## 5 Conclusions  
This paper proposes an improved sparrow search algorithm (ISSA) to solve the multi-objective optimization problem for the steel pipe pile cofferdam construction scheme. Firstly, this paper establishes a multi-objective balanced duration-cost-quality optimization model. Secondly, tent chaotic mapping and rapid non-dominated sorting are introduced into the SSA to solve the model. Finally, the feasibility and effectiveness of the system are verified by using the Changsha Jinan Avenue project as a case study. Based on the results, the following conclusions are drawn:  
1. The multi-objective balanced duration-cost-quality optimization model and the ISSA proposed in this paper are practical. They are suitable for the construction management of steel pipe pile cofferdam projects.
2. As a case study, the Changsha Jinan Avenue project was analyzed by applying the multi-objective balanced duration-cost-quality optimization model, the ISSA, and gray correlation analysis. The results show that the duration of the optimized scheme was shortened by 18 days compared with that of the pre-optimization scheme, a reduction of 21%; the total cost was reduced by CNY 220,000, saving 3% of the cost; and the quality remained at a high level of 0.90 or more, which meets the requirement for the minimum acceptable quality value of 0.8.
3. The multi-objective balanced duration-cost-quality optimization model and the ISSA proposed in this paper are highly adaptable, and construction projects other than steel pipe cofferdam construction projects can also be modeled and solved after decomposing construction processes. If force majeure factors such as delays occur during the construction process, the Pareto solution set can be obtained by inputting data such as the minimum and maximum values of the duration of the set process, and then based on the gray correlation analysis of the Pareto solution set for selection, the optimal solution for construction can be determined.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scheme** & **Gray Correlation** & **Scheme** & **Gray Correlation** \\ \hline A15 & 0.6831 & A16 & 0.5725 \\ A17 & 0.5731 & A18 & 0.6047 \\ \hline \hline \end{tabular}
\end{table}
Table 4: _Cont._  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Scheme** & **Duration/Days** & **Cost/(CNY 10k)** & **Quality** \\ \hline Scheme before optimization & 87 & 738.50 & 0.9663 \\ Scheme after optimization (A2) & 69 & 716.50 & 0.9370 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Duration, cost, and quality with the original and final schemes.  
**Author Contributions:** Conceptualization, Z.J., C.Y. and H.Y.; methodology, Z.J. and C.Y.; software, C.Y.; validation, Z.J. and C.Y.; formal analysis, Z.J. and C.Y.; investigation, Z.J., C.Y. and H.Y.; resources, Z.J., C.Y. and H.Y.; data curation, C.Y.; writing--original draft preparation, C.Y. and H.Y.; writing--review and editing, Z.J., C.Y. and H.Y.; visualization, H.Y.; supervision, Z.J.; project administration, Z.J. and H.Y.; funding acquisition, Z.J. and H.Y. All authors have read and agreed to the published version of the manuscript.  
**Funding:** This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
**Data Availability Statement:** Data are contained within the article.  
**Conflicts of Interest:** The authors declare no conflicts of interest.