An Enhanced Lightning Attachment Procedure Optimization Algorithm
Abstract
:1. Introduction
2. The Lightning Attachment Procedure Optimization Algorithm
2.1. Initialize the Population
2.2. Downward Leader Movement
2.3. Upward Leader Movement
2.4. Branch Fading
2.5. Enhancement of the Performance
3. The Enhanced Lightning Attachment Procedure Optimization Algorithm
3.1. Improved Downward Leader Movement
3.2. Improvement of Upward Leader Movement
3.3. The Improved Enhancement Performance
3.4. The Pseudo Code of the ELAPO Algorithm
Initialize the first population of test points randomly in the specific range |
Calculate the fitness of test points |
while the end criterion is not achieved |
Set the test point with the worst fitness as Xw |
for j = 1:D |
end |
if the fitness of is better than the fitness of Xw |
end |
Obtain Xave which is the mean value of all the test points |
Calculate the fitness of Xave as Fave |
for i = 1:NP (each test point) |
Select randomly which is not equal to |
Set the test point with the best fitness as |
for j = 1:D (number of variables) |
Update the variables of based on Equation (6), as |
Check the boundary. If the particle exceeds the boundary value, it is generated randomly within the boundary. |
end |
Calculate the fitness of |
if the fitness of is better than |
end |
end |
for i = 1:NP (each test point) |
for j = 1:D (number of variables) |
Update the variables of based on Equation (8) as |
Check the boundary. If the particle exceeds the boundary value, it is generated randomly within the boundary. |
end |
Calculation the fitness of |
if the fitness of is better than |
end |
end |
XBEST = the test point with the best fitness |
FBEST = the best fitness |
end |
return XBEST, FBEST |
4. Analysis of the Simulation Results
5. Conclusions and Future Research
Author Contributions
Funding
Conflicts of Interest
References
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Name | Function | Range | Optimal |
---|---|---|---|
f1 sumPower | [−1, 1] | 0 | |
f2 Sphere | [−100, 100] | 0 | |
f3 SumSquares | [−10, 10] | 0 | |
f4 Step | [−1.28, 1.28] | 0 | |
f5 Schwefel 2.22 | [−10, 10] | 0 | |
f6 Schwefel 1.2 | [−100, 100] | 0 | |
f7 Schwefel2.21 | [−100, 100] | 0 | |
f8 Schwefel1.2 with Noise | [−100, 100] | 0 | |
f9 Rastrigin | [−5.12, 5.12] | 0 | |
f10 Shifted Rotated Rastrigin’s | [−5.12, 5.12] | 0 | |
f11 Griewank | [−600, 600] | 0 | |
f12 Ackley | [−32, 32] | 0 | |
f13 Weierstrass | [−0.5, 0.5] | 0 | |
f14 Himmelblau | [−5, 5] | −78.3323 | |
f15 Cross-in-tray | [−10, 10] | −2.0626 | |
f16 Six-hump Camel | [−5.12, 5.12] | −1.0316 |
Algorithms | Parameter |
---|---|
ADN-RSN-PSO | W = 0.7298, c1 = c2 = 2.05 |
ABCADE | SN = 50, limit = 200, m = 5, n = 10, c 1 = 0.9, c 2 = 0.999 |
