# Simulated Annealing with Exploratory Sensing for Global Optimization
## 4 Numerical Simulation
### Results and Discussion
In the second experiment, we investigate the diversification index's performance and how it could help the SA-based algorithms gain better performance. The average values of this index are reported in Figures 5 and 6 over 25 independent runs using the classical and hard test functions, respectively, with the same conditions stated in the first experiment. In most of the used test \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Criterion** & **Compared Methods** & **No. of Beats** & \(\mathbf{R}^{+}\) & \(\mathbf{R}^{-}\) & \(\mathbf{p}\)**-Value** & **Best Method** \\ \hline Average Errors & SA & SAES\({}_{w}\) & 8/17 & 215 & 110 & 0.8614 & – \\  & SA & SAES & 2/23 & 320 & 5 & 0.0049 & SAES \\  & SAES\({}_{w}\) & SAES & 1/24 & 324 & 1 & 0.0049 & SAES \\ \hline Minimum Errors & SA & SAES\({}_{w}\) & 12/12 & 187.5 & 137.5 & 0.9459 & – \\  & SA & SAES & 8/17 & 259 & 66 & 0.0036 & SAES \\  & SAES\({}_{w}\) & SAES & 6/19 & 288 & 37 & 0.0030 & SAES \\ \hline Processing Time & SA & SAES\({}_{w}\) & 21/4 & 23.5 & 301.5 & 0.7636 & – \\  & SA & SAES & 24/1 & 1 & 324 & 0.0052 & SA \\  & SAES\({}_{w}\) & SAES & 24/1 & 1 & 324 & 0.0052 & SAES\({}_{w}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Wilcoxon rank-sum test for the results of SA methods using the hard test functions. Figure 4: Averages of processing time in seconds using the hard test functions. \begin{table}
\begin{tabular}{l l l l l l l l} \hline \hline
**Criterion** & **Compared Methods** & **No. of Beats** & \(\mathbf{R}^{+}\) & \(\mathbf{R}^{-}\) & \(\mathbf{p}\)**-Value** & **Best Method** \\ \hline Average Errors & SA & SAES\({}_{w}\) & 9/14 & 205.5 & 119.5 & 0.9536 & – \\  & SA & SAES & 3/22 & 300 & 25 & 0.0062 & SAES \\  & SAES\({}_{w}\) & SAES & 2/23 & 306 & 19 & 0.0066 & SAES \\ \hline Minimum Errors & SA & SAES\({}_{w}\) & 7/15 & 222 & 103 & 0.8996 & – \\  & SA & SAES & 3/22 & 303 & 22 & 0.0025 & SAES \\  & SAES\({}_{w}\) & SAES & 3/22 & 304 & 21 & 0.0026 & SAES \\ \hline Processing Time & SA & SAES\({}_{w}\) & 9/9 & 177 & 174 & 1.0000 & – \\  & SA & SAES & 26/0 & 0 & 351 & 0.6018 & – \\  & SAES\({}_{w}\) & SAES & 26/0 & 0 & 351 & 0.6341 & – \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Wilcoxon rank-sum test for the results of SA methods using the classical test functions. functions, the diversification phase could help the SA-based algorithms explore the search space more broadly. Figures 7 and 8 show examples of the progress of the best solution and diversification index improvements over generations. This shows how the diversification phase has improved the quality of obtained solutions bv an SA-based method. In the last experiment, we compare the proposed SAES with the other state-of-the-art methods abbreviated as follows, where 14 of comparative methods are used. * BLX-GL50 [81]: Hybrid real-coded genetic algorithms with female and male differentiation. * BLX-MA [82]: Adaptive local search parameters for real-coded memetic algorithms. * COEVO [83]: Real-parameter optimization using the mutation step co-evolution. * DE [84]: Real-parameter optimization with differential evolution. * DMS-L-PSO [85]: Dynamic multi-swarm particle swarm optimizer with local search. Figure 5: Diversification indices of classical test functions. Figure 6: Diversification indices of classical hard functions. * EDA [86]: A simple continuous estimated distribution algorithm. * IPOP-CMA-ES [87]: Restart with increasing population size Covariance Matrix Adaptation Evolution Strategy (CMA-ES). * LR-CMA-ES [88]: Local restart CMA-ES. * K-PCX [89]: A population-based, steady-state procedure for real-parameter optimization. * L-SaDE [90]: Self-adaptive differential evolution algorithm. * SPC-PNX [91]: Scaled probabilistic crowding genetic algorithm with parent centric normal crossover. * cHS [92]: Cellular harmony search algorithm. * HC [93]: Hill climbing. * \(\beta\)HC [94]: \(\beta\)-Hill climbing. The results of the benchmark mentioned above methods are reported in Table 7 along with the SAES results using the hard test functions with \(n=10\).