Refereed, Geographical review of Japan, Series B., The Association of Japanese Geographers, 立地-配分モデルによるクリスタラー中心地理論の定式化の試み, ISHIZAKI Kenji, This paper reconstructs the marketing principle of Christaller's central place theory using the location-allocation model. According to the marketing principle, the system of central places is constructed in terms of the upper limit of the range by satisfying the following constraints :
(a) All consumers must be supplied with all goods from central places.
(b) The central places at a higher-level offer not only same order goods but also all the other goods of a lower-level (i. e., successively inclusive).
In this way, Christaller's requirement organizes the system of central places which is based upon the condition of a minimum number of central places (Saey, 1973). Therefore, the marketing principle is formulated as the set-covering problem which minimizes the number of central places required to satisfy the demand for coverage subject to the maximum distance constraint. The model can be represented as a hierarchical programming (Daskin and Stern, 1981),
miniZ
m=WΣx
jmj+ΣjS
im (1)
subject to
Σja
ijmx
jm-S
im=1 (2)
x
jm-x
j(m-1)_??_0 (m=2, ……, L) (3)
S
im_??_0 (4)
where d
ij=the distance from demand node i to potential center site j; R
m=the range of good m, W=a very large positive weight;a
ijm=1 if the distance from i to j is less than or equal to R
m, 0 otherise; x
jm=1 if good m is offered from node j, 0 otherwise; S
im=number of additional central places capable of serving node i with good m.
The objective function (1) has two hierarchical objectives : the primary objective is to minimize the number of central places; the secondary objective is to minimize the extent of market overlap. Constraints (2) and (4) define the above constraint (a) in Christaller's theory. Constraint (3) insures that the hierarchy is successively inclusive (i. e., the above constraint (b)).
The model is applied to a hypothetical lattice network (Fig. 5) which is composed of two parts : an inner network where the entry of central places is possible and all nodes must be served from central places; and an outer network where nodes are not potential center sites but demand nodes that do not have, to covered with all goods. To build the hierarchy, both the top-down method and the bottom-up method (Fig. 3) are applied. Results are summarized as follows :
(1) As can be seen in Fig. 8 and Table 3, the system of central places generated by the bottom-up method is that of the K =3 system developed by Christaller. The solution result using the top-down method (Fig. 7 and Table 2), however, clearly does not fit Christaller's K =3 system. The system generated by the latter method reveals a complicated configuration of central places and has a number of hierarchical levels.
(2) As compared with the result by the top-down method, we notice that the system built using the bottom-up method is characterized by a concentration of functions (Fig. 9) and, therefore, shows efficient hierarchical organization.
Consequently, the above model is valid only if it is constructed by the bottom-up method. In order to generate the K=3 system by the top-down method, it is necessary to reinterpret Christaller's objective function., Oct. 1992, 65A, 10, 747-768, 768