IDEAS home Printed from https://ideas.repec.org/a/inm/ortrsc/v42y2008i2p166-174.html
   My bibliography  Save this article

The Stochastic Eulerian Tour Problem

Author

Listed:
  • Srimathy Mohan

    (School of Global Management and Leadership, Arizona State University, Phoenix, Arizona 85069)

  • Michel Gendreau

    (Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT), Université de Montréal, Montréal, Québec, H3C 3J7 Canada)

  • Jean-Marc Rousseau

    (Interuniversity Research Centre on Enterprise Networks, Logistics and Transportation (CIRRELT), Université de Montréal, Montréal, Québec, H3C 3J7 Canada)

Abstract

This paper defines the stochastic Eulerian tour problem (SETP) and investigates several characteristics of this problem. Given an undirected Eulerian graph G = ( V, E ), a subset R (| R | = n ) of the edges in E that require service, and a probability distribution for the number of edges in R that have to be visited in any given instance of the graph, the SETP seeks an a priori Eulerian tour of minimum expected length. We derive a closed-form expression for the expected length of a given Eulerian tour when the number of required edges that have to be visited follows a binomial distribution. We also show that the SETP is NP-hard, even though the deterministic counterpart is solvable in polynomial time. We derive further properties and a worst-case ratio of the deviation of the expected length of a random Eulerian tour from the expected length of the optimal tour. Finally, we present some of the desirable properties in a good a priori tour using illustrative examples.

Suggested Citation

  • Srimathy Mohan & Michel Gendreau & Jean-Marc Rousseau, 2008. "The Stochastic Eulerian Tour Problem," Transportation Science, INFORMS, vol. 42(2), pages 166-174, May.
    • Handle: RePEc:inm:ortrsc:v:42:y:2008:i:2:p:166-174
    DOI: 10.1287/trsc.1080.0232
    as

    Download full text from publisher

    File URL: http://dx.doi.org/10.1287/trsc.1080.0232
    Download Restriction: no
    File URL: https://libkey.io/10.1287/trsc.1080.0232?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    References listed on IDEAS

