A Model of Optimal Spatial Allocation of Branch Offices from the Viewpoint of Communication Cost

Abstract

In the previous literature, the urban system is a result of corporate hierarchy. In reality, however, the urban system is also a determinant of allocation of branch offices. In this paper, the optimal allocation of branch offices is analyzed within a given urban system.In preparation for the analysis, an urban system is defined in section II. There are two major models of urban systems, that is, Christaller's and Pred's urban systems. However, these systems do not conflict, because Christaller's system is comprised in Pred's. Although Pred's urban system is more general, it is not capable of “identifying”the existing urban system as a particular system. Therefore, in this paper, it is supposed that urban system is Christaller's system, which is identifiable. It is necessary to suppose such a condition, in order to get the optimal locations of branch offices from the optimal number of branch offices.In section III, optimal number n* and locations of branch offices are analyzed. Suppose that the manufacturing firm has a nationwide market area of goods, a head office in the highest-ranked central place, and n(≥0) branch offices in the country. Each branch office is of the same scale. The territories of the branch offices necessarily correspond with the complementary areas of central places with a certain rank, and never intersect. And branch offices have the budgets, have no rank, and cannot contact without intervention of the head office.The roles of branch office are as follows, that is, ‘communication to head’ and ‘communication to market.’ TR and TM, respectively represent the communication costs corresponding to those two roles, and C represents operational cost of offices. Then, TR, TM and C are defined as the functions of n. And it is supposed that the behavioral principle of the firm is cost minimizing. In the case that C is a linearly increasing function of n, there are two cases about the value of n*, [see figure 1];•when the three functions are all increasing, n*=0 (named the centralized type)•when some decreasing functions are contained, n*>0.Next, this model is linked to central place theory. In the central place rank, i=0, 1, …, m, i=0 represents the highest-ranked central place, the larger i becomes, the lower the central place. Because branch offices can only be located at central place, there is a relation, that is, n=ki-1 (k means Christaller's k, k>1), between n and i. By this formula, we can decide the optimal i, that is, the arrangement of branch offices [see figure 2]. Although the value of i corresponding to n* may not be an integer, it is an optimal i that a firm usually decides in advance. The value of n* is decided on optimali later.Central place rank i indicates totally, not only the number of central places, but also their scale or arrangement. The largest merit of the approach, in which the optimal arrangement of branch offices is solved as the equation of i, is that i gives much information.