An Improved Multiobjective Particle Swarm Optimization Based on Culture Algorithms
Abstract
:1. Introduction
2. Preliminary
2.1. Cultural Algorithms
2.2. Multiobjective Optimization Problems
s.t.g(x) ≤ 0, h(x) = 0
2.3. Particle Swam Optimization
Algorithm 1. The Procedure of Particle Swarm Optimization (PSO) |
(1) Calculate the fitness function of each particle. |
(2) Update p and p. |
(3) Update the velocity and position of each particle. |
(4) If the stop criterion is not met, go to (1), else p is the best position |
3. The Proposed Algorithm
3.1. Three Kinds of Knowledge Based on Population
3.2. Personal Guide
Algorithm 2. The Detailed Procedure of A Personal Guide |
For the d-th dimension of the i-th particle, |
(1) bn+1 = max({pbestid|i = 1, 2, …, n}); |
b1 = min({pbestid|i = 1, 2, …, n}); |
bi = b1 + (bn+1 – b1)·(i – 1)/n, i = 2, 3, …, n |
(2) si = rand(bi, bi+1); |
(3) Replace x with si and get a new particle |
(4) If the newly generated particle dominates xi, then it replaces xi; break; |
if not, the newly generated particle is discarded |
3.3. Global Guide
3.4. Dominate Criteria Based on the Hypercube
3.5. Mutation Operator
3.6. Updating the Repository
Algorithm 3. The Update Function of the Dominance Archive |
Suppose element f’ is a new element that is to be judged whether to enter the repository (rep). |
(1) Input rep, f |
(2) If ¬f’ rep:hypercube(f)hypercube(f’) then rep = rep∪{f}E |
(3) If f’:hypercube(f’) = hypercube(f)∩fε + f’ then rep = rep∪{f}f’ |
(4) If ¬f’:hypercube(f’) = hypercube(f)∪hypercube(f’)hypercube(f), then rep = rep∪{f} |
(5) Output rep |
4. Experiments
4.1. Quantitative Performance Evaluations
4.2. Comparison of the Results Using Hypercube-Based -Domination and Normal Domination Hypervolume
4.3. Comparison of the Result with and without the Mutation Operator
5. Conclusions
Acknowledgments
Author Contributions
Conflicts of Interest
References
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Test Functions | MOPSO | IC MOPSO | Cultural MOQPSO | Cultural MOPSO | NSGA-II | CD MOPSO | |
---|---|---|---|---|---|---|---|
ZDT1 | mean | 5.56 × 10−1 | 8.73 × 10−1 | 8.42 × 10−1 | 8.66 × 10−1 | 8.11 × 10−1 | 8.74 × 10−1 |
standard deviation (std) | 9.03 × 10−2 | 5.38 × 10−3 | 8.01 × 10−2 | 7.00 × 10−3 | 8.46 × 10−3 | 4.76 × 10−3 | |
p-value | 8.42 × 10−2 | 3.72 × 10−2 | 8.82 × 10−5 | 6.65 × 10−40 | 6.25 × 10−1 | ||
ZDT2 | mean | 2.88 × 10−1 | 5.38 × 10−1 | 4.72 × 10−1 | 4.20 × 10−1 | 1.20 × 10−1 | 5.37 × 10−1 |
std | 1.19 × 10−1 | 7.88 × 10−3 | 1.07 × 10−1 | 1.91 × 10−1 | 2.22 × 10−2 | 6.43 × 10−3 | |
p-value | 1.69 × 10−16 | 1.40 × 10−3 | 1.31 × 10−5 | 6.53 × 10−66 | 8.25 × 10−1 | ||
ZDT3 | mean | 1.00 | 1.01 | 9.39 × 10−1 | 7.61 × 10−1 | 7.60 × 10−1 | 1.01 |
std | 5.17 × 10−3 | 4.01 × 10-3 | 1.43 × 10−1 | 4.00 × 10−1 | 1.31 × 10−2 | 4.37 × 10−3 | |
p-value | 1.72 × 10−7 | 0.0110 | 1.35 × 10−3 | 2.35 × 10−66 | 8.76 × 10−1 | ||