    as
    1. H. A. Eiselt & Michel Gendreau & Gilbert Laporte, 1995. "Arc Routing Problems, Part II: The Rural Postman Problem," Operations Research, INFORMS, vol. 43(3), pages 399-414, June.
    2. Patrick Jaillet, 1988. "A Priori Solution of a Traveling Salesman Problem in Which a Random Subset of the Customers Are Visited," Operations Research, INFORMS, vol. 36(6), pages 929-936, December.
    3. Laporte, Gilbert & Louveaux, Francois & Mercure, Helene, 1989. "Models and exact solutions for a class of stochastic location-routing problems," European Journal of Operational Research, Elsevier, vol. 39(1), pages 71-78, March.
    4. Bertsimas, Dimitris & Howell, Louis H., 1993. "Further results on the probabilistic traveling salesman problem," European Journal of Operational Research, Elsevier, vol. 65(1), pages 68-95, February.
    5. G Fleury & P Lacomme & C Prins & W Ramdane-Chérif, 2005. "Improving robustness of solutions to arc routing problems," Journal of the Operational Research Society, Palgrave Macmillan;The OR Society, vol. 56(5), pages 526-538, May.
    6. H. A. Eiselt & Michel Gendreau & Gilbert Laporte, 1995. "Arc Routing Problems, Part I: The Chinese Postman Problem," Operations Research, INFORMS, vol. 43(2), pages 231-242, April.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Si Chen & Bruce Golden & Richard Wong & Hongsheng Zhong, 2009. "Arc-Routing Models for Small-Package Local Routing," Transportation Science, INFORMS, vol. 43(1), pages 43-55, February.
    2. Mohan, Srimathy & Gendreau, Michel & Rousseau, Jean-Marc, 2010. "Heuristics for the Stochastic Eulerian Tour Problem," European Journal of Operational Research, Elsevier, vol. 203(1), pages 107-117, May.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Si Chen & Bruce Golden & Richard Wong & Hongsheng Zhong, 2009. "Arc-Routing Models for Small-Package Local Routing," Transportation Science, INFORMS, vol. 43(1), pages 43-55, February.
    2. Ann M. Campbell & Barrett W. Thomas, 2008. "Probabilistic Traveling Salesman Problem with Deadlines," Transportation Science, INFORMS, vol. 42(1), pages 1-21, February.
    3. Gendreau, Michel & Laporte, Gilbert & Seguin, Rene, 1996. "Stochastic vehicle routing," European Journal of Operational Research, Elsevier, vol. 88(1), pages 3-12, January.
    4. Astrid S. Kenyon & David P. Morton, 2003. "Stochastic Vehicle Routing with Random Travel Times," Transportation Science, INFORMS, vol. 37(1), pages 69-82, February.
    5. Wen-Huei Yang & Kamlesh Mathur & Ronald H. Ballou, 2000. "Stochastic Vehicle Routing Problem with Restocking," Transportation Science, INFORMS, vol. 34(1), pages 99-112, February.
    6. Corberan, A. & Sanchis, J. M., 1998. "The general routing problem polyhedron: Facets from the RPP and GTSP polyhedra," European Journal of Operational Research, Elsevier, vol. 108(3), pages 538-550, August.
    7. John Sniezek & Lawrence Bodin, 2006. "Using mixed integer programming for solving the capacitated arc routing problem with vehicle/site dependencies with an application to the routing of residential sanitation collection vehicles," Annals of Operations Research, Springer, vol. 144(1), pages 33-58, April.
    8. Grunert, Tore & Sebastian, Hans-Jurgen, 2000. "Planning models for long-haul operations of postal and express shipment companies," European Journal of Operational Research, Elsevier, vol. 122(2), pages 289-309, April.
    9. Arbib, Claudio & Servilio, Mara & Archetti, Claudia & Speranza, M. Grazia, 2014. "The directed profitable location Rural Postman Problem," European Journal of Operational Research, Elsevier, vol. 236(3), pages 811-819.
    10. Akbari, Vahid & Salman, F. Sibel, 2017. "Multi-vehicle synchronized arc routing problem to restore post-disaster network connectivity," European Journal of Operational Research, Elsevier, vol. 257(2), pages 625-640.
    11. Barbara De Rosa & Gennaro Improta & Gianpaolo Ghiani & Roberto Musmanno, 2002. "The Arc Routing and Scheduling Problem with Transshipment," Transportation Science, INFORMS, vol. 36(3), pages 301-313, August.
    12. Ghiani, Gianpaolo & Improta, Gennaro, 2000. "An efficient transformation of the generalized vehicle routing problem," European Journal of Operational Research, Elsevier, vol. 122(1), pages 11-17, April.
    13. Lawrence Bodin & Aristide Mingozzi & Roberto Baldacci & Michael Ball, 2000. "The Rollon–Rolloff Vehicle Routing Problem," Transportation Science, INFORMS, vol. 34(3), pages 271-288, August.
    14. Dewil, R. & Vansteenwegen, P. & Cattrysse, D. & Van Oudheusden, D., 2015. "A minimum cost network flow model for the maximum covering and patrol routing problem," European Journal of Operational Research, Elsevier, vol. 247(1), pages 27-36.
    15. Angel Corberán & Gustavo Mejía & José M. Sanchis, 2005. "New Results on the Mixed General Routing Problem," Operations Research, INFORMS, vol. 53(2), pages 363-376, April.
    16. Albareda-Sambola, Maria & Fernandez, Elena & Laporte, Gilbert, 2007. "Heuristic and lower bound for a stochastic location-routing problem," European Journal of Operational Research, Elsevier, vol. 179(3), pages 940-955, June.
    17. Jesica Armas & Peter Keenan & Angel A. Juan & Seán McGarraghy, 2019. "Solving large-scale time capacitated arc routing problems: from real-time heuristics to metaheuristics," Annals of Operations Research, Springer, vol. 273(1), pages 135-162, February.
    18. Soumia Ichoua & Michel Gendreau & Jean-Yves Potvin, 2006. "Exploiting Knowledge About Future Demands for Real-Time Vehicle Dispatching," Transportation Science, INFORMS, vol. 40(2), pages 211-225, May.
    19. Chefi Triki, 2017. "Solving the Periodic Edge Routing Problem in the Municipal Waste Collection," Asia-Pacific Journal of Operational Research (APJOR), World Scientific Publishing Co. Pte. Ltd., vol. 34(03), pages 1-13, June.
    20. Kelley, Jason & Kuby, Michael & Sierra, Rodrigo, 2013. "Transportation network optimization for the movement of indigenous goods in Amazonian Ecuador," Journal of Transport Geography, Elsevier, vol. 28(C), pages 89-100.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inm:ortrsc:v:42:y:2008:i:2:p:166-174. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Chris Asher (email available below). General contact details of provider: https://edirc.repec.org/data/inforea.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.