# A Pareto-Based Hybrid Whale Optimization Algorithm with Tabu Search for Multi-Objective Optimization

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## Abstract

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## 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, $EL$) |

Select $\eta $ solutions randomly from the swarm S. for$m=1,\dots ,\eta $do Select a random solution ${\overrightarrow{X}}_{rand}$ from EL. Update $\alpha $ random features in current whale ${\overrightarrow{X}}_{m}$ with $\alpha $ features selected from ${\overrightarrow{X}}_{rand}$. end forReturn the updated swarm S. |

#### 4.2.2. Diversification by Crossover

Algorithm 2 Diversification_Crossover(S) |

Select $\eta $ solutions randomly from the swarm S. for$m=1,\dots ,\eta $do Select a random solution ${\overrightarrow{X}}_{rand}$ from the current swarm other than current whale ${\overrightarrow{X}}_{m}$. Update $\alpha $ random features in the current whale ${\overrightarrow{X}}_{m}$ with the $\alpha $ random features selected from ${\overrightarrow{X}}_{rand}$. end forReturn the updated swarm S. |

#### 4.2.3. MOWOATS Algorithm

Algorithm 3 Pseudocode for MOWOATS algorithm |

Initialization.Set of particles in swarm $Np$, number of features of each object d, empty $EL$, set $Max\_NonImprove$ to be maximum number of iterations without improvement, and initialize the whale algorithm parameters. for$i=1,\dots ,Np$do 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 $EL$ according to Pareto dominance criterion. end forMain Loop.for$t=1,\dots ,MaxIt$dofor $i=1,\dots ,Np$ do Update WOA parameters $a,A,C,l,p$. if $(p>0.2)$ thenif $\left(\right|A|<1)$ then Select a random whale ${\overrightarrow{X}}^{\ast}\left(t\right)$ from EL. Update position of the current whale ${\overrightarrow{X}}_{i}\left(t\right)$ applying next equation.
$$\overrightarrow{D}=|C\xb7{\overrightarrow{X}}^{\ast}\left(t\right)-\overrightarrow{X}\left(t\right)|.$$
$$\overrightarrow{X}(t+1)=\overrightarrow{X}\left(t\right)+(A/4)\xb7\overrightarrow{D}.$$
else if $\left(\right|A|\ge 1)$ then Select a random whale ${\overrightarrow{X}}_{rand}\left(t\right)$ from current swarm. Update position of current whale ${\overrightarrow{X}}_{i}\left(t\right)$ applying next equation.
$$\overrightarrow{D}=|C\xb7{\overrightarrow{X}}_{rand}\left(t\right)-\overrightarrow{X}\left(t\right)|.$$
$$\overrightarrow{X}(t+1)=\overrightarrow{X}\left(t\right)+(A/4)\xb7\overrightarrow{D}$$
end ifelse if $(p\le 0.2)$ then Select a random whale ${\overrightarrow{X}}^{\ast}\left(t\right)$ from EL. Update position of current whale ${\overrightarrow{X}}_{i}\left(t\right)$ applying next equation
$$\overrightarrow{D}=|{\overrightarrow{X}}^{\ast}\left(t\right)-\overrightarrow{X}\left(t\right)|,$$
$$\overrightarrow{X}(t+1)=\overrightarrow{D}\xb7{e}^{bl}\xb7cos\left(2\pi l\right)+\overrightarrow{{X}^{\ast}}\left(t\right)$$
end if Compute the objective value of current whale ${\overrightarrow{X}}_{i}\left(t\right)$ according to the objective functions that describe the problem. Update solutions in $EL$ according to Pareto dominance criterion. end forif (number of iterations without improvement ≥$Max\_NonImprove$) then Set $\tau $ to a random value. if $(\tau <0.5)$ then Apply intensification procedure Intensification_Crossover(S,EL) (Algorithm 1). else Apply diversification procedure Diversification_Crossover(S) (Algorithm 2). end ifend ifend forReturn 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 |
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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${}^{-\mathbf{5}}$ | 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 | $1.0431\times 4$ | 1.22008 | NA |

**Table 3.**Mean of IGD values for MOWOATS, Cultural MOQPSO, and some population-based algorithms over CEC2009 test problems.

