## 6 Conclusions  
This study proposes a hierarchical maximal accessibility equality model for optimizing the locations of hierarchical healthcare facilities to improve the equality in accessibility to them. It extends the maximal accessibility equality model, which aims to minimize the variance of spatial accessibility to facilities by accounting for the hierarchical features of healthcare facilities. The minimal variance in spatial accessibility is pursued at each respective level. The Gaussian-based 2 SFCA method is applied to measure the spatial accessibility to healthcare facilities at each level. The optimization model is solved by using the PSO algorithm. The empirical results demonstrate that healthcare facilities at each level need to be more dispersedly distributed to achieve maximal accessibility equality in Shenzhen. Compared to the current distribution, the accessibility equality of tertiary and secondary healthcare facilities in optimal solutions can be improved by 40% and 38%, respectively, which proves the validity of the proposed optimization model. Both newly added facilities and adjustments of existing facilities are needed to achieve equal healthcareaccessibility. Furthermore, the optimization results are quite different for facilities at different levels, which highlights the importance of considering hierarchy structure in the optimization of healthcare facilities. The findings provide evidence-based suggestions for the policymaking in Shenzhen to improve the accessibility to hierarchical healthcare facilities. All in all, this study provides transferable methods for the equality-oriented spatial optimization of hierarchical facilities.  
Conceptualization, Z.T., Q.W. and W.H.; methodology, Z.T.; software, Z.T.; validation, Z.T. and W.H.; formal analysis, Z.T.; investigation, Z.T., Q.W. and W.H.; resources, Z.T., Q.W. and W.H.; data curation, Z.T.; writing--original draft preparation, Z.T. and W.H.; writing--review and editing, Z.T., Q.W. and W.H.; visualization, Z.T.; project administration, Z.T.; funding acquisition, Z.T. All authors have read and agreed to the published version of the manuscript.  
This work was funded by the National Natural Science Foundation of China (grant number 42101189) and the Fundamental Research Funds for the Central Universities. The funding bodies had no direct role in the design of the study or the collection, analysis, and interpretation of data, or in writing the manuscript.  
Not applicable.  
The healthcare facility data can be obtained from the website of Shenzhen Municipal Health Commission ([[http://wjw.sz.gov.cn/bmfw/wycx/fwyl/yycx/index.html](http://wjw.sz.gov.cn/bmfw/wycx/fwyl/yycx/index.html)]([http://wjw.sz.gov.cn/bmfw/wycx/fwyl/yycx/index.html](http://wjw.sz.gov.cn/bmfw/wycx/fwyl/yycx/index.html)) accessed on 26 October 2021). The travel time data were estimated using the public web API of Baidu Map ([[http://lbsyun.baidu.com/index.php?title=jspopular/guide/routeplam](http://lbsyun.baidu.com/index.php?title=jspopular/guide/routeplam)]([http://lbsyun.baidu.com/index.php?title=jspopular/guide/routeplam](http://lbsyun.baidu.com/index.php?title=jspopular/guide/routeplam)) accessed on 26 October 2021). The Particle Swarm Optimization Toolbox is available at [https://www2 mathworks.cn/matlabcentral/fileexchange/7506-particle-swarm-optimization-toolbox](https://www2 mathworks.cn/matlabcentral/fileexchange/7506-particle-swarm-optimization-toolbox) accessed on 26 October 2021.  
We would like to thank the editors and anonymous reviewers for their insightful suggestions.  
The authors declare no conflict of interest.