## 4 Proposed Algorithm
\begin{tabular}{l l} \hline \hline \multicolumn{2}{l}{Simulated Annealing algorithm framework \(SA()\)} \\ \hline
1 & \(S^{uw}\gets ISG()\) ; \\
2 & \(S^{current}\gets S^{uw}\) ; \\
3 & **for**\(\varphi=t\); \(\varphi>1\); \(\varphi\gets 0.9\times\varphi\)**do** \\
4 & **for**\(\tau=0\); \(\tau<1000\); \(\tau\leftarrow\tau+1\)**do** \\
5 & Swapping two random disjoint subsequences of \(S^{current}\) to generate \(S^{new}\); \\
6 & **if**\(Fit(S^{uw})<Fit(S^{current})\) **then**/* \(Fit(S^{uw})\) is the fitness of \(S^{new}\) */ \\
7 & \(S^{current}\gets S^{uw}\) ; \\
8 & **if**\(Fit(S^{current})<Fit(S^{uw})\) **then** \\
9 & \(S^{uw}\gets S^{current}\) ; \\
10 & **else** \\
11 & \(p\gets e^{-\frac{1+\omega(1,2)}{\omega B\varphi}}\) ;/* Accept an inferior solution with the probability \(p\) */ \\
12 & \(A\) random number \(pr\) is generated between 0 and 1; \\
13 & **if**\(pr<p\)**then** \\
14 & \(S^{current}\gets S^{uw}\) ; \\
15 & **return**\(S^{uw}\); \\ \hline \hline \end{tabular} 

### Variable Neighborhood Search Algorithm Framework

Variable Neighborhood Search (VNS) is a metaheuristic with the systematic change of neighborhood in a search. VNS was first proposed for solving the traveling salesman problem and its effectiveness is illustrated in [21]. Since TSP can be taken as a subproblem of the considered problem in this paper, we chose VNS to improve our previous work. VNS can escape from the localoptimum by restructuring the neighborhood. Different neighborhoods supply different candidate solutions, thus it is possible to find a better solution. ISG is used to generate the initial solution (Line 2). The approximate optimal solution to the scheduling scheme \(S^{uw}\) is calculated by iteration (Lines 4-12). In the iterative process, the neighborhood is obtained by shaking (Line 5), the local optimum \(S^{uw}\)' is calculated by iteration (Lines 6-7), and the global optimal solution is recorded (Lines 8-9). \begin{tabular}{l l} \hline \hline Variable Neighborhood Search algorithm framework & \(VNS()\) \\ \hline
1 & Select the set of neighborhood structures \(N_{k}\), \(k=1,\ldots,k_{max}\); \\
2 & \(S^{uw}\gets ISG()\) ; \\
3 & Set \(k\)=1; \\
4 & **While \(k\leq k_{max}\)do** \\
5 & & Generate a point \(S^{uw}\), randomly from \(N_{k}\) (\(S^{uw}\)); /* Shaking Step*/ \\
6 & **While** not Termination Criterion **do** \\  & /* Obtain local optimum \(S^{uw}\)' from an initial solution \(S^{uw}\)*/ \\
7 & & Generation of candidate neighbor \(S^{uw}\)'; \\
8 & **if \(Fit(S^{uw})<Fit(S^{uw})\)then** \\
9 & & \(S^{uw}\gets S^{uw}\)', \(k=1\) ; \\
10 & **else** \\
11 & & \(k=k+1\); \\
12 & **return \(S^{uw}\);** \\ \hline \hline \end{tabular} 

As our algorithm is a hybrid of Initial Solution Generation Algorithm (ISG) and heuristic algorithms, we first provide a brief description about ISG algorithm as follows. ### Time Complexity Analysis

(1) Since the time complexity of the ISG algorithm is related to the number of packages \(n\), the number of UAVs \(h\) and the number of stations \(|S|\), it can be denoted as \(O(n+h*|S|)\). (2) The time complexity of the SA algorithm is related to the ISG algorithm, the initial temperature \(t\) and the number of iterations \(\tau\). Therefore, the time complexity is \(O\Big{(}\Big{(}\log_{0,9}\frac{1}{t}\Big{)}*\tau*(n+h*|S|)\Big{)}\). (3) The time complexity of the VNS algorithm is related to the ISG algorithm, the number of neighbors \(|N_{k}|\), and the number of neighbors \(I_{k}\) per exploration. Therefore, the time complexity is \(O(\sum_{k=1}^{k=k_{max}}I_{k}*|N_{k}|*(n+h*|S|))\). ## 5 Simulated Experiments

In this section, numerical results are reported. We first calibrate the important parameter initial temperature \(t\) of the SA algorithm framework. Then, the robustness of the algorithm is analyzed for all instance parameters. All experiments are coded in Java and run on a PC with an Intel(R) Core(TM) i5-7500 CPU @3.40 Ghz, 8 GB of RAM. The version of Integrated Development Environment (IDE) is eclipse Jee Oxygen April 2018 \(\times\)64.