Continuous Location Models

Abstract

Models discussed in this chapter fall into various classes, based on the dimensionality of the space under consideration. The simplest such type of problem involves a line, a one-dimensional market, along which retail market outlets are to be located, or, similarly, a train route along which stations are located. More common and realistic are location problems in the two-dimensional Cartesian plane. It is usually assumed that the plane is unbounded and that the location of demand points and facilities can be described by their coordinates. A few studies have dealt with continuous spaces other than the plane; most prominent among them is the surface of a simple sphere. Locating facilities in three or more dimensions is also possible. An application of a location problem in a d-dimensional space is a clustering problem involving biological genotypes; for details see Cooper (1973). An example of this type of problem outside of clustering and pattern recognition is the positioning of satellites. Typical problems involve the number and position of satellites in order to optimize global surveillance or communication. A microscopic application relates to medicine where a radiological device is implanted to kill a tumor. The best position for such a device can be structured as a three-dimensional location problem.