## 4 Results and Discussion  
This study presents a method for obtaining an optimal solution using the DP algorithm, wherein a cost function with multiple weights is applied. To obtain an optimal train-running speed profile that satisfies the existing travel target time, the process and time of the repeated weight were changed and the calculations were shortened. We, thus, propose a method for calculating the optimal speed profile, the running time, and the energy consumption, which can be simultaneously operated. The train-operation information such as moving distance, gradient, and speed limit used in this study is based on actual Seoul Metropolitan Rapid Transit route operation data. The specifications of the computer used for the calculation are Intel core i5 processor with four cores, a clock speed of 3.4 GHz, and 16 GB RAM of memory. To study the processing performance of the proposed method, the number of weights was increased, and the computation time was checked accordingly. In Table 4, the calculation time (a) is the minimum-cost-search computation time, excluding the cost-setting time of each node and edge in the DP algorithm. The calculation time (b) results from the total computation time, including the cost setting and minimum-cost search of each edge and node in the DP algorithm. Figure 2 displays a plot of the results presented in Table 4. This shows that significantly more time is consumed when setting the cost of the computation space than the time (a) required to find the optimal solution. Here, the case of one weight of the calculation time (a) is the same as the time taken for one calculation in the existing method. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Number of Weights** & **1** & **2** & **10** & **20** & **40** & **80** \\ \hline Computation time (s) (a) & 1.568 & 1.631 & 1.985 & 2.284 & 3.260 & 5.359 \\ Computation time (s) (b) & 48.445 & 78.732 & 81.390 & 84.794 & 91.130 & 104.074 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Processing result according to number of weights. If the conventional algorithm finds the optimal speed profile of the target travel time after only 10 weight corrections [15], according to the total calculation time (b), it will take 10 times the search time, that is, 48.445 \(\times\) 10 = 484 s. In [15], the tolerance for the target travel time is 0.5 s. Assuming that the computational space is sufficiently discretized and achieves the same purpose as that of the proposed method, the target result can be obtained within two trials. First, 80 weights in the range of 0-1 are applied, and the calculation is performed once. In this result, if there is no target travel time result within the tolerance, the range is limited to the weights of the front and back areas, including the target travel time, and the same calculation is performed by applying 80 redivided weights. Thus, it will be possible to obtain a velocity profile that satisfies the target travel time. In the test results presented in Table 5, the maximum travel-time difference between weights was within 8.6 s. If the calculation space has a sufficient discrete space, and it is calculated using 80 redivided weights within this range of adjacent weights, the target velocity profile within the target tolerance of 0.5 s can be obtained. The number of weight divisions may change according to the desired degree of error. In this case, when the proposed method is applied, the solution can be found in approximately 104 \(\times\) 2 = 208 s, and multiple velocity profiles can be obtained by a batch-processing method within 1/2 h of the existing method. If this result is integrated as a table, it is possible to immediately apply the desired travel-time speed profile without performing calculations when the target travel time changes, when considering a train operation delay. Figure 3 shows each velocity profile generated by applying 80 weights. The total travel distance is 806.25 m, and the maximum speed is 80 km/h.