UF1 | UF2 | UF3 | UF4 | UF5 | UF6 | UF7 | UF8 | UF9 | UF10 | |
---|---|---|---|---|---|---|---|---|---|---|

GDE3 | 5.34 × 10${}^{-2}$ | 1.20 × 10${}^{-2}$ | 1.06 × 10${}^{-1}$ | 2.65 × 10${}^{-2}$ | 3.93 × 10${}^{-2}$ | 2.51 × 10${}^{-1}$ | 2.52 × 10${}^{-2}$ | 2.49 × 10${}^{-1}$ | 8.25 × 10${}^{-2}$ | 4.33 × 10${}^{-1}$ |

MOEADGM | 6.20 × 10${}^{-3}$ | 6.40 × 10${}^{-3}$ | 4.90 × 10${}^{-2}$ | 4.76 × 10${}^{-2}$ | 1.79 | 5.56 × 10${}^{-1}$ | 7.60 × 10${}^{-3}$ | 2.45 × 10${}^{-1}$ | 1.88 × 10${}^{-1}$ | 5.65 × 10${}^{-1}$ |

MTS | 6.46 × 10${}^{-3}$ | 6.15 × 10${}^{-3}$ | 5.31 × 10${}^{-2}$ | 2.36 × 10${}^{-2}$ | 1.49 × 10${}^{-2}$ | 5.92 × 10${}^{-2}$ | 4.08 × 10${}^{-2}$ | 1.13 × 10${}^{-1}$ | 1.14 × 10${}^{-1}$ | 5.53 × 10${}^{-1}$ |

DMOEA-DD | 1.04 × 10${}^{-2}$ | 6.79 × 10${}^{-3}$ | 3.34 × 10${}^{-2}$ | 4.27 × 10${}^{-2}$ | 3.15 × 10${}^{-1}$ | 6.67 × 10${}^{-2}$ | 1.03 × 10${}^{-2}$ | 6.84 × 10${}^{-2}$ | 4.90 × 10${}^{-2}$ | 3.22 × 10${}^{-1}$ |

NSGA-II-LS | 1.15 × 10${}^{-2}$ | 1.24 × 10${}^{-2}$ | 1.06 × 10${}^{-1}$ | 5.84 × 10${}^{-2}$ | 5.66 × 10${}^{-1}$ | 3.10 × 10${}^{-1}$ | 2.13 × 10${}^{-2}$ | 8.63 × 10${}^{-2}$ | 7.19 × 10${}^{-2}$ | 8.45 × 10${}^{-1}$ |

OWMOsaDE | 1.22 × 10${}^{-2}$ | 8.10 × 10${}^{-3}$ | 1.03 × 10${}^{-1}$ | 5.13 × 10${}^{-2}$ | 4.30 × 10${}^{-1}$ | 1.92 × 10${}^{-1}$ | 5.85 × 10${}^{-2}$ | 9.45 × 10${}^{-2}$ | 9.83 × 10${}^{-2}$ | 7.43 × 10${}^{-1}$ |

Clustering MOEA | 2.99 × 10${}^{-2}$ | 2.28 × 10${}^{-2}$ | 5.49 × 10${}^{-2}$ | 5.85 × 10${}^{-2}$ | 2.47 × 10${}^{-1}$ | 8.71 × 10${}^{-2}$ | 2.23 × 10${}^{-1}$ | 2.38 × 10${}^{-1}$ | 2.93 × 10${}^{-1}$ | 4.11 × 10${}^{-1}$ |

AMGA | 3.59 × 10${}^{-2}$ | 1.62 × 10${}^{-2}$ | 7.00 × 10${}^{-2}$ | 4.06 × 10${}^{-2}$ | 9.41 × 10${}^{-2}$ | 1.29 × 10${}^{-1}$ | 5.71 × 10${}^{-2}$ | 1.71 × 10${}^{-1}$ | 1.89 × 10${}^{-1}$ | 3.24 × 10${}^{-1}$ |

MOEP | 5.96 × 10${}^{-2}$ | 1.89 × 10${}^{-2}$ | 9.90 × 10${}^{-2}$ | 4.27 × 10${}^{-2}$ | 2.25 × 10${}^{-1}$ | 1.03 × 10${}^{-1}$ | 1.97 × 10${}^{-2}$ | 4.23 × 10${}^{-1}$ | 3.42 × 10${}^{-1}$ | 3.62 × 10${}^{-1}$ |

OMOEA-II | 8.56 × 10${}^{-2}$ | 3.06 × 10${}^{-2}$ | 2.71 × 10${}^{-1}$ | 4.62 × 10${}^{-2}$ | 1.69 × 10${}^{-1}$ | 7.34 × 10${}^{-2}$ | 3.35 × 10${}^{-2}$ | 1.92 × 10${}^{-1}$ | 2.32 × 10${}^{-1}$ | 6.28 × 10${}^{-1}$ |

Cultural | 1.11 × 10${}^{-2}$ | 2.15 × 10${}^{-2}$ | 3.75 × 10${}^{-2}$ | 5.98 × 10${}^{-2}$ | 1.23 × 10${}^{-1}$ | 1.66 × 10${}^{-1}$ | 1.13 × 10${}^{-2}$ | 1.18 × 10${}^{-1}$ | 1.16 × 10${}^{-1}$ | 8.29 × 10${}^{-1}$ |

