## 5 Results and Discussion
### Radom Walk Analysis: The Comparison of the Results to Each Other
The higher the fitness value of a solution, obviously the farther away from the best solution. According to Figure 10b the hamming distances range from 2 to 38 (hamming distance unit). The greater the fitness value of a solution, the greater the hamming distance from the best solution. According to Figure 11, the basic swap sequence distances are between 1-32 (basic swap sequence distance unit). The higher the fitness value of a solution, the greater the distance of the basic swap from the best solution. Figure 10: The fitness values and their fitness distances (**a**) and hamming distances (**b**) from the best of the solutions in case of 2-opt operator. In the following, the results of the order crossover operator are detailed, where distances are calculated from the best element in the solution set. The higher the fitness value of a solution, obviously the farther away from the best solution. The hamming distances range from 2 to 40 (hamming distance unit). The greater the fitness value of a solution, the greater the hamming distance from the best solution. The basic swap sequence distances are between 2-34 (basic swap sequence distance unit). The higher the fitness value of a solution, the greater the distance of the basic swap from the best solution. The results of the cycle crossover operator and a partially matched crossover are detailed, where distances are calculated from the best element of the solution set. These analyzes also show that the higher the fitness value of a solution, the greater the distance of the fitness from the best solution, and the hamming and basic swap sequence distances also increase as a function of the fitness value. For a cycle crossover, the hamming distance is 0-38 (hamming distance unit), while the basic swap sequence distance is 0-29 (basic swap sequence distance unit), and for a partially matched crossover, the hamming distance is 2-40 (hamming distance unit) and the basic swap sequence distance is 1-34 (basic swap sequence distance unit). In the following, the distance from the best solution and the average of the distances from the other solutions are presented. According to Figure 12a, this is a parabolic function for 2-opt operator, so for large and small distances the average distances are large, while for medium distances the average distances are small for random walk and 2-opt. This shows that the extreme case is not in the middle of the distribution, it is located close to the elements in the center. According to Figure 12b the distance from the best solution during hamming distances does not affect the average distance here. Figure 13 shows the results for the basic swap sequence, here too the average basic swap sequence distance is not affected by the distance from the best solution to the given result. In the case of order crossover, we also get a parabolic function for fitness distances. The results for cycle crossover and partially matched crossover are similar to those for 2-opt and cycle crossover. Figures 14 and 15 show the cost of density results, i.e., how many solutions with the same fitness value are present between each solution. Based on the results, in the case of 2-opt, almost every solution has a unique fitness value. In the case of order crossover and cycle crossover, some solutions have the same fitness value, while in the case of partially matched crossover, each solution also has a unique fitness value. Table 3, illustrates the evaluation methods of the results of random walk analysis, and Table 4. presents the summary of the results of the random walk where the solutions are compared to each other. Figure 11: The basic swap sequence distances from the best of the solutions in case of 2-opt operator. Figure 14: The cost of density in case of 2-opt (**a**) and order crossover operator (**b**). Figure 12: The distance from the best solution and the average of the fitness distances (**a**) and the hamming distances (**b**) from the other solutions in case of 2-opt operator.