3.2.3 Calculating the shortest route using A\* algorithm

The shortest route between each nursing facility and each area will be calculated using A\* algorithm. A representative search method for finding the shortest route is the Dijkstra method (Dijkstra, 1959) [30]. However, in the present study, the latitude and longitude are used for the coordinates of the representative points in each area as well as the nursing facilities, and the estimate value of the shortest routes between the two points are available in advance. Therefore, A\* algorithm, which is the improved version of the Dijkstra method that can effectively calculate the shortest routes, is used.

In general, when considering the shortest route which starts at the start node, goes through node n, and ends at the end node, the route is expressed as shown in Eq. (3):

$$f^\*(n) = \mathbf{g}^\*(n) + h^\*(n) \tag{3}$$

where f ∗ ð Þ <sup>n</sup> is the shortest route distance (m), <sup>g</sup><sup>∗</sup>ð Þ <sup>n</sup> is the shortest distance from the start node to n (m), and h<sup>∗</sup> ð Þ n is the shortest distance from n to the end node (m).

Though f ∗ ð Þ <sup>n</sup> can be easily obtained if the values of <sup>g</sup><sup>∗</sup>ð Þ <sup>n</sup> and <sup>h</sup><sup>∗</sup> ð Þ n are already known, in reality, the values of <sup>g</sup><sup>∗</sup>ð Þ <sup>n</sup> and <sup>h</sup><sup>∗</sup> ð Þ n are impossible to obtain beforehand. Therefore, the method of replacing f ∗ ð Þ n with the estimated f nð Þ is called A\* algorithm, and it is expressed as shown in Eq. (4):

$$f(n) = \mathbf{g}(n) + h(n) \tag{4}$$

where f nð Þ is the estimate value of the shortest route, g nð Þ is the estimate value of the shortest route from the start node to n, and h nð Þ is the estimate value of the shortest route from n to the end node.

An example of actually searching for the shortest route using A\* algorithm on a computer is shown in Figure 3. In this example, the shortest route from coordinates (2,2) to (5,5) is obtained. The gray cells are set as impassable, while F, G, and H correspond to f nð Þ,g nð Þ, and h nð Þ of Eq. (4). However, G is the actual travel distance from (2,2) to the current cell, and the distance for moving one cell over in

is shorter, as shown in the weighting pattern diagram in Figure 1. If a user from a certain area selects a nursing facility that requires him or her to go through a transit area with a large demand, there is a high chance that there will be competition among users, as those from the transit area will most likely select the same nursing facility. Therefore, the weighting is conducted as mentioned above with the

The specialization coefficient of the population aging rate is used as the weighting coefficient. The specialization coefficient of the population aging rate is the value indicating how high the population aging rate is in a certain area. Besides the specialization coefficient of the population aging rate, the population aging rate and the aging population can both be considered as a weighting coefficient. However, because the population aging rate is the ratio with the value always being 1 or less, the distance is always calculated to be shorter. Additionally, the difference in the aging population among the different areas is substantial enough to greatly affect the distance. On the other hand, the specialization coefficient of the population aging rate does not make the value too big or too small; hence, it is suitable as a

3.2.2.2 Calculating the specialization coefficient of the population aging rate

over 75 in area i (people), and pi is the total population of area i (people).

The specialization coefficient of the population aging rate in each area is calculated. In order to calculate the specialization coefficient, the population aging rate must be calculated first. The population aging rate and the specialization coefficient of the population aging rate are, respectively, calculated using Eqs. (1) and (2):

> Ai <sup>¼</sup> <sup>p</sup>75<sup>i</sup> pi

> SCi <sup>¼</sup> Ai

where SCi is the specialization coefficient of the population aging rate in area i, Ai is the population aging rate of area i (%), A is the population aging rate of all

The specialization coefficient of the population aging rate calculated in Eq. (2) is added to the road network map using PostgreSQL. This is done by multiplying the distance between nodes by the specialization coefficient of the population aging

3.2.2.3 Adding the specialization coefficient of the population aging rate to the road

where Ai is the population aging rate of area i (%), p75<sup>i</sup> is the population of those

(1)

<sup>A</sup> (2)

assumption that psychological distance of the users will be longer.

weighting coefficient.

Figure 1.

Weighting pattern diagram.

Geographic Information Systems and Science

areas (%).

28

network map
