# New Insights into Fuzzy Genetic Algorithms for Optimization Problems
## 4 Experiments and Discussion
### Initial Experiments on Popular Benchmarks
In all the cases, except for the Ackley function, with any dimension, the highest standard deviation was observed for the GFGA-NNC. For the EFGA-NNC, it was found that the standard deviation was generally comparable to that related to the GA or even less in a couple of cases (both for D = 5 and D = 1000), suggesting better stability of the EFGA-NNC with respect to the GFGA-NNC. The \(f_{2}\) function, with \(D=5\), was also considered in [22] and the results by a fuzzy logic controlled genetic algorithm (FCGA) [7], an improved fuzzy genetic algorithm (IFGA) [13]  
Figure 5: First set of benchmark functions. Average fitness value for the following: (a) D = 5; (b) D = 1000.  
and another kind of FGA [23] were compared. The latter outperformed the other methods, providing an average fitness value of around 100 after 14 generations, while IFGA provided the closest value to the optimum in the least average number of generations, i.e., 1205. The authors used a population of 20 individuals. With the same population size and after 14 generations, we obtained 85 and 68 by GFGA-NNC and EFGA-NNC, respectively. In Figure 6a,b, one can see the effect of the type of MF, t-norm, and t-conorm on the result by EFGA-NNC, where \(m=N_{fitness}\), for the sake of simplicity. One can observe that the best result was obtained by using the min t-norm and max t-conorm (max-min on the plot). Even though there is not much difference due to the type of MF, it is possible to see that the best result was obtained with \(m=5\) and pi-shaped MF. Similar behavior was observed over the other benchmarks. The benchmark functions \(f_{1}\), \(f_{2}\), and \(f_{4}\) with the same domains and \(D=1000\) have been considered in [24], where the same number of runs was also adopted (i.e., 20). The au  
Figure 6: Rastrigin function (D = 5). Average fitness value after 500 generations with the following: (**a**) Pi-shaped MF; (**b**) Trapezoidal MF. thors of [24] used a standard GA as a baseline but with a population size of 50 and a number of iterations of 1000. They performed the experiments with their approach, i.e., the adaptive dimensionality reduction genetic algorithm (ADRGA), the hybrid genetic and simulated annealing algorithm (HGSA), and the adaptive genetic algorithm (AGA). We consider their published results for a fair comparison, reporting our results after 1000 generations. In Figure 7, the mean and standard deviation (sd) for the above-mentioned benchmark functions are shown. For the Ackley function, the results by EFGA-NNC and GFGA-NNC are quite close to that by ADRGA, also considering that the standard deviation for the latter is 20 times greater than that of EFGA and GFGA. A similar situation can be observed for the Shwefel function, with a standard deviation for the ADRGA, which is almost 5 and 4 times greater than that by EFGA-NNC and GFGA-NNC, respectively. For the Rastrigin function, the ADRGA outperforms. Unfortunately, in [24], there is no information about the computational cost of ADRGA or runtime. Figure 7: Comparison against state-of-the-art approaches in [24] (\(D=1000\); 1000 generations) for the following: (**a**) The Ackley function; (**b**) The Rastrigin function; (**c**) The Shwefel function.