## 4 Experimental Results
### Statistical Comparisons Using Mean and Standard Deviation  
Since ACO algorithms are non-deterministic stochastic metaheuristics, 50 independent runs were executed on the same set of random seed numbers to perform statistical comparisons. Tables 1 and 2 shows the mean and standard deviation of the \(\bar{P}_{offline}\) results obtained by the \(\mathcal{MM}\)AS and P-ACO algorithms for the DTSP with node and weight changes, respectively. In addition, the statistical test comparisons are reported using the Wilcoxon rank-sum statistical test with a significance level of 0.05. Figures 2-4 show plots of \(\bar{P}_{offline}\) against the environmental changes, for kroA100, rat575, and pr1002, respectively. From the comparisons, the following observations can be drawn.  
Figure 4: Offline performance results (averaged over 50 runs) of \(\mathcal{M}\mathcal{M}\)AS and P-ACO for each environmental change on pr1002 with node (**left**) and weight (**right**) changes.  
Figure 3: Offline performance results (averaged over 50 runs) of \(\mathcal{M}\mathcal{M}\)AS and P-ACO for each environmental change on rat575 with node (**left**) and weight (**right**) changes.  
Figure 2: Offline performance results (averaged over 50 runs) of \(\mathcal{M}\mathcal{M}\)AS and P-ACO for each environmental change on kroA100 with node (**left**) and weight (**right**) changes.  
First, P-ACO performs significantly better than \(\mathcal{MM}\)AS in most DTSPs with node changes. This is because the P-ACO has been designed to address DTSPs with node changes. In particular, when ants are currently stored in the population list when a dynamic change occurs, they will be repaired heuristically using change-related information (i.e., the inserted and removed nodes). Therefore, a constant amount of pheromones will be removed directly from the outdated trails associated with the arcs connecting the nodes to be removed. At the same time, the pheromone trails associated with the arcs connecting the nodes to be inserted will receive a constant amount of pheromone. This specific pheromone update will have a direct effect on the solution construction of the next iteration. On the other hand, the pheromone evaporation of \(\mathcal{MM}\)AS requires more iterations to express its effect (e.g., more than the frequency used in these experiments [10]). Also, the removal of outdated pheromone trails is not applied locally to the affected areas, as in the P-ACO, because the evaporation is applied globally on all arcs.  
Second, \(\mathcal{MM}\)AS performs significantly better than P-ACO in most DTSPs with weight changes. This is because repairing the ants stored in the population list has no effect when a dynamic change of this type occurs. In particular, the feasibility of the stored solutions is not affected as in the case of the DTSP with node changes. Only the solution quality will change (which will be re-evaluated); however, because a constant pheromone update is performed, it has no effect on the P-ACO's behavior. However, since the pheromone trails in P-ACO are removed directly because of the constant pheromone removal, there is a higher risk of erasing knowledge from previous environments. On the other hand, the pheromone evaporation in \(\mathcal{MM}\)AS is able to better adapt to these types of dynamic changes because the decrease in the pheromone trails is performed gradually.