## 3 Proposed Method
### Improved SMA
### Benchmark Function Validation and Parameter Settings

In this study, the test set for the DQOBLSMA comparison experiment was the 23 classical test functions that had been used in the literature [34]. The details are shown in Table 2. These classical test functions are divided into unimodal functions, multimodal functions, and fixed-dimension multimodal functions. The unimodal functions (F1-F7) have only one local solution and one optimal global solution and are usually used to evaluate the local exploitation ability of the algorithm. Multimodal functions (F8-F13) are often used to test the exploration ability of the algorithm. F14-F23 are fixed-dimensional multimodal functions with many local optimal points and low dimensionality, which can be used to evaluate the stability of the algorithm. The DQOBLSMA has been compared to the original SMA and five other algorithms: the slime mold algorithm improved by opposition-based learning and Levy flight distribution (OBLSMAL) [48], the equilibrium slime mold algorithm (ESMA) [49], the equilibrium optimizer with a mutation strategy (MEO) [50], the adaptive differential evolution with an optional external archive (JADE) [51], and the gray wolf optimizer based on random walk (RWGWO) [52]. The parameter settings of each algorithm are shown in Table 3, and the experimental parameters for all optimization algorithms were chosen to be the same as those reported in the original works. In order to maintain a fair comparison, each algorithm was independently run 30 times. The population size (\(N\)) and the maximum function evaluation times (\(FEs\)) of all experimental methods were fixed at 30 and 15,000, respectively. The comparative experiment was run under the same test conditions to keep the experimental conditions consistent. The proposed method was coded in Python3.8 and tested on a PC with an AMD R5-4600 Hz, 3.00 GHz of memory, 16 GB of RAM, and the Windows 11 operating system. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Function Type** & **Function** & **Name** & **Dimension** & **Range** & **Theoretical Value** \\ \hline \multirow{8}{*}{Unimodal test functions} & F1 & Sphere & 30 & \([-100,100]\) & 0 \\  & F2 & Schwefel 2.22 & 30 & \([-10,10]\) & 0 \\  & F3 & Schwefel 1.2 & 30 & \([-100,100]\) & 0 \\  & F4 & Schwefel 2.21 & 30 & \([-100,100]\) & 0 \\  & F5 & Rosenbrock & 30 & \([-30,30]\) & 0 \\  & F6 & Step & 30 & \([-100,100]\) & 0 \\ \hline \multirow{8}{*}{Multimodal test functions} & F7 & Quartic & 30 & \([-1.28,1.28]\) & 0 \\  & F8 & Schwefel 2.26 & 30 & \([-500,500]\) & \(-418.9829\times D\) \\  & F9 & Rastrigin & 30 & \([-5.12,5.12]\) & 0 \\  & F10 & Ackley & 30 & \([-32,32]\) & 0 \\  & F11 & [PERSON] & 30 & \([-600,600]\) & 0 \\  & F12 & Penalized & 30 & \([-50,50]\) & 0 \\  & F13 & Penalized2 & 30 & \([-50,50]\) & 0 \\ \hline \multirow{8}{*}{Fixed-dimension multimodal test functions} & F14 & Foxholes & 2 & \([-65,65]\) & 0.998004 \\  & F15 & [PERSON] & 4 & \([-5,5]\) & 0.0003075 \\  & F16 & Six-Hump Camel Back & 2 & \([-5,5]\) & \(-1.03163\) \\  & F17 & [PERSON] & 2 & \([-5,5]\) & 0.398 \\  & F18 & Goldstein Price & 2 & \([-2,2]\) & 3 \\  & F19 & [PERSON] 3 & 3 & \([-1,2]\) & \(-3.8628\) \\  & F20 & [PERSON] 6 & 6 & \([0,1]\) & \(-3.322\) \\  & F21 & [PERSON] 5 & 4 & \([0,10]\) & \(-10.1532\) \\  & F22 & [PERSON] 7 & 4 & \([0,10]\) & \(-10.4028\) \\  & F23 & [PERSON] 10 & 4 & \([0,10]\) & \(-10.5363\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The classic benchmark functions. \begin{table}
\begin{tabular}{c c} \hline \hline
**Algorithm** & **Parameter** \\ \hline OBLSMAL & \(z=0.03,p_{1}=0.5,p_{2}=0.5\) \\ ESMA & \(z=0.03\) \\ MEO & \(a_{1}=2,a_{2}=1,GP=0.5\) \\ JADE & \(\mu F=0.5,\mu CR=0.5,p=0.1,c=0.1\) \\ RWGWO & Control parameter \(a,b\) decrease linearly from 2 to 0 \\ SMA & \(z=0.03\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Parameter settings for the comparative algorithms. ### Impacts of Components

In this section, different versions of the improvement are investigated.