# Revised Gravitational Search Algorithms Based on Evolutionary-Fuzzy Systems
## 2 The Gravitational Search Algorithm
However, there is an improvement of FGSA-GA at the end of the iterations. Algorithm 5 uses GA to optimize the membership functions of the fuzzy system. However, the parameter \(\alpha\) of GSA can be optimized with the help of other optimization methods such as PSO and DE. In FGSA, the idea is to define the membership functions range of \(\alpha\), taking into account the optimal value of \(\alpha\) obtained with PSO and DE. The procedure is described in Algorithm 7. We compare the results of GSA and FGSA optimized by PSO (GSA-PSO and FGSA-PSO, respectively). By observing Figure 9, we can note that the best-so-far solution of FGSA-PSO is better than GSA-PSO for the test function \(F1\). In fact, from Iteration 700-1000, the FGSA curve is below the GSA curve. There is a good improvement for the test function \(F7\) (see Figure 10) where FGSA-PSO overcomes the performances of GSA-PSO. For the other test functions, the results are about the same. Table 2.1 Comparison of the performance of GSA and FGSA-GA for minimization of \(F12\). Table 2.2 Comparison of the performance of FGSA-PSO and GSA-PSO for minimization of \(F7\). Table 2.3 Comparison of performance of FGSA-PSO and GSA-PSO for minimization of \(F1\). Table 7 shows the comparison of the results achieved by [22; 26; 27; 35] with the performances of the algorithms FGSA-GA, FGSA-PSO and FGSA-DE over 30 runs. This table does not contain the results of all 13 test functions, but it resumes the more significant improvements for certain benchmark functions. The last three columns of Table 7 contain the average of the best solution in the last iteration of Algorithms 6-7. FGSA-GA gives better results than the GSA proposed by [22] for all of the functions of Table 2.4 The maximum improvement is 19 orders of magnitude for the function \(F12\). Moreover, FGSA-GA is better than the BGSA proposed in [35] for all of the functions shown in Table 2.5 The best improvement is of 22 orders of magnitude for the function \(F1\). FGSA-GA improves the GSA designed in [27] for the functions \(F2\), \(F3\), \(F4\), \(F11\) and \(F12\). In particular, for the function \(F12\), an improvement of 20 orders of magnitude is achieved. The comparison between our FGSA and FGSA proposed by [PERSON] [26] can be accomplished only for the functions \(F1-F4\). Except for the function \(F3\), our algorithm has better solutions than FGSA in [26]. FGSA-PSO is better than [22] for the functions \(F1\), \(F2\), \(F3\), \(F10\) and \(F12\). The best improvement is of 14 orders of magnitude for the function \(F1\). Note that, by comparing the FGSA-PSO results with [35], FGSA-PSO provides better outcomes except for \(F3\) and \(F11\). Only for the function \(F12\), FGSA-PSO overcomes the GSA proposed in [27]. FGSA-DE improves the results of [22; 26; 27; 35] for all of the functions in Table 7, except for \(F3\) of [26]. Table 8 shows the results of FGSA-GA, FGSA-PSO and FGSA-DE versus GSA of [28] over 16 runs. FGSA-GA improves the results of [28] for all of the functions, providing a relevant improvement for \(F1\) of 20 orders of magnitude. The provided solutions by FGSA-PSO are better than GSA [28] only for the function \(F1\). FGSA-DE is better than GSA [28] for the functions \(F1-F4\) and \(F10\). In this case, an improvement of 18 orders of magnitude for \(F1\) is achieved. Finally, the results show that, both over 30 and 16 runs, FGSA-DE is optimal for unimodal functions, whereas FGSA-GA is good for multimodal functions.