A Pareto-Based Hybrid Whale Optimization Algorithm with Tabu Search for Multi-Objective Optimization
Abstract
:1. Introduction
- Design a simple and hybrid multi-objective optimization algorithm that attains more accurate solutions than common SI-based optimization heuristics. It uses the EL memory component to store non-dominated solutions. These solutions are used to guide swarm members during the search process, which eliminates the need for local search methods.
- A new diversification step is proposed to ensure an effective coverage of search space. The method ensures the balance between local optima and global optimization.
- MOWOATS uses the Pareto dominance criterion to evaluate the solutions. This method allows MOWOATS to optimize the whole objectives simultaneously and ensures obtaining solutions that are evenly distributed over solution space.
- MOWOATS is tested over different benchmark multi-objective test functions, such as Zitzler-Deb Thiele (ZDT) [20] test functions proposed by Zitzler et al., Deb-Thiele-Laumanns Zitzler (DTLZ) [20] test functions proposed by Deb et al., and CEC2009 test functions proposed by Zhang et al. [21]. The Inverted Generational Distance (IGD) metric [13] is used for the comparison.
2. Related Work
3. Background
3.1. Multi-Objective Problems
3.2. Tabu Search
3.3. Whale Optimization Algorithm
4. Methodology
4.1. Multi-Objective Whale Optimization Algorithm Combined with Tabu Search (MOWOATS)
4.2. The Algorithm and Its Components
4.2.1. Intensification by Crossover
Algorithm 1 Intensification_Crossover(S, ) |
Select solutions randomly from the swarm S. fordo Select a random solution from EL. Update random features in current whale with features selected from . end for Return the updated swarm S. |
4.2.2. Diversification by Crossover
Algorithm 2 Diversification_Crossover(S) |
Select solutions randomly from the swarm S. fordo Select a random solution from the current swarm other than current whale . Update random features in the current whale with the random features selected from . end for Return the updated swarm S. |
4.2.3. MOWOATS Algorithm
Algorithm 3 Pseudocode for MOWOATS algorithm |
Initialization. Set of particles in swarm , number of features of each object d, empty , set to be maximum number of iterations without improvement, and initialize the whale algorithm parameters. fordo Generate initial solutions randomly from the dataset. Compute the objective value of current solution according to objective functions that describe the problem. Update solutions in according to Pareto dominance criterion. end for Main Loop. fordo for do Update WOA parameters . if then if then Select a random whale from EL. Update position of the current whale applying next equation.