ZDT4 | mean | 0.00 | 8.71 × 10−1 | 8.67 × 10-1 | 0.00 | 3.66 × 10−1 | 0.00 |
std | 0.00 | 6.08 × 10−3 | 5.18 × 10−3 | 0.00 | 1.77 × 10−1 | 0.00 | |
p-value | 1.80 × 10−118 | 1.59 × 10−2 | 1.80 × 10−118 | 1.80 × 10−118 | 1.80 × 10−118 | ||
ZDT6 | mean | 4.94 × 10−1 | 5.04 × 10−1 | 4.65 × 10−1 | 5.03 × 10−1 | 0.00 | 5.05 × 10−1 |
std | 7.43 × 10−3 | 4.88 × 10−3 | 1.12 × 10−1 | 7.63 × 10−3 | 0.00 | 5.77 × 10−3 | |
p-value | 5.97 × 10−08 | 6.31 × 10−2 | 8.61 × 10−1 | 3.40 × 10−110 | 8.76 × 10−1 | ||
DTLZ1 | mean | 0.00 | 1.30 | 1.30 | 0.00 | 0.00 | 0.00 |
std | 0.00 | 2.02 × 10−3 | 3.07 × 10−3 | 0.00 | 0.00 | 0.00 | |
p-value | 2.90 × 10−156 | 6.44 × 10−1 | 2.90 × 10−156 | 2.90 × 10−156 | 2.90 × 10−156 | ||
DTLZ2 | mean | 6.74 × 10-1 | 7.15 × 10−1 | 4.27 × 10−1 | 2.46 × 10−1 | 6.93 × 10−1 | 7.20 × 10−1 |
std | 1.50 × 10−2 | 9.74 × 10−3 | 2.41 × 10−2 | 1.26 × 10−2 | 8.35 × 10−3 | 8.12 × 10−3 | |
p-value | 4.76 × 10−18 | 3.83 × 10−54 | 1.33 × 10−78 | 3.17 × 10−13 | 3.85 × 10−2 | ||
DTLZ3 | mean | 0.00 | 7.16 × 10−1 | 7.15 × 10−1 | 0.00 | 0.00 | 0.00 |
std | 0.00 | 7.63 × 10−3 | 9.61 × 10−3 | 0.00 | 0.00 | 0.00 | |
p-value | 8.20 × 10−108 | 9.47 × 10−1 | 8.20 × 10−108 | 8.20 × 10−108 | 8.20 × 10−108 | ||
DTLZ4 | mean | 6.82 × 10−1 | 7.29 × 10−1 | 5.46 × 10−1 | 1.67 × 10−1 | 6.65 × 10−1 | 7.22 × 10−1 |
std | 1.56 × 10−2 | 7.45 × 10−3 | 5.70 × 10−2 | 6.76 × 10−2 | 3.11 × 10−2 | 8.74 × 10−3 | |
p-value | 1.55 × 10−21 | 9.53 × 10−25 | 6.11 × 10−47 | 9.22 × 10−16 | 8.66 × 10−4 | ||
DTLZ5 | mean | 4.34 × 10−1 | 4.39 × 10−1 | 4.35 × 10−1 | 2.43 × 10−1 | 4.38 × 10−1 | 4.38 × 10−1 |
std | 6.21 × 10−3 | 4.94×10−3 | 8.00 × 10−3 | 4.96 × 10−3 | 5.74 × 10−3 | 5.86 × 10−3 | |
p-value | 1.65 × 10−3 | 3.72 × 10−2 | 2.13 × 10−77 | 5.64 × 10−1 | 3.14 × 10−1 | ||
DTLZ6 | mean | 4.29 × 10−1 | 4.40×10−1 | 4.21 × 10−1 | 2.36 × 10−1 | 0.00 | 4.39 × 10−1 |
std | 6.99 × 10−3 | 5.97×10−3 | 7.53 × 10−3 | 2.81 × 10−2 | 0.00 | 6.85 × 10−3 | |
p-value | 1.43 × 10−8 | 5.64 × 10−15 | 3.58 × 10−43 | 9.89 × 10−102 | 6.28 × 10−1 | ||
DTLZ7 | mean | 1.27 | 1.43 | 1.21 | 5.44 × 10−1 | 1.47 × 10−1 | 1.38 |
std | 2.93 × 10−2 | 3.20 × 10−2 | 4.10 × 10−2 | 8.36 × 10−2 | 4.43 × 10−2 | 4.25 × 10−2 | |
p-value | 1.19 × 10−28 | 2.91 × 10−31 | 1.98 × 10−51 | 5.57 × 10−73 | 5.54 × 10−6 | ||
Better/similar/worse | 12/0/0 | 9/3/0 | 11/1/0 | 11/1/0 | 5/6/1 |
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Jia, C.; Zhu, H. An Improved Multiobjective Particle Swarm Optimization Based on Culture Algorithms. Algorithms 2017, 10, 46. https://doi.org/10.3390/a10020046
Jia C, Zhu H. An Improved Multiobjective Particle Swarm Optimization Based on Culture Algorithms. Algorithms. 2017; 10(2):46. https://doi.org/10.3390/a10020046
Chicago/Turabian StyleJia, Chunhua, and Hong Zhu. 2017. "An Improved Multiobjective Particle Swarm Optimization Based on Culture Algorithms" Algorithms 10, no. 2: 46. https://doi.org/10.3390/a10020046