Fig 1.
Flow-chart for the proposed technique.
Table 1.
Summary of the properties of FEIW function.
Table 2.
The parameters and properties of six variations of FEIW.
Fig 2.
Six variations of Flexible Exponential Inertia Weight (FEIW) strategy.
(A) FEIW-1. (B) FEIW-2. (C) FEIW-3. (D) FEIW-4. € FEIW-5. (F) FEIW-6.
Table 3.
Benchmark functions for simulation.
Table 4.
Benchmark functions formula.
Table 5.
Comparison of success rate, average and minimum number of iterations of successful runs for considered PSO variants with condition 2, Imax = 1000, D = 10, ε = 10−1 for f2, f3, f4, f10 functions and ε = 10−10 for others (υ > Imax).
Table 6.
Comparison of success rate, average and minimum number of iterations of successful runs for considered PSO variants with condition 2, Imax = 1000, D = 10, ε = 5 for f15 and f20 functions, ε = 10−1 for f19, f21, f24, f25 functions and ε = 10−10 for others (υ > Imax).
Table 7.
Comparison of average, minimum and standard deviation of error for considered PSO variants with condition 1, Imax = 1000 and D = 10.
Table 8.
Comparison of average, minimum and standard deviation of error for considered PSO variants with condition 1, Imax = 1000 and D = 10.
Table 9.
Comparison of average, minimum and standard deviation of error for considered PSO variants with condition 1, Imax = 1000 and D = 10.
Table 10.
Comparison of average, minimum and standard deviation of error for considered PSO variants with condition 1, Imax = 1000 and D = 10.
Table 11.
Comparison of average, minimum and standard deviation of error for considered PSO variants with condition 1, Imax = 500 and D = 50.
Table 12.
Comparison of average, minimum and standard deviation of error for considered PSO variants with condition 1, Imax = 500 and D = 50.
Table 13.
Comparison of average, minimum and standard deviation of error for considered PSO variants with condition 1, Imax = 1000 and D = 50.
Table 14.
Comparison of average, minimum and standard deviation of error for considered PSO variants with condition 1, Imax = 1000 and D = 50.
Table 15.
Best and worst IW strategies for each benchmark function in terms of success rate, average and minimum number of iterations of successful runs according to Table 5.
Table 16.
Best and worst IW strategies for each benchmark function in terms of success rate, average and minimum number of iterations of successful runs according to Table 6.
Table 17.
Best and worst IW strategies for each benchmark function in terms of success rate, average and minimum number of iterations of successful runs according to Table 6.
Table 18.
Best and worst IW strategies for each benchmark function in terms of average, minimum and standard deviation of error according to Tables 7 and 8.
Table 19.
Best and worst IW strategies for each benchmark function in terms of average, minimum and standard deviation of error according to Table 9.
Table 20.
Best and worst IW strategies for each benchmark function in terms of average, minimum and standard deviation of error according to Table 10.
Table 21.
Best and worst IW strategies for each benchmark function in terms of average, minimum and standard deviation of error according to Tables 11 and 12.
Table 22.
Best and worst IW strategies for each benchmark function in terms of average, minimum and standard deviation of error according to Tables 13 and 14.
Table 23.
Table 24.
Wilcoxon-ranks and p-value on the average and minimum number of iterations of successful runs according to Table 5.
Table 25.
Wilcoxon-ranks and p-value on the average and minimum number of iterations of successful runs according to Table 6.
Table 26.
Wilcoxon-ranks and p-value on the average and minimum error according to Tables 7 and 8.
Table 27.
Wilcoxon-ranks and p-value on the average and minimum error according to Tables 9 and 10.
Table 28.
Wilcoxon-ranks and p-value on the average and minimum error according to Tables 11 and 12.
Table 29.
Wilcoxon-ranks and p-value on the average and minimum error according to Tables 13 and 14.
Table 30.
Friedman test based on Table 5.
Table 31.
Friedman test based on Table 6.
Table 32.
Table 33.
Table 34.
Table 35.
Fig 3.
(A) Average iterations based on Table 5. (B) Average iterations based on Table 6. (C) Minimum iterations based on Table 5. (D) Minimum iterations based on Table 6. (E) Success rate based on Table 5. (F) Success rate based on Table 6.
Fig 4.
(A) Average error based on Tables 7 and 8. (B) Average error based on Tables 9 and 10. (C) Minimum error based on Tables 7 and 8. (D) Minimum error based on Tables 9 and 10. (E) Standard deviation of error based on Tables 7 and 8. (F) Standard deviation of error based on Tables 9 and 10.
Fig 5.
(A) Average error based on Tables 11 and 12. (B) Average error based on Tables 13 and 14. (C) Minimum error based on Tables 11 and 12. (D) Minimum error based on Tables 13 and 14. (E) Standard deviation of error based on Tables 11 and 12. (F) Standard deviation of error based on Tables 13 and 14.
Fig 6.
Boxplots of considered PSO variants.
(A) Average iterations based on Table 5. (B) Average iterations based on Table 6. (C) Minimum iterations based on Table 5. (D) Minimum iterations based on Table 6. (E) Success rate based on Table 5. (F) Success rate based on Table 6.
Fig 7.
Boxplots of considered PSO variants.
(A) Average error based on Tables 7 and 8. (B) Average error based on Tables 9 and 10. (C) Minimum error based on Tables 7 and 8. (D) Minimum error based on Tables 9 and 10. (E) Standard deviation of error based on Tables 7 and 8. (F) Standard deviation of error based on Tables 9 and 10.
Fig 8.
Boxplots of considered PSO variants.
(A) Average error based on Tables 11 and 12. (B) Average error based on Tables 13 and 14. (C) Minimum error based on Tables 11 and 12. (D) Minimum error based on Tables 13 and 14. (E) Standard deviation of error based on Tables 11 and 12. (F) Standard deviation of error based on Tables 13 and 14.
Fig 9.
Convergence graph for some PSO variants.
(A) Sphere Function with ε = 10−20. (B) Griewank Function with ε = 10−1. (C) Ackley Function with ε = 10−15. (D) Zakharov Function with ε = 10−200. (E) Schwefel's Problem 2.22 with ε = 10−20. (F) Weierstrass Function with ε = 10−30.