else if then Select a random whale from current swarm. Update position of current whale applying next equation.
end if else if then Select a random whale from EL. Update position of current whale applying next equation
end if Compute the objective value of current whale according to the objective functions that describe the problem. Update solutions in according to Pareto dominance criterion. end for if (number of iterations without improvement ≥) then Set to a random value. if then Apply intensification procedure Intensification_Crossover(S,EL) (Algorithm 1). else Apply diversification procedure Diversification_Crossover(S) (Algorithm 2). end if end if end for Return non-dominated solutions stored in EL |
5. Numerical Experiments
5.1. Parameters Setting
5.2. Results and Discussion
6. Conclusions and Future Work
Author Contributions
Funding
Acknowledgments
Conflicts of Interest
References
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Parameter | Definition | Value |
---|---|---|
MaxIt | Maximum number of iterations | 1000 |
Np | Population size | 100 |
Max_NonImprove | Maximum number of iterations without improvement | 15 |
max_EL | Maximum number of solutions stored in elite list | 100 |
UF1(bi-objective) | UF2(bi-objective) | |||||||
IGD | MOWOATS | MOGWO | MOPSO | MOEA/D | MOWOATS | MOGWO | MOPSO | MOEA/D |
Average | 0.002318 | 0.1144 | 0.1370 | 0.1871 | 0.002213 | 0.0582 | 0.0604 | 0.1223 |
Median | 0.002299 | 0.113 | 0.1317 | 0.1828 | 0.0021 | 0.0577 | 0.0483 | 0.1201 |
STD | 0.0001827 | 0.0195 | 0.0441 | 0.0507 | 0.0004524 | 0.0073 | 0.0276 | 0.0107 |
Worst | 0.002549 | 0.1577 | 0.2278 | 0.2464 | 0.003084 | 0.0732 | 0.1305 | 0.14369 |
Best | 0.00212 | 0.0802 | 0.0899 | 0.1265 | 0.001783 | 0.0498 | 0.0369 | 0.1048 |
UF3(bi-objective) | UF4(bi-objective) | |||||||
IGD | MOWOATS | MOGWO | MOPSO | MOEA/D | MOWOATS | MOGWO | MOPSO | MOEA/D |
Average | 0.009766 | 0.2556 | 0.3139 | 0.2886 | 0.001829 | 0.0586 | 0.1363 | 0.0681 |
Median | 0.009917 | 0.2509 | 0.3080 | 0.2892 | 0.001828 | 0.0586 | 0.1343 | 0.0684 |
STD | 0.0004257 | 0.0807 | 0.0447 | 0.0159 | 4.0192 × 10 | 0.0004 | 0.0073 | 0.0021 |
Worst | 0.01021 | 0.3678 | 0.3773 | 0.3129 | 0.001894 | 0.0593 | 0.1518 | 0.0703 |
