### 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.

Original language | English (US) |
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Pages (from-to) | 166-174 |

Number of pages | 9 |

Journal | Transportation Science |

Volume | 42 |

Issue number | 2 |

DOIs | |

State | Published - May 2008 |

### Keywords

- Arc routing
- Eulerian tour problem
- Stochastic demand

### ASJC Scopus subject areas

- Civil and Structural Engineering
- Transportation

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## Cite this

*Transportation Science*,

*42*(2), 166-174. https://doi.org/10.1287/trsc.1080.0232