A note on the optimal hierarchy system of facilities

This paper treats the minisum type of optimal location-allocation problem for hierarchical system. The purpose is to clarify why a hierarchy structure of point-like facilities emerges in urbanized area. Formulated is the problem of selecting a hierarchy structure and locating each level of facility, such that the sum of the product of distance from a user to his nearest facility and frequency in use for each level of service is minimized, under the budget constraint of facility construction. First, optimal location and hierarchy structure of some numerical examples are calculated by the geographical optimization method (GOM). Optimal locational configuration shows the hierarchy, which is caused by the difference in both frequency in use and construction cost per unit of each service level. The most notable finding is that the configuration is similar to that in the central place theory in the case of that many facilities can be provided. Using this property, an analytical solution of optimal hierarchy is derived by an approximated model, secondly. The above problem becomes one kind of the optimal control problems, and the solution is represented by an optimal number function of facilities, which can explain the solution by GOM. [In Japanese.]