Best | 0.009194 | 0.1295 | 0.2564 | 0.2634 | 0.001781 | 0.0579 | 0.1273 | 0.0646 |
UF5(bi-objective) | UF6(bi-objective) | |||||||
IGD | MOWOATS | MOGWO | MOPSO | MOEA/D | MOWOATS | MOGWO | MOPSO | MOEA/D |
Average | 0.07289 | 0.7970 | 2.2023 | 1.2914 | 0.01039 | 0.2793 | 0.6475 | 0.6881 |
Median | 0.08149 | 0.6994 | 2.1257 | 1.3376 | 0.0106 | 0.2443 | 0.5507 | 0.6984 |
STD | 0.03034 | 0.3785 | 0.5530 | 0.1348 | 0.001497 | 0.1044 | 0.2661 | 0.0553 |
Worst | 0.09776 | 1.7385 | 3.0383 | 1.4674 | 0.01175 | 0.5504 | 1.2428 | 0.7401 |
Best | 0.01682 | 0.4679 | 1.4647 | 1.1230 | 0.007643 | 0.1934 | 0.3793 | 0.5524 |
UF7(bi-objective) | UF8(tri-objective) | |||||||
IGD | MOWOATS | MOGWO | MOPSO | MOEA/D | MOWOATS | MOGWO | MOPSO | MOEA/D |
Average | 0.002116 | 0.1603 | 0.3539 | 0.4552 | 0.003607 | 2.0577 | 0.5367 | NA |
Median | 0.002123 | 0.0734 | 0.3873 | 0.4377 | 0.003225 | 2.3359 | 0.5364 | NA |
STD | 0.0001523 | 0.1391 | 0.2044 | 0.1898 | 0.0008838 | 1.1455 | 0.1825 | NA |
Worst | 0.002331 | 0.4014 | 0.6151 | 0.677 | 0.005344 | 3.8789 | 0.7964 | NA |
Best | 0.001889 | 0.0628 | 0.054 | 0.029 | 0.003021 | 0.4613 | 0.2453 | NA |
UF9(tri-objective) | UF10(tri-objective) | |||||||
IGD | MOWOATS | MOGWO | MOPSO | MOEA/D | MOWOATS | MOGWO | MOPSO | MOEA/D |
Average | 0.001473 | 0.1917 | 0.4885 | NA | 0.005116 | 3.5945 | 1.6372 | NA |
Median | 0.001406 | 0.166 | 0.4145 | NA | 0.005558 | 2.8255 | 1.5916 | NA |
STD | 0.0002609 | 0.0925 | 0.1445 | NA | 0.0008621 | 3.4882 | 0.2988 | NA |
Worst | 0.001814 | 0.4479 | 0.7221 | NA | 0.005628 | 12.9564 | 2.1622 | NA |
Best | 0.001133 | 0.1291 | 0.3336 | NA | 0.003464 | 1.22008 | NA |
UF1 | UF2 | UF3 | UF4 | UF5 | UF6 | UF7 | UF8 | UF9 | UF10 | |
---|---|---|---|---|---|---|---|---|---|---|
GDE3 | 5.34 × 10 | 1.20 × 10 | 1.06 × 10 | 2.65 × 10 | 3.93 × 10 | 2.51 × 10 | 2.52 × 10 | 2.49 × 10 | 8.25 × 10 | 4.33 × 10 |
MOEADGM | 6.20 × 10 | 6.40 × 10 | 4.90 × 10 | 4.76 × 10 | 1.79 | 5.56 × 10 | 7.60 × 10 | 2.45 × 10 | 1.88 × 10 | 5.65 × 10 |
MTS | 6.46 × 10 | 6.15 × 10 | 5.31 × 10 | 2.36 × 10 | 1.49 × 10 | 5.92 × 10 | 4.08 × 10 | 1.13 × 10 | 1.14 × 10 | 5.53 × 10 |
DMOEA-DD | 1.04 × 10 | 6.79 × 10 | 3.34 × 10 | 4.27 × 10 | 3.15 × 10 | 6.67 × 10 | 1.03 × 10 | 6.84 × 10 | 4.90 × 10 | 3.22 × 10 |
NSGA-II-LS | 1.15 × 10 | 1.24 × 10 | 1.06 × 10 | 5.84 × 10 | 5.66 × 10 | 3.10 × 10 | 2.13 × 10 | 8.63 × 10 | 7.19 × 10 | 8.45 × 10 |
OWMOsaDE | 1.22 × 10 | 8.10 × 10 | 1.03 × 10 | 5.13 × 10 | 4.30 × 10 | 1.92 × 10 | 5.85 × 10 | 9.45 × 10 | 9.83 × 10 | 7.43 × 10 |
