When closest is not always the best: The distributed p-median problem

The classical p-median problem assumes that service to customers is always provided by the closest facility, while in practice, customers often interact for a variety of reasons with several of the facilities (not just the closest). In this article, we examine the concept of a distribution rule for modelling a more general case where the demand of a customer is not entirely satisfied by its closest facility, but rather is split into different flows to different facilities according to the given rule. We use this concept to formulate a new class of median problems, which we call the ""distributed"" p-median problem. Different types of distribution rules are investigated leading to some interesting properties. For example, if the weights are increasing (ie, assigned flows are greater to facilities that are further away), the problem can be solved in polynomial time as a 1-median problem. For decreasing weights, we obtain new and efficient generalizations of the standard continuous and discrete p-median models, which in turn lead to a broader interpretation of median points and a generalization of Cooper's well-known locate-allocate heuristic. Some small numerical examples and computational results are given to illustrate the concepts.