###### Abstract  
Line planning and timetabling play important roles in the design of urban rail transportation services. Due to the complexity of the integrated optimization of entire transportation plans, previous studies have generally considered line planning and timetabling design independently, which cannot ensure the global optimality of transportation services. In this study, the integrated design problem of line planning and timetabling was characterized as an equilibrium space-time network design problem and solved with a bi-objective nonlinear integer programming model. The model, in which train overtaking and passenger path choice behavior were considered, adjusted the network topology and link attributes (time and capacity) of the travel space-time network by optimizing the train service frequency, operation zone, stopping pattern, train formation, and train order to minimize the system life cycle cost and total passenger travel time perception. An algorithm was constructed using the non-dominated sorting genetic algorithm \(\Pi\) combined with the self-adaptive gradient projection algorithm to solve the model. A real-world case was considered to evaluate the effectiveness of the proposed model and algorithm. The results showed that the model not only performed well in the trade-off between system cost and passenger travel efficiency, but it could also reduce the imbalance of train and station loads. Pareto front analysis of the model with different parameters showed that more types of trains did not correlate with a better performance, some line-planning strategies had a combination effect, and multi-strategy line planning was more suitable for scenarios with a high imbalance in the temporal and spatial distributions of passenger flow. urban rail transit; timetable; train overtaking; life cycle cost analysis; user equilibrium; integrated optimization; multi-objective optimization model +
Footnote †: journal: Applied Sciences  
stopping pattern are tactical strategies for adapting the spatial imbalance of passenger flow, such as multiple turn-backs [4] and express/local stopping [5] modes. Due to the extremely high complexity of the entire system, most existing studies have focused on quantitatively investigating a single specific issue. Table 1 summarizes the operational effects of four main line-planning strategies [6]. It can be seen that a single line-planning strategy cannot simultaneously deal with the imbalances in the temporal and spatial distributions of passenger flow. Rather, a combination of multiple strategies is required to provide diversified train services. These multi-strategy service modes have been widely applied in many metropolises, including London, Tokyo, Paris, and New York. Taking the Seibu-Shinjuku line in Japan as an example, Figure 1 shows the morning peak timetable on weekdays. There are five train operation zones: from Shinjuku to Kami-Shakuji, Tanashi, Kodaira (branch line to Hajjima), Shi-Tokorozawa, and Hon-Kawagoe, respectively. Each operation zone has different stopping patterns, which include four types of trains: local, semi-express, express, and limited express Koedo-go. Although we can analyze the advantages and disadvantages of this complex line plan with diversified train services and all-stop modes by a quantitative calculation of operation indicators, such as cost and travel efficiency, it is difficult to determine whether the solution is optimal, because the theoretical research on strategy comparison and combination is behind the practical research. Optimal diversified train services must not only have a line plan with multiple line-planning strategies, but also a feasible timetable, which are two sequential processes of the train-planning process, including demand analysis, line planning, train scheduling (i.e., timetabling), rolling stock planning, crew scheduling, and crew rostering [7].