Clustering MOEA | 2.99 × 10 | 2.28 × 10 | 5.49 × 10 | 5.85 × 10 | 2.47 × 10 | 8.71 × 10 | 2.23 × 10 | 2.38 × 10 | 2.93 × 10 | 4.11 × 10 |
AMGA | 3.59 × 10 | 1.62 × 10 | 7.00 × 10 | 4.06 × 10 | 9.41 × 10 | 1.29 × 10 | 5.71 × 10 | 1.71 × 10 | 1.89 × 10 | 3.24 × 10 |
MOEP | 5.96 × 10 | 1.89 × 10 | 9.90 × 10 | 4.27 × 10 | 2.25 × 10 | 1.03 × 10 | 1.97 × 10 | 4.23 × 10 | 3.42 × 10 | 3.62 × 10 |
OMOEA-II | 8.56 × 10 | 3.06 × 10 | 2.71 × 10 | 4.62 × 10 | 1.69 × 10 | 7.34 × 10 | 3.35 × 10 | 1.92 × 10 | 2.32 × 10 | 6.28 × 10 |
Cultural | 1.11 × 10 | 2.15 × 10 | 3.75 × 10 | 5.98 × 10 | 1.23 × 10 | 1.66 × 10 | 1.13 × 10 | 1.18 × 10 | 1.16 × 10 | 8.29 × 10 |
MOQPSO | ||||||||||
MOWOATS | 2.32 × 10 | 2.21 × 10 | 9.77 × 10 | 1.83 × 10 | 7.29 × 10 | 1.04 × 10 | 2.12 × 10 | 3.61 × 10 | 1.47 × 10 | 5.12 × 10 |
F1(ZDT1) | F2(ZDT2) | F3(ZDT3) | F4(ZDT4) | F5(ZDT6) | F6(DTLZ1) | F7(DTLZ2) | F8(DTLZ4) | |
---|---|---|---|---|---|---|---|---|
Sigma MOPSO | 6.07 × 10 | 5.00 × 10 | 5.50 × 10 | 2.64 × 10 | 2.88 × 10 | 2.23 × 10 | 2.84 × 10 | 7.49 × 10 |
Cluster MOPSO | 9.03 × 10 | 9.19 × 10 | 2.92 × 10 | 9.89 | 6.24 × 10 | 1.39 | 1.43 × 10 | 8.93 × 10 |
MOPSO-CD | 1.30 × 10 | 6.60 × 10 | 3.07 × 10 | 1.53 × 10 | 2.93 × 10 | 2.80 | 9.86 × 10 | 6.18 × 10 |
MOQPSO | 2.17 × 10 | 4.25 × 10 | 1.87 × 10 | 8.81 | 3.92 × 10 | 1.38 × 10 | 6.94 × 10 | 1.02 × 10 |
Preference-order | 6.94 × 10 | 4.75 × 10 | 1.90 × 10 | 2.08 | 5.86 × 10 | 1.39 × 10 | 2.29 × 10 | 2.91 × 10 |
based QPSO | ||||||||
Cultural | 6.13 × 10 | 4.87 × 10 | 1.89 × 10 | 5.26 × 10 | 5.38 × 10 | 4.70 × 10 | 6.51 × 10 | 6.70 × 10 |
MOQPSO | ||||||||
MOWOATS | 1.30 × 10 | 7.32 × 10 | 2.87 × 10 | 2.85 × 10 | 8.92 × 10 | 3.76 × 10 | 9.73 × 10 | 1.28 × 10 |
Problem | MOWOATS | AgMOPSO | MMOPSO | MOPSO | EAG-MOEA/D | NSGA-II | |
---|---|---|---|---|---|---|---|
ZDT1 | mean | 1.303 × 10 | 3.701 × 10 | 3.936 × 10 | 1.038 × 10 | 3.757 × 10 | 4.976 × 10 |
std | 7.450 × 10 | 2.83 × 10 | 3.4.56 × 10 | 6.07 × 10 | 1.02 × 10 | 1.73 × 10 | |
ZDT2 | mean | 7.318 × 10 | 3.828 × 10 | 2.414 × 10 | 4.904 × 10 | 2.113 × 10 | 5.102 × 10 |
std | 2.955 × 10 | 3.15 × 10 | 1.11 × 10 | 2.45 × 10 | 8.19 × 10 | 1.79 × 10 | |
ZDT3 | mean | 2.872 × 10 | 4.367 × 10 | 4.413 × 10 | 1.404 × 10 | 3.142 × 10 | 6.408 × 10 |
std | 2.691 × 10 | 5.23 × 10 | 4.28 × 10 | 4.34 × 10 | 3.73 × 10 | 5.41 × 10 | |
ZDT4 | mean | 2.850 × 10 | 7.942 × 10 | 2.342 × 10 | 3.203 | 2.357 × 10 | 7.654 × 10 |
