Optimally locating p facilities to serve customer demand at n>p locations is a simplified description of many locational analysis problems (many of which may be formulated as p-median problems — see further, Section 7.4.2, Larger p-median and p-center problems). If the existence of a network is ignored, and connectivity is assumed to be by direct connection in the plane from each customer to a single, closest, facility then a simple location-allocation problem is defined — selecting locations and then allocating customers or demand to these locations according to some rule (e.g. nearest facility). Frequently, however, a network (e.g. a road network) exists together with a matrix of shortest path/least cost distances between vertices, and this provides a key input into the optimization process.
Real world problems are, of course, far more complex, since there are many other variables to be considered (e.g. availability of suitable sites, cost of sites, size of facility, access to and from sites, regulatory issues, planning controls, availability of suitable labor, timing of developments, etc.) and all of these considerations exist within a dynamic environment that affects these and core variables such as customer demand patterns, materials supply and changes in the technological, commercial and political environment. Hence optimization is only a part of a much larger decision-making process, whose importance will vary depending on the particular problem at hand (for an excellent recent discussion of some of these issues, see O’Kelly, 2008). Obtaining near (provably) optimum solutions to many problems, and understanding their robustness to variation in parameters and locational changes is often at least as important as seeking absolute optima for problems that have been very narrowly defined. Such procedures may also provide a form of benchmarking, identifying the cost associated with the “best” solution, which may then be compared with other options within a broader decision-making framework.