## 4 Analysis of Experimental Results
### Cognitive Radio Application
#### 4.2.1 IGEO Problem Solving Model
The IGEO methodology is supported by all of the findings, feedback, and analyses as being a superior approach that can be used to tackle challenging engineering optimization problems. In future research, IGEO is expected to be applied to more practical spectrum allocation problems in order to investigate the algorithm's additional capabilities. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & & & & & & & & & \\ \cline{2-8}  & **GWO** & **PSO** & **SCA** & **SSA** & **WOA** & **GEO** & **GEO\_DLS** & **IGEO** \\ \hline x1 & 6.021203 & 13.33283 & 5.925464 & 6.018755 & 5.817309 & 5.043958 & 5.733949 & 4.539906 \\ x2 & 5.311574 & 10.6711 & 5.730895 & 5.306471 & 5.746442 & 7.588327 & 3.837676 & 17.7145 \\ x3 & 4.487085 & 36.93033 & 4.41001 & 4.494158 & 4.487412 & 0.01 & 5.845848 & 7.484595 \\ x4 & 3.500028 & 7.705549 & 3.544291 & 3.500494 & 3.493616 & 35.90359 & 3.614627 & 4.238047 \\ x5 & 2.153853 & 26.01948 & 1.968378 & 2.153787 & 2.023312 & 16.6398 & 0.038911 & 5.066114 \\ Best & 1.336526 & 5.891594 & 1.343079 & 1.336521 & 1.342398 & 1.479208 & 1.374305 & 1.179635 \\ Mean & 1.336557 & 7.772976 & 1.380032 & 1.336529 & 1.433402 & 1.642724 & 1.515431 & 1.370618 \\ Std & 2.76 \(\times\) 10\({}^{-5}\) & 0.969485 & 0.022838 & 6.10 \(\times\) 10\({}^{-6}\) & 0.07762 & 0.072901 & 0.066534 & 0.073602 \\ \hline \end{tabular}
\end{table} 
Table 17: Comparison of the best results and the statistical results obtained using different optimizers for the cantilever beam design problem. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline  & & & & & & & & \\ \cline{2-8}  & **GWO** & **PSO** & **SCA** & **SSA** & **WOA** & **GEO** & **GEO\_DLS** & **IGEO** \\ \hline x1 & 0.788599 & 0.768879 & 0.788357 & 0.788649 & 0.789717 & 0.799781 & 0.799896 & 0.765336 \\ x2 & 0.408375 & 0.381687 & 0.408994 & 0.408235 & 0.405222 & 0.269902 & 0.626887 & 0.674236 \\ Best & 263.8915 & 259.8307 & 263.8942 & 263.8915 & 263.8923 & 264.1117 & 264.1828 & 259.8111 \\ Mean & 263.892 & 261.3095 & 265.215 & 263.8915 & 264.3333 & 265.4653 & 266.1265 & 259.8717 \\ Std & 0.000989 & 1.432297 & 4.791442 & 4.60 \(\times\) 10\({}^{-7}\) & 0.683376 & 1.090409 & 1.748286 & 0.035903 \\ \hline \end{tabular}
\end{table} 
Table 18: Comparison of the best results and the statistical results obtained using different optimizers for the three-bar truss design problem. **Author Contributions:** Conceptualization, J.D. ; methodology, J.D. ; software, J.D. ; validation, J.D., D.Z. and Q.H. ; formal analysis, J.D. ; writing--original draft preparation, J.D. ; writing--review and editing, J.D. and L.L. ; resources, D.Z. and Q.H. ; All authors have read and agreed to the published version of the manuscript. **Funding:** This research and the APC were funded by National Natural Science Foundation of China, grant number No 62062021, 61872034, 62166006; Natural Science Foundation of Guizhou Province, grant number [2020]1Y254; Guizhou Provincial Science and Technology Projects, grant number Guizhou Science Foundation-ZK [2021] General 335. **Institutional Review Board Statement:** Not applicable. **Informed Consent Statement:** Not applicable. **Data Availability Statement:** The study did not report any data. **Acknowledgments:** The author is extremely thankful to anonymous referees and the editor for their valuable comments towards improving the manuscript. **Conflicts of Interest:** The authors declare no conflict of interest. ## Appendix B

_Appendix B.1. Piston Rod Design Problem_

Minimize \(\mathrm{F(x)}=\frac{1}{4}\pi x_{3}^{2}(L_{2}-L_{1})\)

Subject to

\(g_{1}=QL\cos\theta-R\times F\leq 0\)

\(g_{2}=Q(L-x_{4})-M_{\mathrm{max}}\leq 0\)

\(g_{3}=1.2(L_{2}-L_{1})-L_{1}\leq 0\)

\(g_{4}=\frac{x_{3}}{2}-x_{2}\leq 0\)

\(L_{1}=\sqrt{(x_{4}-x_{2})^{2}+x_{1}^{2}}\), \(L_{2}=\sqrt{(x_{4}\sin\theta+x_{1})^{2}+(x_{2}-x_{4}\cos\theta)^{2}}\)

\(\theta=45^{\circ}\), \(Q=10,000\)_lbs_, \(L=240\)_in_, \(M_{\mathrm{max}}=1.8\times 10^{6}lbs\) in, \(P=1500\)_psi_

where \(0.05\leq x_{1},x_{2},x_{4}\leq 500,\ 0.05\leq x_{3}\leq 120\).