# Simulated Annealing with Exploratory Sensing for Global Optimization
## 4 Numerical Simulation
### Results and Discussion  
The proposed method was applied to 50 benchmark problems shown with their details in Appendices A and B. The results of the proposed SA versions are compared to each other to analyze their performance. Moreover, the SAES results are also compared to those of other benchmark methods to assess the proposed method's efficiency. In the first experiment, the SA, SAES\({}_{w}\), and SAES methods are reported in Tables 2 and 3 which include the averages and minima of the best solutions obtained by each method using the classical and hard test functions, respectively. The numbers of variables in classical functions \(f_{1}-f_{13}\) and all hard functions are set equal to 30 in this experiment. The numbers of variables in \(f_{24}\) and \(f_{25}\) are 100. The other functions have different numbers of variables, as shown in Appendix A. These results are recorded using 25 independent runs. The average errors between the obtained solutions shown in Tables 2 and 3, and the global optima are reported in Table 4. Moreover, the averages of the processing time used by each method are illustrated in Figures 3 and 4 using the classical and hard test functions, respectively. All these results are statistical analyzed in Table 5 and 6. In general, the results are improved according to the complication of the algorithm. Therefore, the results of the SAES\({}_{w}\) method are better than those of the SA method and the results of the SAES is better than those of the other two methods as shown in Tables 2-4. This proves the efficiency of the main two additive components of the diversification phase and the final intensification. The processing time varies according to the problem dimension and the sophistication of the objective function formula, as shown in Figures 3 and 4. The statistical measures in Tables 5 and 6 indicate the following conclusion:  
* There are no significant differences between the obtained errors in the SA and SAES\({}_{w}\) methods, although the latter method could beat the first method in almost all used test functions. * The SAES is significantly better than the other two methods in terms of obtaining better errors. * The processing time of the SAES method is slightly longer than that of the SA and SAES\({}_{w}\) methods with no significant differences between all methods processing times except in the hard test functions. As a final note on analyzing the SAES performance in terms of its ability to reach optimal solutions, this has been studied using the relative error gap in the following formula:  
\[RE=\frac{|\hat{f}-f^{*}|}{max\{1,|f^{*}|\}},\]  
where \(f^{*}\) is the exact global solution, and \(\hat{f}\) the average solution obtained by the SAES method. The method could reach the global solutions within a \(RE\) gap of \(10^{-3}\) for 8 of the classical test functions and 3 of the hard test functions. If we consider a \(RE\) gap of 1, the SAES hits the vicinity of the global solutions for 11 of the classical test functions and 11 of the hard test functions. The standard SA fails to reach near global solutions for any of the classical functions or the hard functions considering the \(RE\) gap of \(10^{-3}\) or even of 1.