## 9 Conclusions  
This study introduced the Hybrid FVIMDE Algorithm, ingeniously combining the Finite Volume Integration Method (FVIM) with Differential Evolution (DE) to tackle complex optimization challenges across engineering design domains. Our extensive evaluations, conducted over a diverse set of benchmarks, including CEC2017, CEC2022, and an ad \begin{table}
\begin{tabular}{l l l l l l l} \hline \hline  & \multicolumn{3}{c}{**Statistical Measurements**} & \multicolumn{3}{c}{**(XI-X4) for Minimum Value**} \\
**Sheet Name** & **Mean** & **Std** & **Max** & **Best** & **K** & **L** \\ \hline FVIMDE & 2.649E+01 & 2.160E-09 & 2.649E+01 & 2.649E+01 & 5.452E+00 & 2.916E-01 \\ SNS & 2.649E+01 & 3.413E-09 & 2.649E+01 & 2.649E+01 & 5.452E+00 & 2.916E-01 \\ GWO & 2.655E+01 & 3.984E-02 & 2.661E+01 & 2.649E+01 & 5.460E+00 & 2.916E-01 \\ HHO & 2.663E+01 & 8.373E-02 & 2.676E+01 & 2.653E+01 & 5.444E+00 & 2.931E-01 \\ MVO & 2.931E+01 & 1.346E+00 & 3.119E+01 & 2.693E+01 & 5.232E+00 & 3.211E-01 \\ CPO & 2.664E+01 & 9.296E-02 & 2.682E+01 & 2.654E+01 & 5.417E+00 & 2.959E-01 \\ FOX & 3.032E+01 & 1.989E+00 & 3.310E+01 & 2.698E+01 & 5.234E+00 & 3.220E-01 \\ ZOA & 2.663E+01 & 8.875E-02 & 2.679E-01 & 2.649E+01 & 5.428E+00 & 2.939E-01 \\ FOX & 2.762E+01 & 1.409E+00 & 3.111E+01 & 2.653E+01 & 5.473E+00 & 2.906E-01 \\ AVOA & 2.681E+01 & 1.914E-01 & 2.706E+01 & 2.650E+01 & 5.419E+00 & 2.950E-01 \\ SCA & 2.664E+01 & 1.049E-01 & 2.681E+01 & 2.651E+01 & 5.457E+00 & 2.916E-01 \\ MFO & 2.659E+01 & 5.646E-02 & 2.672E+01 & 2.651E+01 & 5.462E+00 & 2.912E-01 \\ WOA & 2.649E+01 & 4.410E-04 & 2.649E+01 & 2.649E+01 & 5.452E+00 & 2.916E-01 \\ \hline \hline \end{tabular}
\end{table} 
Table 22: Results of the tabular column design problem, run = 30, agents = 50, FES = 1000.  
ditional 50 benchmark functions, demonstrate the robustness and superior performance of FVIMDE against traditional optimization algorithms. FVIMDE showed outperformed results in benchmark tests and also showcased remarkable proficiency in addressing real-world structural engineering problems. We compared its efficacy in solving five complex structural engineering tasks against other established algorithms, revealing its potential to outperform in terms of both efficiency and outcome quality. The FVIMDE hybrid algorithm combines the strengths of FVIM for precise exploitation and DE for dynamic exploration, maintaining an effective balance that enhances convergence rates and solution accuracy. The algorithm's design ensures adaptability and robustness across various optimization scenarios. The success of FVIMDE invites further exploration into its applications and enhancements, particularly in expanding its utility to other types of engineering challenges and exploring adaptive parameter tuning mechanisms to further its efficacy. Future studies may also explore diverse hybridization methods for FVIM and tackle clustering or classification problems, large-scale optimizations, and multiobjective optimizations. Conceptualization, H.N.F., A.S.A.-S., A.I., S.N.F., F.H. and S.N.M. ; methodology, H.N.F., A.S.A.-S., A.I., S.N.F, F.H. and S.N.M. ; validation, H.N.F., A.S.A.-S., A.I., S.N.F, F.H. and S.N.M. ; formal analysis, H.N.F., A.S.A.-S., A.I., S.N.F., F.H. and S.N.M. ; investigation, H.N.F., A.S.A.-S., A.I., S.N.F., F.H. and S.N.M. ; resources, H.N.F., A.S.A.-S., A.I., S.N.F., F.H. and S.M. ; data curation, H.N.F., A.S.A.-S., A.I., S.N.F., F.H. and S.N.M. ; writing--original draft preparation, H.N.F., A.S.A.-S., A.I., S.N.F., F.H. and S.N.M. ; writing--review and editing, H.N.F., A.S.A.-S., A.I., S.N.F., F.H. and S.N.M. ; visualization, H.N.F., A.S.A.-S., A.I., S.N.F., F.H. and S.N.M. All authors have read and agreed to the published version of the manuscript. This research received no external funding. Data availability Statement: Data are available upon request from the authors. The authors declare no conflicts of interest.