# Evolutionary Computation for Sparse Synthesis Optimization of CCAAs: An Enhanced Whale Optimization Algorithm Method

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## Abstract

**:**

## 1. Introduction

- We formulate a novel optimization problem to turn off a specific number of antennas while reducing the sidelobes of CCAAs. The problem is challenging because it involves both binary and continuous decision variables and is an NP-hard problem.
- We propose an enhanced whale optimization algorithm (EWOA) to solve our formulated problem. EWOA introduces chaos theory and thereby proposes a novel hybrid solution initialization method. Moreover, we propose a hybrid solution crossover approach to balance the exploitation and exploration abilities of the EWOA. Finally, we propose a hybrid solution update method to handle the specific structure of the problem. After being enhanced, the proposed EWOA is more suitable and efficient in solving our considered problem and has a main difference from the conventional EWOA.
- We conduct extensive simulations to verify the performance of the proposed EWOA. The simulation results show that the proposed EWOA is effective and outperforms other peer algorithms. Moreover, the proposed improved factors are also evaluated and demonstrated.

## 2. Related Works

#### 2.1. Deterministic Methods

#### 2.2. Evolutionary Computation Approaches

## 3. Model and Problem

#### 3.1. CCAA Models

#### 3.2. Inactivated or Activated Antenna Model

#### 3.3. Sidelobe Suppression Model

#### 3.4. Optimization Problem

## 4. Proposed Algorithm

#### 4.1. Overview of Evolutionary Computation

#### 4.2. Conventional Whale Optimization Algorithm

#### 4.2.1. Encircling Prey

#### 4.2.2. Bubble-Net Attacking Method

- (i)
- Shrinking encircling mechanism. This behavior is achieved by reducing the value of parameter a, and the details can be found in [24].
- (ii)
- Spiral updating position. WOA uses a helix-shaped movement of humpback whales to guide the update of the population, which can be shown as follows.

#### 4.2.3. Search for Prey

#### 4.2.4. Shortcomings

#### 4.3. Proposed Enhanced Algorithm

#### 4.3.1. Hybrid Solution Initialization Method

#### 4.3.2. Hybrid Solution Crossover

Algorithm 1: Hybrid Solution Crossover. |

#### 4.3.3. Hybrid Solution Update

Algorithm 2: Hybrid Solution Update. |

#### 4.3.4. Main Steps and Complexity Analysis of EWOA

Algorithm 3: EWOA. |

## 5. Simulations

#### 5.1. Setups

#### 5.2. Simulation Results

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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(DB) | AVE. | MAX. | MIN. |
---|---|---|---|

