# Linear Project-Scheduling Optimization Considering a Reverse Construction Scenario
## 4 Design of a Solving Algorithm for the LSM-CBPS Model
### Improved Grey Wolf Optimization Algorithm Design
In initial-scale case, scenario 1 only allows forward construction of activities, and according to Algorithm 1, its minimum project duration is 112 days. The construction schedule is shown in Figure 9. Scenario 2 allows activities to adopt reverse construction methods. As shown in Figure 10, due to the cancellation of construction direction restrictions, all linear activities have some subactivities (A2, C3, D3, E2, F2, G2, and H2) that adopt reverse construction methods, which successfully shorten the project duration to 91.29 (see Table 12), achieved a 18.5% reduction. In large-scale case, similar conclusions could be obtained. The project duration was shortened from 112.19 days to 98.1 days due to the introduction of reverse construction method. Evidently, in both scaled-up cases, the inclusion of reverse construction scenarios led to further optimization of the results. The reason for the improvement of project duration is that the introduction of reverse construction brought greater spatiotemporal flexibility, such that activities could be arranged more flexibly without violating logical constraints. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Number of Feasible** & **Best** & **Minimum Constraint** \\  & **Solutions Obtained** & **Solution** & **Violation Value** \\ \hline Improved GWO & 9 & 138.67 & — \\ Original GWO & 3 & 192 & — \\ GA & 0 & — & 40.43 \\ PSO & 0 & — & 31.16 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Results of algorithms without the heuristic rule. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Run Times** & **Improved GWO Algorithm** & **Original GWO Algorithm** \\ \hline
1 & 105.38 & 198.90 \\
2 & 100 & 256.57 \\
3 & 98.10 & 196.14 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Results under the large-scale case of algorithms. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scenario** & **Case** & **Whether Reverse** & **Best** \\  & **Selection** & **Construction Is Allowed** & **Solution** \\ \hline
1 & Initial case & No & 112 \\
2 & Initial case & Yes & 91.29 \\
3 & Large-scale case & No & 112.19 \\
4 & Large-scale case & Yes & 98.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: The best solution of the different scenarios. Figure 10: LSM diagram of the best solution of Scenario 2. Figure 9: LSM diagram of the best solution of Scenario 1. ### Managerial Insights

Further analysis on potential positive impact of introducing reverse construction on scheduling in terms of project duration is conducted in this section. Consider the subactivity \(j_{l}\) and its preceding subactivity \(i_{k}\). Site managers may consider the following methods for shortening the duration. (1) If \(tp_{i}=line\), \(tp_{j}=line\), \(SD_{i,k}<SD_{j,l}\), \(ED_{i,k}<ED_{j,l}\), \(ED_{i,k}>SD_{j,l}\), and \(pr_{i,k}<pr_{j,l}\). 1) When both \(i_{k}\) and \(j_{l}\) choose reverse direction, activity \(j_{l}\) achieves an earlier start time compared with when both constructions are forward. 2) When \((SD_{j,l}-SD_{i,k})/pr_{i,k}>(ED_{i,k}-SD_{j,l})/pr_{j,l}\), \(i_{k}\) selects reverse direction and \(j_{l}\) selects forward direction, \(j_{l}\) achieves an earlier start time compared with when both constructions are forward. 3) When \(BD_{j,k}-BD_{i,h}>BD_{i,h}-SD_{j,k}\), \(i_{k}\) selects forward dirtction and \(j_{l}\) selects reverse dirtction, \(j_{l}\) achieves an earlier start time compared with when both constructions are forward. Figure 11: LSM diagram of the best solution of Scenario 3. Figure 12: LSM diagram of the best solution of Scenario 4. (2) If \(tp_{i}=line\), \(tp_{j}=line\), \(SD_{i,k}\geq SD_{j,l}\), \(ED_{i,k}\leq ED_{j,l}\), and \(pr_{i,k}<pr_{j,l}\). 1) When \(ED_{j,l}-SD_{i,k}>ED_{i,k}-SD_{j,l}\), \(i_{k}\) and \(j_{l}\) both choose reverse direction, \(j_{l}\) achieves an earlier start time compared with when both constructions are forward.