IMSaDE | NEP = 7([0.1,0.3]*NP), ST = 3, CRl = 0.3, Cru = 1, Fl = 0.1, Fu = 0.9 |
Function | Statistic | ADN-RSN-PSO | ABCADE | DSTLBO | IMSaDE | LAPO | ELAPO |
---|---|---|---|---|---|---|---|
f1 | Min | 1.9433 × 105 | 1.0604 × 10−122 | 0 | 9.1175 × 10−190 | 1.2787 × 10−140 | 0 |
Mean | 7.0621 × 10−3 | 5.0305 × 10−96 | 0 | 8.4357 × 10−171 | 1.4429 × 10−131 | 0 | |
Max | 6.0864 × 10−2 | 1.5085 × 10−94 | 0 | 8.5523 × 10−170 | 3.7225 × 10−130 | 0 | |
Std | 1.3886 × 10−2 | 2.7540 × 10−95 | 0 | 0 | 6.7837 × 10−131 | 0 | |
Robustness | 0 | 100 | 100 | 100 | 100 | 100 | |
Rank | 6 | 5 | 1.5 | 3 | 4 | 1.5 | |
f2 | Min | 2.7340 × 10−9 | 6.1134 × 10−49 | 0 | 5.6607 × 10−95 | 1.3390 × 10−36 | 0 |
Mean | 4.5246 × 10−1 | 4.3991 × 10−42 | 2.4682 × 10−318 | 4.4691 × 10−88 | 1.2060 × 10−33 | 0 | |
Max | 8.1548 | 6.9662 × 10−41 | 7.2645 × 10−317 | 4.8224 × 10−87 | 1.3582 × 10−32 | 0 | |
Std | 1.5287 | 1.3292 × 10−41 | 0 | 1.2051 × 10−87 | 3.0653 × 10−33 | 0 | |
Robustness | 0 | 100 | 100 | 100 | 100 | 100 | |
Rank | 6 | 4 | 2 | 3 | 5 | 1 | |
f3 | Min | 2.0553× 10−13 | 2.9427× 10−51 | 0 | 2.0072 × 10−95 | 1.1679 × 10−38 | 0 |
Mean | 2.5192 | 1.1781× 10−41 | 1.1660 × 10−321 | 1.1548 × 10−88 | 1.4772 × 10−33 | 0 | |
Max | 4.9549 × 101 | 2.7244× 10−40 | 3.4760 × 10−320 | 3.4405 × 10−87 | 2.4542 × 10−32 | 0 | |
Std | 9.134 | 5.0461× 10−41 | 0 | 6.2801 × 10−88 | 4.8781 × 10−33 | 0 | |
Robustness | 10 | 100 | 100 | 100 | 100 | 100 | |
Rank | 6 | 4 | 2 | 3 | 5 | 1 | |
f4 | Min | 3.3295 | 0 | 4.2081 | 3.0815 × 10−33 | 2.0431 × 10−17 | 0 |
Mean | 5.6636 | 5.9986 × 10−32 | 5.476 | 1.1884 × 10−31 | 2.8720 × 10−16 | 1.0812 × 10−29 | |
Max | 7.5679 | 8.0735 × 10−31 | 7.0288 | 1.5068 × 10−30 | 1.4498 × 10−15 | 4.2786 × 10−29 | |
Std | 1.1575 | 1.4892 × 10−31 | 8.6427 × 10−1 | 2.7273 × 10−31 | 3.6114 × 10−16 | 1.7057 × 10−29 | |
Robustness | 0 | 100 | 0 | 100 | 100 | 100 | |
Rank | 6 | 1 | 5 | 2 | 4 | 3 | |
f5 | Min | 5.2882 × 10−5 | 4.7392 × 10−32 | 5.7955 × 10−169 | 6.2444 × 10−53 | 5.1613 × 10−21 | 2.2875 × 10−175 |
Mean | 2.1283 | 2.9518 × 10−25 | 1.8017 × 10−162 | 2.9148 × 10−49 | 1.8239 × 10−19 | 7.2628 × 10−171 | |
Max | 1.4320 × 101 | 5.0681 × 10−24 | 2.3000 × 10−161 | 3.1393 × 10−48 | 1.0955 × 10−18 | 1.5612 × 10−169 | |
Std | 3.6429 | 9.9107 × 10−25 | 5.8809 × 10−162 | 8.1886 × 10−49 | 2.5217 × 10−19 | 0 | |
Robustness | 0 | 100 | 100 | 100 | 100 | 100 | |
Rank | 6 | 4 | 2 | 3 | 5 | 1 | |
f6 | Min | 7.9614 × 10−11 | 4.9825 × 10−1 | 0 | 1.7910 × 10−13 | 3.6024 × 10−5 | 2.0577 × 10−204 |
Mean | 3.6063 × 101 | 2.1321 × 101 | 1.4778 × 10−317 | 2.2465 × 10−9 | 2.7988 × 10−4 | 3.8434 × 10−188 | |
Max | 9.1658 × 102 | 1.0491 × 102 | 4.4236 × 10−316 | 6.6102 × 10−8 | 8.6964 × 10−4 | 1.1172 × 10−186 | |