MOQPSO | ||||||||||

MOWOATS | 2.32 × 10${}^{-\mathbf{3}}$ | 2.21 × 10${}^{-\mathbf{3}}$ | 9.77 × 10${}^{-\mathbf{3}}$ | 1.83 × 10${}^{-\mathbf{3}}$ | 7.29 × 10${}^{-2}$ | 1.04 × 10${}^{-\mathbf{2}}$ | 2.12 × 10${}^{-\mathbf{3}}$ | 3.61 × 10${}^{-\mathbf{3}}$ | 1.47 × 10${}^{-\mathbf{3}}$ | 5.12 × 10${}^{-\mathbf{3}}$ |

**Table 4.**Mean of IGD values for MOWOATS, Cultural MOQPSO, and some population-based algorithms over (ZDT, DTLZ) test problems.

F1(ZDT1) | F2(ZDT2) | F3(ZDT3) | F4(ZDT4) | F5(ZDT6) | F6(DTLZ1) | F7(DTLZ2) | F8(DTLZ4) | |
---|---|---|---|---|---|---|---|---|

Sigma MOPSO | 6.07 × 10${}^{-1}$ | 5.00 × 10${}^{-1}$ | 5.50 × 10${}^{-1}$ | 2.64 × 10${}^{1}$ | 2.88 × 10${}^{-3}$ | 2.23 × 10${}^{1}$ | 2.84 × 10${}^{-1}$ | 7.49 × 10${}^{-1}$ |

Cluster MOPSO | 9.03 × 10${}^{-3}$ | 9.19 × 10${}^{-2}$ | 2.92 × 10${}^{-1}$ | 9.89 | 6.24 × 10${}^{-3}$ | 1.39 | 1.43 × 10${}^{-1}$ | 8.93 × 10${}^{-1}$ |

MOPSO-CD | 1.30 × 10${}^{-1}$ | 6.60 × 10${}^{-3}$ | 3.07 × 10${}^{-1}$ | 1.53 × 10${}^{1}$ | 2.93 × 10${}^{-3}$ | 2.80 | 9.86 × 10${}^{-2}$ | 6.18 × 10${}^{-1}$ |

MOQPSO | 2.17 × 10${}^{-2}$ | 4.25 × 10${}^{-2}$ | 1.87 × 10${}^{-1}$ | 8.81 | 3.92 × 10${}^{-1}$ | 1.38 × 10${}^{1}$ | 6.94 × 10${}^{-2}$ | 1.02 × 10${}^{-1}$ |

Preference-order | 6.94 × 10${}^{-3}$ | 4.75 × 10${}^{-1}$ | 1.90 × 10${}^{-1}$ | 2.08 | 5.86 × 10${}^{-3}$ | 1.39 × 10${}^{1}$ | 2.29 × 10${}^{-1}$ | 2.91 × 10${}^{-1}$ |

based QPSO | ||||||||

Cultural | 6.13 × 10${}^{-3}$ | 4.87 × 10${}^{-3}$ | 1.89 × 10${}^{-1}$ | 5.26 × 10${}^{-3}$ | 5.38 × 10${}^{-3}$ | 4.70 × 10${}^{-2}$ | 6.51 × 10${}^{-2}$ | 6.70 × 10${}^{-2}$ |

MOQPSO | ||||||||

MOWOATS | 1.30 × 10${}^{-\mathbf{3}}$ | 7.32 × 10${}^{-\mathbf{4}}$ | 2.87 × 10${}^{-\mathbf{3}}$ | 2.85 × 10${}^{-\mathbf{4}}$ | 8.92 × 10${}^{-\mathbf{4}}$ | 3.76 × 10${}^{-\mathbf{4}}$ | 9.73 × 10${}^{-\mathbf{4}}$ | 1.28 × 10${}^{-\mathbf{3}}$ |

**Table 5.**Mean of IGD values for MOWOATS, AgMOPSO, and some population-based algorithms over (ZDT, DTLZ) test problems, averaged for 30 different runs.

Problem | MOWOATS | AgMOPSO | MMOPSO | ${\mathit{D}}^{2}$MOPSO | EAG-MOEA/D | NSGA-II | |
---|---|---|---|---|---|---|---|

ZDT1 | mean | 1.303 × 10${}^{-\mathbf{3}}$ | 3.701 × 10${}^{-3}$ | 3.936 × 10${}^{-3}$ | 1.038 × 10${}^{-2}$ | 3.757 × 10${}^{-3}$ | 4.976 × 10${}^{-3}$ |

std | 7.450 × 10${}^{-4}$ | 2.83 × 10${}^{-\mathbf{5}}$ | 3.4.56 × 10${}^{-5}$ | 6.07 × 10${}^{-3}$ | 1.02 × 10${}^{-4}$ | 1.73 × 10${}^{-4}$ | |

