A General Framework for Local Search Applied to the Continuous p-Median Problem

The basic aim of continuous location problems is to generate facility sites in a given continuous space, usually the Euclidean plane, in order to optimize some performance measure relative to a set of given points (customers). From a mathematical perspective, we may attribute the start of continuous location theory to Fermat who suggested the problem of locating a point in the plane that minimizes the sum of distances to three given points. The idea was generalized in an applied sense by Weber (1909) a few centuries later by extending the problem to any number “n” of given (or fixed) points representing markets and associating weights (or demands) to these points. The objective function, a weighted sum of distances from the facility to the given markets, now measured the cost of delivering goods to the markets as a function of the facility location. This function is known to be convex for any distance norm and hence amenable to solution by local descent methods. One such method developed for Euclidean distance, the well-known single-point iterative scheme by Weiszfeld (1936), has received much attention in the literature, including for example, seminal papers by Kuhn (1973) and Katz (1974), which studied the global and local convergence properties of this method. For further reading on the rich history of the continuous single-facility minisum (or 1-median) location problem, see Wesolowsky (1993) and Drezner et al. (2002).