# An Improved Variable Neighborhood Search for the Reconfigurable Assembly Line Reconfiguring Problem
## 5 Proposed IVNS-RALRP
### Solution Evaluation
To further assess the reconfiguration cost obtained by both algorithms, the reconfiguration cost from 10 computations are displayed as box plots in Figure 6. From these box plots, it is evident that IVNS-RALRP performs better than VNS in terms of optimal objective value and computation stability. Moreover, as the problem size increases, the computation stability becomes more pronounced. Hence, given the above analysis, it can be ascertained that the contribution of the shaking and local search procedure is substantial in improving the performance of proposed approach. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**No.** & **IVNS-RALRP** & **IVNS-WDS** & **Decrease * \\ \hline
1 & 6.52 & 7.80 & \(-16\%\) \\
2 & 6.85 & 8.38 & \(-18\%\) \\
3 & 7.06 & 8.60 & \(-18\%\) \\
4 & 12.99 & 16.70 & \(-22\%\) \\
5 & 13.40 & 17.32 & \(-23\%\) \\
6 & 13.70 & 17.89 & \(-23\%\) \\ \hline \hline \end{tabular} (*): Decrease in the results of IVNS-RALRP compared with IVNS-WDS.

\end{table} 
Table 4: The computation speed metric \(\overline{t}\) comparison of IVNS-RALRP and IVNS-WDS. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{4}{c}{**IVNS-RALRP**} & \multicolumn{4}{c}{**VNS**} \\ \cline{2-9}  & \(R\) & \(\overline{r}\) & \(S\) & \(\overline{t}\) & \(R\) & \(\overline{r}\) & \(S\) & \(\overline{t}\) \\ \hline
1 & **31** & 31.3 & 0.21 & 6.52 & **31** & 32.3 & 0.61 & 4.33 \\
2 & **38** & 39.0 & 0.80 & 6.85 & **38** & 39.1 & 2.29 & 4.36 \\
3 & **43** & 43.8 & 0.56 & 7.06 & 44 & 45.2 & 1.56 & 4.47 \\
4 & **108** & 109.5 & 1.45 & 12.99 & **108** & 111.8 & 13.56 & 6.00 \\
5 & **113** & 114.1 & 1.69 & 13.40 & 114 & 116.9 & 7.09 & 6.15 \\
6 & **121** & 121.7 & 1.01 & 13.70 & 122 & 125.8 & 8.56 & 6.35 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The computational results of IVNS-RALRP and VNS. ### Comparison with Other Heuristic Algorithms

IVNS-RALRP is further compared to two other well-known heuristic algorithms, which are GA, TS, SA, and PSO. To make a fair comparison of the results, the maximum iteration is set to 1000 for all the compared algorithms. To determine the optimal parameters of GA, we carry out the Taguchi method of Design of Experiments (DOE). The detailed process of parameter calibration is given in Appendix A.2. The experimental results for the four performance metrics are tabulated in Table 6. It is evident that compared to GA, TS, SA and PSO, IVNS-RALRP achieves equivalent or superior performance in terms of the \(R\), \(\overline{r}\), and \(S\) metrics. In terms of solution time, IVNS-RALRP does not exhibit superiority; however, the time difference of seconds can be ignored in the reconfiguration stage of the assembly line. In addition, the reconfiguration cost from 10 computations are illustrated as box plots in Figure 7. These box plots clearly demonstrate that IVNS-RALRP outperforms GA, TS, SA, and PSO in terms of optimal objective value and computation stability. Moreover, as the problem size increases, the computation stability becomes more pronounced. This underscores that IVNS-RALRP effectively maintains superior results while enhancing solution stability.