Study on a hybrid algorithm combining enhanced ant colony optimization and double improved simulated annealing via clustering in the Traveling Salesman Problem (TSP)

Strategy one

The optimal path between a colony’s exploration of a nest and a food source relies heavily on the information transmitted through pheromones. If the ant colony already knows the starting city $a$ and the ending city $b$ (the ending city is the city before the ant colony returns to its starting point), the ant colony tends to choose the city closer to the ending city, leading to a biased selection process for the next visited city during exploration. Utilizing the best pheromone information between the starting city $a$ and the ending city $b$, the ant colony can efficiently determine the appropriate direction to explore, thereby accelerating the search process. To address this, this article proposes an improved pheromone update strategy in conjunction with the EACO. Furthermore, all edges are initialized based on the distance between the starting city $a$ and the ending city $b$. The concentration of pheromones in the initialized ant colony is determined as follows:

(11) $${\tau }_{ij}\left(0\right)=\frac{d_{ab}}{d_{aj}}+d_{jb}$$

${\tau }_{ij}\left(0\right)$ is the initial pheromone concentration between the current city $i$ and the next city $j$, $d_{ab}$ is the linear distance between the starting city $a$ and the ending city $b$, $d_{aj}$ is the distance from the starting city $a$ to the next city $j$, $d_{jb}$ is the distance from the next city $j$ to the ending city $b$.

This strategy changes the initial pheromone concentration of the EACO and focuses on the distance between the current city and the ending city, which provides directional guidance for the initial ant colony and avoids blind searches of the ant colony, thus improving the speed of solution and accuracy of the solution.

Strategy two

The method for ants to select the next city from the current city is primarily based on the roulette wheel betting method. This method ensures a well-balanced algorithm, where ants with higher fitness values are more likely to be selected, while ants with lower fitness values still have a chance to be chosen. This approach allows the ant colony to explore and experiment within the solution space, allowing all unvisited cities to be selected. However, in the actual solution process, it is important to avoid consecutively visiting two cities that are particularly far apart. If the ant colony selects a distant city as the next destination from its current location, it would result in wasted iterations, leading to increased solution time for the algorithm. More critically, it could significantly affect the overall direction of the algorithm’s solution and potentially trap it in a local optimal solution. Therefore, this article restricts the city selection process to the roulette wheel betting method. Ants are not allowed to choose a more distant city as the next visited destination. This constraint is expressed mathematically as follows:

(12) $$R=\frac{\lambda {\sum }_{r\notin tab{u}_k}{d}_{ir}}{Numberofcitiesnotvisited}$$

(13) $$x_{ir}=\{\begin{array}{c}1d_{ir}<R\hfill \\ 0d_{ir}\ge R\hfill \end{array}$$

R is the maximum distance from other cities that can be visited, $\lambda$ is parameter [1,2], $d_{ir}$ is the current city $i$ to a city $r$ that has not been visited, Eq. (13) indicates whether the unvisited city $r$ can be added to the roulette match, 1 is acceptance and 0 indicates no acceptance, $x_{ir}$ is the decision variable for the current city $i$ to the unvisited city $r$.

Our algorithm called the Adaptive Elite Ant Colony Optimization (AEACO) is an efficient optimization-seeking algorithm for small-scale TSP instances, which will automatically adjust the number of iterations and the number of ants for different small-scale instances. Meanwhile, a comparison test with the ACO and the AEACO is run to evaluate the performance of the AEACO. The AEACO, ACO, and AEACO are put to the test 30 times, with each algorithm’s optimal solution, average error rate, and solution time is provided. Table 1, Figures 1 and 2 show the experimental results, Meanwhile, $Time$ is the solution time for the algorithm to solve the TSP instance 30 times, SD is the error rate and calculated by Eq. (14), which is the difference between the optimal solution (denoted as Opt) obtained by the algorithm and the known optimal solution (denoted as KopS) of TSPLIB, $S{D}_{avg}$ is the average error of the solved result at the end of the every process after 30 runs, $S{D}_{best}$ is the error of the best solution after 30 runs and $Best$ is best optimal solution after 30 runs.

(14) $$SD=\left(\frac{Opt-KopS}{Kops}\right)\times 100$$

Parameter setting of ACO: $\alpha =1$, $\beta =5$, $Q=1$, $\rho =1$, the number of ants $m$ is the number of cities.

