Furthest Neighbour Distance Distribution Function: An Application to Evaluate the Relationship Between the Density of City Benches and the Required Continuous Walking Distance Distribution

In order to help elderly people walk in the city, their continuous walking distance, defined as a person’s maximum walking distance without resting, should be considered when planning the locations of city benches. This measure of the furthest neighbour is uniquely determined for any city bench and the furthest neighbour distance (FND) can be regarded as the required continuous walking distance to any city bench (simply called the required continuous walking distance ). The FND distribution is an important measure with which to understand the relationship between the required continuous walking distance distribution and the number of city benches randomly distributed in a district. An FND distribution function has not been analytically derived, however, and so the objective of this study is to derive the FND distribution function and evaluate the relationship between the density of randomly distributed city benches and the required continuous walking distance distribution. Estimating the FND distribution according to the density of city benches in the districts around Tokyo Central Station, where there is a pronounced lack of city benches, demonstrates that the present location of city benches requires elderly people to complete continuous walking distances ranging from 100 to 350 m. The optimal number of city benches in the districts around Tokyo Central Station is 681, obtained as the solution to a bi-objective optimisation problem which efficiently minimises the probability that the FND is greater than 100 m at the expense of increasing the probability that the nearest neighbour distance is shorter than 10 m. The former and latter probabilities are 0.01 and 0.33, respectively.