## 3 Solution Approach
### Re-Optimization Stage and VNS-TA Algorithm  
#### 3.2.1 Idea and Process  
To more fully develop and improve the pre-optimized initial scheme, the VNS-TA is designed. The VNS algorithm is realized through the shaking process and the local search process [44; 45]. First, the neighborhood structures are sequentially used to perform neighborhood transformation operations on the current solution to obtain the neighborhood solution. Second, a local search process is performed on the neighborhood solution to obtain the local optimal solution. Then, the two processes are repeated to realize the exploration of the current solution space. Finally, the current solution continues to approach the global optimal solution after several iterations. The process of the TA algorithm functions to gradually narrow the threshold acceptance range and to determine whether to accept the new solution. There are two cases for accepting a new solution. One is when the value of the objective function of the new solution is lower than that of the current optimal solution. The other is that the objective function value of the new solution is higher than that of the current optimal solution, but it does not exceed the threshold acceptance range. Otherwise, reject the new solution. The VNS-TA algorithm proceeds as follows. First, five improved neighborhood structures are designed based on the traditional neighborhood operators. The five improved neighborhood structures are successively used to perform neighborhood transformation operations on the pre-optimized initial scheme to obtain the neighborhood solution. This design increases the diversity of the neighborhood solutions and improves the global exploration ability. Second, a neighbor path selection mechanism is designed, which combines local search operators to optimize the neighborhood solution locally. The local search operators consist of 2-opt*, swap (exchange), insert and 2-opt. This design reduces the number of invalid searches, speeds up the convergence and improves the local optimization effect. Then, the TA algorithm is effectively embedded in the VNS algorithm. The tolerance coefficient of the phased decline is designed to realize the gradual and slow reduction of the threshold acceptance range. The solution acceptance process is oriented by the current optimal solution and benchmarked by the current threshold acceptance range. This design can perform a breadth search for solutions in the early stage of iteration and depth search for solutions in the late stage of iteration. Finally, the stopping criterion of the maximum number of iterations is used to stop the iteration loop, and the re-optimized scheme is obtained and output. A \"breadth search for solutions\" means that, in the early stage of iteration, the threshold acceptance range is large, and the tolerance for poor-quality solutions is large. The poor-quality solutions are appropriately retained to increase the diversity of solutions and expand the search space of solutions. A \"depth search for solutions\" means that, in the later stage of iteration, the threshold acceptance range is small, and the tolerance for poor-quality solutions is small. Only solutions with a small gap from the current optimal solution are retained, and these solutions are constantly optimized to improve the quality of the solutions. The algorithm process is as Algorithm 2. ```
Tolerance coefficient rho, decline rate of tolerance coefficient phi and neighborhood structure set \(N_{k}(k=1,2,\ldots,k_{max})\), where \(k_{max}=5\), and the maximum number of iterations \(p_{max}=500\). The current solution \(x_{current}=x_{initial}\), and the current optimal solution \(x_{current\_optimal}=x_{initial}\). //Loop: for\(p=1\)to\(p_{max}\)do \(k=1\) while\(k\leq k_{max}\)do  Shaking process: select the neighborhood structure \(N_{k}\) to perform the neighborhood transformation operation on \(x_{current}\) sequentially and obtain the neighborhood solution \(x_{neighborhood}\).