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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
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    References listed on IDEAS

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    Cited by:

    1. 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.
    2. 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.

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