# Gaslike Social Motility: Optimization Algorithm with Application in Image Thresholding Segmentation
## 3 GSM Algorithm Performance Results
For unimodal and separable functions such as \(F_{2}\), \(F_{3}\), and \(F_{4}\), algorithms like AHA, AHA-AO, and SNS exhibit better performance in high-dimensional spaces, indicating their ability to converge quickly on smooth functions. However, in low-dimensional settings, the GSM algorithm shows a slight improvement over these algorithms. Functions \(F_{7}\) and \(F_{8}\) are unimodal, continuous, differentiable, separable, and scalable. For \(F_{7}\), the AHA, AHA-AO, and SNS algorithms achieve the best results. In the case of \(F_{8}\), the GSM, AHA, AHA-AO, and SNS \begin{table}
\begin{tabular}{c c c} \hline \hline
**Algorithm** & **Parameters** & **Reference** \\ \hline GSM & \(R=0.1\), \(\varepsilon=0.0001\) & \\  & \(\gamma=0.001\)\(bs=5\) & \\ \hline PSO & \(c_{1}=2.0\), \(c_{2}=2.0\), and & \\  & \(\omega=0.9-0.2\) & [23] \\ \hline DE & \(F=0.9\) and \(cr=0.2\) & [24] \\ \hline BA & \(f_{min}=0\),\(f_{max}=1\), \(A_{0}=1\), \\  & \(r_{0}=0.1\), \(\alpha=0.9\),\(\gamma=0.9\) & [25] \\ \hline ABC & Food sources=10 & [20] \\ \hline AHA & \(Mig_{trigger}=20\), Flight Step & \\  & size \(F_{sz}=[-1,1]\) and & [8] \\  & \(r\in[0,1]\) & \\ \hline CBO & no parameters & [9] \\ \hline ECBO & \(Pro=0.5\) & [10] \\ \hline AHA-AO & \(\alpha=0.1\),\(\delta=0.1\),\(\omega=0.005\) & \\  & and \(r\in[0,1]\) & [11] \\ \hline SNS & no parameters & [12] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters of the algorithms. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**No** & **Function (F)** & **Type** & \(D\) & **Range** & **Formulation** & \(p\) & \(I\) \\ \hline \(F_{1}\) & BOOTH & U, C, Di, NS, NSc & 2 & \([-10,10]\) & \(f(x)=(x_{1}+2x_{2}-\) & 50 & 100 \\  & & & \(7)^{2}+(2x_{1}+x_{2}-5)^{2}\) & & & \\ \hline \(F_{2}\) & SPHERE & U, S & 2 & \([-5.12,5.12]\) & \(f(x)=\sum_{i=1}^{d}x_{i}^{2}\) & 50 & 50 \\ \hline \(F_{3}\) & SPHERE & U, S & 10 & \([-5.12,5.12]\) & \(f(x)=\sum_{i=1}^{d}x_{i}^{2}\) & 150 & 100 \\ \hline \(F_{4}\) & SPHERE & U, S & 20 & \([-5.12,5.12]\) & \(f(x)=\sum_{i=1}^{d}x_{i}^{2}\) & 150 & 150 \\  & & & & \(f(x)=\) & & \\ \hline \(F_{5}\) & BEALE & U, C, Di, NS, NSc, S & 2 & \([-4.5,4.5]\) & \((1.5-x_{1}+x_{1}x_{2})^{2}+\) & 50 & 100 \\  & & & & \((2.25-x_{1}+x_{2}^{2})^{2}\) & & \\ \hline \(F_{6}\) & ROTATED & U, C, Di, NS, NSc & 2 & \([-65.536,65.536]\) & \(f(x)=\sum_{i=1}^{d}\sum_{j=1}^{i}x_{j}^{2}\) & 50 & 250 \\  & HYPER-ELLIPSOID & & & & & \\ \hline \(F_{7}\) & SUM SQUARES & U, C, Di, S, Sc & 20 & \([-5.12,5.12]\) & \(f(x)=\sum_{i=1}^{d}x_{i}^{2}\) & 200 & 500 \\ \hline \(F_{8}\) & QUARTIC & U, C, Di, S, Sc & 2 & \([-1.28,1.28]\) & \(f(x)=f(x_{1},...,x_{n})=\) & \\  & & & & \(\frac{\sum_{i=1}^{d}x_{i}^{2}}{random[0,1)}\) & 50 & 100 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Unimodal benchmark functions used in experiments \(F\), the number of iterations \(I\), population \(P\), and dimension \(D\). algorithms perform similarly, making it difficult to determine which algorithm is superior for this function. On the other hand, functions \(F_{9}\) and \(F_{18}\) are multimodal, continuous, differentiable, non-separable, and non-scalable, featuring multiple local minima. For these functions, the GSM algorithm achieves the best results with a low standard deviation, indicating high precision. Functions \(F_{10}\) and \(F_{14}\) are multimodal, continuous, differentiable, non-separable, and scalable. For \(F_{10}\), the GSM algorithm performs slightly better than the other algorithms. However, for \(F_{14}\), the AHA and AHA-AO algorithms demonstrate superior performance. Functions \(F_{11}\), \(F_{12}\), and \(F_{13}\) are multimodal and separable, posing challenges for all algorithms in locating the global optimum. For \(F_{11}\) and \(F_{13}\), the GSM algorithm produces results closest to the optimum, while for \(F_{12}\), the DE algorithm outperforms the others. For the non-scalable function \(F_{15}\), the GSM algorithm surpasses all competing algorithms. Similarly, for \(F_{16}\), a multimodal and non-separable function, the GSM algorithm proves to be the best.