8. Continuous Space Coverage

An important distinction in location analysis and modeling has long been discrete versus continuous approaches. In previous chapters, for the most part, the reviewed coverage problems have been discrete in the sense that the places at which a facility may be sited are known and finite in number, and the demand locations to be served are also known and finite. This has enabled discrete integer programming formulations of models to be developed, allowing for efficient and exact solution in many cases. In some circumstances, however, neither potential facility sites nor demand locations are necessarily known and finite. Thus, one aspect of a continuous space location model is that facilities may be sited anywhere. An example is depicted in Fig. 8.1 where all locations within the region are feasible. The implication is that there are an infinite number of potential facility locations to be considered, in contrast to an assumed finite set of potential locations in discrete approaches (see Chap. 2). Another aspect of a continuous space problem is that demand too is not limited to a finite set of locations. Rather, demand is assumed to be continuously distributed across geographic space, varying over a study area. One example of this is shown in Fig. 8.2, where the height of the surface reflects demand for service. As is evident in the figure, some level of demand can be observed everywhere and this varies across space. Another example is given in Fig. 8.3. The Census unit color reflects the amount of demand in each area. Within a unit the demand varies in some manner, but given limited knowledge, a small geographic area and relative homogeneity, it is often thought that demand is uniformly distributed in the unit. Thus, Fig. 8.3 reflects discontinuities in demand variability, but it remains varying across space. Irrespective of representation, the implication is that demand for service is everywhere, and in some cases can possibly be defined/described by a mathematical function. From a practical standpoint, the study area could be a demand region, collection of areal units, set of line segments, or any group of spatial objects. This clearly makes dealing with demand and its geographic variability particularly challenging, especially when compared to a discrete representation view based on points. This contrasting view can be observed in Fig. 8.4 as centroids for each Census unit are identified, and demand is assumed to occur at these precise points in traditional modeling approaches.