## 3 Data and Methods
### Implementation of the HMAE Model
If the minimum travel time between a demand node and tertiary facilities was larger than the catchment size, it could be considered to be an underserved demand node of tertiary facilities. The same procedure was executed for secondary facilities. The distribution of underserved demand nodes (communities) is shown in Figure 3.  
The underserved demand nodes of tertiary facilities are concentrated in the east part of Shenzhen, one in Pingshan District, and the others in Dapeng District. Population density is relatively low in these areas. Therefore, one new candidate location was added in each district. Furthermore, even though all demand nodes in the west part of Shenzhen (i.e., Bao'an and Guangming districts) are covered by existing tertiary facilities, most of these demand nodes are quite far away from existing tertiary facilities. The closest tertiary facility is in the southernmost area of Bao'an District. Therefore, two new candidate locations were added in the northern Bao'an and Guangming Districts. The underserved demand nodes of secondary facilities are mainly concentrated in Longhua district, where moderate population density presents. There is another underserved demand node in Luohu District. Although Luohu is one of the central districts, population density in this subdistrict is relatively low, where the highest mountain, Wutong mountain, is located. Therefore, new candidate locations were only selected in Longhua District, on the basis of the distribution pattern of underserved demand nodes. Taken together, as shown in Figure 3, there are 23 and 39 candidate locations for tertiary facilities and secondary facilities, respectively. The average supply size of each facility was set as the same with existing facilities at each level. The total numbers of physicians at the two levels were found to be 9700 and 7100, respectively. Table 2 summarizes the setting of all parameters. Following existing studies [29; 49], the HMAE model was solved by using the particle swarm optimization (PSO) algorithm developed by [PERSON] and [PERSON] [50]. PSO specifies a fitness function to evaluate the performance of each possible solution, which is represented as the total accessibility difference calculated by Equation (1). It provides an \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Facility Level** & **Catchment Size** & **Number of Underserved Communities** & **Number of Candidate Locations** & **Total Physicians** \\ \hline Secondary facilities & 40 min & 23 & 39 & 7100 \\ Tertiary facilities & 70 min & 9 & 23 & 9700 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters at each level. Figure 3: Underserved demand nodes of existing healthcare facilities. efficient approach to solving optimization problems by simultaneously considering a wide range of possible solutions and moving towards the optimal solution in an evolutionary manner. Each solution is termed as a particle, which consists of the facility sizes at all candidate locations. In each iteration, the performance of current solutions is evaluated by the fitness function and compared with the previous iteration. If the current solution generates better performance, it would be retained and the solutions in the next iteration would be determined on the basis of the trend. These solutions are expected to converge to a global optimal solution after a certain number of iterations. PSO was first introduced by [PERSON] et al. [29] into the maximal equality model. PSO can be operationalized with a toolbox in MATLAB developed by [PERSON] [51]. The implementation of PSO is required for set-up of a few parameters, among which range of X (i.e., the size of facility) and dimension of particles are related to the specific case. Dimension of particles was set as the number of candidate locations, i.e., 39 and 23 for tertiary and secondary facilities, respectively. Range of X is defined by upper bound and lower bound. The lower bound was set as 0.