## 4 Numerical Tests and Results
### Results of the Test Runs
In the literature on rescheduling, quite narrow time horizons (e.g., less than one hour) are commonly found. In this study, the length of the time horizon was, however, more than two hours from initial departure time of the first train to the resolution of the last conflict (See Figure 6). Furthermore, as previously stated, we tested our algorithm on instances with relatively short headways for a single-track railway, averaging between 10-20 min in one direction, to see the worst-case performance of the algorithm. In addition, in these instances, we introduced the trains into the system in an order that maximized the number of following and overtaking conflicts. To the best of our knowledge, no other study analyzes such a worst-case performance. Thus, 6 + 5 is quite a high number of trains to be rescheduled on a single-track railway line, given the worst case approach (small headways and order of the trains specially chosen to maximize the number of intertrain conflicts to be resolved) used. Lastly, we considered quite a long stretch of a single-track railway, with 17 stations included. Recent entries in the literature have been generally focused on double-track lines in which there are no meetings and the number of secondary lines is not a major problem, often not considered at all. However, there are still single-track railways all over the world. We consider these facts as the significance and contribution of our study. Like all studies in this field, our study has some limitations, assumptions and simplifications. Conflict resolution often brings unplanned stops to some trains, and these increase minimum running times due to the typically slow acceleration and braking capabilities of rail vehicles. Our model assumes that minimum running times are independent from acceleration and braking. Indeed, acceleration and braking effects are partially and implicitly reflected in the numerical values of the minimum headways. Moreover, in most cases, waiting times (due to conflict resolutions) are a lot longer than the running time increases that acceleration and braking can cause; therefore, we conclude that this limitation would not be a real performance deficit. We note that, on the other hand, neglecting time losses due to acceleration and braking is quite a common practice in the literature, as incorporating them makes the underlying mathematical model nonlinear. Another limitation of the study is its pure focus on rescheduling. Although we present a generic model that can be extended to train scheduling at a tactical level (timetabling), this is not the main purpose of the study. The study was designed with rescheduling in mind. Therefore, no timetable was prepared in the process, and no timetable quality factors (e.g., meeting the passenger or freight demand, periodicity, obedience to the desired train timing intervals, timetable robustness, energy efficiency, etc.) have been considered. We did not consider other tactical planning issues such as crew scheduling, rolling stock scheduling, etc. One of the unique features of our study is that we used a two-objective model, i.e., trying to minimize both the weighted maximum delay and the weighted total delay. Minimizing delays over such objectives would certainly help avoid losing feasibility in rolling stock circulation and crew circulation. Future work on this model will include adopting cleverer decomposition algorithms. In our mathematical model, we defined meetings in stations with more than one secondary track by two binary variables per station per train couple. This puts a huge computational workload onto the model, and it was unnecessary most cases. Such variables are needed only when sliding capacity constraints for more than one eastbound train and more than one westbound train will be binding at a particular station. Thus, future work will include such decomposition algorithms that these variables may also be \"lazy\". Conceptualization, G.A. and I.S. ; methodology, G.A and I.S. ; software, G.A. ; validation, G.A. and I.S. ; formal analysis, G.A.