## 4 Proposed Algorithm
### Pheromone Update Strategy  
The pheromone update strategy guides the algorithm update through a positive feedback mechanism; however, if the positive feedback mechanism is too strong, local convergence problems can easily occur, and if the mechanism is too weak, the algorithm will converge more slowly. To improve the convergence and diversity of the algorithms, this paper proposes a pheromone update strategy that combines the nondominated ranks with the elite routing strategy and the max-min pheromone update strategy. In this strategy, the pheromones of the routes are updated according to their nondominated rank, which comprehensively considers the influence of multiple objectives. Then, the elite paths with better nondominated ranks are updated twice to improve the convergence of the algorithm. Finally, to avoid the algorithm falling into a local optimum due to an excessively large difference between pheromone concentrations, the max-min pheromone strategy is adopted to limit the range of pheromone concentrations.  
The update strategy is as follows:  
(1) The pheromone update process is shown in Equations (33)-(35):  
\[\tau_{ij}(t+1)=(1-\rho)\tau_{ij}(t)+\Delta\tau_{ij}(t,t+1)+\Delta\tau_{ij}^{*} (t,t+1) \tag{33}\]  
\[\Delta\tau_{ij}(t,t+1)=\sum_{k=1}^{m}\Delta\tau_{ij}^{k}(t,t+1) \tag{34}\]  
\[\Delta\tau_{ij}^{*}(t,t+1)=\sum_{k=1}^{m}\Delta\tau_{ij}^{*k}(t,t+1) \tag{35}\]where \(\rho\) (\(0<\rho<1\)) represents the pheromone volatilization rate. \(\Delta\tau_{ij}(t,t+1)\) represents the pheromone increment of path \((i,j)\) as the sum of the pheromone increments left by all ants that traverse path \((i,j)\) in the cycle. \(\Delta\tau_{ij}^{k}\) represents the amount of pheromone released by ant \(k\) to edge \((i,j)\). \(\Delta\tau_{ij}^{*k}\) represents the pheromone increment of the elite path \((i,j)\). \(\Delta\tau_{ij}^{*k}\) represents the amount of pheromone released by elite ant \(*k\) to edge \((i,j)\).  
(2) Common ant pheromone update strategy: \(\Delta\tau_{ij}^{k}\) is updated for all routings according to the nondominance rank.  
\[\Delta\tau_{ij}^{*k}(t,t+1)=\left\{\begin{array}{l}C_{P}*C_{N}/n,\;n=1,\;2,3,4\\ 0,other\end{array}\right. \tag{36}\]  
where \(k\) represents the \(kth\) ant. \(C_{P}\) represents the pheromone constant, \(C_{N}\) represents the nondominant constant, and \(n\) represents the nondominated rank of the routing in this iteration.  
(3) Elite routing pheromone update strategy: \(\Delta\tau_{ij}^{*k}\) updated pheromone concentrations for elite routing with a nondominance rank of 1, 2.  
\[\Delta\tau_{ij}^{*k}(t,t+1)=\left\{\begin{array}{l}(6-n)*C_{P}*C_{N}/n,\;n=1, 2\\ 0,other\end{array}\right. \tag{37}\]  
(4) Max-min pheromone limits: Since there are various nondominance ranks, a large difference in pheromone concentration tends to occur between paths after the pheromone is updated. A too large difference in pheromone concentration can easily lead to premature stalling of the search, increasing susceptibility of the algorithm to falling into local optima. To avoid this, explicit limits are imposed on the minimum and maximum pheromone concentrations in the pheromone update strategy, as shown in Equation (38). After each iteration, it must be ensured that the pheromone concentration adheres to the limits.  
\[\tau_{ij}(t+1)=\left\{\begin{array}{l}\log_{u}\left(\frac{\tau_{ij}(t+1)}{2 \sigma}\right),\tau_{ij}\geq\tau_{min}\\ \tau_{min},other\end{array}\right. \tag{38}\]  
\[\tau_{min}=\frac{1}{2*cusum*(1-\rho)} \tag{39}\]  
where \(a\) represents the base of the logarithm, which is used to control the range of pheromone concentrations; \(\sigma\) represents the limiting factor; and \(cusum\) represents the number of nodes.