## 7 Comments and conclusions  
Some considerations can be done about the results shown above. First of all, they all are some rather existent problems in laminate design, and ALE-PSO has been able to find numerical solutions that are very close to an exact one. A second remark concerns the difference in the order of magnitude of the residuals: it is very small in example 2, while it increases in example 3 and even more in example 1. This is intrinsic to the problem: whilst the case treated in example 2 uses continuous variables and has 12 plies, so that solutions with a residual extremely close to zero can be found, in example 3 there are discrete variables and only 8 layers. These two circumstances contribute to render much more intricate the problem; in fact, it is well known that the difficulty in satisfying the condition of **B** null increases when the number of plies decreases, and in addition the requirement that the plies can take only discrete values diminishes the number of solutions. For example 1, as already said, the objective is so difficult that it is taken as a sort of benchmark in problems of laminate design. Nevertheless, several solutions have been found, in a very short computing time: all the examples run for no more than 3.5 seconds on a common laptop (the author and co-workers had used in the past a genetic algorithm to treat analogous problems, and example 1 was solved using a population of 2000 individuals and 3000 iterations, for a time of about 30 minutes to be solved with the same order of magnitude of the residual; the best solution found by ALE-PSO has used a population of 100 particles and 300 iterations, _i.e._ 200 times less evaluations of the objective function; for more details on the use of genetic algorithms in the global design of laminates, see [15], [21]. More interesting is to compare the trajectories of the parameters to determine the most performing one. To this purpose it is worth to consider the cases of the expected and best value of the residual, for the swarm average and best particle; this is done in Tab. 7, where the trajectory giving the best result for each quantity is indicated, using the results of Tab.s 3, 4 and 6. \begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline  & \(E_{x}\) & \(E_{x}\) & \(G_{xy}\) & \(\
u_{xy}\) & \(C_{xx}\) & \(C_{xy}\) & \(C_{ss}\) & \(C_{xy}\) & \(C_{xx}\) & \(C_{ss}\) \\ \hline A* & 63.82 & 77.27 & 26.32 & 0.26 & 69.60 & 84.26 & 26.32 & 22.05 & 0 & 0 \\ \hline
**B* & - & - & - & - & - & 0 & 0 & 0 & 0.056 & 0.006 \\ \hline
**D* & 112.31 & 66.21 & 13.06 & 0.13 & 113.47 & 66.89 & 13.06 & 8.79 & 0 & 0 \\ \hline \end{tabular}
\end{table} 
Table 6: Example 3: technical, Cartesian and polar constants of the best solution laminate; \(C_{ij}\) is a generic Cartesian component (GPa). It can be remarked that T3 is the most suitable trajectory to find the best value of the solution, in all the cases. For the average of the best value, the three trajectories T3, T4 and T5 are equally performing. For the expected value, T5 seems to be the best choice. In any case, trajectories T3, T4 or T5 give better results than the cases of T1 and T2, _i.e._ when parameters are fixed. But, more important, is another fact: the parameters in T1 and T2 have been accurately chosen after a long series of numerical tests and, especially for the case of T1, the values of the parameters are so precise (0.729 for \(c_{0}\), 1.49 for \(c_{1}\) and \(c_{2}\)) that it is doubtful their effectiveness for the whole objective functions. In other words, if the value of the parameters must be determined to within a precision of the order \(10^{-2}\div 10^{-3}\), it can be a problem to determine the best choice of the parameters for a given problem, or, which is the same, how much changes of these parameters condition the result? So, the choice to use evolving parameters can be a better strategy, as the fact that their value changes along the iterations do not force the user to a precise choice and ensures him to have a good compromise between exploration and exploitation.