std | 7.729 × 10 | 2.23 × 10 | 4.42 × 10 | 2.06 | 3.18 × 10 | 2.45 × 10 | |
ZDT6 | mean | 8.922 × 10 | 2.997 × 10 | 3.635 × 10 | 1.423 × 10 | 3.132 × 10 | 9.088 × 10 |
std | 3.476 × 10 | 9.51 × 10 | 2.31 × 10 | 8.14 × 10 | 2.13 × 10 | 1.00 × 10 | |
DTLZ1 | mean | 3.760 × 10 | 2.183 × 10 | 2.754 × 10 | 1.515 | 2.582 × 10 | 2.544 × 10 |
std | 2.735 × 10 | 1.229 × 10 | 2.56 × 10 | 2.22 | 3.14 × 10 | 3.02 × 10 | |
DTLZ2 | mean | 9.731 × 10 | 5.133 × 10 | 6.354 × 10 | 6.078 × 10 | 5.93 × 10 | 6.725 × 10 |
std | 7.122 × 10 | 2.50 × 10 | 1.82 × 10 | 1.53 × 10 | 1.73 × 10 | 2.74 × 10 | |
DTLZ3 | mean | 2.242 × 10 | 3.619 × 10 | 1.929 | 4.505 × 10 | 1.505 × 10 | 1.525 × 10 |
std | 6.181 × 10 | 5.79 × 10 | 1.61 | 2.49 × 10 | 1.78 × 10 | 2.52 × 10 | |
DTLZ4 | mean | 1.286 × 10 | 3.304 × 10 | 6.325 × 10 | 6.261 × 10 | 1.872 × 10 | 6.181 × 10 |
std | 1.325 × 10 | 4.63 × 10 | 4.55 × 10 | 3.36 × 10 | 1.47 × 10 | 6.24 × 10 | |
DTLZ5 | mean | 4.365 × 10 | 3.868 × 10 | 3.825 × 10 | 6.072 × 10 | 3.876 × 10 | 5.217 × 10 |
std | 1.507 × 10 | 8.12 × 10 | 9.45 × 10 | 1.12 × 10 | 8.97 × 10 | 2.69 × 10 | |
DTLZ6 | mean | 1.858 × 10 | 3.670 × 10 | 3.756 × 10 | 1.392 × 10 | 3.730 × 10 | 1.733 × 10 |
std | 1.400 × 10 | 1.653 × 10 | 1.57 × 10 | 2.06 × 10 | 1.05 × 10 | 1.38 × 10 | |
DTLZ7 | mean | 2.751 × 10 | 7.712 × 10 | 3.756 × 10 | 8.312 × 10 | 4.100 × 10 | 7.405 × 10 |
std | 6.663 × 10 | 4.40 × 10 | 1.57 × 10 | 5.29 × 10 | 2.49 × 10 | 3.00 × 10 |
PF Shape | Problem | Obj. | MOWOATS | CA-MOEA | MOEA/D | EMyO/C | RVEA* | NSGA-II | NSGA-III |
---|---|---|---|---|---|---|---|---|---|
Irregular | DTLZ4 | 3 | 1.286 × 10 | 5.4805 × 10 | 3.385 × 10 | 5.617 × 10 | 2.942 × 10 | 1.119 × 10 | 1.275 × 10 |
1.325 × 10 | 6.98 × 10 | 3.14 × 10 | 7.04 × 10 | 2.85 × 10 | 1.96 × 10 | 1.78 × 10 | |||
DTLZ5 | 3 | 1.834 × 10 | 4.426 × 10 | 3.369 × 10 | 4.583 × 10 | 6.891 × 10 | 5.754 × 10 | 1.267 × 10 | |
1.360 × 10 | 8.96 × 10 | 8.32 × 10 | 6.03 × 10 | 3.91 × 10 | 2.28 × 10 | 1.58 × 10 | |||
DTLZ6 | 3 | 2.038 × 10 | 4.2269 × 10 | 3.381 × 10 | 4.619 × 10 | 7.146 × 10 | 5.891 × 10 | 1.868 × 10 | |
1.653 × 10 | 4.09 × 10 | 1.74 × 10 | 1.32 × 10 | 5.55 × 10 | 5.11 × 10 | 3.31 × 10 | |||
DTLZ7 | 2 | 2.751 × 10 | 4.7253 × 10 | 1.631 × 10 | 6.312 × 10 | 4.852 × 10 | 5.324 × 10 | 6.903 × 10 | |
1.663 × 10 | 8.78 × 10 | 2.14 × 10 | 3.47 × 10 | 1.35 × 10 | 2.47 × 10 | 1.21 × 10 | |||
DTLZ7 | 3 | 6.314 × 10 | 5.8727 × 10 | 1.416 × 10 | 7.861 × 10 | 8.952 × 10 | 1.263 × 10 | 7.546 × 10 | |
5.678 × 10 | 1.41 × 10 | 9.73 × 10 | 6.44 × 10 | 9.04 × 10 | 1.70 × 10 | 2.34 × 10 | |||
