In contrast to private cars, rail transit systems are a more effective way to deal with the emerging challenges in cities with high population densities, such as congestion, air pollution, and traffic emissions. Rail transit systems, however, are commonly costly, due to substantial investments in construction and maintenance. It is thus necessary to design the candidate rail transit systems carefully to ensure public transport accessibility and sustainability, with consideration of the space-time correlation of population densities. In this paper, the space-time correlations of population densities are incorporated into the design of a candidate rail transit line over years. A closed-formed mathematical programming model is proposed, with an optimisation objective of social welfare budget maximisation. The social welfare budget is defined as the summation of the expected social welfare and social welfare margins. The model decision variables include rail line length, rail station number, and project start time of the candidate rail transit line. The analytical solutions for the proposed rail design model are given explicitly for different scenarios with various constraints.

中文翻译：

---

多年来将人口密度的时空相关性纳入候选轨道交通线的设计中

与私家车相比，轨道交通系统是应对人口密度高的城市（如交通拥堵，空气污染和交通排放）中新出现的挑战的更有效方法。然而，由于在建设和维护上的大量投资，铁路运输系统通常很昂贵。因此，有必要在考虑人口密度的时空相关性的前提下，仔细设计候选的轨道交通系统，以确保公共交通的可达性和可持续性。本文将人口密度的时空相关性纳入了多年来的候选轨道交通线的设计中。提出了一种封闭式数学规划模型，其优化目标是实现社会福利预算的最大化。社会福利预算定义为预期社会福利和社会福利边际的总和。模型决策变量包括铁路线长度，火车站号和候选铁路运输线的项目开始时间。针对具有各种约束的不同场景，明确给出了所提出的铁路设计模型的解析解决方案。