## 4 Results and Discussion
### Average Departure Time and Travel Time  
This section examines the average departure times and travel times of the person agents when going to work and getting off work. Figure 7a-c show the results in every iteration for the different step lengths of departure time adjustment. Overall, as the number of iterations increases, the average travel time of the person agents decreases by approximately 3-5 min. It decreases from 45 min down to 40 min when going to work and from 40 min down to 37 min when getting off work. The average departure times of the person agents become earlier by approximately 15 min when going to work and later by approximately 5 min when getting off work. Moreover, when the iteration ends, the latest average departure time with no risk of being late when going to work is approximately 8:04, and the earliest average departure time with no risk of missing metro trips when getting off work is approximately 17:50. Interestingly, the average travel time of the person agents initially increases and  
Figure 5: The numbers of person agents who are late or encounter congestion in each iteration. HToW: home to work; WToH: work to home.  
Figure 6: The average travel scores of the person agents in each iteration. HToW, home to work; WToH, work to home.  
then decreases. Although the average travel time displays the same trend for person agents going to work and getting off work, these trends result from different underlying causes. When getting off work, the increase in travel time is caused by the differences in metro capacity among the iterations. Therefore, the incidence of congestion at metro stops is so high that the travel time of the person agents in the second iteration is obviously longer than that in the first iteration. In the remainder of the iterations, since the person agents that experience adjusted departure times experienced congestion in the previous iteration and the direction of adjustment of the departure time is backward, their travel time must decrease as the number of iterations increases. On the other hand, when going to work, travel time rises in the second iteration for reasons that are similar to those that drive the increase in travel time when getting off work, but a different and somewhat complex cause drives the increase in travel time in the other iterations. Note that the person agents with adjusted departure times arrived late in the previous iteration, and the adjustment direction of the departure time is forward. At the beginning of the iteration, there are so many person agents who arrive late that an adjustment causes only a fraction of the person agents to arrive at their workplaces before the obligatory time. However, the arrival times of most of the person agents do not change, and their travel time is extended by the step length. As the number of iterations increases, the number of person agents who arrive late gradually decreases, resulting in reductions in the number of person agents that receive adjustments, and the travel times of greater numbers of person agents are shortened, due to the decrease in congestion at the metro stops. Thus, the average travel time begins to decrease, due to the objectives we employ in the dynamic optimization of departure time.  
Figure 7: _Cont._  
Figure 7: **(a)** The average departure times and travel times of the person agents when going to work and getting off work in each iteration for a 1-min step length (\(\eta=1\) min); **(b)** The average departure times and travel times of the person agents when going to work and getting off work in each iteration for a 2-min step length (\(\eta=2\) min); **(c)** The average departure times and travel times of the person agents when going to work and getting off work in each iteration for a 4-min step length (\(\eta=4\) min).