## 5 Discussion  
Rational distribution and equalization of healthcare services are critical for the improvement of health and well-being. Equality-oriented spatial optimization models can act as scientific tools for healthcare facilities planning and policymaking. This study developed a hierarchical maximal equality model that sought to maximize the equality in healthcare accessibility. The case study of Shenzhen proves that the model can significantly improve the equality in healthcare accessibility at each level as compared to the status quo. Considering the ubiquitous inequality in healthcare accessibility and the popularity of equalization of healthcare services as a key policy goal in different contexts, this study provides replicable procedures and methods for promoting the equalization of healthcare accessibility. Furthermore, the model can be applied or adapted for analyzing other hierarchical facilities (e.g., educational facilities and public parks) because the equal accessibility to various public services is a common public policy goal. Compared to the original maximal accessibility equality model, the hierarchical maximal accessibility equality model developed in this study further incorporates hierarchical features of healthcare facilities. It specifies different catchment area sizes of, and distance friction effects for facilities at various levels. The empirical analyses demonstrate that the numbers and distribution of healthcare facilities at various levels are quite different. Existing studies have reported significant differences between healthcare accessibility at various levels [32; 33; 34]. The model developed in this study can better quantify the accessibility to hierarchical healthcare facilities across levels and optimize the equality of such accessibility. It highlights that the hierarchy structure of healthcare facilities should be carefully considered in the spatial optimization of public facilities. Spatial optimization of hierarchical facilities is a classic and recurrent topic, as mentioned previously. A set of hierarchical location-allocation models have been developed in the past several decades [24; 36; 37]. These existing studies highlight the needs for taking into account the hierarchy structure of healthcare facilities in the optimization but fail to address the equality issue and comprehensively measure accessibility. This study contributes to the efforts in this respect by extending the maximal equality model into a hierarchical location-allocation model. Compared to existing hierarchical location-allocation models, it improves the measurement of accessibility, namely, interactions between demand and supply, by using a Gaussian-based 2 SFCA method. Furthermore, this study is one of the first studies that has attempted to account for both existing facility locations and newly added locations in the location-allocation analysis. The results reveal that not only existing facilities should be adjusted, but more  
Figure 9: The distribution of optimized accessibility to secondary healthcare facilities. new facilities need to be added in areas that are not well served by existing facilities. Considering that substantial fixed resources have been invested in existing facilities, such setting of candidate facility locations can better reflect the actual base of optimization or planning. Therefore, the procedures developed in this study can provide more feasible solutions of facility planning. Our findings suggest a dispersal strategy to improve spatial equality in healthcare accessibility in Shenzhen by reallocating the existing healthcare resources and supplying extra resources. Generally, the sizes of existing facilities in central areas should be decreased, while the sizes of existing facilities should be increased, and more new facilities should be constructed in periphery areas. Considering that the space and land resources in central areas are in great shortage, the dispersal strategy is feasible to be put into implementation.