## 7 Conclusions and Future Work

Table 13: Optimization result of the composite functions tested, CEC 2017 Test Suite [43]; \(n=10\). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**BenFun** & **VLE** & **Opt** & \((VLE-Opt)^{2}\) & \((PSO-Opt)^{2}\) & \((SGA-Opt)^{2}\) & \((DE-Opt)^{2}\) & \((WOA-Opt)^{2}\) \\ \hline \(f_{1}\) & \(4.4899\times 10^{-7}\) & \(0.0000\) & \(2.0240\times 10^{-12}\) & \(1.8496\times 10^{-3}\) & \(6.4009\times 10^{-32}\) & \(6.7240\times 10^{-27}\) & \(1.9881\times 10^{-60}\) \\ \(f_{2}\) & \(3.0840\times 10^{-6}  
execution, the minimum acceptable difference between the current value of the objective function and the best-found value so as not to become stuck in local optima, and the probability of acceptance of poor solutions. Regarding the disadvantages, we have the following: (1) Between one movement and the next, VLE performs \((\alpha-1)D\) evaluations of the objective function, \(D\) being the number of problem decision variables. In other words, VLE creates a population of \((\alpha-1)D\) solutions whose fitness must be evaluated. The greater the number of decision variables of the problem, the greater the memory used by VLE to perform the search and the greater the time it takes to perform an effective movement; (2) The determination of bubble and dew temperatures requires additional iterative calculations, which translates into an increase in computational cost. However, in this version, the calculations have been reduced to a minimum since they are carried out using the bisection numerical method. The use of this particular method also guarantees the determination of said temperatures; (3) Since it truly simulates the vapour-liquid equilibrium of binary mixtures that form ideal solutions, the search is restricted to the possible equilibrium states of the simulated system. This restriction, of course, can translate into a certain degree of loss of flexibility near a local optimum. At the least, disadvantages 2 and 3 will be eliminated in a future version of VLE. Second, it is important to comment on and highlight the results obtained with our algorithm, applied to the fifteen classical reference functions, and compared with the corresponding results obtained by four well-known and robust optimization algorithms. As we said, the results were analysed using two techniques: (1) by performing a thorough comparative analysis of the average values obtained by each function and each metaheuristic; and (2) by using the root-mean-square error, or RMSE. Based on the first technique of analysis, we may say that VLE had competitive performance among PSO, GSA, DE, and WOA, with a weighted average performance of 44.83%, and that VLE occupied the third place among the five compared algorithms. Using the second technique of analysis, we may affirm that, in general, and in terms of datasets, the average dataset obtained by VLE is better adjusted to the true values of optimum values of the reference functions than any of the published data sets for PSO, GSA, DE, and WOA. This analysis allows us first to corroborate that there is no metaheuristic that is more capable at solving optimization problems than any other metaheuristic and second, to say that to establish a significant difference among these algorithms, it is necessary to considerer a wider set of reference functions. This will be considered in future work. According to the results obtained, we believe that the vapour-liquid equilibrium is a good idea as a source of inspiration to develop a novel and robust metaheuristic. In addition, we affirm that the search of a local optimal can also be performed by using simple mathematical models that simulate the phenomenon that has been the inspiration source of the algorithm. As a future work, we would like to include more benchmark functions of the four types considered in this research. Furthermore, we will increase the number of dimensions from 30 to 50 and 100 to analyse the performance of the procedure with more decision variables.