# Pressure Vessel Design Problem Using Improved Gray Wolf Optimizer Based on Cauchy Distribution
## 6 Conclusions  
In this study, we introduce a refined Gray Wolf Optimization algorithm--CGWO--that was innovatively enhanced using the Cauchy distribution. This advancement is primarily aimed at overcoming the limitations of traditional GWO, such as its susceptibility to local optima and subpar optimization accuracy. Capitalizing on the long -tail characteristic of the Cauchy distribution, the CGWO algorithm incorporates an initialization strategy to broaden the search space from the outset. Moreover, by integrating a dynamic, nonlinear inertial weight strategy derived from both the Cauchy distribution and logarithmic functions, CGWO adeptly adapts to the distinct stages of the optimization process. This approach ensures a more effective balance between global and local searches, thereby mitigating the likelihood of converging on local optima and facilitating a swifter rate of convergence. A key feature of CGWO is the implementation of a Cauchy mutation strategy, guided by dynamically changing inertia weights that serve as probabilities for mutation. In the algorithm's later stages, this strategy enhances population diversity, harnessing a survival \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{4}{c}{**Optimal Variables**} & \multirow{2}{*}{**Optimum**} \\ \cline{2-2} \cline{4-5}  & \(\mathbf{T_{s}}\) & \(\mathbf{T_{h}}\) & \(\mathbf{R}\) & \(\mathbf{L}\) & **Cost** \\ \hline CGWO & 0.7808 & 0.3870 & 40.4466 & 198.3237 & 5885.3327 \\ PSO & 1.0018 & 0.5267 & 50.6469 & 103.1721 & 6321.9003 \\ FA & 1.0409 & 0.5232 & 53.2578 & 82.9511 & 6414.2396 \\ FOA & 0.8063 & 0.9444 & 41.4192 & 191.1033 & 6924.7596 \\ GWO & 0.8233 & 0.4083 & 42.6509 & 179.1444 & 5890.6112 \\ SCA & 0.9229 & 0.4988 & 45.7581 & 152.7993 & 6138.6313 \\ WDO & 1.2770 & 0.6984 & 58.5036 & 55.0970 & 6909.6709 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: The solution results from the pressure vessel design problems. of-the-fittest approach to control the disturbances induced by Cauchy mutations. This aspect significantly aids the algorithm in escaping local optima, thus improving both the accuracy and convergence speed. To validate the enhancements proposed in this paper, CGWO was rigorously tested against 23 standard test functions from various perspectives. Comparative analysis with other optimization algorithms and improved GWO variants revealed that CGWO exhibits superior optimization capabilities and faster convergence, with the Cauchy dynamic strategy outperforming other inertial weight strategies. Further substantiation through the Wilcoxon rank-sum test established CGWO's significant improvements over several competing algorithms, highlighting the efficacy and potential of our proposed enhancements in refining the basic Gray Wolf Optimizer and reducing the risk of entrapment in local optima. Applying CGWO to a classical real-world engineering problem (specifically, pressure vessel design), the algorithm demonstrated notable effectiveness, suggesting its viability for complex engineering challenges. This indicates promising prospects for CGWO in addressing practical optimization problems, underscoring the feasibility and impact of our improvement strategy. Future work will focus on a more comprehensive and in-depth performance evaluation of CGWO. Additionally, exploring its application in areas such as power system energy load prediction, autonomous vehicle fault diagnosis, and optimal path planning presents intriguing avenues for further research. Conceptualization, K.S. ; Data curation, J.L. ; Formal analysis, J.L. ; Funding acquisition, K.S. ; Methodology, J.L. and K.S. ; Project administration, K.S. ; Resources, K.S. ; Software, J.L. ; Supervision, K.S. ; Validation, J.L. ; Visualization, J.L. ; Writing--original draft, J.L. ; Writing--review and editing, K.S. All authors have read and agreed to the published version of the manuscript.