数理最適化手法による立地コストと施設容量を考慮した商業均衡分布の導出

抄録

 The purpose of this research is to derive an equilibrium arrangement of commercial distribution based on the economic principle, considering the influence of customers purchasing behavior on commercial distribution. Explicitly considering the locational cost and facility capacity, we specifically extended the concept of equilibrium in balancing-mechanism and we consider trade-off relationship between purchase and locational cost. By proposing a mathematical programming problem that simultaneously satisfies equilibrium conditions while leaving the purified formulation, we can propose a generalized mathematical model of the market equilibrium. Considering the reality, it is obvious that the building capacity is finite and the rent in the commercial accumulation areas will be higher. Therefore considering such a model is very important.
 First of all, we outline the formulation of the balancing-mechanism. Then, we derive the mathematical programming problem that satisfies balancing-mechanism and discuss its mathematical characteristics. Concretely, we discuss about the convexity of the objective function in the mathematical programming problem. This means the optimum solution for each condition is unique.
 Secondly, we introduce locational cost and facility capacity which are essential concepts when the commercial accumulation is being considered. This model is a comprehensive model on more commercial distribution of the market equilibrium. Then , we indicate that this model can also be represented by a mathematical programming problem satisfying the equilibrium condition and show the discussion on the uniqueness of solutions can be developed by the form of functions of α, other parameters and locational cost. We also show this model include balancing-mechanism and it can be expected to develop into a real space in consideration of locational cost and facility capacity.
 While performing numerical analysis, we change some parameters. For the balancing-mechanism in the repeating calculation and the mathematical optimization method, in the case where α is less than 1, it was the same result. It can be said that the validity of the mathematical programming problem satisfy the balancing-mechanism. On the other hand, in the case where α is more than 1, these results are not the same due to the fact that the objective function of the mathematical programming problem may be a non-convex function. In that case, the local equilibrium solution may appear as a solution. When mathematical programming problem considers locational cost and facility capacity, in the case where there is no locational cost and the facility capacity is not effective, this model shows the same result as the balancing-mechanism. Furthermore, when α is more than 1, the amount of the commercial is large and the facility capacity is effective, it can be seen that commercial distribution is spreading. And the commercial distribution spreads evenly when locational cost is effective. From these results, it can be said that our model closer to real space could be proposed.
 In this research, commercial distribution is derived without changing the distribution of living. In adapting to the real space, further expansion of the urban model of this research, such as a model having interaction including jobs, housing and commercial, is an attractive task in the future. Considering the system dynamics of the city could be an interesting future task. It should also be generalized to grasp more than one equilibrium point and consider thresholds while shifting to another equilibrium.