## 3 Solution Approach
### Re-Optimization Stage and VNS-TA Algorithm
#### 3.2.4 Acceptance Criteria for the Solution  
In the TA algorithm, whether to accept the current solution depends on the comparison between it with the current optimal solution. First, when the fitness function value of the current solution is less than that of the current optimal solution, as shown in Equation (22), the current solution is retained. Second, when the fitness function value of the current solution is greater than that of the current optimal solution but does not exceed the current threshold tolerance range, as shown in Equation (23), the current solution is retained. It should be clarified here that (1) the current solution is the feasible solution obtained under the current number of iterations, (2) the current optimal solution is the optimal feasible solution running the iteration so far and (3) the global optimal solution is the optimal solution of the mathematical model of the problem. The current optimal solution is not the global optimal solution. NP-hard problems are almost impossible to solve by accurate algorithms unless \(P=NP\)[47]. \[fitness(x_{current})-fitness\Big{(}x_{current\_optimal}\Big{)}<0 \tag{22}\]  
\[0<fitness(x_{current})-fitness\Big{(}x_{current\_optimal}\Big{)}<T_{i} \tag{23}\]  
Next, this paper will explain how to obtain the local optimal solution from the perspective of algorithm design. The TA algorithm is effectively embedded in the VNS algorithm, and the threshold acceptance range is used as the benchmark to judge whether the solution is accepted. The TA algorithm is implemented by the threshold acceptance range change formula and the tolerance coefficient change formula, as shown in Equations (24) and (25). Among them, \(T_{i}\) represents the threshold acceptance range of the current solution at the \(i\)th iteration, \(fitness\Big{(}x_{current\_optimal}\Big{)}\) represents the objective function value of the current optimal solution at the \(i\)th iteration, \(rho_{i}\) represents the tolerance coefficient of the \(i\)th iteration, \(rho_{j+1}\) represents the tolerance coefficient of the \(j+1\)th iteration, \(rho_{j}\) represents the tolerance coefficient of the \(j\)th iteration and \(phi\) represents the rate of decline in the tolerance coefficient. Equation (24) shows that, with the increase in the tolerance coefficient and the objective function value of the current optimal solution, the threshold acceptance range will also increase. Equation (25) shows that, as the decline rate of the tolerance coefficient becomes larger, the decline of the tolerance coefficient slows, and the narrowing speed of the threshold acceptance range also slows. \[T_{i}=fitness\Big{(}x_{current\_optimal}\Big{)}\cdot rho_{i} \tag{24}\]  
\[rho_{j+1}=rho_{j}\cdot phi \tag{25}\]  
In addition, a stage decline strategy for the tolerance coefficient (Strategy 1) is designed to achieve the following two points. First, in the current inner loop, the tolerance coefficient is unchanged, and the change in the threshold acceptance range only emerges from the update of the current optimal solution. Second, when entering the next outer cycle from the current outer cycle, the tolerance coefficient decreases once, and the threshold acceptance range achieves a \"jump down\". To verify the effectiveness of the stage decline strategy for the tolerance coefficient, an exponential decline strategy for the tolerance coefficient (Strategy 2) is designed for comparison. An example is given to illustrate the difference between the two strategies. The VNS process is repeated five times, and the shaking process and local optimization process are performed three times in each VNS. The initial tolerance coefficient is 0.5, and the decline rate of the tolerance coefficient is 0.8. The changes in the tolerance coefficients of Strategy 1 and Strategy 2 are shown in Figure 2.  
The framework of the algorithm based on Strategy 1 and Strategy 2 is as Algorithms 3 and 4.