## 4 Results  
### Optimal Distribution of Tertiary Healthcare Facilities  
In addition to the existing 19 tertiary facilities, four new facility locations were added in areas that are quite far away from existing facilities. The newly added locations are located in Bao'an, Guangming, Pingshan, and Dapeng Districts. The optimal sizes of these candidate locations, both the existing facilities and newly added locations, were determined by using the hierarchical maximal equality model, aiming to minimize the variation in the spatial accessibility to facilities for all demand nodes.  
The results are shown in Figure 4. The optimized facilities were classified into small-, middle-, and large-sized facilities (corresponding to facilities with 100-300, 300-800, or 800-1200 physicians, respectively) on the basis of the natural-breaks method. The ratios of three types of facilities were 48%, 39%, and 13%, respectively. Small- and middle-sized facilities are dominant. There are only three large-sized tertiary facilities, which are respectively located in Nanshan, Bao'an and Guangming Districts. By contrast, the tertiary facilities in Ftitian and Luohu Districts, which are regarded as the core of Shenzhen, are middle- or small-sized. However, a relatively large number of existing tertiary facilities are concentrated in Ftitian and Luohu Districts. The tertiary facilities in Longhua, Longgang, and Pingshan Districts, where the distribution of facilities is relatively dispersed, are mainly middle-sized. The only tertiary facility in Dapeng District, which is newly added, is small-sized, due to the low population density in Dapeng and the surrounding areas. Generally, the optimized distribution of tertiary healthcare facilities presents a pattern in that facilities in the central areas are densely distributed but small-to-middle-sized, while facilities in the peripheral areas are middle-to-large-sized but dispersedly distributed.  
The differences between optimized sizes and actual sizes of existing tertiary facilities were also calculated. This can help determine which adjustments of existing facilities are needed to achieve the optimal distribution, which is useful for decision making. As shown in Figure 5, the differences were significant, indicating that large adjustments are needed to materialize healthcare accessibility equality. In other words, the distribution of existing tertiary facilities is poorly performed in terms of providing equal healthcare accessibility.  
Existing tertiary facilities that require positive size adjustments are mainly located in Nanshan, Bao'an, Longgang, and Longhua districts, while most facilities in Luohu and Futian districts need to be cut down in size. The pattern reveals that to achieve equal accessibility, more healthcare resources need to be allocated in the peripheral areas. Note that in the optimized distribution, some downsized (\"negative adjustment\") facilities may be close to upsized (\"positive adjustment\") facilities, e.g., the example marked by a yellow box in Figure 5. In such cases, the positive and negative adjustments close to each other can be counteracted. By doing so, many costs of adjustments can be saved, but with only negligible impacts on resources distribution and healthcare accessibility.