Advertisement

The Stochastic Vehicle Routing Problem for Minimum Unmet Demand

  • Zhihong Shen
  • Fernando Ordòñez
  • Maged M. Dessouky
Chapter
Part of the Springer Optimization and Its Applications book series (SOIA, volume 30)

Summary

In this chapter, we are interested in routing vehicles to minimize unmet demand with uncertain demand and travel time parameters. Such a problem arises in situations with large demand or tight deadlines so that routes that satisfy all demand points are difficult or impossible to obtain. An important application is the distribution of medical supplies to respond to large-scale emergencies, such as natural disasters or terrorist attacks. We present a chance constrained formulation of the problem that is equivalent to a deterministic problem with modified demand and travel time parameters under mild assumptions on the distribution of stochastic parameters and relate it to a robust optimization approach. A tabu heuristic is proposed to solve this MIP and simulations are conducted to evaluate the quality of routes generated from both deterministic and chance constrained formulations. We observe that chance constrained routes can reduce the unmet demand by around 2%-6% for moderately tight deadline and total supply constraints.

Keywords

Tabu Search Vehicle Rout Problem Demand Point Demand Node Unmet Demand 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

This research was supported by the United States Department of Homeland Security through the Center for Risk and Economic Analysis of Terrorism Events (CREATE), grant number EMW-2004-GR-0112. However, any opinions, findings, and conclusions or recommendations in this document are those of the author(s) and do not necessarily reflect views of the U.S. Department of Homeland Security. Also the authors wish to thank Ilgaz Sungur, Hongzhong Jia, Harry Bowman, Richard Larson and Terry O’Sullivan for their valuable input and comments for the improvement of this chapter.