DA | −23.2598 | −22.6900 | −23.7409 |

GWO | −23.2644 | −22.6404 | −24.0011 |

SSA | −23.2434 | −22.6982 | −23.8245 |

WOA | −23.441 | −22.8497 | −24.0907 |

SCA | −21.2753 | −20.1031 | −22.3683 |

EWOA | −23.7883 | −23.2147 | −24.1995 |

(DB) | AVE. | MAX. | MIN. |
---|---|---|---|

DA | 409.3217 | 520.8348 | 286.9482 |

GWO | 271.8296 | 298.7387 | 232.12 |

SSA | 273.4268 | 302.3688 | 222.1255 |

WOA | 297.5742 | 312.5098 | 290.1445 |

SCA | 254.5393 | 290.0058 | 189.7762 |

EWOA | 313.075 | 250.7853 | 331.1821 |

(DB) | AVE. | MAX. | MIN. |
---|---|---|---|

Chebyshev | −23.5074 | −23.0789 | −23.934 |

Circle | −23.4647 | −23.0118 | −23.8412 |

Guass/mouse | −23.4788 | −22.9087 | −24.0222 |

Iterative | −23.5758 | −23.0417 | −23.9682 |

Logistic | −23.4941 | −22.5767 | −23.9068 |

Piecewise | −23.4257 | −22.7275 | −23.8546 |

Sine | −23.4498 | −22.8004 | −24.0941 |

Singer | −23.4784 | −22.9801 | −23.9801 |

Sinusoidal | −23.508 | −23.0097 | −23.9226 |

Tent | −23.4395 | −23.0049 | −23.9017 |

Crossover | −23.7153 | −23.1617 | −24.2604 |

EWOA | −23.7883 | −23.2147 | −24.1995 |

(s) | AVE. | MAX. | MIN. |
---|---|---|---|

Chebyshev | 300.2103 | 311.8449 | 273.2599 |

Circle | 300.4639 | 308.2679 | 264.3575 |

Guass/mouse | 305.5025 | 321.2203 | 285.5261 |

Iterative | 311.7976 | 337.1776 | 182.6852 |

Logistic | 306.6116 | 323.0361 | 290.9548 |

Piecewise | 306.8332 | 328.9165 | 279.5588 |

Sine | 331.7841 | 340.0234 | 322.1776 |

Singer | 333.9409 | 341.0207 | 314.4988 |

Sinusoidal | 337.2349 | 350.9823 | 277.0756 |

Tent | 336.8633 | 354.8735 | 274.9848 |

Crossover | 302.2373 | 316.8397 | 243.6888 |

EWOA | 313.075 | 331.1821 | 250.7853 |

**Table 5.**Statistical results of the objective values obtained by different approach in the case with 10 populations and 440 elements.

(DB) | AVE. | MAX. | MIN. |
---|---|---|---|

DA | −22.492 | −21.281 | −23.5388 |

GWO | −22.7027 | −21.8035 | −23.5196 |

SSA | −22.7403 | −22.0197 | −23.8178 |

WOA | −22.7205 | −21.647 | −23.4629 |

SCA | −20.1068 | −18.9223 | −22.2253 |

EWOA | −23.0437 | −22.5879 | −23.6858 |

**Table 6.**Statistical results of the objective values obtained by different approach in the case with 50 populations and 120 elements.

(DB) | AVE. | MAX. | MIN. |
---|---|---|---|

DA | −8.73719 | −8.39645 | −9.46009 |

GWO | −9.05977 | −8.16812 | −9.57519 |

SSA | −8.99624 | −8.71156 | −9.36257 |

WOA | −9.09705 | −8.32748 | −9.47902 |

SCA | −8.91788 | −8.3524 | −9.36407 |

EWOA | −9.41423 | −9.15111 | −9.6363 |

**Table 7.**Statistical results of the objective values obtained by different approach in the case with 10 populations and 120 elements.

(DB) | AVE. | MAX. | MIN. |
---|---|---|---|

DA | −8.27269 | −7.7332 | −9.08744 |

GWO | −8.68996 | −7.24128 | −9.26944 |

SSA | −8.59402 | −7.72822 | −9.31861 |

WOA | −8.67264 | −7.46379 | −9.38475 |

SCA | −8.56307 | −7.73576 | −9.14937 |

EWOA | −9.1513 | −8.67405 | −9.45595 |

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**MDPI and ACS Style**

Tang, B.; Cai, L.; Yang, S.; Xu, J.; Yu, Y.
Evolutionary Computation for Sparse Synthesis Optimization of CCAAs: An Enhanced Whale Optimization Algorithm Method. *Future Internet* **2022**, *14*, 347.
https://doi.org/10.3390/fi14120347

**AMA Style**

Tang B, Cai L, Yang S, Xu J, Yu Y.
Evolutionary Computation for Sparse Synthesis Optimization of CCAAs: An Enhanced Whale Optimization Algorithm Method. *Future Internet*. 2022; 14(12):347.
https://doi.org/10.3390/fi14120347

**Chicago/Turabian Style**

Tang, Bohao, Lihua Cai, Shuai Yang, Jiaxing Xu, and Yi Yu.
2022. "Evolutionary Computation for Sparse Synthesis Optimization of CCAAs: An Enhanced Whale Optimization Algorithm Method" *Future Internet* 14, no. 12: 347.
https://doi.org/10.3390/fi14120347