# Self-Adaptive Differential Evolution with Gauss Distribution for Optimal Mechanism Design
## 5 Numerical Simulation
### Performance Comparison of G-ISADE with Its Previous Improvements  
In this part, we will assess and compare the effectiveness of the suggested algorithms (G-ISADE) with the original DE [1] and some previous improvements of DE, such as ISADE [10], JOBIDE, and EOBIDE [9]. The comparing criterion are the success rate (SR) and the total count of function evaluations (FE) where SR is calculated by the ratio of the number of the successful findings in total number of samples, NS = 30, one numerical simulation can accomplish if the ending condition is reached (error value, \(\varepsilon\leq 10^{-8}\) ); FE is the most popular parameter for calculating the convergence speed of an optimization algorithm, measured by the number of the function calls to compute the fitness value. In addition, an improvement rate (IR) is defined by improvement percentage ratio on FE of the proposed algorithm in comparison with original DE, this factor is to confirm the effectiveness of the proposed algorithm on the convergence speed while solving the benchmark functions.  
In the experiment, the parameter was set as follows. The number of dimensions, \(D=30\) and the number of populations, \(NP=100\). From original DE algorithm proposed by [PERSON]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Function (30-D)**} & \multicolumn{2}{c}{**IDE**} & \multicolumn{2}{c}{**G-ISADE**} \\ \cline{2-5}  & **Mean** & **Std** & **Mean** & **Std** \\ \hline F1 & 0.00 & 0.00 & **0.00** & **0.00** \\ F2 & 5.00 & 2.82 & \(7.28\times 10^{-5}\) & \(1.11\times 10^{-4}\) \\ F3 & \(3.42\times 10^{-2}\) & \(1.47\times 10^{-2}\) & \(7.73\times 10^{-6}\) & \(1.96\times 10^{-5}\) \\ F4 & 0.00 & 0.00 & **0.00** & **0.00** \\ F5 & 20.9 & \(4.78\times 10^{-2}\) & **0.00** & **0.00** \\ F17 & \(2.08\times 10^{4}\) & \(1.46\times 10^{5}\) & **0.00** & **0.00** \\ F18 & 9.94 & 0.49 & **0.00** & **0.00** \\ F19 & 23.4 & 31.8 & **0.00** & **0.00** \\ \hline \hline \end{tabular}
\end{table}
Table 1: Average of fitness value (Mean) and the standard deviation (Std) of 51 runs.  
and Price [1], we chose \(F=0.5\) and \(Cr=0.9\). All of the benchmarks were independently tested with 30 random initial samples. This was carried out using a random number generator in MATLAB software, version R2015a with a fixed seed. The simulation results are described in Table 2 and Figure 5. As can be seen, G-ISADE and EOBLDE could obtain the optimal solutions for all considered benchmark functions. ISADE was not able to reach the optimal solution for the test function \(f9,f11,f16\), and \(f18\) while DE demonstrated instability in reaching the termination condition of the error value (\(\varepsilon\leq 10^{-8}\)); it was unable to succeed in solving the test function \(f2,f4,f9,f11,f16,f18\), and \(f19\). In terms of comparing the number of function evaluation calls, G-ISADE achieved the optimal solution using fewer function evaluation calls, except for test functions \(f11\) and \(f14\). With the \(11\) test functions \(f1,f3,f5,f6,f7,f8,f10,f12,f13,f15\), and \(f17\), the proposed method demonstrated a significant improvement in terms of the convergence speed.  
Figures 6-22 express the behavior of the convergence of all algorithms for 17 benchmark-test problems except for \(f11\) and \(f14\) because the performance of the proposed algorithm was not as good as EOBLDE for these test functions. On the graph, the number of iterations is shown on the horizontal axis, while the function value is shown on the vertical axis. These graphs show that the proposed algorithm achieved convergence to the global optimum minimum value with \(\varepsilon\leq 10^{-8}\) in fewer iterations when compared to the remaining reference algorithms.