Optimal Parent and Offspring Population Sizes
The next group of experiments focused on determining the optimal sizes of populations of parents and offspring for MAs. Three different combinations of sizes of parent and offspring populations were considered for MA-ES and two combinations for MA-GAs.
Typical results obtained with MA-ES for Problem I are presented in Figure 9. It shows the results of the design experiments in which three combinations of the parent and offspring population sizes were used, including MA-ES(1+5), MA-ES(5+25), and MA-ES(50+250). Mutation and crossover rates were kept the same in all experiments shown in Figure 9 and equal to 0.025 and 0.2, respectively.
It is clear that MA-ES using large population sizes, i. e. MA-ES(50+250), produced inferior results compared to the other two MA-ES with smaller population sizes. On the other hand, it also produced the smallest variance. The other two MA-ES with smaller population sizes achieved almost the same optimization progress in terms of the average best-so-far fitness of the produced designs. However, MA-ES(1+5), i. e. the ‘greedy’ MA-ES preserving only the single best individual to the next generation, exhibited much larger variance compared to MA-ES(5+25) which preserves the top 5 individuals to the next generation. Thus, in this case parallel search conducted by MA-ES(5+25) reduces the variance of the obtained results without decreasing the performance of the algorithm. On the other hand, when the size of populations is increased too much, e. g. as in MA-ES(50+250), the reduction of variance comes at a cost of a substantial decrease of the performance of the algorithm.
The outcomes were again different for MA-GAs. In both cases, i. e. for MA-GA(5,25) and MA – GA(50,50), the performance of the algorithm was almost identical. Figure 10 shows typical results of the design experiments involving MA-GA(5,25) and MA-GA(50,50). The specific results presented in this figure were produced by the two algorithms with the same mutation and crossover rates equal to 0.3 and 0.5, respectively.
The two best-so-far curves shown in Figure 10 are almost identical in their nature. The only difference between the two curves is the reduction of variance for the algorithm with larger population sizes, i. e. for MA-GA(50,50). Similar behavior was also observed for MA-ES (see Figure 9). Figure 11 shows that MA-ES produced exactly the same patterns for Problem II as for Problem I. Good optimization progress was obtained for small and medium population sizes, i. e., MA-ES(1+5) and MA-ES(5+25), while the best results in terms of progress and variance were achieved by MA – ES(5+25).
Concluding, small population sizes seem to be preferred by MA-ES for these problem domains. However, too small population sizes increase the variance of the obtained results. Good results in terms of both performance and variance were produced when moderate sizes of population sizes were employed, e. g. 5 in the case of the parent population and 25 in the case of the offspring population. The impact of the sizes of parent and offspring populations on the performance of MA-GAs seems to be negligible and related only to the reduction of variance of the obtained results. It didn’t influence the actual performance of the algorithms in these problem domains.