Optimal Rates of Mutation and Crossover Operators

Initial experiments focused on finding the optimal rates of mutation and crossover operators understood here as the rates which produced the best progress of MAs. An extensive parameter search was conducted to determine the optimal rates (see Table 3). Obtained results differed for various types of evolutionary algorithms. Typical results for MA-ES are presented in Figure 4 which shows the average best-so-far fitness values and 95% confidence intervals (vertical lines) calculated using Johnson’s modified t test (Johnson, 1978) obtained in a series of design experiments with MA – ES(5+25) for Problem Ia. In these experiments, the rate of uniform crossover was equal to 0.2.

A clear pattern can be identified in Figure 4 regarding the impact of the mutation rate on the fitness of produced designs: lower mutation rates produce better fitness (i. e. lower fitness because it is a minimization problem) of designs produced. This pattern was observed in all design experiments involving MA-ES for various parent and offspring population sizes, and crossover rates, as illustrated in Figure 5. It shows average best-so-far fitness values and corresponding confidence intervals obtained at the end of short-term experiments for Problem Ia. Similar patterns were obtained for Problems Ib-c, and Problem II when ES was used.

Figure 4: Progress of MA-ES for various mutation Figure 5: Impact of mutation rates on MA-ES rates

Different results and patterns were obtained when MA-GAs were employed to optimize topologies of steel structural systems. Figure 6 compares the results produced by two MAs, one employing MA­ES and one utilizing MA-GA. Here, the graphs produced by MA-ES(5,25) (left) are compared to the graphs produced by MA-GA(5,25) (center) and MA-GA(50,50) (right). It is clear that the results produced by MA-GAs are significantly different in terms of patterns produced: higher mutation rates produce better results, particularly when low crossover rates are used. In this case, however, the differences among the results produced by MA-GAs with various rates of mutation are small.

A search for the optimal rate of the crossover operator was conducted by analyzing the results of the design experiments in which various crossover rates were used but the mutation rate was kept the same. Figure 7 presents typical results obtained in the experiments in which MA-ES were used. It shows the average best-so-far fitness values and 95% confidence intervals obtained in the design experiments with MA-ES(5+25) and 3 different rates of crossover, i. e. 0.0, 0.2, and 0.5. The mutation rate was kept the same and equal to 0.025. Figure 7 shows that various crossover rates yielded only small differences in the fitness of produced designs. No clear pattern could be observed, as it was the case with the mutation operator. These observations were further confirmed by the results presented in Figure 8. It shows that there was no trend favoring specific crossover rates. On the contrary, in some cases the best results were achieved with no crossover at all and sometimes the best results were obtained when very high crossover rates are used, i. e. when the rate was equal to 0.5. Figure 8 also shows that even if there were differences among the fitness values obtained with various crossover rates, they were not significant (confidence intervals overlap in all cases). These results were consistent for all design problems reported in the paper. Similarly as in the case of MA-ES, MA-GAs do not exhibit any clear pattern in terms of preferred crossover rates. The graph showing these results was, however, omitted.

Concluding, MA-ES produce the best results when low rates of mutation operator are used, e. g. 0.025. On the contrary, higher rates of mutation seem to be preferred by MA-GAs but the differences in the obtained results are not as significant as in the case of MA-ES. No such patterns were obtained for crossover rates.