# Chaotic Java Approaches to Solving Electromagnetic Optimization Benchmark Problems
## 5 Experimental Study and Discussion
### Results for the Loney's Solenoid  
In the cases that follow, in all algorithm variants, we set the same population size (20) and the same stopping criterion (which was set to 3000 objective function evaluations).  
Table 3 illustrates the optimization results for [PERSON]'s solenoid case of the Jaya and CJ (1-10) approaches, where the numbers 1 through 10 in the CJ approaches represented the chaotic map given by (1) Chebyshev, (2) circle, (3) Gauss, (4) iterative, (5) logistic, (6) piecewise, (7) sine, (8) singer, (9) sinusoidal, and (10) tent mapping, respectively.  
One may notice that there were three different basins of attraction of local minima in the domain of \(f\), with values of \(f(s,l)>4\cdot 10^{-8}\) (high level region), \(3\cdot 10^{-8}<f(s,l)<4\cdot 10^{-8}\) (low level region), and \(f(s,l)<3\cdot 10^{-8}\) (very low level region, the global minimum region).  
As seen from Table 3, CJ (5) outperformed the other tested optimizers in terms of the best objective function value \(f(s,l)\) in 50 runs. The best result (minimum) using CJ (5) is  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Optimizer**} & \multicolumn{2}{c}{**Minimum (10\({}^{-8}\))**} & \multirow{2}{*}{**Mean**} & **Maximum** & **Standard** \\  & **(Bes0)** & & **(Worst)** & **Deviation** \\ \hline Jaya & 3.4564 & 1.74\(\cdot 10^{-7}\) & 9.32\(\cdot 10^{-6}\) & 3.82\(\cdot 10^{-7}\) \\ CJ (1) & 2.4380 & **3.38\(\cdot 10^{-8}\)** & **4.39\(\cdot 10^{-8}\)** & 7.03\(\cdot 10^{-11}\) \\ CJ (2) & 3.1906 & 2.24\(\cdot 10^{-7}\) & 7.93\(\cdot 10^{-6}\) & 2.95\(\cdot 10^{-7}\) \\ CJ (3) & 7.7139 & 9.60\(\cdot 10^{-5}\) & 1.51\(\cdot 10^{-3}\) & 8.95\(\cdot 10^{-5}\) \\ CJ (4) & 3.0942 & 6.94\(\cdot 10^{-8}\) & 1.84\(\cdot 10^{-6}\) & 7.38\(\cdot 10^{-8}\) \\ CJ (5) & **2.0566** & 6.60\(\cdot 10^{-8}\) & 1.93\(\cdot 10^{-6}\) & 3.20\(\cdot 10^{-7}\) \\ CJ (6) & 3.0217 & 2.07\(\cdot 10^{-7}\) & 6.60\(\cdot 10^{-6}\) & 4.19\(\cdot 10^{-7}\) \\ CJ (7) & 2.1721 & 6.48\(\cdot 10^{-8}\) & 1.37\(\cdot 10^{-6}\) & 4.58\(\cdot 10^{-8}\) \\ CJ (8) & 2.2034 & 5.88\(\cdot 10^{-7}\) & 3.42\(\cdot 10^{-5}\) & 1.49\(\cdot 10^{-6}\) \\ CJ (9) & 3.8248 & 1.11\(\cdot 10^{-5}\) & 3.65\(\cdot 10^{-4}\) & 1.04\(\cdot 10^{-5}\) \\ CJ (10) & 3.9749 & 5.96\(\cdot 10^{-5}\) & 8.21\(\cdot 10^{-4}\) & 1.75\(\cdot 10^{-5}\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: Optimization results for the [PERSON]’s solenoid in 50 runs considering the objective function to be minimized, given by \(f(s,l)\). Data in bold font indicates the smaller values.  
\(f(s,l)=2.0567\cdot 10^{-8}\), with \(s=11.4244\) cm and \(l=1.4091\) cm. However, CJ (1) presented the best mean \(f(s,l)\) value of the compared Jaya optimizers in Table 3.  
To verify the performance and effectiveness of the Jaya and CJ methods for the Loney solenoid, a brief analysis of the performance is also provided in Table 4, which shows the comparison with the results reported in [19; 20; 21; 22; 23]. It can be seen that CJ (5) obtained comparative results with those metaheuristics in terms of the best objective function value \(f(s,l)\). On the other hand, the mean \(f(s,l)\) obtained by CJ (1) was competitive with the other optimizers.