## 6 Experimental Results
### Comparative study of the mean values:  
Tables 10 and 11 show the result of the comparative analysis of the performance of VLE compared to that of PSO, GSA, DE and WOA before the 15 reference functions already indicated. The results in Table 10 indicate that VLE shows a performance somewhat superior to that of PSO (60.00%), slightly higher than that of GSA (53.33%), and somewhat lower than that of WOA (40.00%). Since there is no information available for DE regarding functions \(f_{14}\) and \(f_{15}\), if these functions are excluded, it can be seen that VLE has low performance compared to DE (23.08%). Based on these observations, and the  
Figure 17: Convergence graphics for classical functions tested. Figure 18: Distribution graphics for classical functions tested. benchmark functions considered, it can be stated that, in general, and qualitative terms, VLE presents a competitive performance among PSO, GSA, DE, and WOA.I In quantitative terms, this analysis allows us to affirm that VLE has a weighted average performance of 44.83% among this group of functions and algorithms. In other words, considering this group of algorithms, this means that, if we take a population of 100 benchmark functions, VLE will deliver the best average fitness in 45 of these functions. The results of Table 11 show that, in general, the average performance of VLE is 2.6 on a scale of 1 to 5, with 1 being the best performance evaluation. This average global ranking considers that VLE reached first place 4 times, second place 2 times, third place 5 times, fourth place 4 times and that VLE never occupied fifth place. In addition to these results, it can be seen that the VLE ranking by type of benchmark function is 3.5 for the unimodal functions, 3.0 for the multimodal functions, and 2.0 for the multimodal test functions with a fixed number of dimensions. If functions \(f_{14}\) and \(f_{15}\) are excluded because there is no information available for DE, then the estimated global average performance of VLE is 2.85. In summary, if this indicator is rounded to the nearest integer, then VLE ranks third among the five metaheuristics considered and the 15 functions evaluated. This comparative analysis allows us to corroborate that there is no metaheuristic that is capable of solving any optimization problem better than all the others. Some will work better than others for certain problems, while others will not perform as well [2]. In the case of VLE, this outcome may be due to the complexity of the algorithm, which simulates the physical-chemical process that served as inspiration. The real simulation restricts the potential capacity of the algorithm to reach more promising solutions with certain functions. For example, if a chemical species A and chemical species B are very similar to each other, the region enclosed between the saturation curves of the vapour-liquid equilibrium diagram will be very fine and will trend to be horizontal. This will produce more big steps around a good solution, than with a region of the same shape but with a greater incline, i.e., with a strong negative slope. In the limit, when chemical A is equal to chemical B, the system will be constituted by only one chemical species, i.e., will be a pure chemical species, so in this case, the binary diagram will be a horizontal straight line with no steps.