Refereneces

  1. 1.
    C. Archetti, A. Hertz, and M. Speranza. A tabu search algorithm for the split delivery vehicle routing problem. Transportation Science, 40:64–73, 2006.CrossRefGoogle Scholar
  2. 2.
    A. Ben-Tal and A. Nemirovski. Robust convex optimization. Mathematics of Operations Research, 23(4):769–805, 1998.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    R. W. Bent and P. V. Hentenryck. Scenario-based planning for partially dynamic vehicle routing with stochastic customers. Operations Research, 52(6):977–987, 2004.MATHCrossRefGoogle Scholar
  4. 4.
    D. Bertsimas. Probabilistic combinational optimization problems. PhD thesis, Operation Research Center, Massachusetts Institute of Technology, Cambridge, MA, 1988.Google Scholar
  5. 5.
    D. Bertsimas. A vehicle routing problem with stochastic demand. Operations Research, 40(3), May 1992.Google Scholar
  6. 6.
    D. Bertsimas and G. Van Ryzin. A stochastic and dynamic vehicle routing problem in the Euclidean plane. Operations Research, 39(4):601–615, 1991.MATHCrossRefGoogle Scholar
  7. 7.
    D. Bertsimas and G. Van Ryzin. Stochastic and dynamic vehicle routing in the Euclidean plane with multiple capacitated vehicles. Operations Research, 41(1):60–76, 1993.MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    D. Bertsimas and M. Sim. Robust discrete optimization and network flows. Mathematical Programming, 98:49–71, 2003.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    D. Bertsimas and D. Simchi-levi. A new generation of vehicle routing research: robust algorithms, addressing uncertainty. Operations Research, 44(2), March 1996.Google Scholar
  10. 10.
    N. Bianchessi and G. Righini. Heuristic algorithms for the vehicle routing problem with simultaneous pick-up and delivery. Computers and Operations Research, 34:578–594, 2006.CrossRefGoogle Scholar
  11. 11.
    J. Branke, M. Middendorf, G. Noeth, and M. Dessouky. Waiting strategies for dynamic vehicle routing. Transportation Science, 39:298–312, 2005.CrossRefGoogle Scholar
  12. 12.
    G. Calafiore and M.C. Campi. Uncertain convex programs: randomized solutions and confidence levels. Mathematical Programming, 102:25–46, 2005.MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    R. Carraway, T. Morin, and H. Moskowitz. Generalized dynamic programming for stochastic combinatorial optimization. Operations Research, 37(5):819–829, 1989.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    X. Chen, M. Sim, and P. Sun. A robust optimization perspective on stochastic programming. Optimization Online, 2005. http://www.optimization-online.org/DB_HTML/2005/06/1152.html.
  15. 15.
    G. Clarke and J.W. Wright. Scheduling of vehicles from a central depot to a number of delivery points. Operations Research, 12:568–581, 1964.CrossRefGoogle Scholar
  16. 16.
    M. Dror. Modeling vehicle routing with uncertain demands as a stochastic program: properties of the corresponding solution. European Journal of Operational Research, 64:432–441, 1993.MATHCrossRefGoogle Scholar
  17. 17.
    M. Dror, G. Laporte, and P. Trudeau. Vehicle routing with stochastic demands: properties and solution framework. Transportation Science, 23(3), August 1989.Google Scholar
  18. 18.
    M. Dror and P. Trudeau. Stochastic vehicle routing with modified savings algorithm. European Journal of Operational Research, 23:228–235, 1986.MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    E. ErdoĞan and G. Iyengar. Ambiguous chance constrained problems and robust optimization. Mathematical Programming, 107:37–61, 2006.MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    M. Gendreau, A. Hertz, and G. Laporte. A tabu search heuristic for the vehicle routing problem. Management Science, 40:1276–1290, 1994.MATHCrossRefGoogle Scholar
  21. 21.
    M. Gendreau, G. Laporte, and J. Y. Potvin. The Vehicle Routing Problem, chapter Metaheuristics for the Capacitated VRP, pages 129–154. SIAM Monographs on Discrete Mathematics and Applications, SIAM Publishing, 2002.Google Scholar
  22. 22.
    M. Gendreau, G. Laporte, and R. Seguin. An exact algorithm for the vehicle routing problem with stochastic demands and customers. Transportation Science, 29(2), May 1995.Google Scholar
  23. 23.
    M. Gendreau, G. Laporte, and R. Seguin. Stochastic vehicle routing. European Journal of Operational Research, 88:3–12, 1996.MATHCrossRefGoogle Scholar
  24. 24.
    M. Gendreau, G. Laporte, and R. Seguin. A tabu search heuristic for the vehicle routing problem with stochastic demands and customers. Operations Research, 44(3), May 1996.Google Scholar
  25. 25.
    G. Glover and M. Laguna. Tabu Search. Kluwer, Boston, MA, 1997.MATHGoogle Scholar
  26. 26.
    E. Hadjiconstantinou and D. Roberts. The Vehicle Routing Problem, chapter Routing under Uncertainty: an Application in the Scheduling of Field Service Engineers, pages 331–352. SIAM Monographs on Discrete Mathematics and Applications, SIAM Publishing, 2002.Google Scholar
