Configuration of Physical Distribution Networks

Distribution problems, including vehicle routing and warehouse location problems, are usually formulated by considering a finite number of possible locations for the customers, the warehouses, and vehicle stops. The question of selecting which of these points are actually used (and how) is a mixed-integer programming problem which is difficult to solve. Thus, such a discrete formulation results in a problem that has to be solved heuristically; it also entails a large data preparation effort each time a solution has to be developed in response to changing world conditions. The continuous approach used in this paper attempts to circumvent some of these drawbacks. We consider one source and its customers in a service area; customer locations are modeled by a density surface over the service area. With this information, and data about the cost of inventory and transportation, we can determine the number of transhipment points, and the frequency and routing of all the distribution vehicles. An example is given. The continuous approach does not yield a solution. It gives design guidelines, which ensure near minimum total cost. These design guidelines are based on general properties of optimal solutions (discussed at the beginning of the paper) and on the specific characteristics of the case at hand. Implementation of the guidelines to obtain a feasible configuration requires human intervention. While the continuous method involves approximations (the real world is discrete and considerably more complicated than in our model), it yields insight into the structure of logistic systems. This insight should not only help in the design process; it may well also lead to improved heuristic solution methods for discrete formulations. Hybrid methods may eventually emerge.