## 6 Numerical Experiments
### Test Results  
Three new indicators (i.e., Equations (29) and (30)) are established to better illustrate the effectiveness of the strategy and collaborative optimization model. To fully demonstrate the trade off between empty container traffic volume and loaded container transportation volume, the concept of an empty weight ratio is introduced and can be formulated as  
\[\alpha=\text{ave}(\frac{XE_{ij}^{\text{vs}}}{XC_{ij}^{\text{vs}}}*100\%) \tag{29}\]  
There are two sources of empty containers: one is empty container repositioning, and the other is empty container leasing. To show the trade off between empty container \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameters** & **Route1** & **Route2** & **Route3** \\ \hline \(es_{ij}^{p}\) & \(l_{ij}^{1}\star 0.5\) & \(l_{ij}^{2}\star 0.5\) & \(l_{ij}^{3}\star 0.5\) \\ \(ef_{ij}^{p}\) & \(l_{ij}^{1}\star 0.4\) & \(l_{ij}^{2}\star 0.4\) & \(l_{ij}^{3}\star 0.4\) \\ \(cc_{ij}^{c}\) & \(ef_{ij}^{1\star}\) rand (0.5, 0.6) & \(ef_{ij}^{2\star}\) rand (0.5, 0.6) & \(ef_{ij}^{3\star}\)rand (0.5, 0.6) \\ \(cc_{ij}^{c}\) & \(cc_{ij}^{c}\) rand (0.5, 0.6) & \(cc_{ij}^{2\star}\) rand (0.5, 0.6) & \(cc_{ij}^{3\star}\) rand (0.5, 0.6) \\ \(cl_{ij}^{p}\) & rand (150, 300) & rand (150, 300) & rand (150, 300) \\ \(cs_{i}^{p}\) & rand (650, 750) & rand (650, 750) & rand (650, 750) \\ \(c\text{o}^{\text{b}}\) & 3000 & 3000 & 3000 \\ \(DS_{ij}^{p}\) & \(N\big{(}200,2^{2}\big{)}\) & \(N\big{(}200,2^{2}\big{)}\) & \(N\big{(}200,2^{2}\big{)}\) \\ \(DF_{ij}^{p}\) & \(N\big{(}200,2^{2}\big{)}\) & \(N\big{(}200,2^{2}\big{)}\) & \(N\big{(}200,2^{2}\big{)}\) \\ \(DE_{ij}^{p}\) & \(N\big{(}40,2^{2}\big{)}\) & \(N\big{(}40,2^{2}\big{)}\) & \(N\big{(}40,2^{2}\big{)}\) \\ \(IE\mathcal{L}_{i}\) & rand (200, 500) & rand (200, 500) & rand (200, 500) \\ \(cap_{i}^{p}\) & 12,000 & 12,000 \\ \(\lambda s_{ij}^{p}\) & 0.95 + rand (\(-\)0.03, 0.03) & 0.95 + rand (\(-\)0.03, 0.03) & 0.95 + rand (\(-\)0.03, 0.03) \\ \(\lambda f_{ij}^{p}\) & 0.95 + rand (\(-\)0.03, 0.03) & 0.95 + rand (\(-\)0.03, 0.03) & 0.95 + rand (\(-\)0.03, 0.03) \\ \(\gamma f_{ij}^{p}\) & 0.3 + rand (\(-\)0.2, 0.2) & 0.3 + rand (\(-\)0.2, 0.2) & 0.3 + rand (\(-\)0.2, 0.2) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter setting for numerical test. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & & & **CPLEX** & & **BC** & \\ \cline{3-6}
**Case** & **Route** & **Voyage** & **Obj. (USD)** & **Runtimes (s)** & **Obj. (USD)** & **Runtimes (s)** & **Cuts** \\ \hline
1 & 1 & 4 & 300,047,427 & 1.57 & 300,047,427 & 0.06 & 11 \\
2 & 2 & 4 & 290,253,066 & 2.49 & 290,253,066 & 1.17 & 43 \\
3 & 3 & 4 & 188,742,495 & 12.03 & 188,742,495 & 1.09 & 39 \\
4 & 1,2 & 4 & 590,396,489 & 5.31 & 590,396,489 & 2.12  
repositioning and the leasing volume, the concept of an empty container repositioning satisfaction rate \(\beta\) is introduced. This can be formulated as  
\[\beta=\frac{\sum\limits_{s\in S}\sum\limits_{v\in V}\sum\limits_{j\in P}\sum \limits_{i\in P}X\bar{c}_{ij}^{vs}}{\sum\limits_{s\in S}\sum\limits_{v\in V} \sum\limits_{j\in P}\sum\limits_{i\in P}\left(X\bar{c}_{ij}^{vs}+X\bar{c}_{ij}^ {vs}\right)}*100\% \tag{30}\]  
To evaluate the performance of BC, two sets of tests are compared for different problem sizes. The first set uses the CPLEX MIP solver to solve the model under default settings directly. The other group is computed with the BC algorithm. Ten different cases are tested in this paper, as shown in Table 3. Cases 1, 2, and 3 only contain one shipping route and four voyages, while Cases 4, 5, and 6 involve two routes and four voyages. The routes consisting of Cases 7 and 8 are the same, whereas Case 8 includes eight voyages. Evidently, Cases 9 and 10 contain more shipping routes. As shown in Table 3, the exact solution could be obtained by CPLEX in most instances. However, the CPU runtime of CPLEX increases significantly as the scale of the problem grows. While the size of the CSAECRP is enormous, CPLEX cannot obtain the exact solution in an acceptable time frame (10,800 s).