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On the Effectiveness of Set Covering Formulations for the Vehicle Routing Problem with Time Windows

Author

Listed:
  • Julien Bramel

    (Columbia University, New York)

  • David Simchi-Levi

    (Northwestern University, Evanston, Illinois)

Abstract

The Vehicle Routing Problem with Time Windows (VRPTW) is one of the most important problems in distribution and transportation. A classical and recently popular technique that has proven effective for solving these problems is based on formulating them as a set covering problem . The method starts by solving its linear programming relaxation , via column generation, and then uses a branch and bound strategy to find an integer solution to the set covering problem: a solution to the VRPTW. An empirically observed property is that the optimal solution value of the set covering problem is very close to its linear programming relaxation which makes the branch and bound step extremely efficient. In this paper we explain this behavior by demonstrating that for any distribution of service times, time windows, customer loads, and locations, the relative gap between fractional and integer solutions of the set covering problem becomes arbitrarily small as the number of customers increases.

Suggested Citation

  • Julien Bramel & David Simchi-Levi, 1997. "On the Effectiveness of Set Covering Formulations for the Vehicle Routing Problem with Time Windows," Operations Research, INFORMS, vol. 45(2), pages 295-301, April.
    • Handle: RePEc:inm:oropre:v:45:y:1997:i:2:p:295-301
    DOI: 10.1287/opre.45.2.295
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    Citations

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    Cited by:

    1. Nizar Hachemi & Issmail Hallaoui & Michel Gendreau & Louis-Martin Rousseau, 2015. "Flow-based integer linear programs to solve the weekly log-truck scheduling problem," Annals of Operations Research, Springer, vol. 232(1), pages 87-97, September.
    2. Oktay Günlük & Tracy Kimbrel & Laszlo Ladanyi & Baruch Schieber & Gregory B. Sorkin, 2006. "Vehicle Routing and Staffing for Sedan Service," Transportation Science, INFORMS, vol. 40(3), pages 313-326, August.
    3. Irnich, Stefan, 2000. "A multi-depot pickup and delivery problem with a single hub and heterogeneous vehicles," European Journal of Operational Research, Elsevier, vol. 122(2), pages 310-328, April.
    4. Victor Reyes & Ignacio Araya, 2021. "A GRASP-based scheme for the set covering problem," Operational Research, Springer, vol. 21(4), pages 2391-2408, December.
    5. Liu, Fuh-Hwa Franklin & Shen, Sheng-Yuan, 1999. "A route-neighborhood-based metaheuristic for vehicle routing problem with time windows," European Journal of Operational Research, Elsevier, vol. 118(3), pages 485-504, November.
    6. Jütte, Silke & Thonemann, Ulrich W., 2012. "Divide-and-price: A decomposition algorithm for solving large railway crew scheduling problems," European Journal of Operational Research, Elsevier, vol. 219(2), pages 214-223.
    7. Lap Mui Ann Chan & Ana Muriel & David Simchi-Levi, 1998. "Parallel Machine Scheduling, Linear Programming, and Parameter List Scheduling Heuristics," Operations Research, INFORMS, vol. 46(5), pages 729-741, October.
    8. Song, Jiongjiong & Regan, Amelia, 2005. "Approximation algorithms for the bid construction problem in combinatorial auctions for the procurement of freight transportation contracts," Transportation Research Part B: Methodological, Elsevier, vol. 39(10), pages 914-933, December.
    9. L Tansini & O Viera, 2006. "New measures of proximity for the assignment algorithms in the MDVRPTW," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 57(3), pages 241-249, March.
    10. Cochran, Jeffery K. & Marquez Uribe, Alberto, 2005. "A set covering formulation for agile capacity planning within supply chains," International Journal of Production Economics, Elsevier, vol. 95(2), pages 139-149, February.
    11. Stephan Westphal & Sven Krumke, 2008. "Pruning in column generation for service vehicle dispatching," Annals of Operations Research, Springer, vol. 159(1), pages 355-371, March.
    12. Du, Jia Yan & Brunner, Jens O. & Kolisch, Rainer, 2014. "Planning towing processes at airports more efficiently," Transportation Research Part E: Logistics and Transportation Review, Elsevier, vol. 70(C), pages 293-304.
    13. Jian Yang & Patrick Jaillet & Hani Mahmassani, 2004. "Real-Time Multivehicle Truckload Pickup and Delivery Problems," Transportation Science, INFORMS, vol. 38(2), pages 135-148, May.
    14. Song, Jiongjiong & Regan, A C, 2003. "Approximation Algorithms for the Bid Construction Problem in Combinatorial Auctions for the Procurement of Freight Transportation Contracts," University of California Transportation Center, Working Papers qt3j7034z2, University of California Transportation Center.
    15. Vis, Iris F.A., 2006. "Survey of research in the design and control of automated guided vehicle systems," European Journal of Operational Research, Elsevier, vol. 170(3), pages 677-709, May.

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