ZDT2 | mean | 7.318 × 10${}^{-\mathbf{4}}$ | 3.828 × 10${}^{-3}$ | 2.414 × 10${}^{-2}$ | 4.904 × 10${}^{-1}$ | 2.113 × 10${}^{-2}$ | 5.102 × 10${}^{-3}$ |

std | 2.955 × 10${}^{-4}$ | 3.15 × 10${}^{-\mathbf{5}}$ | 1.11 × 10${}^{-1}$ | 2.45 × 10${}^{-1}$ | 8.19 × 10${}^{-2}$ | 1.79 × 10${}^{-4}$ | |

ZDT3 | mean | 2.872 × 10${}^{-\mathbf{3}}$ | 4.367 × 10${}^{-3}$ | 4.413 × 10${}^{-3}$ | 1.404 × 10${}^{-2}$ | 3.142 × 10${}^{-2}$ | 6.408 × 10${}^{-3}$ |

std | 2.691 × 10${}^{-3}$ | 5.23 × 10${}^{-5}$ | 4.28 × 10${}^{-\mathbf{5}}$ | 4.34 × 10${}^{-3}$ | 3.73 × 10${}^{-2}$ | 5.41 × 10${}^{-3}$ | |

ZDT4 | mean | 2.850 × 10${}^{-\mathbf{4}}$ | 7.942 × 10${}^{-3}$ | 2.342 × 10${}^{-2}$ | 3.203 | 2.357 × 10${}^{-2}$ | 7.654 × 10${}^{-3}$ |

std | 7.729 × 10${}^{-\mathbf{5}}$ | 2.23 × 10${}^{-2}$ | 4.42 × 10${}^{-2}$ | 2.06 | 3.18 × 10${}^{-2}$ | 2.45 × 10${}^{-3}$ | |

ZDT6 | mean | 8.922 × 10${}^{-\mathbf{4}}$ | 2.997 × 10${}^{-3}$ | 3.635 × 10${}^{-3}$ | 1.423 × 10${}^{-2}$ | 3.132 × 10${}^{-3}$ | 9.088 × 10${}^{-3}$ |

std | 3.476 × 10${}^{-4}$ | 9.51 × 10${}^{-\mathbf{5}}$ | 2.31 × 10${}^{-4}$ | 8.14 × 10${}^{-3}$ | 2.13 × 10${}^{-4}$ | 1.00 × 10${}^{-3}$ | |

DTLZ1 | mean | 3.760 × 10${}^{-\mathbf{4}}$ | 2.183 × 10${}^{-2}$ | 2.754 × 10${}^{-2}$ | 1.515 | 2.582 × 10${}^{-2}$ | 2.544 × 10${}^{-2}$ |

std | 2.735 × 10${}^{-\mathbf{5}}$ | 1.229 × 10${}^{-4}$ | 2.56 × 10${}^{-2}$ | 2.22 | 3.14 × 10${}^{-3}$ | 3.02 × 10${}^{-3}$ | |

DTLZ2 | mean | 9.731 × 10${}^{-\mathbf{4}}$ | 5.133 × 10${}^{-2}$ | 6.354 × 10${}^{-2}$ | 6.078 × 10${}^{-2}$ | 5.93 × 10${}^{-2}$ | 6.725 × 10${}^{-2}$ |

std | 7.122 × 10${}^{-\mathbf{5}}$ | 2.50 × 10${}^{-4}$ | 1.82 × 10${}^{-3}$ | 1.53 × 10${}^{-3}$ | 1.73 × 10${}^{-3}$ | 2.74 × 10${}^{-3}$ | |

DTLZ3 | mean | 2.242 × 10${}^{-\mathbf{3}}$ | 3.619 × 10${}^{-1}$ | 1.929 | 4.505 × 10${}^{1}$ | 1.505 × 10${}^{-1}$ | 1.525 × 10${}^{-1}$ |

std | 6.181 × 10${}^{-\mathbf{4}}$ | 5.79 × 10${}^{-1}$ | 1.61 | 2.49 × 10${}^{1}$ | 1.78 × 10${}^{-1}$ | 2.52 × 10${}^{-1}$ | |

DTLZ4 | mean | 1.286 × 10${}^{-\mathbf{3}}$ | 3.304 × 10${}^{-2}$ | 6.325 × 10${}^{-2}$ | 6.261 × 10${}^{-2}$ | 1.872 × 10${}^{-1}$ | 6.181 × 10${}^{-2}$ |

std | 1.325 × 10${}^{-\mathbf{4}}$ | 4.63 × 10${}^{-4}$ | 4.55 × 10${}^{-3}$ | 3.36 × 10${}^{-3}$ | 1.47 × 10${}^{-1}$ | 6.24 × 10${}^{-3}$ | |

