## 4 Results and Discussion
A total of 90 nodes were created for each distance and speed in two-dimensional space. From the results in Table 5, a speed profile that satisfies a minimum travel time of 75.56 s and a maximum travel time of 172.5 s \begin{table}
\begin{tabular}{c c c c} \hline
**Index** & **Weighting Factor** & **Travel Time (s)** & **Energy (kwh)** \\ \hline
**1** & **0** & 75.56 & 59,129.12 \\
**2** & **0** & 75.56 & 59,129.12 \\
**3** & **0** & 75.56 & 59,129.12 \\
**4** & **0** & 75.56 & 59,129.12 \\
**5** & **0.01** & **75.6** & **58,516.07** \\ \(\vdots\) & \(\vdots\) & \(\vdots\) & \(\vdots\) \\
**76** & **0.83** & **154.49** & **26,338.68** \\
**77** & **0.87** & **157.7** & **26,279.28** \\
**78** & **0.91** & **157.7** & **26,279.28** \\
**79** & **0.96** & **163.94** & **26,239.95** \\
**80** & **1** & **172.5** & **26,218.99** \\ \hline \end{tabular}
\end{table} 
Table 5: Results of speed profiles obtained for multiple weights. Figure 2: Processing time for number of weights. was obtained, according to the section-operation-restriction conditions (slope and speed limit). The minimum travel time is the time required for the operation when the maximum performance (acceleration and deceleration performance) of the train is exhibited under operating constraints such as limited speed, gradient, travel distance, propulsion, and braking performance. The maximum travel time is the time it takes for a train to travel to its destination while consuming minimum energy under operational constraints. From this result, 49 speed profiles without any overlap were obtained, and 31 profiles with the same velocity were created. This result generates the same velocity profile because the resolution of the distance and velocity axis is not sufficiently generated to obtain a different cost. Therefore, to increase the arrival time resolution, it is necessary to increase the number of nodes on the distance and velocity axes, that is, to improve the resolution. Figure 4 shows the results of the travel time and energy consumption for each weight. The smaller the travel time is, the sharper the energy consumption is. Here, the minimum and maximum times are 75.56 s and 172.5 s, respectively. In the cost-function equation, Equation (14), if the weighting factor 1 with the minimum energy is exceeded, the driving energy is increased again, as shown in Figure 4. This is because of the energy continuously used by the electric equipment in the train, regardless of the driving. Figure 5 shows the travel time for the applied weights. As shown in the figure, the exact weight is duplicated at some arrival times. When the weight resolution is higher than the node resolution, a duplicate velocity profile with the same arrival time is created. Here,  
Figure 4: Pareto curve for multiple weights between energy consumption versus time. Figure 3: Speed profiles by DP with multiple weights. the weight division interval is increased exponentially rather than linearly, and, thus, the velocity profile is evenly created. Figure 6 shows that three of the split weights, [0, 0.15, 0.96], are applied to the single cost function to check whether the processing result of the DP algorithm with the multi-weighted array processing is consistent with the single-weighted processing result. From the results, it can be confirmed that the results are consistent with each other. In Table 6, the calculation results of each method are compared and presented as quantitative values. This was performed to confirm that each weight could yield the same result. This confirmed that the multi-weighted array's DP algorithm could be seamlessly calculated.