# An Improved Variable Neighborhood Search for the Reconfigurable Assembly Line Reconfiguring Problem
## 5 Proposed IVNS-RALRP
### Solution Evaluation
Table 2 illustrates the different scale problems used to evaluate the performance of the algorithms studied in this section. The detailed benchmarking dataset is given in the in Appendix A.1. ### Performance Analysis of Each Component

In this section, the contribution of the proposed data structure and the improvement of VNS are investigated. We employ all the four performance metrics described in Section 6.1 to evaluate the solutions obtained from all algorithms. All algorithms are applied the same maximum iteration (\(maxIter=1000\)) for a fair comparison. #### 6.3.1 Effect of the Data Structure

To assess the effectiveness of the proposed data structure, we consider the IVNS-RALRP without these data structure (IVNS-WDS) and compare it to IVNS-RALRP for all problems. Table 3 shows four performance metrics across 10 independent runs for all test cases. As shown in Table 3, IVNS-RALRP outperforms IVNS-WDS for \(\overline{t}\) metrics, while showing nearly identical performance for the other three metrics for all data sets. The competitive advantage of IVNS-RALRP in computation speed is attributed to the data structure, which enhances search efficiency while maintaining more process information. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Case Size** & **No.** & **Number of Operations** & **Cycle Time (s)** \\ \hline \multirow{3}{*}{small-scale} & 1 & 20 & 200 \\  & 2 & 25 & 200 \\  & 3 & 30 & 200 \\ \hline \multirow{3}{*}{large-scale} & 4 & 70 & 200 \\  & 5 & 75 & 200 \\ \cline{1-1}  & 6 & 80 & 200 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Cases of different sizes. \begin{table}
\begin{tabular}{l c c c c c c c c} \hline \hline \multirow{2}{*}{**No.**} & \multicolumn{5}{c}{**IVNS-RALRP**} & \multicolumn{5}{c}{**IVNS-WDS**} \\ \cline{2-9}  & \(R\) & \(\bar{r}\) & \(S\) & \(\bar{t}\) & \(R\) & \(\bar{r}\) & \(S\) & \(\bar{t}\) \\ \hline
1 & 31 & 31.3 & 0.21 & **6.52** & 31 & 31.3 & 0.41 & 7.80 \\
2 & 38 & 39.0 & 0.80 & **6.85** & 38 & 39.0 & 1.60 & 8.38 \\
3 & 43 & 43.8 & 0.56 & **7.06** & 43 & 43.9 & 0.69 & 8.60 \\
4 & 108 & 109.5 & 1.45 & **12.99** & 108 & 109.3 & 1.21 & 16.70 \\
5 & 113 & 114.1 & 1.69 & **13.40** & 113 & 114.5 & 2.45 & 17.32 \\
6 & 121 & 121.7 & 1.01 & **13.70** & 121 & 121.8 & 0.96 & 17.89 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The computational results of IVNS-RALRP and IVNS-WDS. To further evaluate computation speed, we compare the two algorithms for the same case of \(\overline{t}\). Table 4 presents the comparative results, indicating a significant reduction in computation time for IVNS-RALRP compared to IVNS-WDS as the problem size increases. This improvement is due to the implementation of efficient data structure in the iterative search. On the whole, it can be concluded that the proposed data structure is a vital component of the approach, contributing significantly to the computation speed of IVNS-RALRP. #### 6.3.2 Effect of the Shaking and Local Search Procedure

To evaluate the influence of the shaking and local search procedure described in Sections 5.5 and 5.6, we consider the proposed IVNS-RALRP and compare it to basic VNS. The results across 10 independent runs are shown in Table 5. Concerning the optimal objective value metrics \(R\), IVNS-RALRP searches for lower reconfiguration cost than VNS in three cases. Concerning the average objective value metrics \(\overline{r}\) and the computation stability metrics \(S\) in Table 5, IVNS-RALRP is superior to VNS across all data sets. The notable performance of the IVNS-RALRP for the three metrics can be attributed to the improvement of the shaking and local search procedure. Concerning \(\overline{t}\) metrics, the improved algorithm consumes additional computation time. As the reconfiguration of assembly line is not real-time responsive, allowing for low reconfiguration cost at computation times of a few seconds.