# Solution to Solid Wood Board Cutting Stock Problem
## 4 Experiments and Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline \multirow{2}{*}{_Class_} & \multirow{2}{*}{\(n\)} & \multirow{2}{*}{\(m\)} & \multirow{2}{*}{\(d\)} & \multicolumn{2}{c}{_AC-IGA_} & \multicolumn{2}{c}{_ACA_} & \multicolumn{2}{c}{_GA_} & \multicolumn{1}{c}{_GWO_} & \multicolumn{1}{c}{_PBO_} \\ \cline{5-10}  & & & & \(u\) (\%) & \(u\) (\%) & \(u\) (\%) & \(u\) (\%) & \(u\) (\%) \\ \hline
1 & 1 & 4 & \([20,50)\) & 83.230 & 82.335 & 81.818 & 82.618 & 81.276 \\
2 & 2 & 5 & \([50,100)\) & 95.984 & 90.647 & 90.468 & 90.449 & 87.853 \\
3 & 3 & 5 & \([100,150)\) & 98.574 & 96.316 & 96.003 & 96.199 & 94.360 \\
4 & 3 & 6 & \([150,200)\) & 98.566 & 95.713 & 95.454 & 95.456 & 93.729 \\
5 & 4 & 7 & \([200,250)\) & 98.932 & 95.909 & 95.860 & 95.851 & 94.727 \\
6 & 5 & 8 & \([250,300)\) & 99.212 & 96.695 & 96.631 & 96.727 & 95.695 \\
7 & 6 & 9 & \([300,350)\) & 98.895 & 97.799 & 97.719 & 97.774 & 96.967 \\
8 & 8 & 10 & \([350,400)\) & 98.994 & 97.769 & 97.757 & 97.797 & 97.096 \\
9 & 10 & 15 & \([400,450)\) & 99.060 & 97.501 & 97.409 & 97.458 & 96.727 \\ \hline \hline \multicolumn{5}{c}{_Arg_ (\%)} & \multicolumn{5}{c}{96.828} & 94.520 & 94.346 & 94.481 & 93.159 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters and experimental results. Figure 9: Comparison diagram of the statistical experimental results. Table 4 shows the analysis of the experimental results, where Min, Max, and Mean are the minimum, maximum, and average of the experimental results obtained by the corresponding algorithm, and SD is the standard deviation. Compared with the ACA, GA, GWO, and PBO, the Min obtained by the AC-IGA increased by 3.157%, 3.290%, 3.090%, and 4.801%, respectively, the Max obtained by the AC-IGA increased by 2.974%, 2.771%, 2.839%, and 3.846%, respectively, the Mean obtained by the AC-IGA increased by 2.974%, 3.034%, 2.949%, and 4.063%, respectively, and the SD of the AC-IGA is the smallest, which indicates that the AC-IGA has better optimization ability and robustness. Figure 10 shows the relationship between the highest board utilization rate searched by the five algorithms and the number of iterations; the red solid line is AC-IGA, the blue dashed line is ACA, the green dash-dot line is GA, the yellow solid line is GWO, and the purple dotted line is PBO. The board utilization rate of the AC-IGA converges to 99.240% after twenty-seven iterations and the board utilization rate of the GA converges to 96.469% after thirty-two iterations, which shows that the AC-IGA can find the optimal solution faster, and the \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Algorithm**} & \multicolumn{4}{c}{**Utilization Rate of Board (\%)**} \\ \cline{2-5}  & **Min** & **Max** & **Mean** & **SD** \\ \hline _AC-IGA_ & 98.929 & 99.240 & 99.051 & 0.097 \\ _ACA_ & 95.772 & 96.266 & 96.077 & 0.149 \\ _GA_ & 95.639 & 96.469 & 96.017 & 0.244 \\ _GWO_ & 95.839 & 96.401 & 96.102 & 0.154 \\ _PBO_ & 94.128 & 95.394 & 94.988 & 0.379 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Analysis of the experimental results. Figure 10: Iterative convergence curves. solution quality is better, than the GA. This is mainly because the performance of the GA is excessively dependent on the initial population's quality; the quality of the initial solutions generated randomly is low, which results in low solving efficiency. In this paper, the solutions generated by each generation of IACA are used as the initial population of AC-IGA, which can improve the solution quality and increase the convergence speed. The board utilization rate of the ACA converges to 96.266% after nine iterations, which shows that although the AC-IGA is slower than the ACA in finding the optimal solution, the solution quality of the AC-IGA is better. This is mainly because ants are concentrated on some local solutions earlier under the action of the positive feedback mechanism of the ACA, which reduces the population diversity and makes it difficult for ants to further explore new solutions.