UF6 | 2 | 5.970 × 10 | 1.1464 × 10 | 4.468 × 10 | 1.673 × 10 | 3.052 × 10 | 1.643 × 10 | 1.441 × 10 | |
5.135 × 10 | 9.11 × 10 | 1.51 × 10 | 1.03 × 10 | 1.45 × 10 | 9.00 × 10 | 8.67 × 10 | |||
UF9 | 3 | 2.885 × 10 | 1.143 × 10 | 2.515 × 10 | 9.055 × 10 | 2.272 × 10 | 2.539 × 10 | 2.016 × 10 | |
2.193 × 10 | 0.0427 | 1.36 × 10 | 3.52 × 10 | 8.32 × 10 | 1.18 × 10 | 8.25 × 10 | |||
Regular | DTLZ1 | 2 | 2.065 × 10 | 1.8936 × 10 | 1.811 × 10 | 2.057 × 10 | 1.893 × 10 | 2.231 × 10 | 1.800 × 10 |
5.054 × 10 | 1.80 × 10 | 5.21 × 10 | 7.75 × 10 | 2.76 × 10 | 7.02 × 10 | 2.33 × 10 | |||
DTLZ1 | 3 | 3.760 × 10 | 2.0269 × 10 | 2.060 × 10 | 2.097 × 10 | 2.119 × 10 | 2.728 × 10 | 2.057 × 10 | |
2.735 × 10 | 1.23 × 10 | 5.06 × 10 | 3.44 × 10 | 2.27 × 10 | 1.34 × 10 | 1.67 × 10 | |||
DTLZ2 | 2 | 1.244 × 10 | 4.2168 × 10 | 3.966 × 10 | 4.473 × 10 | 4.133 × 10 | 5.032 × 10 | 3.969 × 10 | |
5.449 × 10 | 6.48 × 10 | 2.95 × 10 | 6.54 × 10 | 3.06 × 10 | 1.84 × 10 | 6.95 × 10 | |||
DTLZ2 | 3 | 9.731 × 10 | 5.3461 × 10 | 5.447 × 10 | 5.640 × 10 | 5.534 × 10 | 6.887 × 10 | 5.448 × 10 | |
7.122 × 10 | 3.21 × 10 | 2.75 × 10 | 5.65 × 10 | 3.65 × 10 | 2.73 × 10 | 1.43 × 10 | |||
DTLZ3 | 2 | 2.406 × 10 | 4.7153 × 10 | 4.391 × 10 | 4.950 × 10 | 4.345 × 10 | 5.193 × 10 | 4.195 × 10 | |
1.010 × 10 | 4.17 × 10 | 4.61 × 10 | 1.23 × 10 | 2.12 × 10 | 2.63 × 10 | 2.74 × 10 | |||
DTLZ3 | 3 | 7.847 × 10 | 5.7776 × 10 | 5.487 × 10 | 5.611 × 10 | 5.548 × 10 | 6.836 × 10 | 5.4833 × 10 | |
2.989 × 10 | 4.00 × 10 | 7.43 × 10 | 7.69 × 10 | 5.92 × 10 | 3.97 × 10 | 2.84 × 10 | |||
DTLZ4 | 2 | 2.374 × 10 | 4.1222 × 10 | 3.362 × 10 | 4.497 × 10 | 7.793 × 10 | 7.879 × 10 | 3.983 × 10 | |
1.531 × 10 | 1.65 × 10 | 3.77 × 10 | 9.98 × 10 | 2.27 × 10 | 2.27 × 10 | 5.48 × 10 | |||
DTLZ5 | 2 | 6.643 × 10 | 4.2387 × 10 | 3.385 × 10 | 4.461 × 10 | 4.132 × 10 | 5.128 × 10 | 3.968 × 10 | |
2.612 × 10 | 3.08 × 10 | 3.14 × 10 | 9.99 × 10 | 5.18 × 10 | 1.77 × 10 | 1.22 × 10 | |||
DTLZ6 | 2 | 2.752 × 10 | 4.2160 × 10 | 3.966 × 10 | 4.434 × 10 | 4.060 × 10 | 5.730 × 10 | 3.966 × 10 | |
7.603 × 10 | 2.17 × 10 | 4.66 × 10 | 8.91 × 10 | 3.57 × 10 | 3.08 × 10 | 4.74 × 10 |
Problem | MOWOATS | R2HMOPSO | R2HMOPSO1 | MOEA/D | NSGA-II | dMOPSO | R2MOPSO | |
---|---|---|---|---|---|---|---|---|
ZDT1 | mean | 1.849 × 10 | 3.904 × 10 | 3.943 × 10 | 7.544 × 10 | 4.929 × 10 | 3.899 × 10 | 3.928 × 10 |
std | 2.648 × 10 | 6.209 × 10 | 8.593 × 10 | 8.473 × 10 | 2.043 × 10 | 3.839 × 10 | 1.091 × 10 | |
