoit.chappert@eleves.enpc.fr; de-la-ta@eleves.enpc.fr; frederic.meunier@enpc.fr; ludovic.robinet@eleves.enpc.fr
Ludovic Robinet
Affiliation:
Université Paris Est, ENPC, 6-8 avenue Blaise Pascal, Cité Descartes Champs-sur-Marne, 77455 Marne-la-Vallée Cedex 2, France. mike.benchimol@eleves.enpc.fr; benoit.chappert@eleves.enpc.fr; de-la-ta@eleves.enpc.fr; frederic.meunier@enpc.fr; ludovic.robinet@eleves.enpc.fr
Get access

Abstract

This paper is motivated by operating self service transport systems that flourish nowadays. In cities where such systems have been set up with bikes, trucks travel to maintain a suitable number of bikes per station. It is natural to study a version of the C-delivery TSP defined by Chalasani and Motwani in which, unlike their definition, C is part of the input: each vertex v of a graph G=(V,E) has a certain amount xv of a commodity and wishes to have an amount equal to yv (we assume that $\sum_{v\in V}x_v=\sum_{v\in V}y_v$ and all quantities are assumed to be integers); given a vehicle of capacity C, find a minimal route that balances all vertices, that is, that allows to have an amount yv of the commodity on each vertex v. This paper presents among other things complexity results, lower bounds, approximation algorithms, and a polynomial algorithm when G is a tree.

Type
Research Article
Copyright
© EDP Sciences, ROADEF, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

S. Anily and J. Bramel, Approximation algorithms for the capacitated traveling salesman problem with pickups and deliveries. Nav. Res. Logist. (1997).
Anily, S. and Hassin, R., The swapping problem. Networks 22 (1992) 419433. CrossRef
Augerat, P., Belenguer, J.M., Benavent, E., Corberán, A. and Naddef, D., Seprating capacity constraints in the CRVP using tabu search. Eur. J. Oper. Res. 106 (1998) 546557. CrossRef
Chalasani, P. and Motwani, R., Approximating capacited routing and delivery problem. SIAM J. Comput. 28 (1999) 21332149. CrossRef
Charikar, M., Khuller, S. and Raghavachari, B., Algorithms for capacitated vehicle routing. SIAM J. Comput. 31 (2001) 665682. CrossRef
N. Christofides, Worst-case analysis for a new heuristic for the Traveling Salesman problem, in Symposium on New Directions and Recent Results in Algorithms and Complexity, edited by J.F. Traub, Academic Press (1976).
M.R. Garey and D.S. Johnson, Computers and intractability: a guide to the theory of NP-completness. W.H. Freeman (1979).
H. Hernández-Pérz and J.-J. Salazar-González, The one-commodity pickup-and-delivery travelling salesman problem, in Lect. Notes Comput. Sci. 2570 (2002) 89–104. CrossRef
Hernández-Pérz, H. and Salazar-González, J.-J., A branch-and-cut algorithm for a traveling salesman problem with pickup and delivery. Discr. Appl. Math. 145 (2004) 126139. CrossRef
Hoogeveen, J.A., Analysis of christofides'heuristic: some paths are more difficult than cycles. Oper. Res. Lett. 10 (1991) 291295. CrossRef
König, D., Uber Graphen und ihre Anwendung auf Determinantentheorie une Mengenlehre. Math. Ann. 77 (1916) 453465. CrossRef
A. Lim, F. Wang and Z. Xu, The capacitated traveling salesman problem with pickups and deliveries on a tree, in Lect. Notes Comput. Sci. Algorithms and Computation. Springer Berlin/Heidelberg 3827 (2005) 1061–1070.