## 6 Case Study
### Performance of Different Strategies  
The ESTNDM is a general model, which considers multiple line-planning strategies and can be compatible with a single strategy (by setting non-optimized variables as constants). It provides convenience for us to compare the line plans of different strategies in one model with different parameters instead of developing different models. In this section, we compare the optimization results of different strategies to reflect the advantages of the multi-strategy line planning. We number the methods using the decimal number of the quaternion 0-1 array, which denotes whether the train service frequency, operation zone, stopping pattern, and train formation are optimized. For example, the model that optimizes train service frequency only is numbered 8, i.e., (1, 0, 0, 0), and the model proposed in Section 4 is numbered 15, i.e., (1, 1, 1). The Pareto fronts of models 1, 8, and 9 are obtained by the enumeration method due to the small decision space, because the service frequency and train formation value range are small, while other models are solved by the algorithm in Section 5. All these models have the same parameters as model 15; unused strategies are taken according to the real timetable; and \(K=3\) (except \(K=1\) in model 8 because different types of trains cannot be distinguished by only optimizing the service frequency). The Pareto fronts of all 15 models and the real timetable are shown in Figure 21. Theoretically, the Pareto front of model 15 is optimal, which is located at the lower left of the other curves, but the algorithm proposed in this paper obtains the approximate optimal solution in a limited time, so some solutions for model 6 (optimizing the operation zone and stopping pattern) are better than those of model 15. One possible reason for this is that the optimization of the operation zone and stopping pattern has a great impact on the quality of the solution, and a model with a fixed service frequency and train formation will have a small solution space to search for a better solution in the same generations. In region III, the \(\mathbf{\Psi}^{(39)}\) for model 15 has the best TTTP with 7.4% optimization and the lowest SIT and IS. The \(\mathbf{\Psi}^{(191)}\) for model 9 has the lowest TIT. The \(\mathbf{\Psi}^{(24)}\) for model 6 has the lowest cost difference, and the cost of variable items is optimized by 14.9%. The theoretical curve of the Pareto front of each model exhibits a good or bad relationship. For example, the Pareto front of model 8 is the Pareto front of the subspace composed of the special solutions, whose operation zone, stopping pattern, and train formation are the same as the real timetable of model 15. Therefore, all models can be divided into four categories (Classes A, B, C, and D) according to the number of line-planning strategies and rank, as shown in Figure 22, where the arrow points from good to bad models. There are no obvious advantages or disadvantages of the models in the same category or without continuous arrow connections, such as models 13 and 2. The distribution of the four categories in region III is shown in Figure 21.  
Class A only contains model 15, and the Pareto front is optimal, i.e., at the bottom left. Models 7, 11, and 14 in Class B, models 3, 6, and 10 in Class C, and model 2 in Class D have obvious Pareto front aggregation, which is difficult to distinguish. Compared with other models, the biggest feature of these models is the optimization of the stopping pattern. This shows that strategies that contain stopping patterns are very effective. In particular, model 6 can achieve the performance of model 15 by only optimizing the operation zone and stopping pattern when the TTTP is greater than 2.42 \(\times\) 107 s. The next aggregation area on the upper right contains models 12 and 13. Of the remaining models, only these two optimize the service frequency and operation zone simultaneously. Model 13 is better than model 12 in general as it also optimizes the train formation.