## 5 Numerical Experiments
### Overall Computational Results
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \multicolumn{1}{c}{**Gurbeli**} & \multicolumn{6}{c}{**AINS**} \\ \hline
**Instance** & **Warchouses** & **OPT** & **T\_G** & **BEST\_SO11** & **GAP1** & **T\_B** & **INT\_SO1** & **GAP2** & **T\_B\_I** & **BEST\_SO12** & **GAP3** & **T\_A** \\
**Instance** & **ID** & **OPT** & **T\_G** & **BEST\_SO11** & **GAP1** & **(6)** & **INT\_SO12** & **GAP2** & **(6)** & **BEST\_SO12** & **GAP3** & **T\_A** \\
**BST\_SO12** & **GAP2** & **T\_A** & **BST\_SO12** & **GAP1** & **(6)** & **BST\_SO12** & **(6)** & **BST\_SO12** & **GAP3** & **(6)** \\ \hline \hline \multicolumn{1}{c}{**Set 2**} & \multicolumn{6}{c}{**AINS**} \\ \hline E-0224+s10-14 & 3,11,13,20 & - & 0.44 & - & - & 142,24 & - & - & 1.07 & - & - & 1.95 \\ E-0224+s11-12 & 2,8 15,17 & - & 0.2  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \multicolumn{1}{c}{**Gurboi**} & \multicolumn{8}{c}{**BB**} & \multicolumn{3}{c}{**AINS**} \\ \hline
**Instance** & **Warehouses** & **OPT** & **T,G** **(**6**) & **BEST\_SO11** & **GAP1** & **T,B** (**6**) & **INIT\_SO1** & **GAP2** & **T,B,I** (**6**) & **BEST\_SO12** & **GAP3** & **T,A** (**6**) \\ \hline \multicolumn{10}{c}{**Set 3**} \\ \hline E63-34+1622 & 4,11,20,28 & - & 0.35 & - & - & 205.54 & - & - & 0.03 & - & - & 2.55 \\ E63-34+1624 & 5,11,  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c} \hline \hline \multicolumn{2}{c}{**Cavobi**} & \multicolumn{4}{c}{**Bb**} & \multicolumn{4}{c}{**AINS**} \\ \hline
**Instance** & **Warehouses** & **OPT** & **T\_G** & **gG** & **BST\_SO1.1** & **GAP1** & **T\_B** & **INIT\_SO1** & **GAP2** & **T\_B\_1** & **BST\_SO2** & **GAP3** & **T\_A** \\  & **ID** & **(s)** & & & & & & & & & & \\ \hline \multicolumn{2}{c}{**Set 4**} & \multicolumn{4}{c}{**Set 4**} & \multicolumn{4}{c}{} \\ \hline Instance0-13 & 9, 28, 31, 34, 44 & - & 0.66 & - & - & 553,71 & - & - & 0.05 & - & - & 46.38 \\ Instance0-14 & 23, 24, 27, 28, 38, 4  
\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \multicolumn{2}{c}{**Gurebul**} & \multicolumn{1}{c}{**ABB**} & \multicolumn{3}{c}{**AINS**} \\ \hline
**Instance** & **Warbhouses** & **T.G** & **REST\_SO1.1** & **GAP1** & **T.B** & **INIT\_SO1.1** & **GAP2** & **T.B.J** & **BST\_SO1.2** & **GAP3** & **T.A** \\  & **ID** & **(a)** & **REST\_SO1.1** & **GAP1** & **(a)** & **INIT\_SO1.1** & **GAP2** & **(a)** & **REST\_SO1.2** & **GAP3** & **(a)** \\ \hline \multicolumn{2}{c}{**Set 4**} & \multicolumn{1}{c}{} & & & & & & & & & \\ \hline Instance00-47 & 10.16, 31, 39, 47, 50 & 3,165,681.32 & 71.95 & - & - & 600.76 & - & - & 0.31 & - & - & 108.09  
According to the results of Set 2 and Set 3, branch-and-bound algorithm can obtain optimal solutions for most instances, the gap is less than 9% for some hard instances, and the mean gap is 2%. In the computational time, Gurobi solver (561.15 s for Set 2 and 217.85 s for Set 3) is much more efficient than the branch-and-bound (0.83 s for Set 2 and 0.31 s for Set 3). However, we noticed that the initial solution of branch-and-bound is tight, the mean gaps are 10% and 6%, and average running times are 0.55 s and 0.04 s. For some instances, even initial solutions are optimal (E-n22-k4-s19-21, E-n33-k4-s19-26, E-n51-k5-13-19, and E-n51-k5-41-42). Results indicate the heuristic lower bound is relative loose, even though the initial solution has good quality, branch-and-bound algorithm needs more branches to converge. Based on this behavior, we design the ALNS with the efficient initial solution and results also show the benefits of heuristics from the computational point of view, presenting mean values of 3.67 s and 2.48 s, and average gap of 3% and 0%. For instances E-n51-k5-s11-19-27-47, E-n51-k5-s2-4-17-46, and E-n51-k5-s6-12-32-37 in Set 2, both the solver and branch-and-bound need spend more time than other instances, which indicates the locations of warehouses and DCs can affect the complexity of the model.