Coverage of Network-Based Structures: Paths, Tours and Trees

Generally speaking, location analysis and modeling is often characterized as a network based problem. Classic location books have reinforced this characterization, such as that by Handler and Mirchandani (1979) titled Location on Networks and that of Daskin (1995) titled Network and Discrete Location, as well as many other books and papers (the seminal work of Hakimi 1965, likely influenced this too). In fact, location modeling tools in ArcGIS, as an example, are part of the Network Analyst toolbox and strictly require an underlying network in order to structure and solve associated location (coverage) models. However, as reflected in the much of this book, there is no inherent need or assumption that location decision making be restricted to a network, even if attribute data is derived from a corresponding network. This chapter deviates from much of the book in this respect, and assumes the underlying representation of an analysis region, including demand and potential facility locations, is based on a network. Most network location problems involve the positioning of one or more facilities at nodes or along arcs in order to optimize a level of service or access. Often, however, site selection is restricted to the nodes of the network. In the early 1980s Morgan and Slater (1980), Current (1981), and Slater (1982) broke new ground within location science when they independently suggested that facilities may be represented by some type of network structure. Examples include a path, a tree, a tour, or even the network itself. Slater (1982) suggested that facilities “… can be of an extended nature, rather than occupying a single point of a network.” They suggested, for example, that facilities may be modeled as network paths representing railroad lines, pipelines, or transit routes. Slater (1982) proposed that four classes of locational problems should be considered within the context of network location: point-serves-point, point-serves-structure, structure-serves-point, and structure-serves-structure (see also Slater 1981, 1983). In this chapter we address the problem of designing a structure to serve points. One of the first examples in the literature is that of Morgan and Slater (1980) who defined the “core” of a tree network as the shortest path connecting two endpoints of the tree, minimizing the sum of distances to all other nodes of the tree.