DTLZ5 | mean | 4.365 × 10${}^{-\mathbf{5}}$ | 3.868 × 10${}^{-3}$ | 3.825 × 10${}^{-3}$ | 6.072 × 10${}^{-3}$ | 3.876 × 10${}^{-3}$ | 5.217 × 10${}^{-3}$ |

std | 1.507 × 10${}^{-\mathbf{5}}$ | 8.12 × 10${}^{-5}$ | 9.45 × 10${}^{-5}$ | 1.12 × 10${}^{-3}$ | 8.97 × 10${}^{-5}$ | 2.69 × 10${}^{-4}$ | |

DTLZ6 | mean | 1.858 × 10${}^{-\mathbf{4}}$ | 3.670 × 10${}^{-3}$ | 3.756 × 10${}^{-3}$ | 1.392 × 10${}^{-2}$ | 3.730 × 10${}^{-3}$ | 1.733 × 10${}^{-2}$ |

std | 1.400 × 10${}^{-4}$ | 1.653 × 10${}^{-4}$ | 1.57 × 10${}^{-4}$ | 2.06 × 10${}^{-3}$ | 1.05 × 10${}^{-\mathbf{4}}$ | 1.38 × 10${}^{-2}$ | |

DTLZ7 | mean | 2.751 × 10${}^{-\mathbf{3}}$ | 7.712 × 10${}^{-2}$ | 3.756 × 10${}^{-3}$ | 8.312 × 10${}^{-2}$ | 4.100 × 10${}^{-1}$ | 7.405 × 10${}^{-2}$ |

std | 6.663 × 10${}^{-4}$ | 4.40 × 10${}^{-3}$ | 1.57 × 10${}^{-\mathbf{4}}$ | 5.29 × 10${}^{-3}$ | 2.49 × 10${}^{-1}$ | 3.00 × 10${}^{-3}$ |

**Table 6.**A comparison among MOWOATS, CA-MOEA and some multi-objective evolutionary computing methods over ZDT and DTLZ test functions according to IGD criterion.

PF Shape | Problem | Obj. | MOWOATS | CA-MOEA | MOEA/D | EMyO/C | RVEA* | NSGA-II | NSGA-III |
---|---|---|---|---|---|---|---|---|---|

Irregular | DTLZ4 | 3 | 1.286 × 10${}^{-3}$ | 5.4805 × 10${}^{-3}$ | 3.385 × 10${}^{-1}$ | 5.617 × 10${}^{-2}$ | 2.942 × 10${}^{-1}$ | 1.119 × 10${}^{-1}$ | 1.275 × 10${}^{-1}$ |

1.325 × 10${}^{-4}$ | 6.98 × 10${}^{-4}$ | 3.14 × 10${}^{-1}$ | 7.04 × 10${}^{-4}$ | 2.85 × 10${}^{-1}$ | 1.96 × 10${}^{-1}$ | 1.78 × 10${}^{-1}$ | |||

DTLZ5 | 3 | 1.834 × 10${}^{-4}$ | 4.426 × 10${}^{-3}$ | 3.369 × 10${}^{-2}$ | 4.583 × 10${}^{-3}$ | 6.891 × 10${}^{-3}$ | 5.754 × 10${}^{-3}$ | 1.267 × 10${}^{-2}$ | |

1.360 × 10${}^{-4}$ | 8.96 × 10${}^{-5}$ | 8.32 × 10${}^{-5}$ | 6.03 × 10${}^{-5}$ | 3.91 × 10${}^{-4}$ | 2.28 × 10${}^{-4}$ | 1.58 × 10${}^{-3}$ | |||

DTLZ6 | 3 | 2.038 × 10${}^{-4}$ | 4.2269 × 10${}^{-3}$ | 3.381 × 10${}^{-2}$ | 4.619 × 10${}^{-3}$ | 7.146 × 10${}^{-3}$ | 5.891 × 10${}^{-3}$ | 1.868 × 10${}^{-2}$ | |

1.653 × 10${}^{-4}$ | 4.09 × 10${}^{-5}$ | 1.74 × 10${}^{-4}$ | 1.32 × 10${}^{-4}$ | 5.55 × 10${}^{-4}$ | 5.11 × 10${}^{-4}$ | 3.31 × 10${}^{-3}$ | |||

DTLZ7 | 2 | 2.751 × 10${}^{-3}$ | 4.7253 × 10${}^{-3}$ | 1.631 × 10${}^{-1}$ | 6.312 × 10${}^{-3}$ | 4.852 × 10${}^{-2}$ | 5.324 × 10${}^{-3}$ | 6.903 × 10${}^{-3}$ | |

1.663 × 10${}^{-3}$ | 8.78 × 10${}^{-5}$ | 2.14 × 10${}^{-1}$ | 3.47 × 10${}^{-4}$ | 1.35 × 10${}^{-1}$ | 2.47 × 10${}^{-4}$ | 1.21 × 10${}^{-4}$ | |||