Parameter setting of EACO: $\alpha =1$, $\beta =5$, $Q=1$, $\rho =1$, $e=0.5$, the number of ants $m$ is the number of cities.

Parameter setting of AEACO: $\alpha =7$, $\beta =10$, $\rho =0.1$, $Q=1$, $e=0.5$, $\lambda =1.2$, the number of ants $m$ is the number of cities, the number of iterations of the algorithm is $0.5$ times the number of cities.

From Fig. 1, the AEACO has demonstrated superior performance in terms of speed and accuracy compared to the ACO and EACO, it is evident that the AEACO converges to the optimal solution with minimal time and number of iterations, the convergence also speed is also remarkably fast as depicted in Fig. 2 and Table 2, it can be observed that the AEACO produces better results. Nevertheless, as the scale increases, the algorithm’s accuracy diminishes, and the solution time considerably lengthens.

Strategy one

The traditional simulated annealing, which primarily employs perturbation operations to perturb the solution sequence, uses a random exchange of the positions of a specific pair of cities as its perturbation mechanism in the TSP instances, which is the primary cause of the algorithm’s slow convergence. The perturbation mechanism, however, plays a critical role in the algorithm’s superiority. As the perturbation approach for the SA process cannot be overly complicated, we will employ three conventional perturbation operations to change the old solution sequence.

(1) Swap method: randomly swap two positions in the solution sequence.

(2) Random insert method: randomly swap two adjacent positions in the solution sequence.

(3) 2-opt method: two positions in the solution sequence are randomly selected and arranged in reverse order from these two positions.

Strategy two

The traditional SA typically employ a sufficiently large initial temperature to enhance the search performance. However, there is no universally recommended method for determining the initial temperature. Selecting an inappropriate initial temperature not only results in wasted time but also hampers the effectiveness of the solution search. According to the method described in literature (Lin, Kao & Hsu, 1993), we employed a specific initial temperature approach tailored to the unique characteristics of our algorithm:

(16) $$T=-\frac{E\left(X_{avg}\right)-E\left(X_{min}\right)}{a\ln p}$$

$E_{avg}$ and $E_{min}$ are the expected average and minimum values, respectively, of the objective function for N randomly selected feasible solutions within the solution space, $p$ is parameter [0,1], $a$ is a parameter value that mainly prevents the temperature starting point from being too high, resulting in slow convergence of the algorithm.

The MSA-1 is randomly selected to perturb the solution sequence by the swap method, random insertion method, and 2-opt method. The MSA-1 means that the MSA uses only strategy 1 and not strategy 2. In this article, three instances of pr76, tsp225, and pcb1173 are used to test the MSA-1 for four different proportions of perturbations of the swap method, random insertion method, and 2-opt method. The experimental results are shown in Table 2. In the table, K is the number of temperature changes, $Errors$ is the average error of the solution after 30 experiments. Meanwhile, $Agiter$ is the average of the number of iterations in which the optimal result emerges during the solution of the TSP instances after 30 times. $S{D}_{avg}$ is the average error rate of solving 30 times.

As shown in Table 2, when the $swapmethod:randominsertionmethod:2-optmethod=1:1:2$, the solution effect is better. A larger proportion of the 2-opt method is beneficial for obtaining better solutions and reducing the number of iterations.

By utilizing the swap method, random insertion method, and 2-opt method to enhance perturbation in the SA. The perturbation capabilities of the SA can be improved, which can leading to more accurate solutions. Using strategy 2 to improve the initialization temperature of the SA has the advantage of avoiding too high or too low a temperature that would cause the algorithm to fall into a local optimal solution. In this study, comparison tests between the MSA-1 and the conventional the SA are carried out using the same initial solution sequence, initial temperature, termination temperature, cooling factor, and the maximum number of iterations. The MSA does not use the same initial temperature. A total of 30 tests are carried out, and the experimental results are displayed in Table 3 and Fig. 3.

The parameters were set as follows: initial temperature: 300, termination temperature 1, cooling factor: 0.998, and the maximum number of iterations: 100.

