# 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
\[pval_{j,h}=Pst_{j,h}-st_{j,h}, \tag{78}\]3

If the maximum difference \(mpval_{h,m}\) is less than or equal to 0, it indicates that the adjusted start time of each subactivity satisfies both the minimum and maximum time constraints simultaneously, and the final start time \(st_{j,h}\) is obtained. \[pval_{j,h}=Pst_{j,h}-st_{j,h}, \tag{78}\]4

## 5 Case Study

### Case Description

In this study, the subgrade project from Xiaohuangshan to Wucaiwan of the Wuzhun Railway in China was used. This case is a classic example in the field of linear project scheduling optimization. This railway project spans 7 km and consists of eight activities. Detailed information regarding the starting and finishing distances, construction modes, and time and space constraints for each activity can be found in the literature [25; 53]. To comprehensively test the effectiveness of the model and algorithm, the following supplements were made. 1. To validate the model's capability for multicrew and subsection construction and account for the nonuniform characteristics of section lengths in actual construction projects due to varying geological conditions and activity types, each activity was divided into sections based on the specifications provided in \[pval_{j,h}=Pst_{j,h}-st_{j,h}, \tag{78}\]5 The number of optional construction crews for each activity corresponds to the number of sections;
2. The constraint values for daily resource usage were set to 190 persons/day, resource leveling constraint value was set to 690, indirect cost was set to 2000 yuan/day, and cost constraint value was set to 9,197,389 yuan. ### Algorithm Effectiveness Analysis

To assess the performance of the improved GWO algorithm, a comparative analysis with the original GWO, genetic, and particle swarm algorithms was conducted. These additions were made to ensure a comprehensive evaluation of the model's and algorithm's effectiveness in the real-world scenarios. All four algorithms were evaluated using the same population size and number of iterations (50 and 500, respectively). After reaching the number of iterations, the algorithm will terminate. Table 8 provides details of the parameter settings used. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Activity** & **Number of Sections** & **Starting Location of Each Section** \\ \hline Ground shaping & 3 & DK53+000, DK56+000, DK58+000 \\ Treatment of weak foundation & 1 & DK53+350 \\ General embankment filling & 4 & DK53+000, DK54+500, DK56+000; DK58+000 \\ Embankment filling for base of foundation & 4 & DK53+000, DK54+500, DK55+500, DK58+000 \\ Embankment filling for foundation surface & 3 & DK53+000, DK55+500, DK57+500 \\ Trim of subgrade & 3 & DK53+000, DK55+500, DK57+500 \\ Ancillary works for subgrade & 3 & DK53+000, DK55+500, DK57+500 \\ Trim and ending & 3 & DK53+000, DK55+500, DK57+500 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Segments of activities. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Assessed Factors** & **Related Case** & **Related Section** \\ \hline  & Initial-scale case described & \\  & in Section 5.1 & Section 5.2 \\ Performance of GWO & Large-scale case (expanded & \\  & five times by initial-scale & Section 5.2 \\  & \multicolumn{1}{c}{} & \\  & Initial-scale case without & \\  & reverse construction & Section 5.3 \\  & \multicolumn{1}{c}{} & \\  & Initial-scale case with & \\  & reverse construction & Section 5.3 \\ Performance of LSM-CBPS & Large-scale case without & \\  & reverse construction & \\  & Large-scale case with & \\  & reverse construction & Section 5.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Key experimental design. To ensure a fair comparison and highlight the differences in the algorithm frameworks, the personalized improvement strategies implemented in the improved GWO algorithm, including population initialization, heuristic rules, and dynamic constraint processing methods, were also applied to the original GWO, genetic algorithm, and particle swarm optimization algorithms.