DTLZ7 | 3 | 6.314 × 10${}^{-2}$ | 5.8727 × 10${}^{-2}$ | 1.416 × 10${}^{-1}$ | 7.861 × 10${}^{-2}$ | 8.952 × 10${}^{-2}$ | 1.263 × 10${}^{-1}$ | 7.546 × 10${}^{-2}$ | |

5.678 × 10${}^{-3}$ | 1.41 × 10${}^{-3}$ | 9.73 × 10${}^{-4}$ | 6.44 × 10${}^{-2}$ | 9.04 × 10${}^{-2}$ | 1.70 × 10${}^{-1}$ | 2.34 × 10${}^{-3}$ | |||

UF6 | 2 | 5.970 × 10${}^{-2}$ | 1.1464 × 10${}^{-1}$ | 4.468 × 10${}^{-1}$ | 1.673 × 10${}^{-1}$ | 3.052 × 10${}^{-1}$ | 1.643 × 10${}^{-1}$ | 1.441 × 10${}^{-1}$ | |

5.135 × 10${}^{-2}$ | 9.11 × 10${}^{-2}$ | 1.51 × 10${}^{-1}$ | 1.03 × 10${}^{-1}$ | 1.45 × 10${}^{-1}$ | 9.00 × 10${}^{-2}$ | 8.67 × 10${}^{-2}$ | |||

UF9 | 3 | 2.885 × 10${}^{-2}$ | 1.143 × 10${}^{-1}$ | 2.515 × 10${}^{-1}$ | 9.055 × 10${}^{-2}$ | 2.272 × 10${}^{-1}$ | 2.539 × 10${}^{-1}$ | 2.016 × 10${}^{-1}$ | |

2.193 × 10${}^{-2}$ | 0.0427 | 1.36 × 10${}^{-2}$ | 3.52 × 10${}^{-2}$ | 8.32 × 10${}^{-2}$ | 1.18 × 10${}^{-1}$ | 8.25 × 10${}^{-2}$ | |||

Regular | DTLZ1 | 2 | 2.065 × 10${}^{-5}$ | 1.8936 × 10${}^{-3}$ | 1.811 × 10${}^{-3}$ | 2.057 × 10${}^{-3}$ | 1.893 × 10${}^{-3}$ | 2.231 × 10${}^{-3}$ | 1.800 × 10${}^{-3}$ |

5.054 × 10${}^{-6}$ | 1.80 × 10${}^{-5}$ | 5.21 × 10${}^{-5}$ | 7.75 × 10${}^{-5}$ | 2.76 × 10${}^{-5}$ | 7.02 × 10${}^{-5}$ | 2.33 × 10${}^{-5}$ | |||

DTLZ1 | 3 | 3.760 × 10${}^{-4}$ | 2.0269 × 10${}^{-2}$ | 2.060 × 10${}^{-2}$ | 2.097 × 10${}^{-2}$ | 2.119 × 10${}^{-2}$ | 2.728 × 10${}^{-2}$ | 2.057 × 10${}^{-2}$ | |

2.735 × 10${}^{-5}$ | 1.23 × 10${}^{-4}$ | 5.06 × 10${}^{-5}$ | 3.44 × 10${}^{-4}$ | 2.27 × 10${}^{-4}$ | 1.34 × 10${}^{-3}$ | 1.67 × 10${}^{-5}$ | |||

DTLZ2 | 2 | 1.244 × 10${}^{-4}$ | 4.2168 × 10${}^{-3}$ | 3.966 × 10${}^{-3}$ | 4.473 × 10${}^{-3}$ | 4.133 × 10${}^{-3}$ | 5.032 × 10${}^{-3}$ | 3.969 × 10${}^{-3}$ | |

5.449 × 10${}^{-5}$ | 6.48 × 10${}^{-5}$ | 2.95 × 10${}^{-7}$ | 6.54 × 10${}^{-5}$ | 3.06 × 10${}^{-5}$ | 1.84 × 10${}^{-4}$ | 6.95 × 10${}^{-6}$ | |||

DTLZ2 | 3 | 9.731 × 10${}^{-4}$ | 5.3461 × 10${}^{-2}$ | 5.447 × 10${}^{-2}$ | 5.640 × 10${}^{-2}$ | 5.534 × 10${}^{-2}$ | 6.887 × 10${}^{-2}$ | 5.448 × 10${}^{-2}$ | |

7.122 × 10${}^{-5}$ | 3.21 × 10${}^{-4}$ | 2.75 × 10${}^{-6}$ | 5.65 × 10${}^{-4}$ | 3.65 × 10${}^{-4}$ | 2.73 × 10${}^{-3}$ | 1.43 × 10${}^{-5}$ | |||

