Determining the vulnerability of urban areas is one of the requirements that must be met before urban development to manage urban growth efficiently. This study aims to assess the vulnerability of urban areas using the data envelopment analysis method. We determined the physical growth of Yazd City in central Iran, utilizing the satellite images of 1976, 1984, 1993, 2006, and 2013. We studied the city’s growth in three districts as well as four different geographical directions in different years. Afterwards, this study identified the geomorphological indicators influential on the development of the districts. The constant return to scale additive model was designed and proposed to investigate different districts’ vulnerability and identify the areas with the highest and lowest vulnerability degrees. Finally, we introduced the districts with the least vulnerability as a pattern for the sustainable growth of the city. The modelling results show that District 2 in 1976 had the highest vulnerability (Z0∗=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{Z}}_{0}^{*} = 0$$\end{document}). Despite having the smallest area (input), this district has the highest unsuitable performance in terms of the geomorphological indicators. The performance of other units was found appropriate. We compared Districts 2 and 3 with District 1 from the year 2013. Thus, the decision-making units reveals that Districts 2 and 3 in 2013 have the rank of 14 and 13, respectively, while District 1 has 8 with the lowest vulnerability of growth. It recognized that District 1 has the highest suitability performance and better sustainability than two other districts. Therefore, we recommend it to the government for the sustainable and smart growth of the city considering Sustainable Development Goal (SDG)-11.

中文翻译：

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使用数据包络分析对城市增长进行脆弱性评估和建模

确定城市地区的脆弱性是城市发展有效管理城市增长之前必须满足的要求之一。本研究旨在使用数据包络分析方法评估城市地区的脆弱性。我们利用 1976 年、1984 年、1993 年、2006 年和 2013 年的卫星图像确定了伊朗中部亚兹德市的物理增长。我们研究了该城市在不同年份三个地区以及四个不同地理方向的增长。随后，本研究确定了影响区域发展的地貌指标。设计并提出了规模报酬不变可加模型来调查不同地区的脆弱性，并确定脆弱度最高和最低的地区。最后，我们将脆弱性最低的地区作为城市可持续发展的模式。建模结果表明，1976 年的 District 2 漏洞最高（Z0∗=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy } \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{Z}}_{0}^{*} = 0$$\end{document} ）。尽管面积（输入）最小，但该地区在地貌指标方面的不适宜表现最高。其他单位的表现被认为是合适的。我们将2013年的2区和3区与1区进行了比较，因此决策单位显示2013年2区和3区的排名分别为14和13，而第 1 区有 8 个增长脆弱性最低。它认识到，与其他两个区相比，第 1 区具有最高的适宜性表现和更好的可持续性。因此，考虑到可持续发展目标 (SDG)-11，我们向政府推荐它，以实现城市的可持续和智能增长。