# An Integrated Optimization Method of High-Speed Railway Rescheduling Problem at the Network Level
## 4 NSGA-III Algorithm
### Initial Populations and Infeasible Solution Adjustment
The method is shown as follows:

Step 1: Create set \(CP\), add all the cross-line passenger demands \(x_{c}\) which need be served to \(CP\). Create set \(NP\), add all the other passenger demands \(x_{n}\) which need be served to \(NP\). Create set \(BT\), add all the standby train-sets \(b\) to \(BT\). Create set \(OT\), add all the other affected trains \(t\) to \(OT\). Step 2: For demand \(x_{c}\) in \(CP\), judge whether there is train \(t\) in \(OT\) can serve it. If train \(t\) can serve demand \(x_{c}\), three conditions need to be satisfied: (1) The number of passengers cannot be more than the seating capacity of train \(t\) in any sections. (2) The scheduled departure time of demand \(x_{c}\) should be later than the departure time of train \(t\) at the start station. (3) train \(t\) must stop at the departure station and arrival station of demand \(x_{c}\). If yes, let train \(t\) serve demand \(x_{c}\), and remove \(x_{c}\) from set \(CP\); if not, go to step 3. Step 3: Judge whether all the demands in \(CP\) are chosen and judged. If yes, go to step 4; if not, go to step 2. Step 4: For each demand \(x_{c}\) in \(CP\), judge whether there is standby train-set \(b\) in \(BT\) can serve it. If standby train-sets \(b\) can serve demand \(x_{c}\), four conditions need to be satisfied: (1) The number of passengers cannot be more than the seating capacity of standby train-set \(b\) in any sections. (2) The scheduled departure time of demand \(x_{c}\) should be later than the departure time of standby train-set \(b\) at the start station. (3) standby train-set \(b\) must stop at the departure station and arrival station of demand \(x_{c}\). (4) The arrival time of demand \(x_{c}\) plus the servicing time should be earlier than the departure time of the train that standby train-set \(b\) executed next, so as to make sure the next train can depart at rescheduled time. If yes, let standby train-set \(b\) serve demand \(x_{c}\), and remove \(x_{c}\) from set \(CP\); if not, go to step 5. Step 5: Judge whether all the demands in \(CP\) are chosen and judged. If yes, go to step 6; if not, go to step 4. Step 6: For demand \(x_{n}\) in \(NP\), judge whether there is train \(t\) in \(OT\) can serve it. The conditions are the same as that in step 2. If yes, let train \(t\) serve demand \(x_{n}\), and remove \(x_{n}\) from set \(NP\); if not, go to step 7. Step 7: Judge whether all the demands in \(NP\) are chosen and judged. If yes, go to step 8; if not, go to step 6. Step 8: For demand \(x_{n}\) in \(NP\), judge whether there is standby train-set \(b\) in \(OT\) can serve it. The conditions are the same as that in step 4. If yes, let train \(t\) serve demand \(x_{n}\), and remove \(x_{n}\) from set \(NP\); if not, go to step 7. Step 9: Judge whether all the demands in \(NP\) are chosen and judged. If yes, go to step 10; if not, go to step 8. Step 10: Cancel the remaining demands in \(CP\) and \(NP\). After the process of the method, the variables \(tr^{c,t,x}_{\ell^{\prime},t^{\prime}}\) and \(tr^{c,t,x}_{b}\) are determined till now, all the decision variables mentioned in Section 3.1 are determined. ### Fitness Function

The fitness functions are shown in (41)-(43), which are additional operation cost (\(f_{1}\)), total delay (\(f_{2}\)) and the transfer number of affected passengers (\(f_{3}\)). \[f_{1}=\sum_{c\in C^{eff}}\sum_{b\in B}\sum_{t\in T}h^{c,t}_{b}\times(WM^{c,t}_ {b}+ADD^{c,t}_{b})\times OC_{b} \tag{43}\]

\[f_{2}=\sum_{c^{\prime}\in C^{eff}}\sum_{t^{\prime}\in T^{eff}}\sum_{c\in C^{eff }}\sum_{t\in T^{eff}}\sum_{x\in X^{eff}} \begin{array}{c}tr^{c,t,x}_{\ell^{ \prime},t^{\prime}}\times PN^{c,t,x}\times\epsilon\epsilon^{c,t}\times\\ (aat^{c,t^{\prime},t^{\prime}}_{TMS^{c,t^{\prime}}}-PAT^{c,t}_{TMS^{c,t}})\\ +\sum_{b\in B}\sum_{c\in C^{eff}}\sum_{t\in T^{eff}}\sum_{x\in X^{eff}}bf^{c,t,x}_{b}\times PN^{c,t,x}\times\epsilon\epsilon^{c,t}\times(aat^{b}_{TMS^{c,t}} -PAT^{c,t}_{TMS^{c,t}})\\ +\sum_{c\in C^{eff}}\sum_{t\in T^{eff}}\sum_{x\in X^{eff}}PN^{c,t,x}\times(1- can^{c,t})\times PCT^{c,t}_{CAN}\end{array}