## 5 Numerical Experiment
### Scenario Setting
The crossover probability parameter k3 is adjusted in the range [0.5,1] in steps of 0.05. The experimental results are shown in Table 11.  
In general, either k3 = 0.5 or k3 = 0.75 is desirable, but considering that k3 is the crossover probability, which determines the global search ability of the algorithm, and 0.5 is weaker than 0.75, k3 = 0.75 is finally chosen. The crossover probability k1 is then adjusted to [0.5, 1] in steps of 0.1, and the results are shown in Table 12. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Experiment ID**} & \multicolumn{2}{c}{**Number of Repeated Experiments**} & \multirow{2}{*}{**Crossover Possibility**} & \multirow{2}{*}{**Mutation Probability**} & \multicolumn{1}{c}{**Average Running Time(s)**} & \multicolumn{1}{c}{**Objective Function Value**} \\  & & & & & & \\ \hline
1 & 5 & 1 & 0.5 & 0.2 & 0.2 & 145.6 & 43,252 \\
2 & 5 & 1 & 0.55 & 0.2 & 0.2 & 151.3 & 40,137 \\
3 & 5 & 1 & 0.60 & 0.2 & 0.2 & 124.7 & 39,751 \\
4 & 5 & 1 & 0.65 & 0.2 & 0.2 & 154.6 & 42,348 \\
5 & 5 & 1 & 0.70 & 0.2 & 0.2 & 143.4 & 43,022 \\
6 & 5 & 1 & 0.75 & 0.2 & 0.2 & 144.9 & 43,894 \\
7 & 5 & 1 & 0.80 & 0.2 & 0.2 & 157.5 & 41,275 \\
8 & 5 & 1 & 0.85 & 0.2 & 0.2 & 136.6 & 40,388 \\
9 & 5 & 1 & 0.90 & 0.2 & 0.2 & 160.7 & 41,294 \\
10 & 5 & 1 & 0.95 & 0.2 & 0.2 & 179.8 & 44,276 \\
11 & 5 & 1 & 1 & 0.2 & 0.2 & 197.5 & 43,023 \\ \hline \hline \end{tabular}
\end{table} 
Table 11: Tuning result of k3. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Experiment ID**} & \multicolumn{2}{c}{**Number of Repeated Experiments**} & \multirow{2}{*}{**Crossover Possibility**} & \multirow{2}{*}{**Mutation Probability**} & \multicolumn{1}{c}{**Average Running Time(s)**} & \multicolumn{1}{c}{**Objective Function Value**} \\  & & & & & & \\ \hline
1 & 5 & 1 & 0.75 & 0.2 & 0.2 & 144.5 & 44,015 \\
2 & 5 & 0.9 & 0.75 & 0.2 & 0.2 & 153.7 & 43,277 \\
3 & 5 & 0.8 & 0.75 & 0.2 & 0.2 & 140.9 & 43,501 \\
4 & 5 & 0.7 & 0.75 & 0.2 & 0.2 & 142.6 & 44,203 \\
5 & 5 & 0.6 & 0.75 & 0.2 & 0.2 & 154.8 & 43,108 \\
6 & 5 & 0.5 & 0.75 & 0.2 & 0.2 & 174.4 & 43,229 \\ \hline \hline \end{tabular}
\end{table} 
Table 12: Tuning result of k1. From the data in Table 12, the best optimization results and the smallest program running time are obtained when the crossover probability \(k1=0.7\). Therefore, \(k1=0.7\) is taken. The parameters of the adaptive genetic algorithm are determined as follows: population size \(N=100\), number of iterations 100, crossover probability (0.7,0.75), and mutation probability (0.2,0.2). We also solve the problem using the commercial solver GAMS, classical GA (genetic algorithm) and improved adaptive GA under the same maximum number of iterations of 100. For other parameters in classical GA, the crossover probability \(P_{c}\) is set to 0.75 and the mutation probability \(P_{m}\) is set to 0.2. All numerical experiments were completed on a Windows 10 personal computer with Intel(R) Core (TM) i7-6700 CPU 3.40 and 16.0 GB of RAM. The comparison results are shown in Table 13.  
As can be seen from the data in Table 13, the adaptive genetic algorithm outperforms the GAMS software ([[https://www.gams.com/](https://www.gams.com/)]([https://www.gams.com/](https://www.gams.com/)), accessed on 18 April 2023) in terms of solution time but is slightly inferior to the GAMS solution in terms of the quality of the objective function. When compared to the classical genetic algorithm, it outperforms the classical genetic algorithm in terms of solution time and quality.