  27. 27.
    P. Jaillet. Stochastics in Combinatorial Optimization, chapter Stochastic Routing Problem. World Scientific, New Jersey, 1987.Google Scholar
  28. 28.
    P. Jaillet and A. Odoni. Vehicle Routing: Methods and Studies, chapter The Probabilistic Vehicle Routing Problem. North-Holland, Amsterdam, 1988.Google Scholar
  29. 29.
    A. Jèzèquel. Probabilistic vehicle routing problems. Master’s thesis, Department of Civil Engineering, Massachusetts Institute of Technology, 1985.Google Scholar
  30. 30.
    H. Jula, M. M. Dessouky, and P. Ioannou. Truck route planning in non-stationary stochastic networks with time-windows at customer locations. IEEE Transactions on Intelligent Transportation Systems, 2005. to appear.Google Scholar
  31. 31.
    E. Kao. A preference order dynamic program for a stochastic travelling salesman problem. Operations Research, 26:1033–1045, 1978.MATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    O. Klopfenstein and D. Nace. A robust approach to the chance-constrained knapsack problem. Optimization Online, 2006. http://www.optimization-online.org/DB_HTML/2006/03/1341.html.
  33. 33.
    V. Lambert, G. Laporte, and F. Louveaux. Designing collection routes through bank branches. Computers and Operations Research, 20:783–791, 1993.CrossRefGoogle Scholar
  34. 34.
    G. Laporte. The vehicle routing problem: an overview of exact and approximate algorithms. European Journal of Operational Research, 59:345–358, 1992.MATHCrossRefGoogle Scholar
  35. 35.
    G. Laporte, F. Laporte, and H. Mercure. Models and exact solutions for a class of stochastic location-routing problems. European Journal of Operational Research, 39:71–78, 1989.MATHCrossRefMathSciNetGoogle Scholar
  36. 36.
    G. Laporte, F. Louveaux, and H. Mercure. The vehicle routing problem with stochastic travel times. Transportation Science, 26(3), August 1992.Google Scholar
  37. 37.
    R. C. Larson. The McGraw-Hill Handbook of Homeland Security, chapter Decision Models for Emergency Response Planning. The McGraw-Hill Companies, 2005.Google Scholar
  38. 38.
    R. C. Larson, M. Metzger, and M. Cahn. Emergency response for homeland security: lessons learned and the need for analysis. Interfaces, 2005. To appear.Google Scholar
  39. 39.
    I. H. Osman. Metastrategy simulated annealing and tabu search algorithms for the vehicle routing problem. Annals of Operations Research, 41:421–451, 1993.MATHCrossRefGoogle Scholar
  40. 40.
    J. D. Papastavrou. A stochastic and dynamic routing policy using branching processes with state dependent immigration. European Journal of Operational Research, 95:167–177, 1996.MATHCrossRefGoogle Scholar
  41. 41.
    N. Secomandi. A rollout policy for the vehicle routing problem with stochastic demands. Operations Research, 49(5):796–802, 2001.MATHCrossRefGoogle Scholar
  42. 42.
    W. Stewart and B. Golden. Stochastic vehicle routing: a comprehensive approach. European Journal of Operational Research, 14:371–385, 1983.MATHCrossRefGoogle Scholar
  43. 43.
    I. Sungur, Fernando Ordòñez, and Maged M. Dessouky. A robust optimization approach for the capacitated vehicle routing problem with demand uncertainty. Technical report, Deniel J. Epstein Department of Industrial and Systems Engineering, University of Southern California, 2006.Google Scholar
  44. 44.
    M. R. Swihart and J. D. Papastavrou. A stochastic and dynamic model for the single-vehicle pick-up and delivery problem. European Journal of Operational Research, 114:447–464, 1999.MATHCrossRefGoogle Scholar
  45. 45.
    E. D. Taillard, P. Badeau, M. Gendreau, F. Guertin, and J. Y. Potvin. A tabu search heuristic for the vehicle routing problem with soft time windows. Transportation Science, 31:170–186, 1997.MATHCrossRefGoogle Scholar
  46. 46.
    F. Tillman. The multiple terminal delivery problem with probabilistic demands. Transportation Science, 3:192–204, 1969.CrossRefGoogle Scholar
  47. 47.
    P. Toth and D. Vigo. The granular tabu search (and its application to the vehicle routing problem). Technical Report OR/98/9, DEIS, Universitá di Bologna, Italy, 1998.Google Scholar
  48. 48.
    P. Toth and D. Vigo. The Vehicle Routing Problem. SIAM Monographs on Discrete Mathematics and Applications, SIAM Publishing, 2002.Google Scholar
  49. 49.
    J. A. G. Willard. Vehicle routing using r-optimal tabu search. Master’s thesis, The Management School, Imperial College, London, 1989.Google Scholar
  50. 50.
    J. Xu and J. P. Kelly. A network flow-based tabu search heuristic for the vehicle routing problem. Transportation Science, 30:379–393, 1996.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag US 2009

Authors and Affiliations

  • Zhihong Shen
    • 1
  • Fernando Ordòñez
    • 1
  • Maged M. Dessouky
    • 1
  1. 1.Department of Industrial and Systems EngineeringUniversity of Southern CaliforniaLos Angeles

Personalised recommendations