Std | 1.6698 × 102 | 2.3468 × 101 | 0 | 1.2061 × 10−8 | 2.1707 × 10−4 | 0 | |
Robustness | 3.33 | 0 | 100 | 83.33 | 0 | 100 | |
Rank | 6 | 5 | 1 | 3 | 4 | 2 | |
f7 | Min | 1.4991 × 10−5 | 5.4163 | 5.0526 × 10−162 | 1.9698 × 10−2 | 1.1755 × 10−21 | 1.4981 × 10−122 |
Mean | 2.2665 × 10−1 | 1.2694 × 101 | 9.8249 × 10−155 | 1.7505 × 10−1 | 1.5051 × 10−19 | 2.1015 × 10−116 | |
Max | 2.6331 | 2.1428 × 101 | 1.5939 × 10−153 | 5.8210 × 10−1 | 1.4967 × 10−18 | 3.4242 × 10−115 | |
Std | 5.3499 × 10−1 | 3.9118 | 3.6036 × 10−154 | 1.4560 × 10−1 | 2.8717 × 10−19 | 6.6285 × 10−116 | |
Robustness | 0 | 0 | 100 | 0 | 100 | 100 | |
Rank | 6 | 5 | 1 | 4 | 3 | 2 | |
f8 | Min | 9.1385 × 10−7 | 3.7190 × 10−13 | 1.9123 × 10−315 | 4.2590 × 10−51 | 7.2925 × 10−18 | 1.5064 × 10−298 |
Mean | 4.2904 × 102 | 6.4605 × 10−6 | 4.1842 × 10−298 | 6.1752 × 10−30 | 7.6451 × 10−17 | 1.7716 × 10−288 | |
Max | 4.1982 × 103 | 7.5741 × 10−5 | 7.4227 × 10−297 | 1.8141 × 10−28 | 3.1112 × 10−16 | 3.2124 × 10−287 | |
Std | 1.0583 × 103 | 1.8898 × 10−5 | 0 | 3.3101 × 10−29 | 6.9792 × 10−17 | 0 | |
Robustness | 0 | 26.67 | 100 | 100 | 100 | 100 | |
Rank | 6 | 5 | 1 | 3 | 4 | 2 | |
f9 | Min | 8.0008 × 10−7 | 0 | 0 | 6.9647 | 0 | 0 |
Mean | 1.3883 × 101 | 1.9899 × 10−1 | 0 | 1.9655 × 101 | 4.9931 | 1.368 | |
Max | 1.4700 × 102 | 9.9496 × 10−1 | 0 | 7.1757 × 101 | 9.8831 × 101 | 1.5045 × 101 | |
Std | 3.7141 × 101 | 4.0479 × 10−1 | 0 | 1.4266 × 101 | 2.0014 × 101 | 4.1857 | |
Robustness | 0 | 80 | 100 | 0 | 93.33 | 90 | |
Rank | 5 | 2 | 1 | 6 | 4 | 3 | |
f10 | Min | 3.8725 × 10−13 | 1.7764 × 10−15 | 0 | 0 | 5.7384 × 10−2 | 0 |
Mean | 4.2506 × 10−2 | 2.8422 × 10−15 | 0 | 1.5395 × 10−15 | 1.1063 | 0 | |
Max | 4.4505 × 10−1 | 1.7764 × 10−14 | 0 | 1.7764 × 10−15 | 4.6539 | 0 | |
Std | 1.0456 × 10−1 | 3.0447 × 10−15 | 0 | 6.1417 × 10−16 | 1.0863 | 0 | |
Robustness | 6.67 | 100 | 100 | 100 | 0 | 100 | |
Rank | 5 | 4 | 1.5 | 3 | 6 | 1.5 | |
f11 | Min | 2.8111 × 101 | 0 | 3.1916 × 101 | 0 | 6.2061 × 10−14 | 0 |
Mean | 5.2552 × 101 | 1.1102 × 10−16 | 5.4567 × 101 | 1.3323 × 10−16 | 2.7950 × 10−3 | 0 | |
Max | 7.8919 × 101 | 2.2204 × 10−16 | 7.1519 × 101 | 4.4409 × 10−16 | 4.9323 × 10−3 | 0 | |
Std | 1.1032 × 101 | 4.1233 × 10−17 | 7.837 | 7.9313 × 10−17 | 2.4859 × 10−3 | 0 | |
Robustness | 0 | 100 | 0 | 100 | 43.33 | 100 | |
Rank | 5 | 2 | 6 | 3 | 4 | 1 | |
f12 | Min | 2.8188 × 10−5 | 7.1054 × 10−15 | 0 | 3.5527 × 10−15 | 6.2061 × 10−14 | 3.5527 × 10−15 |
Mean | 7.2291 × 10−1 | 1.4211 × 10−14 | 2.7237 × 10−15 | 1.0725 | 2.7950 × 10−3 | 3.9080 × 10−15 | |
Max | 6.5883 | 3.1974 × 10−14 | 3.5527 × 10−15 | 5.3162 | 4.9323 × 10−3 | 7.1054 × 10−15 | |
Std | 1.6003 | 7.6368 × 10−15 | 1.5283 × 10−15 | 1.1266 | 2.4859 × 10−3 | 1.0840 × 10−15 | |
