## 3 Methods
These TSPs are from the TSPLIB [48], which includes vertices ranging from 48 to 783, and are as follows: KroA100, eil51, bier127, pr439, KroA200, lin318, pr144, att532, rat783, a280, and att48. The experiments included using the IAMTSP+ seeding method alongside the Multi-Linear Regression-Based Technique (MLRBT) [13], which was proven to be superior to the Regression-Based Technique (RBT) [22], as well as the random and NN approaches. ### Experimental Results and Discussion

The experiment was carried out ten times for each instance, using the same set of parameters for the two seeding methods: the MLRBT [13] and the IAMTSP+ seeding method. The IAMTSP+ seeding method was found to be more efficient than the MLRBT [13] across all instances. Additionally, Table 2 and Figure 4 demonstrate that, in terms of the optimal solution, the IAMTSP+ technique is superior in every case. Moreover, it can be said that this approach is more effective than random, NN, and RBT [22] methods, as the suggested technique achieved good performance and produced a result that was close to the optimum. When investigating various initialization methods, parameters or factors related to performance (which have been recognized as measurements), such as the error percentage, average convergence rate, and convergence diversity, must be considered. To obtain a nearly optimal solution, these factors are used to evaluate the quality of the acquired population by assessing the effect of the initial population approach on GA performance. For any given problem, the error rate is the percentage difference between the value of the solution's fitness and the known optimal solution [23]. The error rate can be computed using Equation (1). The average convergence rate of a solution is defined as the percentage of fitness reached by the solution considering the known optimal solution to the problem [32]. It can be stated as in Equation (2). \[Error\ Rate=\frac{Fitness-Optimal\ fitness}{fitness}\times 100\% \tag{1}\]

\[Average\ Convergence=1-\frac{Average\ Fitness-Optimal\ fitness}{Average\ Fitness}\times 100\% \tag{2}\]

The experimental results of the error rate are given in Table 3, which includes the error rate of the best solution for every instance. Table 3 shows that the suggested approach produced the lowest possible error rate. Therefore, the results indicate that the individuals generated by the IAMTSP+ method for GAs fit the quality metrics more closely than the individuals generated by the MLRBT. It is noteworthy that the error rates for instances att532 and att48 are substantial for all methods. The best-known solutions for both cases are substantially higher, with 86,729 and 33,522 route lengths, respectively, according to recent research [54; 55; 56]. Therefore, if we adopted these values as the real optimal solutions, the error rates would be relatively small, 5.3% and 3.9% for att532 and att48, respectively. Since we wanted to be safe, we went with the smallest optimal solutions reported, regardless of their correctness, as stated in [57; 58; 59]. It is important to note, nevertheless, that these chosen numbers seem to be far lower than the generally agreed-upon values using Euclidean distances. It is possible that measurements other than the Euclidean distance were used to calculate the reported route lengths for these two instances. However, investigating the correctness of the optimal solutions for these two instances is beyond the scope of this paper. Figure 5 shows the error rates of the initial population methods for all instances. Table 4 and Figure 6 show the average convergence of the two population initialization methods and show how efficient the genetic algorithm is in converging to the optimal solution. As shown in Figure 6, the IAMTSP+ method achieved a larger convergence rate than the MLRBT.