## 6 Case Study
### Computational Result Analyses  
The case study was implemented in MATLAB R2020b at the Center for High Performance Computing at Beijing Jiaotong University with an Intel Xeon Gold 6240R @ 2.4 GHz CPU and 192 GB RAM. The solutions were obtained in 3260 min according to the maximum iteration number. The long solution time is due to energy consumption, STN construction, and equilibrium traffic assignment for each solution of the model, which is reflected in the multi-layer nesting of the algorithm. The optimization process of NSGA-II is shown in Figure 15, which shows that the Pareto front is convergent. The Pareto front and other feasible solutions are presented in Figure 16. The Pareto front presents nonlinear and non-convex relationships between two objects, indicating that the level of optimization of TTTP obtained by increasing the unit cost gradually decreased; that is, to further improve the service quality by a given amount, the additional costs increase. Figure 14: Trajectories of trains and passenger loading rate in the real timetable. Figure 13: Numbers of passengers entering and leaving and OD flow of Beijing Subway Line L.  
Similarly, when enterprises reduce operating costs, the service quality decline will accelerate. Therefore, when implementing a line plan, enterprises should comprehensively consider their cost affordability, expected service quality, and line conditions to compare and select suitable Pareto front solutions. This ability is difficult to realize with other single-objective solution methods, such as the linear weighting method. According to the relation of the objective function value size between feasible solutions and real timetable, we divided all feasible solutions into four categories: I, II, III, and IV, as shown in Figure 16. The solutions in region I are the inferior solutions dominated by the real timetable. The solutions in regions II and IV do not dominate the real timetable, so they can replace the real timetable if needed. The solutions in region III dominate the real timetable, which means that their cost and TTTP are both better than those of the real timetable. The obtained 72 Pareto front solutions (numbered in descending order of cost and marked with the superscript (\(n\))) belong to regions II, III, and IV. The first solution, noted as \(\mathbf{\Psi}^{(1)}\) in region II, has the optimal TTTP of 20,126,617 s, which is 19.2% better than the 24,901,940 s of the real timetable, saving 325 s per capita. Solution \(\mathbf{\Psi}^{(2)}\) in region II has the optimal travel time of 16,069,045 s, which is 10.1% better than the 17,864,865 s of the real timetable, saving 122 s per capita. Solution \(\mathbf{\Psi}^{(72)}\) in region IV has the optimal cost difference of CNY \(-\)7668.92, where the cost of variable items is CNY 18,525.41, which is 29.3% better than the CNY 26,194.33 of the real timetable. Figure 16: Feasible solution and Pareto front. Figure 15: NSGA-II optimization process. Among the objective function values of the Pareto optimal solutions in region III, the TTTP of \(\boldsymbol{\Psi}^{(39)}\), which is 23,062,247 s with 7.4% optimization, is the smallest. This is because the travel time, congestion penalty time, and fatigue penalty time are significantly reduced. Although the waiting time increases by 39.8%, passengers can still obtain fast and comfortable travel services due to the low weight of the waiting time in the TTTP. The total cost of variable items of \(\boldsymbol{\Psi}^{(53)}\), which is CNY 22,427.17 with 14.4% optimization, is the smallest, realized by increasing turn-back stations (low weight of cost) and decreasing vehicle and staff costs (high weight of cost). In addition, this study uses three indexes to measure the TIT, SIT, and the imbalance of station load (IS), which are the standard deviation (SD) of the average train loading rate, section average loading rate, and station flow-service train capacity ratio. Higher values of these indexes reflect a greater imbalance, which is unexpected for managers, passengers, and this paper.