## 5 Discussion  
This study focuses on a novel inventory routing problem in which the lead time is dependent on the routing and has two modes of replenishment. Considering the transportation time in the lead time is of great significance for application scenarios with high timeliness requirements. For example, in emergency rescue, the central depot replenishes materials to shelters, and fresh food retailers replenish goods from the central depot to the front warehouse, etc. Supply of goods to demand locations more precisely can reduce waste and contribute to sustainability.  
A robust model and an adaptive variable neighborhood search algorithm are proposed for the problem. There are five attractive features of our work: (1) the joint decision on inventory and transportation is investigated by considering replenishment-route-dependent lead time, which can better highlight the relationship between the arrival time and the related replenishment quantity; (2) a robust optimization model based on the rolling horizon framework is proposed, which is more appropriate to describe the features of the application scenarios under uncertainty, such as emergency logistics and perishable goods supply chains. (3) A closed-form optimal solution of replenishment quantity is derived under the box uncertainty set. (4) A planning horizon crossing strategy is proposed for balancing  
\begin{table}
\begin{tabular}{l c c c c c c} \hline \hline \multirow{2}{*}{**Instances**} & \multicolumn{3}{c}{**3 Cycles**} & \multicolumn{3}{c}{**7 Cycles**} \\ \cline{2-7}  & **Crossing** & **No Crossing** & **Gap** & **Crossing** & **No Crossing** & **Gap** \\ \hline n32 & 163,211.08 & 164,938.34 & 1.06\% & 327,308.35 & 356,441.07 & 8.90\% \\ n33a & 163,960.52 & 169,704.94 & 3.50\% & 303,855.91 & 348,547.88 & 14.71\% \\ n33b & 163,045.56 & 166,200.35 & 1.93\% & 298,161.20 & 347,464.29 & 16.54\% \\ n34 & 174,358.00 & 180,087.03 & 3.29\% & 334,671.20 & 382,719.74 & 14.36\% \\ n36 & 188,342.72 & 202,710.57 & 7.63\% & 375,903.20 & 452,502.22 & 20.38\% \\ n37a & 204,748.18 & 223,892.86 & 9.35\% & 399,978.16 & 465,360.22 & 16.35\% \\ n37b & 206,469.35 & 215,343.30 & 4.30\% & 398,508.10 & 473,262.75 & 18.76\% \\ n38 & 195,769.57 & 210,628.85 & 7.59\% & 393,328.47 & 469,974.37 & 19.49\% \\ n39a & 221,019.42 & 236,318.88 & 6.92\% & 449,661.71 & 524,091.30 & 16.55\% \\ n39b & 227,172.40 & 240,192.50 & 5.73\% & 463,826.21 & 523,908.45 & 12.95\% \\ n44 & 263,389.99 & 282,210.79 & 7.15\% & 996,281.28 & 1,212,032.23 & 21.66\% \\ n45a & 257,700.96 & 273,531.88 & 6.14\% & 566,026.06 & 631,478.46 & 11.56\% \\ n45b & 239,301.02 & 255,655.97 & 6.83\% & 503,419.06 & 574,148.90 & 14.05\% \\ n46 & 237,234.85 & 254,382.18 & 7.23\% & 533,961.97 & 585,141.99 & 9.58\% \\ n48 & 269,601.58 & 294,109.66 & 9.09\% & 621,412.08 & 657,455.90 & 5.80\% \\ n53 & 405,822.68 & 420,274.01 & 3.56\% & 1,024,807.46 & 1,264,783.87 & 23.42\% \\ n54 & 413,459.23 & 459,495.61 & 11.13\% & 1,130,646.78 & 1,557,692.75 & 37.77\% \\ n55 & 336,885.68 & 359,488.04 & 6.71\% & 1,258,609.13 & 1,612,273.16 & 28.10\% \\ n60 & 385,908.80 & 409,793.86 & 6.19\% & 800,523.92 & 885,149.57 & 10.57\% \\ n61 & 361,645.48 & 388,489.22 & 7.42\% & 786,827.68 & 848,578.68 & 7.85\% \\ n62 & 478,530.30 & 506,094.86 & 5.76\% & 1,003,965.14 & 1,084,640.69 & 8.04\% \\ n63a & 432,636.94 & 459,679.05 & 6.25\% & 1,091,721.73 & 1,141,790.13 & 4.59\% \\ n63b & 378,847.95 & 408,041.14 & 7.70\% & 898,821.89 & 1,151,628.25 & 12.71\% \\ n64 & 430,339.57 & 455,573.92 & 5.86\% & 1,145,910.45 & 1,268,798.23 & 10.72\% \\ n65 & 449,687.04 & 461,608.99 & 2.65\% & 1,197,400.02 & 1,400,670.03 & 16.98\% \\ n69 & 665,654.17 & 735,229.66 & 10.45\% & 1,781,092.36 & 2,003,821.23 & 12.51\% \\ n80 & 947,864.67 & 1,032,533.93 & 8.93\% & 2,710,575.82 & 3,140,257.61 & 15.85\% \\ Average & - & - & 6.31\% & - & - & 15.21\% \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparing the model with/without planning horizon crossing.