# Two-Stage Metaheuristic Algorithms for Order-Batching and Routing Problems
## 5 Computational Experiment
### Evaluation Analysis  
This experiment was conducted to understand the solving capability and solution quality of the algorithms applied to medium- and large-scale problems. According to the types of initial solution, two types of algorithm were applied. This section presents a comparison of the effectiveness of the VNS and SA algorithms in solving BFD initial solutions derived with 40 and 80 orders, respectively. \begin{table}
\begin{tabular}{c} **Objective Value When** \\ \end{tabular} \\ \hline
1 & 1–3 & 1–3 & 10 & 524.7 & 544.0 \\
2 & 1–5 & 1–3 & 10 & 717.3 & 726.7 \\
3 & 1–3 & 1–3 & 20 & 416.0 & 433.3 \\
4 & 1–5 & 1–3 & 20 & 566.0 & 584.0 \\
5 & 1–3 & 1–5 & 10 & 670.0 & 680.7 \\
6 & 1–5 & 1–5 & 10 & 808.7 & 838.0 \\
7 & 1–3 & 1–5 & 20 & 482.7 & 487.3 \\
8 & 1–5 & 1–5 & 20 & 616.0 & 655.3 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Mean objective value of \(L_{max}\) associated with 40 orders (unit: m). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \begin{tabular}{c} **Scenario** \\ \end{tabular}  & \begin{tabular}{c} **Number of Items** \\ \end{tabular}  & \begin{tabular}{c} **Capacity of** \\ \end{tabular}  & \begin{tabular}{c} **Vehicle Capacity** \\ \end{tabular}  & \ \begin{tabular}{c} **Perturbation Factor = 0.2** \\ **(Objective Value/Solving Time)** \\ \end{tabular}  & \begin{tabular}{c} **Perturbation Factor = 0.4** \\ **(Objective Value/Solving Time)** \\ \end{tabular}  \\ \hline
1 & 1–3 & 1–3 & 10 & 462.7 & 0.44 & 462.7 & 0.43 \\
2 & 1–5 & 1–3 & 10 & 544.7 & 0.80 & 544.7 & 0.84 \\
3 & 1–3 & 1–3 & 20 & 320.7 & 0.58 & 320.7 & 0.58 \\
4 & 1–5 & 1–3 & 20 & 380.7 & 0.84 & 380.7 & 0.79 \\
5 &As shown in Figure 5, the computation time trend associated with the 40 orders revealed that the VNS algorithm converged within 1 s and quickly achieved stability, whereas the SA algorithm demonstrated a marked convergence tendency within 5 s and slower converging processes afterward. Regarding the solving trend associated with the 80 orders, the VNS algorithm converged within 1 s; by contrast, the SA algorithm converged within 4-5 and 18-20 s. Therefore, the SA algorithm converged more slowly, and its objective value was less favorable than that of the VNS algorithm. The SA algorithm converged quickly when the number of orders was low. The higher the order number was, the slower the solving process achieved stability; specifically, the convergence proceeded rather slowly. Next, we conducted another comparison between the VNS and SA algorithms by using the various BFD initial solutions obtained for the 40 and 80 orders in each of the eight scenarios. Seeds 10, 20, and 30 were also used in the test. The objective value used in this study was the optimal picking distance. We assumed that the objective value was BV, the optimal mean value of the other algorithm was PV, and the error ratio formula was \(\frac{(PV-BV)}{BV}\). Figure 6 illustrates a comparison of the SA and VND algorithms, revealing that the difference between these algorithms was lower in Scenarios 1 and 5. For the 80 orders, Scenario 5 exhibited the lowest error ratio, followed by Scenario 1. Moreover, as shown in Figure 7, Scenarios 4 and 8 demonstrated high error ratios. According to the aforementioned experimental results, we drew the following conclusion: In batching and picking problems, the VNS algorithm is more favorable than the SA algorithm for medium- and small-scale problems. Moreover, for medium- and large-scale problems, the VNS algorithm demonstrates higher solution quality and a faster solving speed than does the SA algorithm. In the solving process, the VNS algorithm can achieve convergence rapidly. In addition, when the problem becomes more complex, the advantages of the VNS algorithm become more salient, and this is attributable to the algorithm's rapid convergence when solving problems and its capability to determine batches and picking routes swiftly. Therefore, this study confirms that the VNS algorithm is the optimal method.