Changes of travel distances by facility opening/closing and the robust regular location

The purpose of this paper is to find the robust facility location in consideration of facility opening and closing. For evaluating the robustness, we formulate a simple model of the regular facility locations (square lattice, triangular lattice, hexagonal lattice) and analyze the changes of travel distances to the nearest facility. We also question the accepted theory in the Central Place Theory that the triangular lattice is the optimal. First, we consider the case where some existing facilities are closed at random. This assumes the situation that some facilities suffer damage from disasters, or that the planner closes facilities without considering the spatial relationship. We theoretically derive the probability density function of kth-nearest-neighbor distance of the three regular locations and show that the triangular lattice is the optimal when 69% or more of facilities survive. Then we apply the model to school locations in northern Tsukuba and confirm that the model is applicable to real problems. Second, we consider the case where some existing facilities are closed according to a plan. This assumes the situation that the planner controls facility closing in order not to decrease the efficiency of movement. In addition, the case where some new facilities are opened is considered. We derive the average travel distance and the standard deviation of travel distances as a function of the rate of closing (opening) and demonstrate that the hexagonal lattice is the optimal when the rate of closing (opening) is not less than 38% (27%). Comparing the above two cases, we show the effectiveness of planned closing. When closing 50% of facilities, the planner can reduce the average travel distance by 13% in the square lattice, 8% in the triangular lattice, and 15% in the hexagonal lattice.