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Complexity and Reducibility of the Skip Delivery Problem

Author

Listed:
  • C. Archetti

    (Department of Quantitative Methods, University of Brescia, C.da S. Chiara 50, I-25122 Brescia, Italy)

  • R. Mansini

    (Department of Electronics for Automation, University of Brescia, via Branze 38, I-25123 Brescia, Italy)

  • M. G. Speranza

    (Department of Quantitative Methods, University of Brescia, C.da S. Chiara 50, I-25122 Brescia, Italy)

Abstract

In the skip delivery problem (SDP), a fleet of vehicles must deliver skips to a set of customers. Each vehicle has a maximum capacity of two skips, and has to start and end its tour at a central depot. The demand of each customer can be greater than the capacity of the vehicles. The objective is to minimize the cost of the total distance traveled by the vehicles to serve all the customers. We show that the SDP is solvable in polynomial time, while its generalization to the case where all vehicles have a capacity greater than two, known as the split delivery vehicle routing problem (SDVRP), is shown to be NP-hard, even under restricted conditions on the costs. We also show that, if the costs are symmetrical and satisfy the triangle inequality, the SDP is reducible in polynomial time to a problem of possibly smaller size, where each customer has unitary demand. This property allows a remarkable simplification of the problem.

Suggested Citation

  • C. Archetti & R. Mansini & M. G. Speranza, 2005. "Complexity and Reducibility of the Skip Delivery Problem," Transportation Science, INFORMS, vol. 39(2), pages 182-187, May.
    • Handle: RePEc:inm:ortrsc:v:39:y:2005:i:2:p:182-187
    DOI: 10.1287/trsc.1030.0084
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    References listed on IDEAS

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    1. Éric Gourdin & Martine Labbé & Gilbert Laporte, 2000. "The Uncapacitated Facility Location Problem with Client Matching," Operations Research, INFORMS, vol. 48(5), pages 671-685, October.
    2. Moshe Dror & Pierre Trudeau, 1989. "Savings by Split Delivery Routing," Transportation Science, INFORMS, vol. 23(2), pages 141-145, May.
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    Cited by:

    1. C. Archetti & M. G. Speranza & A. Hertz, 2006. "A Tabu Search Algorithm for the Split Delivery Vehicle Routing Problem," Transportation Science, INFORMS, vol. 40(1), pages 64-73, February.
    2. Alberto Ceselli & Giovanni Righini & Matteo Salani, 2009. "A Column Generation Algorithm for a Rich Vehicle-Routing Problem," Transportation Science, INFORMS, vol. 43(1), pages 56-69, February.
    3. Fleming, Christopher L. & Griffis, Stanley E. & Bell, John E., 2013. "The effects of triangle inequality on the vehicle routing problem," European Journal of Operational Research, Elsevier, vol. 224(1), pages 1-7.
    4. José-Manuel Belenguer & Enrique Benavent & Nacima Labadi & Christian Prins & Mohamed Reghioui, 2010. "Split-Delivery Capacitated Arc-Routing Problem: Lower Bound and Metaheuristic," Transportation Science, INFORMS, vol. 44(2), pages 206-220, May.
    5. Fernanda Menezes & Oscar Porto & Marcelo L. Reis & Lorenza Moreno & Marcus Poggi de Aragão & Eduardo Uchoa & Hernán Abeledo & Nelci Carvalho do Nascimento, 2010. "Optimizing Helicopter Transport of Oil Rig Crews at Petrobras," Interfaces, INFORMS, vol. 40(5), pages 408-416, October.
    6. Jan Pelikán & Jan Fábry, 2012. "Heuristics for routes generation in pickup and delivery problem," Central European Journal of Operations Research, Springer;Slovak Society for Operations Research;Hungarian Operational Research Society;Czech Society for Operations Research;Österr. Gesellschaft für Operations Research (ÖGOR);Slovenian Society Informatika - Section for Operational Research;Croatian Operational Research Society, vol. 20(3), pages 463-472, September.
    7. Claudia Archetti & M. Grazia Speranza & Martin W. P. Savelsbergh, 2008. "An Optimization-Based Heuristic for the Split Delivery Vehicle Routing Problem," Transportation Science, INFORMS, vol. 42(1), pages 22-31, February.
    8. Claudia Archetti & Martin W. P. Savelsbergh & M. Grazia Speranza, 2006. "Worst-Case Analysis for Split Delivery Vehicle Routing Problems," Transportation Science, INFORMS, vol. 40(2), pages 226-234, May.
    9. U Derigs & B Li & U Vogel, 2010. "Local search-based metaheuristics for the split delivery vehicle routing problem," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 61(9), pages 1356-1364, September.

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