Experiments show that the solution accuracy of the MSA-1 is much higher than that of the SA in the same environment. The main reason is that Strategy 1 improves the exploration pattern of the SA. Comparing the MSA-1 and the MSA, MSA has strategy 2 which provides an effective initial temperature, leading to a high convergence accuracy of the MSA. But MSA of the average time is longer than the SA, resulting in the inability of the SA to improve the solution accuracy and reduce the solution time, mainly because the MSA is not sufficient in local search capability.

SE (Starting-Ending) strategy

As illustrated in the Fig. 7, the algorithm presented in this article primarily utilizes the clustering method to partition cities into distinct clusters. Subsequently, it aims to determine the optimal sequence for connecting these clusters. Hence, the most crucial consideration lies in selecting cities that are interconnected within their respective clusters. After the perturbation, it is necessary to determine the cities at the starting and ending points within the perturbed part based on the cluster. To achieve this, pheromones and distances from surrounding clusters to cities in the perturbed part are taken into account. Selection probabilities are established using these factors and are combined with the roulette pair method to select the starting city and ending city. The probability of selection of interconnected cities in the current cluster selection and other clusters is given by the formula:

(17) $${\tilde{p}}_{a\overline{k}}^k=\{\begin{array}{c}\frac{{\sum }_{b\in cluste{r}_{\overline{k}}}{\left[{\nu }_{ab}\right]}^{{\alpha }_1}{\left[{\omega }_{ab}\right]}^{{\beta }_1}}{{\sum }_{\mathrm{r}\in cluste{r}_k}{\sum }_{a\in cluste{r}_{\overline{k}}}{\left[{\nu }_{rb}\right]}^{{\alpha }_1}{\left[{\omega }_{rb}\right]}^{{\beta }_1}}a\in cluste{r}_{\mathrm{k}}\\ 0otherwise\end{array}$$

Pheromone update of SE strategy:

(18) $${\tau }_{ij}\left(t+1\right)=\left(1-l_1\right){\tau }_{ij}\left(t\right)+\frac{U_1}{L_1{}^{best}}(t)$$

$a$ denotes a city in the cluster $k$ in the disturbed part, ${\nu }_{ab}$ denotes the heuristic factor between the city $a$ and the city $b$, ${\omega }_{ab}$ is the pheromone concentration between the city $i$ and the city $b$, $cluste{r}_k$ denotes the cluster of the disturbed part, $cluste{r}_{\overline{k}}$ denotes disturbed parts of the surrounding cluster, ${\alpha }_1$ denotes the pheromone factor, ${\beta }_1$ denotes the heuristic factor. $l_1$ is representative of the rate of volatilization, $U_1$ is a constant; $L_1^{best}$ is the current loop optimal solution of MSA.

Steps to optimize the sequence of clusters:

Step 1: Parameter initialization: temperature initialization $T_0$ operation with Eq. (16), the maximum number of iterations $L_0$, the termination temperature $T_{end}$, and the cooling factor ${\rho }_0\left(1<{\rho }_0<0\right)$.

Step 2: Initialization of pheromones: set the same pheromone on each path.

Step 3: The initialization process of the cluster sequence: perform the internal cluster search for each cluster, and build up the optimal solution table of each cluster and the internal sequence table of each cluster according to the sequence of clusters.

Step 4: Perturbation of cluster sequence: perturbation of cluster sequence using swap method, random insertion method and 2-opt method with a probability of $1:1:2$.

Step 5: Calculate the results after the perturbation:

Using SE strategy to find the starting city and the ending city of the perturbed part of the clusters.

Using an intra-cluster optimization search for the clusters in the perturbed part.

Update the optimal solution table and the internal sequence table of the clusters.

Step 6: The Metropolis criterion: use the Metropolis criterion to determine whether to receive a new cluster sequence.

Step 7: Determine if the maximum number of iterations has been reached: if so, exit the current loop, otherwise return to Step 3.

Step 8: Pheromone update: leave pheromones on each ant’s passing edge and reward elite ants with a certain amount of extra pheromones on their passing edge.

Step 9: Cooling: $T_0={\rho }_0{T}_0$.

Step 10: Stop condition: determine whether the termination temperature $T_{end}$ is reached, if $T_{now}=T_{end}$, the algorithm ends. Otherwise return to Step 3.

Step 11: Formation of a quality solution: connect all clusters.