Generalization of Central Place Theory as Location Problems of Single Good Using Mathematical Programming

Abstract

The purpose of this paper is to reinterpret central place theory using mathematical programming to clarify the ambiguities of explanations by Christaller (1933) and Lösch (1940). First, the model of Lösch’s theory is developed based on a location-allocation model by Kuby (1989). The model is reformulated as a total demand maximization problem, as opposed to Kuby’s (1989) objective function, which is a maximization of the number of firms. Equation (1) is an objective function where a_i is the population of demand node i, q_ij is the demand cone with elasticity, and X_ij is the fraction of demand of the ith consumer allocated to the jth firm. The different points from Kuby’s model take into consideration the closest assignment constraint in Eq. (5) where Y_j is 1 or 0 if a firm locates at node j or not, and the indifferent principle in Eqs. (6) and (7) shows that consumers will equally allocate to equidistant firms. The model is tested on hypothetical linear market (Fig. 4) and plane market (Fig. 5) with uniform and nonuniform distributions of population (Fig. 3). Solutions with different market areas are generated by changing threshold values. In particular, there are notable solutions to nonuniform population distributions (Fig. 4-c, d and Fig. 5-d) which are similar to the cobweb-like central place system of Isard (1956).

Second, the generalized model of the total demand maximization problem and total profit maximization problem as the objective opposite to the former is developed by multiobjective programming. Three objectives of the generalized model imply the maximal covering location problem, the median problem, and the set cover problem (fixed charge model) in Eq. (16), where Q is the maximal demand, β is the elasticity of the demand curve, d_ij is the distance from node i to node j, w is the weight for two objectives, and t is the threshold. The generalized model is classified as combinational models of the three conditions of 1) all demand points covered by the range of good, 2) the elasticity of the demand cone, and 3) the weighted value of multiobjective programming.

In conclusion, we can reinterpret Lösch’s locational principle of the single good as a total demand maximization problem that consists of the maximal covering and the median problem, and Christaller’s principle as a generalized median problem with the two objectives of minimizing total travel distance and minimizing the number of firms can be formulated by multiobjective programming. Modelling of central place theory using mathematical programming leads to reading between the lines of original theories, with a better understanding of the deduced process of the central place system.