# 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 the genetic algorithm, a uniform crossover mode was used, and the variation mode of the four decision variables aligned with that of the improved GWO algorithm. In the particle swarm optimization (PSO), the dynamic constraint processing compared the global optimal solution of the population with the historical optimal solution of each individual, thereby influencing the update speed and position of the individuals. The algorithm programs were implemented using MATLAB and executed on a PC with an Intel Core i5 processor and 12 GB of RAM. To mitigate the impact of randomness on the optimization results, each algorithm was independently run 20 times. The optimal solutions, average values, and standard deviations obtained from the four algorithms are presented in Table 9. According to the results in Table 9, the improved GWO algorithm achieved lower average values and standard deviations of the optimal solution were compared with the other three algorithms. This indicates its superior search ability, higher solution precision, and better stability. These findings confirm the effectiveness of the proposed algorithm improvement strategies. To validate the effectiveness of the heuristic rule in Section 3, it was removed from each algorithm, and the case was solved again. Each algorithm was independently run 20 times, and the results are shown in Table 10. A comparison of the results in Tables 9 and 10 reveals that the algorithm utilizing the heuristic rule easily obtained feasible and improved solutions. This further demonstrates that designing heuristic rules based on problem characteristics is an effective approach to enhance algorithm performance for optimization problems with complex constraints. Moreover, Table 10 reaffirms the advantages of the improved GWO algorithm over the other three algorithms, even without the heuristic rule. \begin{table}
\begin{tabular}{c c} \hline
**Algorithm** & **Parameter Setting** \\ \hline Improved GWO & The crossover probability is 0.2, the mutation probability is 0.2, and the control parameters linearly decrease from 2 to 0. The control parameter decreases linearly from 2 to 0. \\ GA & Cross probability is 0.8 and mutation probability is 0.08. \\  & The inertia weight is linearly reduced from 0.9 to 0.5 and the acceleration factor sum is 2. \\ PSO & The maximum speed is 30 in the start time dimension and 2 in the other dimensions \\ \hline \end{tabular}
\end{table} 
Table 8: Other parameter settings of the algorithm. \begin{table}
\begin{tabular}{c c c c} \hline
**Algorithm** & **Best** & **Mean** & **Standard** \\  & **Solution** & **Value** & **Deviation** \\ \hline Improved GWO & 91.29 & 106.01 & 6.39 \\ Original GWO & 110 & 117.02 & 7.12 \\ GA & 130.57 & 176.33 & 22.71 \\ PSO & 129.43 & 179.08 & 22.48 \\ \hline \end{tabular}
\end{table} 
Table 9: Results of the various algorithms. To evaluate the algorithm's capability to handle large-scale problems, the case size was expanded five times, and both the improved and original GWO algorithms were applied to solve the problem. The results were obtained from three independent runs, as presented in Table 11. The findings demonstrate the superiority of the improved algorithm in tackling large-scale linear project-scheduling optimization problems. ### Model Validity Analysis

In the classical LSM, linear activities typically follow forward construction. However, this study aimed to develop a model that can optimize scheduling considering reverse construction activities. To validate the effectiveness of the model and analyze the effect of introducing reverse construction activities in the construction schedule, the following analysis scenario was designed. For the two scaled-up cases mentioned above, four scenarios were designed based on whether reverse construction activities were allowed, and the improved GWO algorithm was employed to solve them 20 times. The best solutions obtained are listed in Table 12, and the corresponding linear schedules for the four results are depicted in Figures 9-12.