# Optimal Route Planning for Truck-Drone Delivery Using Variable Neighborhood Tabu Search Algorithm
## 5 Numerical Experiment
### Results  
We set (1) model parameters: \(M\) = 10,000, \(t_{\rm L}\) = 1 min, \(t_{\rm R}\) = 1 min, \(s^{\rm T}\) = 2 min, \(s^{\rm D}\) = 2 min, \(e\) = 50 min; (2) algorithm parameters: \(H_{1}\) = 100, \(H_{2}\) = 1000. The procedure is coded in MATLAB R2020a, and all experiments are conducted on a Windows Server 2019 server with an Intel Core i9-10900X CPU (4.7 GHZ) and 128 GB DDR4 RAM. Table 2 demonstrates three delivery routes of the three truck-drone teams, including \(u_{j}^{n}\), \(u_{j}^{n^{\prime}}\), \(z_{i}^{n}\), \(\pi_{i}^{n}\), and \(\tau\). The unit of \(u_{j}^{n}\), \(u_{j}^{n^{\prime}}\), \(\pi_{i}^{n}\) and \(\tau\) is the minute. On route A, the truck and drone jointly served eight customer points. The truck route was 0\(\rightarrow\)1\(\rightarrow\)16\(\rightarrow\)17\(\rightarrow\)2\(\rightarrow\)0, i.e., the truck served four customer points. The drone's routes were 0\(\rightarrow\)7\(\rightarrow\)1, 1\(\rightarrow\)8\(\rightarrow\)16, 16\(\rightarrow\)9\(\rightarrow\)17, 17\(\rightarrow\)10\(\rightarrow\)2. After the drone finished the last delivery task at customer point 10, it was collected by the truck at customer point 2, then the truck carried the drone back to the warehouse. The delivery process took 187 min. On route B, the truck and drone also visited eight customer points. The truck route was 0\(\rightarrow\)11\(\rightarrow\)18\(\rightarrow\)20\(\rightarrow\)13\(\rightarrow\)0. The drone routes included 0\(\rightarrow\)3\(\rightarrow\)11, 11\(\rightarrow\)12\(\rightarrow\)18, 18\(\rightarrow\)19\(\rightarrow\)20, 20\(\rightarrow\)21\(\rightarrow\)13, i.e., the drone visited four customer points. The drone returned to the truck at customer point 13 after customer point 21 was visited. The total delivery time was 167 min. On route C, the truck route was 0\(\rightarrow\)4\(\rightarrow\)23\(\rightarrow\)15\(\rightarrow\)0, i.e., the truck delivered parcels to three customer points. The drone flew along three routes: 0\(\rightarrow\)5\(\rightarrow\)4, 4\(\rightarrow\)22\(\rightarrow\)23, 23\(\rightarrow\)14\(\rightarrow\)15, 15\(\rightarrow\)6\(\rightarrow\)0. The drone and the truck returned to the warehouse separately, and the drone arrived at the warehouse 1 min earlier than the truck. The total delivery time was 172 min. According to Equation (1), the total delivery time \(\tau\) was \(max\{187,167,172\}\)\(=\) 187 min. Figure 4 illustrates the three optimal truck-drone delivery routes. As shown in Figure 5, on the above three routes, truck-drone delivery can save 47 min, 48 min and 38 min respectively compared with pure-truck delivery. The whole delivery process can save up to 20.1% of the time. Obviously, truck-drone delivery can save total delivery time, but delivery enterprises must purchase drones, which is a large investment (or cost). In reality, there is a possibility that the benefits of saving delivery time do not offset this cost. Hence, the tradeoff between saving delivery time and purchasing drones should be considered when enterprises determine whether to implement truck-drone delivery.