# Binary Spring Search Algorithm for Solving Various Optimization Problems
## 7 Simulation Results  
The performance of the \(BSSA\) was evaluated on 23 benchmark fitness test functions [72], as defined in Tables 2-4. The performance of the \(BSSA\) was compared with other algorithms, such as Binary Genetic Algorithm (BGA) [73], Binary Particle Swarm Optimization (BPSO) [65], Binary Gravitational Search Algorithm (BESA) [74], Binary Bat Algorithm (BBA) [75], Binary Dragonfly Algorithm (BDA) [76], and Binary Grasshopper Optimization Algorithm (BGOA) [77]. Each optimization algorithm was applied independently 20 times, and the results were averaged. As can be seen from Tables 5-7, the \(BSSA\) provides better results for most functions. \begin{table}
\begin{tabular}{l l} \hline \hline \(F_{1}(x)=\sum_{i=1}^{m}x_{i}^{2}\) & \(\left[-100,100\right]^{m}\) \\ \hline \(F_{2}(x)=\sum_{i=1}^{m}\left|x_{i}\right  
The seven objective functions F1-F7 are suitable for evaluating the exploitation rate. Based on the optimization results presented, Table 5 shows that the proposed algorithm is the best. The objective functions F8-F23 were selected to study and analyze the scan index. The optimization of the objective functions in Tables 6 and 7 shows the exploitability of the algorithm. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline  & & _BSAA_ & **BGOA** & **BMOA** & **BBA** & **BDA** & **BGSA** & **BPSO** & **BGA** \\ \hline \hline \multirow{2}{*}{F\({}_{1}\)} & Ave & **6.74E\(-\)35** & 5.71E\(-\)28 & 4.61E\(-\)23 & 7.86E\(-\)10 & 2.81E\(-\)01 & 1.16E\(-\)16 & 4.98E\(-\)09 & 1.95E\(-\)12 \\ \cline{2-10}  & std & 9.17E\(-\)36 & 8.31E\(-\)29 & 7.37E\(-\)23 & 8.11E\(-\)09 & 1.11E\(-\)01 & 6.10E\(-\)17 & 1.40E\(-\)08 & 2.01E\(-\)11 \\ \hline \multirow{2}{*}{F\({}_{2}\)} & Ave & **7.78E\(-\)45** & 6.20E\(-\)40 & 1.20E\(-\)34 & 5.99E\(-\)20 & 3.96E\(-\)01 & 1.70E\(-\)01 & 7.29E\(-\)04 & 6.53E\(-\)18 \\ \cline{2-10}  & std & 3.48E\(-\)45 & 3.32E\(-\)40 & 1.30E\(-\)34 & 1.11E\(-\)17 & 1.41E\(-\)01 & 9.29E\(-\)01 & 1.84E\(-\)03 & 5.10E\(-\)17 \\ \hline \multirow{2}{*}{F\({}_{3}\)} & Ave & **2.63E\(-\)25** & 2.05E\(-\)19 & 1.00E\(-\)14 & 9.19E\(-\)05 & 4.31E\(+\)01 & 4.16E\(+\)02 & 1.40E\(+\)01 & 7.70E\(-\)10 \\ \cline{2-10}  & std & 9.83E\(-\)27 & 9.17E\(-\)20 & 4.10E\(-\)14 & 6.16E\(-\)04 & 8.97E\(+\)01 & 1.56E\(+\)02 & 7.13E\(+\)00 & 7.36E\(-\)09 \\ \hline \multirow{2}{*}{F\({}_{4}\)} & Ave & **4.65E\(-\)26** & 4.32E\(-\)18 & 2.02E\(-\)14 & 8.73E\(-\)01 & 8.80E\(-\)01 & 1.12E\(+\)00 & 6.00E\(-\)01 & 9.17E\(+\)01 \\ \cline{2-10}  & std & 4.68E\(-\)29 & 3.98E\(-\)19 & 2.43E\(-\)14 & 1.19E\(-\)01 & 2.50E\(-\)01 & 9.89E\(-\)01 & 1.72E\(-\)01 & 5.67E\(+\)01 \\ \hline \multirow{2}{*}{F\({}_{5}\)} & Ave & **5.41E\(-\)01** & 5.07E\(+\)00 & 2.79E\(+\)01 & 8.91E\(+\)02 & 1.18E\(+\)02 & 3.85E\(+\)01 & 4.93E\(+\)01 & 5.57E\(+\)02 \\ \cline{2-10}  & std & 5.05E\(-\)02 & 4.90E\(-\)01 & 1.84E\(+\)00 & 2.97E\(+\)02 & 1.43E\(+\)02 & 3.47E\(+\)01 & 3.89E\(+\)01 & 4.16E\(+\)01 \\ \hline \multirow{2}{*}{F\({}_{6}\)} & Ave & **8.03E\(-\)24** & 7.01E\(-\)19 & 6.58E\(-\)01 & 8.18E\(-\)17 & 3.15E\(-\)01 & 1.08E\(-\)16 & 9.23E\(-\)09 & 3.15E\(-\)01 \\ \cline{2-10}  & std & 5.22E\(-\)26 & 4.39E\(-\)20 & 3.38E\(-\)01 & 1.70E\(-\)18 & 9.98E\(-\)02 & 4.00E\(-\)17 & 1.78E\(-\)08 & 9.98E\(-\)02 \\ \hline \multirow{2}{*}{F\({}_{7}\)} & Ave & **3.33E\(-\)08** & 2.71E\(-\)05 & 7.80E\(-\)04 & 5.37E\(-\)01 & 2.02E\(-\)02 & 7.68E\(-\)01 & 6.92E\(-\)02 & 6.79E\(-\)04 \\ \cline{2-10}  & std & 1.18E\(-\)06 & 9.26E\(-\)06 & 3.85E\(-\)04 & 1.89E\(-\)01 & 7.43E\(-\)03 & 2.77E\(+\)00 & 2.87E\(-\)02 & 3.29E\(-\)03 \\ \hline \end{tabular}
\end{table} 
Table 6: Results for the _BSAA_ and other algorithms in multimodal test functions.