Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-27T08:27:48.271Z Has data issue: false hasContentIssue false
  • English
  • Français

Probabilistic Analysis of Unit-Demand Vehicle Routeing Problems

Part of: Operations research and management science Algorithms - Computer Science

Published online by Cambridge University Press:  14 July 2016

Agustín Bompadre ,
Moshe Dror  and
James B. Orlin
Agustín Bompadre*
Affiliation:
Operations Research Center, MIT
Moshe Dror*
Affiliation:
Operations Research Center, MIT
James B. Orlin*
Affiliation:
Sloan School of Management, MIT
*
∗ Current address: Optiant Inc., 4 Van de Graaff Drive, Burlington, MA 01803, USA. Email address: abompadr@alum.mit.edu
∗∗ Current address: Department of Management Information Systems, Eller College of Management, University of Arizona, 430M McClelland Hall, 1130 E. Helen Street, PO Box 210108, Tucson, AZ 85721-0108, USA.
∗∗∗ Postal address: Sloan School of Management, MIT, 50 Memorial Drive, Cambridge, MA 02142, USA.
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We analyze the unit-demand Euclidean vehicle routeing problem, where n customers are modeled as independent, identically distributed uniform points and have unit demand. We show new lower bounds on the optimal cost for the metric vehicle routeing problem and analyze them in this setting. We prove that there exists a constant ĉ > 0 such that the iterated tour partitioning heuristic given by Haimovich and Rinnooy Kan (1985) is a (2 - ĉ)-approximation algorithm with probability arbitrarily close to 1 as the number of customers goes to ∞. It has been a longstanding open problem as to whether one can improve upon the factor of 2 given by Haimovich and Rinnooy Kan. We also generalize this, and previous results, to the multidepot case.

Keywords

Vehicle routeing problemheuristicprobabilistic analysis of algorithms

MSC classification

Primary: 68W40: Analysis of algorithms
Secondary: 90B06: Transportation, logistics
Type
Research Article
Information
Journal of Applied Probability , Volume 44 , Issue 1 , March 2007 , pp. 259 - 278
Copyright
Copyright © Applied Probability Trust 2007 

References

Altinkemer, K. and Gavish, B. (1990). Heuristics for delivery problems with constant error guarantees. Transportation Sci. 24, 294–297.Google Scholar
Arora, S. (1996). Polynomial time approximation schemes for Euclidean TSP and other geometric problems. In Proc. 37th Annual Symp. Foundations Comput. Sci., IEEE Computer Society Press, Los Alamitos, CA, pp. 2–11.Google Scholar
Asano, T., Katoh, N., Tamaki, H. and Tokuyama, T. (1999). Covering points in the plane by k-tours: towards a polynomial time approximation scheme for general k. In Proc. STOC '97, ACM, New York, pp. 275–283.Google Scholar
Beardwood, J., Halton, J. L. and Hammersley, J. M. (1959). The shortest path through many points. Proc. Camb. Phil. Soc. 55, 299–327.Google Scholar
Blum, A. et al. (1994). The minimum latency problem. In Proc. STOC '94, ACM, New York, pp. 163–171.CrossRefGoogle Scholar
Bompadre, A., Dror, M. and Orlin, J. B. (2006). Improved bounds for vehicle routeing solutions. Discrete Optimization 3, 299–316.Google Scholar
Few, L. (1955). The shortest path and the shortest path through n points. Mathematika 2, 141–144.Google Scholar
Haimovich, M. and Rinnooy Kan, A. H. G. (1985). Bounds and heuristics for capacitated routeing problems. Math. Operat. Res. 10, 527–542.Google Scholar
Haimovich, M., Rinnooy Kan, A. H. G. and Stougie, L. (1988). Analysis of heuristics for vehicle routing problems. In Vehicle Routing: Methods and Studies, eds Golden, B. L. and Assad, A. A., Elsevier, Amsterdam, pp. 47–61.Google Scholar
Li, C. L. and Simchi-Levi, D. (1990). Worst-case analysis of heuristics for the multidepot capacitated vehicle routeing problems. ORSA J. Comput. 2, 64–73.Google Scholar
Mitchell, J. S. B. (1996). Guillotine subdivisions approximate polygonal subdivisions: a simple new method for the geometric k-MST problem. In Proc. 7th ACM-SIAM Symp. Discrete Algorithms, ACM, New York, pp. 402–408.Google Scholar
Papadimitriou, C. H. (1977). The probabilistic analysis of matching heuristics. In Proc. 15th Annual Allerton Conf. Commun. Control Comput., University of Illinois Press, Champaign, IL, pp. 368–378.Google Scholar
Stougie, L. (1987). Design and Analysis of Algorithms for Stochastic Integer Programming (CWI Tract 37). Stichting Mathematisch Centrum, Amsterdam.Google Scholar