# New Insights into Fuzzy Genetic Algorithms for Optimization Problems
## 4 Experiments and Discussion  
### Initial Experiments on Popular Benchmarks  
Experiments were run on the same initial population (ensured with the random seed) through 500 generations. The population size was 100, as in [15]. For any considered benchmark function, there were 20 runs, considering two possible dimensions, i.e., \(D\in\{5,1000\}\). The minimum type t-norm and the product type t-norm were considered. Similarly, the maximum type and the sum type t-conorm were employed. Regarding the membership functions (MFs), two options were explored, i.e., pi-shaped and trapezoid. The aim here was to explore variants of the original EFGA and GFGA, where only the triangular MFs and the minimum t-norm were used. For EFGA, it was chosen \(N_{fitness}=\{3,5,7\}\). The same values for \(N_{age}\) are in GFGA. A larger number of terms would not be beneficial for interpretability. Four typical benchmarks for algorithms in optimization tasks were considered:  
* The Schwefel function \[f_{1}(\mathbf{x})=418.9829d-\sum_{i=1}^{D}x_{i}\sin(\sqrt{|x_{i}|}),\quad x_{ i}\in[-500,500],\] (8)
* The Rastrigin function \[f_{2}(\mathbf{x})=10D+\sum_{i=1}^{D}\Big{(}x_{i}^{2}-10\cos(2\pi x_{i})\Big{)},\quad x_{i}\in[-5.12,5.12],\] (9)
* The Griewank function \[f_{3}(\mathbf{x})=\sum_{i=1}^{D}\frac{x_{i}^{2}}{4000}-\prod_{i=1}^{D}\cos( \frac{x_{i}}{\sqrt{i}})+1,\quad x_{i}\in[-40,40],\] (10)
* The Ackley function \[f_{4}(\mathbf{x})=-a\,e^{\Big{(}-b\sqrt{\frac{1}{D}\sum_{i=1}^{D}x_{i}^{2}} \Big{)}}-e^{\Big{(}\frac{1}{D}\sum_{i=1}^{D}\cos(c\,\,x_{i})\Big{)}}+a+e,\quad x _{i}\in[-32.768,32.768].\] (11)  
The choice of the benchmark functions and the dimension has been motivated by the presence of relevant results in the literature, as will be discussed in the following. The different MFs did not affect the results significantly: a difference of less than \(5\%\) on average was observed. Regarding t-norm or t-conorm types, we observed that the best results were obtained by using the combination of min t-norm and max t-conorm, and this combination was kept for the rest of the experiments, including those discussed in the next sections. We observed that the results between the two types of implementation (i.e., the original one and that with NNC) had no significant differences, i.e., not greater than \(3\%\). Similarly, assuming \(N_{fitness}>5\) for EFGA-NNC and \(N_{age}>5\) for GFGA-NNC did not produce any significant changes in the results and even worsened them. Regarding the MFs, we observed a slightly better performance by using pi-shaped MFs. Hence, we fixed it for all the remaining experiments. Instead, the NNC strategy allowed to significantly shorten the runtime. The same experiments with the same fitness function and parameters were run several times, as described before, and mean runtime and runtime standard deviation were compared. The ratio \(s\) between the runtime with and without NNC in the case of EFGA is \(s=0.077\pm 0.022\) and \(s=0.014\pm 0.007\) for GFGA. Comparing the runtime of EFGA-NNC and GFGA-NNC to that of the standard GA, the latter is still more competitive, with 30-50% shorter runtimes on average, where the worst case is the high-dimensional one (i.e., D = 1000). This computational burden may be seen as the cost of explainability [16]. The results using the different techniques are depicted in Figure 4(a),b. Both for \(D=5\) and \(D=1000\), there is no significant difference between the results by EFGA-NNC andGAs, except for the case of the Shwefel function. In that case, the GA outperforms the other approaches, especially for \(D=5\). At any rate, the advantage over the standard GA is boosting explainability. Regarding the GFGA-NNC, it performs better for problems with higher dimensions, being comparable to the other approaches. The PSO outperforms the other approaches for the Rastrigin and Griewank functions. There is no significant difference among all the approaches in the case of the Ackley function, both for \(D=5\) and \(D=1000\).