## 5 Numerical Experiments
### Real-World Instances  
We also carry out the real-world case study based on the JD Logistics' distribution network in Beijing. As shown in Figure 19, we choose a subset of facilities from the real-world network as the input data, where 40 warehouses store commodities and 10 DCs provide urban distribution services for 100 delivery stations. The demand of stations is 1000 times of the volume of goods in the original dataset. The demand between warehouses and stations is generated according to the EP principle. The capacity of each DC is evenly configured according to the total demand. The distances of links are specified using to the real-world road network.  
For this real-world case, the Gruobi solver could not obtain a feasible solution within a time limit of 5 h. ALNS with an efficient initial solution finally converged in 18 min. Figure 20 shows the convergence curve of the solution. As the initial solution of the algorithm is obtained by adopting the GLB-oriented lower bound, the heuristic algorithm proposed in this paper can roughly estimate the quality of the solution. In this case, the GLB-oriented lower bound is 11,120.16, the initial upper bound is 142,226.97, and the initial GAP is 21%. The final solution obtained when the algorithm terminates is 13,402.43. If  
Figure 19: Warehouses, DCs, and delivery stations in the network.  
still compared with the initial lower bound, the GAP of the final solution should be less than 17%.  
Table 15 presents the details of the final solution, that is, the allocation of warehouses to supply the corresponding DCs and the demand coverage decision from DCs to stations. Figure 21 shows the spatial visualization of the optimization results. According to the final solution, DCs 145 and 146 assume the main distribution function, covering 40% of the total number of warehouses and 59% of the total number of stations. Affected by the spatial distribution of nodes, DC 143 only receives the goods transferred from other DCs and sends them to the station, while DC 148 is only responsible for handling the goods from warehouses and transferring them to other DCs. The flow in the logistics network is shown in Figure 22. In the figure, the red line represents lateral-transshipments flow between DCs, the blue line represents the non-transfer flow, and the width of the line reflects the size of the flow. In this real-world case, the total demand is 35,610 and the lateral-transshipments volume is 6126.  
\begin{table}
\begin{tabular}{l l l} \hline \hline
**DC** & **Warehouses Assigned to DC** & **Stations Covered by DC** \\ \hline
141 & 101; 102; 107; 108; 115; 129 & 4; 8; 37; 43; 57; 92; 94 \\
142 & 110; 117; 121; 128; 132; 138 & 5; 34; 55; 68; 69; 87 \\
143 & - & 12; 39; 51; 77 \\
144 & 124; & 73 \\
145 & 105; 119; 127; 134; 136; 137 & 2; 6; 7; 9; 10; 11; 17; 18; 25; 27; 28; 29; 30; 32; 33; \\
146 & 106; 109; 111; 112; 116; 118; & 1; 3; 13; 14; 16; 19; 20; 21; 22; 31; 35; 48; 49; 50; 52; \\
147 & 104; 133 & 56; 60; 61; 62; 67; 70; 72; 75; 76; 83; 89; 90; 96; 99; 100 \\
148 & 114; 125 & - \\
149 & 120; 135 & 15; 24; 38; 79; 80; 84; 95 \\
150 & 103; 113; 123; 126; 131 & 23; 42; 44; 47; 53; 59; 63; 66; 71; 74; 78; 86; 88; 91 \\ \hline \hline \end{tabular}
\end{table}
Table 15: Details of the final solution.  
Figure 20: Convergence curve of the lower bound and upper bound of the real-world case.