*Forma,* Vol. 23 (No. 2), pp. 89-95, 2008

*Original Paper*

## Analysis of Facility Location Using Ordered Rectilinear Distance in Regular Point Patterns

Masashi Miyagawa

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

E-mail address: mmiyagawa@yamanashi.ac.jp

(Received May 26, 2008; Accepted October 27, 2008)

**Abstract. **
This paper deals with the kth nearest rectilinear distance of two regular point patterns: square and diamond lattices. The probability density functions of the kth nearest rectilinear distance are theoretically derived for *k* = 1, 2, . . . , 8. Upper and lower bounds of the kth nearest distance are also derived. As an application of the kth nearest distance, we consider a facility location problem with closing of facilities. The objective is to find the best configuration of facilities that minimizes the average rectilinear distance from residents to their nearest open facility when some existing facilities are closed. Assuming that facilities are closed independently and at random, we show that the diamond lattice is the best if at least 73% of facilities are open.

**Keywords: **
kth Nearest Distance, Regular Point Patterns, Facility Location, Average Distance

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