# An Algorithm for Rescheduling of Trains under Planned Track Closures
## 6 Simulation Experiments
### Network of Tracks Case  
The second part of experiments was performed on a multi-station network. The considered network is based on the part of the Polish Railway Network from the region of Kielce consisting of 167 tracks connecting 149 stations as it is depicted in Figure 8, and there were originally 95 train runs in the official timetable for one day. The simulations have been performed for different numbers of train runs \(TR\) (i.e., for \(TR=10,11,12,13,14,15,20,30\)) using the same railway network. Ten different sub-sets of the original set of train runs selected at random according to the uniform probability distribution have been generated for a given number of train runs. Each train run was assumed as obligatory. The standing times have been set to \(T^{\rm DL}=15\), \(T^{\rm D}_{i,p}=1\) for all runs \(p\) and stations \(i\). We also assumed that there were three closures in the railway network. Each closure was also assumed as obligatory. Unless otherwise stated, the closures' duration times have been set to 2h (i.e., \(T^{\rm C}_{n}=2h\) for all \(n\)). For each simulation, the closures' tracks have been randomly chosen according to the uniform distribution and the closures' beginning times have also been chosen analogously from the interval \((0,T^{\rm H}-2h)\). These assumptions correspond to the periodic and weekly closure planning when short-term closures caused by the regular maintenance works are taken into consideration. We have limited our consideration to the determination of the plan for one day assuming that the train runs timetable and the closures plan should be repeatable. There stations' capacities have been unlimited. As for a single track case, the overall time interval was also limited to a single day, i.e., \(T^{\rm H}=1440\) min. The results are briefly presented in Figures 9-12 and in corresponding Tables 8-11. Dots, as well as error bars, represent the average value, as well as minimal and maximal values, respectively. The dimensionless performance index \(\widehat{Q}=\frac{Q}{Q}\), where \(\widetilde{Q}\) stands for the best value of (17) found for the particular problem instance along with the computation time \(\widetilde{T}\), evaluates the heuristic algorithm. Analogously as in Section 6.1, we denote the average, minimum, and maximum values of \(\widehat{Q}\) by \(\widehat{Q}_{avg}\), \(\widehat{Q}_{min}\), \(\widehat{Q}_{max}\), respectively. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{\(\boldsymbol{\eta}\)} & \multicolumn{3}{c}{\(TR=10\)} & \multicolumn{3}{c}{\(TR=11\)} & \multicolumn{3}{c}{\(TR=12\)} & \multicolumn{3}{c}{\(TR=13\)} \\ \cline{2-13}  & \(\widehat{Q}_{avg}\) & \(\widehat{Q}_{min}\) & \(\widehat{Q}_{max}\) & \(\widehat{Q}_{avg}\) & \(\  
Figure 8: The considered part of the Polish Railway Network in the area of Kielce (only selected main stations are indicated by dots). Figure 10: Dependence of \(\widetilde{T}\) (seconds) on the number of groups for selected numbers of train runs. Figure 9: Dependence of \(\widehat{Q}\) on the number of groups for selected numbers of train runs. Figure 11: Dependence of \(\widehat{Q}\) on the number of train runs for selected numbers of groups. We can observe in Figures 9 and 11 that the growing number of groups \(\eta\) has the significant influence on the deterioration of solution quality, and it does not improve much the computation time (Figures 10 and 12), i.e., the computation time for a fixed number of train runs does not decrease significantly with increasing number of groups. The increase in the number of groups from 2 to 20 deteriorates the quality of solution up to 320%, with only about 26% saving of the computational time. However, when the aim is to check fast the existence of any feasible solution, and the quality of the solution is meaningless, the decomposition into smaller groups may be justified. The growth of the number of groups (railway sub-networks) \(\eta\) deteriorates the obtained solutions for the same number of train runs \(TR\) (Figure 13 and Table 12).