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Rectilinear distance to a facility in the presence of a square barrier

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Abstract

This paper derives analytical expressions for the rectilinear distance to a facility in the presence of a square barrier. The distribution of the barrier distance is derived for two regular patterns of facilities: square and diamond lattices. This distribution, which provides all the information about the barrier distance, will be useful for facility location problems with barriers and reliability analysis of facility location. The distribution of the barrier distance demonstrates how the location and the size of the barrier affect the barrier distance. A numerical example shows that the total barrier distance increases as the barrier gets closer to a facility, whereas the maximum barrier distance increases as the barrier becomes greater in size.

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Acknowledgements

This work was supported by Grant-in-Aid for Young Scientists (B) (22710140). I am grateful to two anonymous reviewers for helpful comments and suggestions.

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Authors and Affiliations

  1. Department of Ecosocial System Engineering, University of Yamanashi, 4-3-11 Takeda, Kofu, Yamanashi, 400-8511, Japan

    Masashi Miyagawa

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  1. Masashi Miyagawa
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Correspondence to Masashi Miyagawa.

Appendix

Appendix

1.1 A.1 Distribution of the shortest distance

Square lattice

(13)

where

Diamond lattice

(14)

where

1.2 A.2 Distribution of the barrier distance with flexible allocation

Square lattice

(15)

where

Diamond lattice

(16)

where

1.3 A.3 Distribution of the barrier distance with fixed allocation

Square lattice

(17)

where

Diamond lattice

(18)

where

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Miyagawa, M. Rectilinear distance to a facility in the presence of a square barrier. Ann Oper Res 196, 443–458 (2012). https://doi.org/10.1007/s10479-012-1063-z

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