*4.1. Parameter Setting*

Since our focus is to conduct an experimental comparison between different MOEAs and different SOEAs, it was necessary to carry out an exhaustive process to adjust and analyze the ideal parameters for each algorithm. This section provides all the details on the algorithm configurations and the experimental set-up. It is important to note that all the algorithms were implemented in Java using the jMetal [54] framework (the source code used in the current work, as well as the results and graphics extracted from them, can be found through https://github.com/Tomas-Morph/knp-tspjournal-mathematic). We also used the irace package [55] to set the automatic parameters in all of the algorithms implemented. For each problem—KNP and TSP—we defined a common solution encoding for all the algorithms implemented. We also decided to apply some standard and basic operators for all the algorithms implemented (and in the same way for all of them). To set the automatic parameters, a personalized adjustment was made for each approach. The set of configuration parameters that were automatically tuned—for each algorithm—are as follows:


As previously indicated, the parameters listed were automatically tuned using the irace package. We first ran irace with the set of input parameters described in Table 3. For this initial tuning process, we selected a subset of representative instances (different type and sizes) for each problem. The best configuration obtained by this automatic process for each pair problem-algorithm after training for a few hours is shown in Table 4. Considering these parameter settings and in order to achieve statistically significant results, a total of 100 independent runs were executed for each pair (algorithm, problem instance) . In order to statistically support the conclusions, the following statistical testing procedure, which was used in a previous work by the authors [56], was applied to compare the results obtained by the different algorithmic schemes. First, a Shapiro–Wilk test was performed to check whether the values of the results followed a normal (Gaussian) distribution. If so, the Levene test checked for the homogeneity of the variances. If the samples had equal variance, an ANOVA test was done; if not, a Welch test was performed. For non-Gaussian distributions, the non-parametric Kruskal–Wallis test was used. For all the tests, a significance level of *α* = 0.05 was considered.

Finally, it is important to note that this study is not intended to offer a comparison of the best-performing algorithms existing in the related literature; the main goal is to analyze the suitability of MOEAs for optimizing single-objective problems.



**Table 4.** Parameter settings for each problem-algorithm pair.