DTLZ3 | 2 | 2.406 × 10${}^{-4}$ | 4.7153 × 10${}^{-3}$ | 4.391 × 10${}^{-3}$ | 4.950 × 10${}^{-3}$ | 4.345 × 10${}^{-3}$ | 5.193 × 10${}^{-3}$ | 4.195 × 10${}^{-3}$ | |

1.010 × 10${}^{-4}$ | 4.17 × 10${}^{-4}$ | 4.61 × 10${}^{-4}$ | 1.23 × 10${}^{-3}$ | 2.12 × 10${}^{-4}$ | 2.63 × 10${}^{-4}$ | 2.74 × 10${}^{-4}$ | |||

DTLZ3 | 3 | 7.847 × 10${}^{-3}$ | 5.7776 × 10${}^{-2}$ | 5.487 × 10${}^{-2}$ | 5.611 × 10${}^{-2}$ | 5.548 × 10${}^{-2}$ | 6.836 × 10${}^{-2}$ | 5.4833 × 10${}^{-2}$ | |

2.989 × 10${}^{-3}$ | 4.00 × 10${}^{-3}$ | 7.43 × 10${}^{-4}$ | 7.69 × 10${}^{-4}$ | 5.92 × 10${}^{-4}$ | 3.97 × 10${}^{-3}$ | 2.84 × 10${}^{-4}$ | |||

DTLZ4 | 2 | 2.374 × 10${}^{-4}$ | 4.1222 × 10${}^{-3}$ | 3.362 × 10${}^{-1}$ | 4.497 × 10${}^{-3}$ | 7.793 × 10${}^{-2}$ | 7.879 × 10${}^{-2}$ | 3.983 × 10${}^{-3}$ | |

1.531 × 10${}^{-4}$ | 1.65 × 10${}^{-1}$ | 3.77 × 10${}^{-1}$ | 9.98 × 10${}^{-5}$ | 2.27 × 10${}^{-1}$ | 2.27 × 10${}^{-1}$ | 5.48 × 10${}^{-5}$ | |||

DTLZ5 | 2 | 6.643 × 10${}^{-4}$ | 4.2387 × 10${}^{-3}$ | 3.385 × 10${}^{-3}$ | 4.461 × 10${}^{-3}$ | 4.132 × 10${}^{-3}$ | 5.128 × 10${}^{-3}$ | 3.968 × 10${}^{-3}$ | |

2.612 × 10${}^{-4}$ | 3.08 × 10${}^{-3}$ | 3.14 × 10${}^{-1}$ | 9.99 × 10${}^{-5}$ | 5.18 × 10${}^{-5}$ | 1.77 × 10${}^{-4}$ | 1.22 × 10${}^{-6}$ | |||

DTLZ6 | 2 | 2.752 × 10${}^{-4}$ | 4.2160 × 10${}^{-3}$ | 3.966 × 10${}^{-3}$ | 4.434 × 10${}^{-3}$ | 4.060 × 10${}^{-3}$ | 5.730 × 10${}^{-3}$ | 3.966 × 10${}^{-3}$ | |

7.603 × 10${}^{-5}$ | 2.17 × 10${}^{-5}$ | 4.66 × 10${}^{-7}$ | 8.91 × 10${}^{-5}$ | 3.57 × 10${}^{-5}$ | 3.08 × 10${}^{-4}$ | 4.74 × 10${}^{-4}$ |

Problem | MOWOATS | R2HMOPSO | R2HMOPSO1 | MOEA/D | NSGA-II | dMOPSO | R2MOPSO | |
---|---|---|---|---|---|---|---|---|

ZDT1 | mean | 1.849 × 10${}^{-3}$ | 3.904 × 10${}^{-3}$ | 3.943 × 10${}^{-3}$ | 7.544 × 10${}^{-3}$ | 4.929 × 10${}^{-3}$ | 3.899 × 10${}^{-3}$ | 3.928 × 10${}^{-3}$ |

std | 2.648 × 10${}^{-3}$ | 6.209 × 10${}^{-5}$ | 8.593 × 10${}^{-5}$ | 8.473 × 10${}^{-4}$ | 2.043 × 10${}^{-4}$ | 3.839 × 10${}^{-5}$ | 1.091 × 10${}^{-4}$ | |

ZDT2 | mean | 8.951 × 10${}^{-4}$ | 3.834 × 10${}^{-3}$ | 2.260 × 10${}^{-1}$ | 2.503 × 10${}^{-2}$ | 4.900 × 10${}^{-3}$ | 6.442 × 10${}^{-2}$ | 3.825 × 10${}^{-3}$ |

std | 1.549 × 10${}^{-3}$ | 4.907 × 10${}^{-6}$ | 2.969 × 10${}^{-1}$ | 1.104 × 10${}^{-1}$ | 2.074 × 10${}^{-4}$ | 1.849 × 10${}^{-1}$ | 3.336 × 10${}^{-5}$ | |

