## 5 Comparative Experimental Analysis  
A set of comparative experiments were designed and performed to verify the accuracy and effectiveness of the proposed hybrid chaotic whale optimization algorithm for solving the min\(L\) in the case study. The basic particle swarm optimization (PSO) algorithm and standard whale optimization algorithm (WOA) are introduced as the competitive algorithms, where the former is a classical representative meta-heuristic algorithm and the latter is an emerging prevailing one. The evolution curves of the fitness value for min\(L\) are illustrated in Figure 9, where the superiority of HCWOA with both considerable convergence speed and excellent convergence accuracy for solving min\(L\) is demonstrated. In the phase of population initialization, the Tent chaotic mapping strategy is introduced to HCWOA, which achieved the least fitness value at the first iteration compared with other rivals and led to a fast convergence. Significantly, in the period of iteration 200 to 300, the HCWOA hybridized with the Levy flight strategy never failed to keep the momentum in perusing further convergence, while the standard WOA was convergent, which demonstrates that the Levy flight strategy is capable of preventing WOA from trapping in local optimum and improving the convergence accuracy substantially. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Layer**} & \multirow{2}{*}{**Item**} & \multicolumn{8}{c}{**Hierarchical Weight**} \\ \cline{3-13}  & \(Y\) & \(Y_{1}\) & & & & & & \(Y_{2}\) & & & \(Y_{3}\) & \\ \hline \multirow{3}{*}{Criterion} & \(w\) & 0.40 & & & & & 0.40 & & & 0.20 & \\  & \(Z\) & \(Z_{1}\) & \(Z_{2}\) & \(Z_{3}\) & \(Z_{4}\) & \(Z_{5}\) & \(Z_{6}\) & \(Z_{7}\) & \(Z_{8}\) & \(Z_{9}\) & \(Z_{10}\) & \(Z_{11}\) & \(Z_{12}\) \\ \cline{1-1}  & \(w\) & 0.45 & 0.07 & 0.07 & 0.12 & 0.12 & 0.17 & 0.24 & 0.62 & 0.14 & 0.25 & 0.50 & 0.25 \\ \cline{1-1}  & \(I\) & 85 & 80 & 80 & 90 & 90 & 85 & 80 & 85 & 80 & 90 & 90 & 90 \\ \cline{1-1}  & \(P\) & 0.854 & & & & & & & & & & & \\ \cline{1-1}  & \(I\) & 90 & 90 & 85 & 85 & 80 & 85 & 90 & 80 & 75 & 80 & 80 & 80 \\ \cline{1-1}  & \(P\) & 0.835 & & & & & & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 14: Performance coefficients determination of replacement plans. Corresponding to the optimal fitness value of each optimization algorithm, the modified comparison matrices are generated automatically, and the results of consistency tests are gained as follows:  
\[Y_{1,\text{HCWOA}}=\left( \begin{array}{cccccc}1&6.1514&6.1457&3.8134&3.9181&2. 7247\\ 0.1626&1&0.9832&0.2600&0.2600&0.3507\\ 0.1627&1.0170&1&0.2600&0.2600&0.4333\\ 0.2622&3.8462&3.8462&1&1.0591&0.3250\\ 0.2552&3.8462&3.8462&0.9442&1&0.3250\\ 0.3670&2.8514&2.3079&3.0769&3.0769&1\end{array} \right)\]  
\[CI_{Y_{1},\text{HCWOA}}=0.082\quad RI_{Y_{1},\text{HCWOA}}=1.24\quad CR_{Y_{1 },\text{HCWOA}}=0.066<0.10\]  
\[Y_{1,\text{WOA}}=\left( \begin{array}{cccccc}1&6.1156&6.1009&4.1316&3.7958&2. 8451\\ 0.1635&1&1.1734&0.2056&0.2030&0.3003\\ 0.1639&0.8522&1&0.2047&0.2355&0.3941\\ 0.2420&4.8638&4.8852&1&0.9961&0.3250\\ 0.2634&4.9261&4.2463&1.0039&1&0.2854\\ 0.3515&3.3300&2.5374&3.0769&3.5039&1\end{array} \right)\]  
\[CI_{Y_{1},\text{WOA}}=0.105\quad RI_{Y_{1},\text{WOA}}=1.24\quad CR_{Y_{1}, \text{WOA}}=0.085<0.10\]  
\[Y_{1,\text{PSO}}=\left( \begin{array}{cccccc}1&6.1623&6.1083&3.7820&2.8000&2. 4107\\ 0.1623&1&0.9884&0.2450&0.2087&0.3278\\ 0.1637&1.0117&1&0.1982&0.2361&0.3471\\ 0.2644&4.0816&5.0454&1&0.9020&0.2431\\ 0.3571&4.7916&4.2355&1.1086&1&0.2459\\ 0.4148&3.0506&2.8810&4.1135&4.0667&1\end{array} \right)\]  
\[CI_{Y_{1},\text{PSO}}=0.112\quad RI_{Y_{1},\text{PSO}}=1.24\quad CR_{Y_{1}, \text{PSO}}=0.090<0.10\]  
Figure 9: Evolution curves of the fitness value for min\(L\).