## 3 Basic Algorithm
### Max-Min Ant System  
The ant colony optimization (ACO) algorithm was first applied to traveling salesman problem (TSP). It has the advantages of simple implementation, positive feedback and parallel distributed computing. However, the ACO algorithm also has the disadvantages of a long search time and easy restriction to local optimal solutions [27; 28; 29; 30]. In view of the above disadvantages, [PERSON] and [PERSON] proposed the MMAS in 2000 [31]. It is improved based on the ACO algorithm, which effectively improves the performance of the ACO algorithm. The difference between the MMAS and ant system (AS) is mainly reflected in the following aspects:  
1. The MMAS first adopts an elite strategy to update pheromones. The MMAS updates pheromones by using the iterative optimal solution compared with the AS, so the global optimization capability of the algorithm is improved and premature convergence is avoided. To ensure the effectiveness of pheromones, it is necessary for the MMAS to update the pheromones according to Equation (6), where \(i\) and \(j\) represent the optimal paths for this iteration; \[\left\{\begin{array}{ll}\tau_{i,j}(t+1)=(1-\rho)\tau_{i,j}(t)+\Delta\tau_{i, j},&i\in m,j\in m\\ \tau_{i,j}=\frac{Q}{FQ_{all}}\\ FQ_{all}=\sum_{i=1}^{m}\sum_{j=1}^{m}FQ,&i\in m,j\in m\end{array}\right.\] (6)
2. \([\tau_{max},\tau_{min}]\) is the limit interval of the pheromone value, where \(\tau_{max}\) is the upper bound of the pheromone and \(\tau_{min}\) represents the lower bound. Due to this improvement measurement, the possibility of the algorithm falling into a stagnation state in the later stage is reduced, and the algorithm's global optimization capability is also improved;
3. The MMAS initializes the pheromone to \(\tau_{max}\) and sets a smaller volatilization rate \(\rho\);
4. To avoid premature convergence of the MMAS algorithm to the global optimal solution, the smoothing mechanism of pheromones is used to enhance performance.  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Basic Operations** & **Methods** \\ \hline Fitness function & The quality of feasible solutions depends on the fitness value: \(fit(i)=QoS(i)\). \\ Following probability & The followed bees use roulette to select the nectar sources and calculate the following probability based on equation \(P(i)=\frac{fit(i)}{\sum_{i=1}^{m}\pi(i)}\), where \(FN\) is the total number of nectar sources. \\ Detection & It is necessary to investigate new nectar sources when the mining time of a nectar source reaches the maximum and the nectar source is exhausted. Therefore, the detection equation is \\  & \(x(i+1)=x(i)+rand(x_{max}-x_{min})\). \\ \hline \hline \end{tabular}
\end{table}
Table 2: ABC design method.  
The operation steps of the MMAS include parameter initialization, pheromone initialization, a heuristic function, a state transition rule, a fitness function, a pheromone update, etc. The specific operations are shown in Table 3. After the improvement, the MMAS has several advantages, and the following are the main advantages:  
1. The algorithm has three advantages: strong robustness, high operating efficiency and parallelism;
2. The global searching capability is strengthened, and stagnation is avoided.