### Abstract

Different strategies can be used to find a straight cable buried underground. The original problem considered by Faber et al. focused on a telephone company that wishes to dig a trench to locate a straight cable. The cable is known to pass within a given distance, a, from a marker erected above the putative location of the cable. Faber et al. showed that the shortest simply connected curve guaranteed to find the cable is a U-shaped curve whose length is about 18% less than that of a circular trench of radius a. This problem can be regarded as minimizing the maximum length that a trench digger must dig. In reality, once the cable is found, digging can stop. So far, however, no attempt has been made to evaluate the trench shape on characteristics other than the maximum trench length. In this paper, we present geometric probability models to analytically derive the distribution of trench length and calculate the expected value and variance for both the short-length (U-shaped) trench and a circular trench. Our main result is that the expected digging length is about 5% less for the circular trench than for the U-shaped trench.

Original language | English |
---|---|

Pages (from-to) | 400-417 |

Number of pages | 18 |

Journal | Journal of the Operations Research Society of Japan |

Volume | 60 |

Issue number | 3 |

Publication status | Published - 2017 Jul 1 |

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### Keywords

- Applied probability
- Beam detector
- Digging length distribution
- Geometric probability
- Urban infrastructure
- Urban operations research

### ASJC Scopus subject areas

- Decision Sciences(all)
- Management Science and Operations Research

### Cite this

*Journal of the Operations Research Society of Japan*,

*60*(3), 400-417.

**Geometric probability models to analyze strategies for finding a buried cable.** / Tanaka, Kenichi; Shiina, Kana.

Research output: Contribution to journal › Article

*Journal of the Operations Research Society of Japan*, vol. 60, no. 3, pp. 400-417.

}

TY - JOUR

T1 - Geometric probability models to analyze strategies for finding a buried cable

AU - Tanaka, Kenichi

AU - Shiina, Kana

PY - 2017/7/1

Y1 - 2017/7/1

N2 - Different strategies can be used to find a straight cable buried underground. The original problem considered by Faber et al. focused on a telephone company that wishes to dig a trench to locate a straight cable. The cable is known to pass within a given distance, a, from a marker erected above the putative location of the cable. Faber et al. showed that the shortest simply connected curve guaranteed to find the cable is a U-shaped curve whose length is about 18% less than that of a circular trench of radius a. This problem can be regarded as minimizing the maximum length that a trench digger must dig. In reality, once the cable is found, digging can stop. So far, however, no attempt has been made to evaluate the trench shape on characteristics other than the maximum trench length. In this paper, we present geometric probability models to analytically derive the distribution of trench length and calculate the expected value and variance for both the short-length (U-shaped) trench and a circular trench. Our main result is that the expected digging length is about 5% less for the circular trench than for the U-shaped trench.

AB - Different strategies can be used to find a straight cable buried underground. The original problem considered by Faber et al. focused on a telephone company that wishes to dig a trench to locate a straight cable. The cable is known to pass within a given distance, a, from a marker erected above the putative location of the cable. Faber et al. showed that the shortest simply connected curve guaranteed to find the cable is a U-shaped curve whose length is about 18% less than that of a circular trench of radius a. This problem can be regarded as minimizing the maximum length that a trench digger must dig. In reality, once the cable is found, digging can stop. So far, however, no attempt has been made to evaluate the trench shape on characteristics other than the maximum trench length. In this paper, we present geometric probability models to analytically derive the distribution of trench length and calculate the expected value and variance for both the short-length (U-shaped) trench and a circular trench. Our main result is that the expected digging length is about 5% less for the circular trench than for the U-shaped trench.

KW - Applied probability

KW - Beam detector

KW - Digging length distribution

KW - Geometric probability

KW - Urban infrastructure

KW - Urban operations research

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UR - http://www.scopus.com/inward/citedby.url?scp=85026481978&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:85026481978

VL - 60

SP - 400

EP - 417

JO - Journal of the Operations Research Society of Japan

JF - Journal of the Operations Research Society of Japan

SN - 0453-4514

IS - 3

ER -