Generalization of Central Place Theory in Hierarchical Structures Using Mathematical Programming

Abstract

This paper examines the optimality of Lösch's central place system and the difference between the objectives of Lösch (1940) and Christaller (1933) in a hierarchical structure. Superposition of hexagonal networks of the Lösch system can be formulated as combinatorial optimization using mathematical programming. Beavon and Mabin (1975) demonstrated that there are two objects of the maximization of coincidence of supply points and the production of city-rich and city-poor sectors in the Lösch system. The multiobjective function of Eq. (1) contains the above two objects, and the model seeks the optimal combination of hexagonal networks from an enormous number of possible central place systems. Compared with the Lösch system and the solution of the model in the case of 150 market areas, it was shown that the Lösch system is not optimal to produce city-rich sectors. This is because of a locally optimal solution of hierarchical structure that was obtained by a heuristic method for superimposing hexagonal networks.

Lösch's original purpose was to maximize the degree to which firms agglomerate, and the production of city-rich sectors was simply a means to an end. In this paper, the objective function of the agglomeration effect is formulated so as to minimize the number of missing goods of lower order than a hierarchical marginal good. The result of application to the case of 55 market areas indicates that the system derived from a model has a higher effect of agglomeration than the Lösch system. On the other hand, the Christaller system according to the marketing principle is regarded as an optimal solution that agglomerates all goods at each supply point because of a successively inclusive hierarchy.

This paper develops a generalized model for the hierarchical structure of the central place system in Eq. (17), which consists of the agglomeration effect and the locational principle of a single good. Based on the generalized model, we can reinterpret the priority objective of the Christaller model as the former in contrast to the Lösch system which is oriented toward individual entrepreneurial behavior as the latter objective. Although both systems are considered as exceptional in a hierarchical structure, the real world is full of central place systems based on a variety of hierarchical structures that are not only successively inclusive and successively exclusive. The generalized model using multiobjective programming in this paper will be able to derive non-inferior solutions consisting of various hierarchical structures of the central place system.