ZDT3 | mean | 5.586 × 10${}^{-3}$ | 6.123 × 10${}^{-2}$ | 8.460 × 10${}^{-3}$ | 1.212 × 10${}^{-2}$ | 7.254 × 10${}^{-3}$ | 1.064 × 10${}^{-2}$ | 1.017 × 10${}^{-2}$ |

std | 3.631 × 10${}^{-3}$ | 1.583 × 10${}^{-1}$ | 6.058 × 10${}^{-4}$ | 5.585 × 10${}^{-3}$ | 7.634 × 10${}^{-3}$ | 7.098 × 10${}^{-4}$ | 6.508 × 10${}^{-4}$ | |

ZDT4 | mean | 9.756 × 10${}^{-5}$ | 4.518 × 10${}^{-3}$ | 5.093 × 10${}^{-1}$ | 2.452 × 10${}^{-1}$ | 8.420 × 10${}^{-3}$ | 5.972 | 8.374 × 10${}^{-2}$ |

std | 4.180 × 10${}^{-5}$ | 3.733 × 10${}^{-3}$ | 2.356 × 10${}^{-1}$ | 1.571 × 10${}^{-1}$ | 2.731 × 10${}^{-3}$ | 4.477 | 9.787 × 10${}^{-2}$ | |

ZDT6 | mean | 9.519 × 10${}^{-4}$ | 1.888 × 10${}^{-3}$ | 1.868 × 10${}^{-3}$ | 1.892 × 10${}^{-3}$ | 2.696 × 10${}^{-3}$ | 1.879 × 10${}^{-3}$ | 1.865 × 10${}^{-3}$ |

std | 6.917 × 10${}^{-4}$ | 4.334 × 10${}^{-5}$ | 2.227 × 10${}^{-5}$ | 1.179 × 10${}^{-5}$ | 4.111 × 10${}^{-5}$ | 8.549 × 10${}^{-6}$ | 3.484 × 10${}^{-5}$ | |

DTLZ1 | mean | 3.760 × 10${}^{-4}$ | 1.994 × 10${}^{-2}$ | 1.724 | 1.842 × 10${}^{-2}$ | 1.935 × 10${}^{-2}$ | 1.529 × 10${}^{1}$ | 1.745 × 10${}^{1}$ |

std | 2.735 × 10${}^{-5}$ | 2.080 × 10${}^{-3}$ | 9.906 × 10${}^{-1}$ | 4.773 × 10${}^{-4}$ | 6.011 × 10${}^{-4}$ | 1.180 × 10${}^{1}$ | 3.133 | |

DTLZ2 | mean | 9.731 × 10${}^{-4}$ | 4.368 × 10${}^{-2}$ | 5.555 × 10${}^{-2}$ | 4.597 × 10${}^{-2}$ | 5.315 × 10${}^{-2}$ | 4.716 × 10${}^{-2}$ | 2.250 × 10${}^{-1}$ |

std | 7.122 × 10${}^{-5}$ | 1.203 × 10${}^{-3}$ | 3.302 × 10${}^{-3}$ | 1.218 × 10${}^{-3}$ | 2.373 × 10${}^{-3}$ | 1.489 × 10${}^{-3}$ | 1.526 × 10${}^{-2}$ | |

DTLZ4 | mean | 1.286 × 10${}^{-3}$ | 5.198 × 10${}^{-2}$ | 2.505 × 10${}^{-1}$ | 5.160 × 10${}^{-2}$ | 5.417 × 10${}^{-2}$ | 1.175 × 10${}^{-1}$ | 4.343 × 10${}^{-1}$ |

std | 1.325 × 10${}^{-4}$ | 2.011 × 10${}^{-3}$ | 1.262 × 10${}^{-1}$ | 1.948 × 10${}^{-3}$ | 1.727 × 10${}^{-3}$ | 7.627 × 10${}^{-2}$ | 5.111 × 10${}^{-2}$ | |

DTLZ7 | mean | 6.314 × 10${}^{-2}$ | 7.036 × 10${}^{-2}$ | 6.459 × 10${}^{-2}$ | 1.271 × 10${}^{-1}$ | 7.129 × 10${}^{-2}$ | 1.265 × 10${}^{-1}$ | 8.729 × 10${}^{-2}$ |

std | 5.678 × 10${}^{-3}$ | 8.439 × 10${}^{-3}$ | 9.521 × 10${}^{-3}$ | 5.106 × 10${}^{-3}$ | 5.101 × 10${}^{-2}$ | 6.728 × 10${}^{-3}$ | 8.431 × 10${}^{-3}$ |

© 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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**MDPI and ACS Style**

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

**AMA Style**

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 Style**

AbdelAziz, 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