ZDT2 | mean | 8.951 × 10 | 3.834 × 10 | 2.260 × 10 | 2.503 × 10 | 4.900 × 10 | 6.442 × 10 | 3.825 × 10 |
std | 1.549 × 10 | 4.907 × 10 | 2.969 × 10 | 1.104 × 10 | 2.074 × 10 | 1.849 × 10 | 3.336 × 10 | |
ZDT3 | mean | 5.586 × 10 | 6.123 × 10 | 8.460 × 10 | 1.212 × 10 | 7.254 × 10 | 1.064 × 10 | 1.017 × 10 |
std | 3.631 × 10 | 1.583 × 10 | 6.058 × 10 | 5.585 × 10 | 7.634 × 10 | 7.098 × 10 | 6.508 × 10 | |
ZDT4 | mean | 9.756 × 10 | 4.518 × 10 | 5.093 × 10 | 2.452 × 10 | 8.420 × 10 | 5.972 | 8.374 × 10 |
std | 4.180 × 10 | 3.733 × 10 | 2.356 × 10 | 1.571 × 10 | 2.731 × 10 | 4.477 | 9.787 × 10 | |
ZDT6 | mean | 9.519 × 10 | 1.888 × 10 | 1.868 × 10 | 1.892 × 10 | 2.696 × 10 | 1.879 × 10 | 1.865 × 10 |
std | 6.917 × 10 | 4.334 × 10 | 2.227 × 10 | 1.179 × 10 | 4.111 × 10 | 8.549 × 10 | 3.484 × 10 | |
DTLZ1 | mean | 3.760 × 10 | 1.994 × 10 | 1.724 | 1.842 × 10 | 1.935 × 10 | 1.529 × 10 | 1.745 × 10 |
std | 2.735 × 10 | 2.080 × 10 | 9.906 × 10 | 4.773 × 10 | 6.011 × 10 | 1.180 × 10 | 3.133 | |
DTLZ2 | mean | 9.731 × 10 | 4.368 × 10 | 5.555 × 10 | 4.597 × 10 | 5.315 × 10 | 4.716 × 10 | 2.250 × 10 |
std | 7.122 × 10 | 1.203 × 10 | 3.302 × 10 | 1.218 × 10 | 2.373 × 10 | 1.489 × 10 | 1.526 × 10 | |
DTLZ4 | mean | 1.286 × 10 | 5.198 × 10 | 2.505 × 10 | 5.160 × 10 | 5.417 × 10 | 1.175 × 10 | 4.343 × 10 |
std | 1.325 × 10 | 2.011 × 10 | 1.262 × 10 | 1.948 × 10 | 1.727 × 10 | 7.627 × 10 | 5.111 × 10 | |
DTLZ7 | mean | 6.314 × 10 | 7.036 × 10 | 6.459 × 10 | 1.271 × 10 | 7.129 × 10 | 1.265 × 10 | 8.729 × 10 |
std | 5.678 × 10 | 8.439 × 10 | 9.521 × 10 | 5.106 × 10 | 5.101 × 10 | 6.728 × 10 | 8.431 × 10 |
© 2019 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).
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AbdelAziz, A.M.; Soliman, T.H.A.; Ghany, K.K.A.; Sewisy, A.A.E.-M. A Pareto-Based Hybrid Whale Optimization Algorithm with Tabu Search for Multi-Objective Optimization. Algorithms 2019, 12, 261. https://doi.org/10.3390/a12120261
AbdelAziz AM, Soliman THA, Ghany KKA, Sewisy AAE-M. A Pareto-Based Hybrid Whale Optimization Algorithm with Tabu Search for Multi-Objective Optimization. Algorithms. 2019; 12(12):261. https://doi.org/10.3390/a12120261
Chicago/Turabian StyleAbdelAziz, Amr Mohamed, Taysir Hassan A. Soliman, Kareem Kamal A. Ghany, and Adel Abu El-Magd Sewisy. 2019. "A Pareto-Based Hybrid Whale Optimization Algorithm with Tabu Search for Multi-Objective Optimization" Algorithms 12, no. 12: 261. https://doi.org/10.3390/a12120261