Robustness | 0 | 100 | 100 | 36.67 | 43.33 | 100 | |
Rank | 5 | 3 | 1 | 6 | 4 | 2 | |
f13 | Min | 7.2301 × 10−1 | 0 | 0 | 4.3238 × 10−2 | 0 | 0 |
Mean | 3.9722 | 1.0394 × 10−3 | 0 | 9.1739 × 10−1 | 0 | 0 | |
Max | 6.0091 | 3.1181 × 10−2 | 0 | 3.4394 | 0 | 0 | |
Std | 2.4463 | 5.6928 × 10−3 | 0 | 9.1925 × 10−1 | 0 | 0 | |
Robustness | 0 | 96.67 | 100 | 0 | 100 | 100 | |
Rank | 6 | 4 | 2 | 5 | 2 | 2 | |
f14 | Min | −5.3861 × 101 | −7.8332 × 101 | −5.3644 × 101 | −7.7390 × 101 | −6.1359 × 101 | −7.8332 × 101 |
Mean | −4.6574 × 101 | −7.8332 × 101 | −4.7669 × 101 | −7.4720 × 101 | −5.8293 × 101 | −6.8451 × 101 | |
Max | −3.8905 × 101 | −7.8332 × 101 | −4.2338 × 101 | −7.2678 × 101 | −5.6084 × 101 | −5.3228 × 101 | |
Std | 3.1993 | 1.3964 × 10−14 | 2.4718 | 1.5473 | 1.6251 | 4.6141 | |
Robustness | 0 | 100 | 0 | 0 | 0 | 6.67 | |
Rank | 6 | 1 | 5 | 2 | 4 | 3 | |
f15 | Min | −2.0626 | −2.0626 | −2.0626 | −2.0626 | −2.0626 | −2.0626 |
Mean | −2.0626 | −2.0626 | −2.0626 | −2.0626 | −2.0626 | −2.0626 | |
Max | −2.0626 | −2.0626 | −2.0626 | −2.0626 | −2.0626 | −2.0626 | |
Std | 1.0856 × 10−12 | 4.2561 × 10−4 | 2.8300 × 10−6 | 4.0365 × 10−6 | 6.0550 × 10−8 | 9.0336 × 10−16 | |
Robustness | 100 | 100 | 100 | 100 | 100 | 100 | |
Rank | 2 | 6 | 4 | 5 | 3 | 1 | |
f16 | Min | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 |
Mean | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | |
Max | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | |
Std | 1.3596 × 10−5 | 1.3074 × 10−4 | 2.5463 × 10−12 | 4.5168 × 10−16 | 1.4600 × 10−7 | 4.5168 × 10−16 | |
Robustness | 100 | 100 | 100 | 100 | 100 | 100 | |
Rank | 5 | 6 | 3 | 2 | 4 | 2 |
Fridman | Holm | ||||
---|---|---|---|---|---|
i | Algorithm | Rank mean (Ri) | pi | ||
1 | ADN-RSN-PSO | 5.4375 | 5.4808 | 0.0000 | 0.01 |
2 | LAPO | 4.0625 | 3.4019 | 0.0009 | 0.0125 |
3 | ABCADE | 3.8125 | 3.0238 | 0.0035 | 0.0166 |
4 | IMSaDE | 3.5 | 2.5514 | 0.011 | 0.025 |
5 | DSTLBO | 2.4375 | 0.9450 | 0.3221 | 0.05 |
6 | ELAPO | 1.8125 | / | / | / |
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Wang, Y.; Jiang, X. An Enhanced Lightning Attachment Procedure Optimization Algorithm. Algorithms 2019, 12, 134. https://doi.org/10.3390/a12070134
Wang Y, Jiang X. An Enhanced Lightning Attachment Procedure Optimization Algorithm. Algorithms. 2019; 12(7):134. https://doi.org/10.3390/a12070134
Chicago/Turabian StyleWang, Yanjiao, and Xintian Jiang. 2019. "An Enhanced Lightning Attachment Procedure Optimization Algorithm" Algorithms 12, no. 7: 134. https://doi.org/10.3390/a12070134
APA StyleWang, Y., & Jiang, X. (2019). An Enhanced Lightning Attachment Procedure Optimization Algorithm. Algorithms, 12(7), 134. https://doi.org/10.3390/a12070134