# Location of Logistics Distribution Center Based on Improved Bald Eagle Algorithm

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## Abstract

**:**

## 1. Introduction

## 2. Related Work

#### 2.1. Original Bald Eagle Search Algorithm

#### 2.1.1. Select the Search Space

#### 2.1.2. Search Space Prey

#### 2.1.3. Subduction Stage

#### 2.2. Location of Logistics Center

- (1)
- There is regard for the size of logistics centers and other economic issues;
- (2)
- The distribution centers must meet the requirements of all locations, i.e., the sum of the product of the distance between the center and each node and the volume of goods is the minimum;
- (3)
- There is no regard to other costs

## 3. Algorithm Improvements

#### 3.1. Chaos Map Initialization

#### 3.2. Improvements to the Search Phase

## 4. Experiments and Analysis

#### 4.1. Test Algorithm Description

#### 4.2. Test Functions

#### 4.3. Test Environment Setting

## 5. Application in Location of Logistics Center

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Serial Number | Function | Dimension | Search Range | Minimum |
---|---|---|---|---|

1 | ${f}_{1}\left(x\right)={\displaystyle \sum _{i=1}^{n}}{x}_{i}^{2}$ | 50 | [−100, 100] | 0 |

2 | ${f}_{2}\left(x\right)={\displaystyle \sum _{i=1}^{n}}\left|{x}_{i}\right|+{\displaystyle \prod _{i=1}^{n}}\left|{x}_{i}\right|$ | 50 | [−10, 10] | 0 |

3 | ${f}_{3}\left(x\right)={\displaystyle \sum _{i=1}^{30}}({\displaystyle \sum _{j=1}^{i}}{x}_{i}{)}^{2}$ | 50 | [−100, 100] | 0 |

4 | ${f}_{4}\left(x\right)=\underset{i}{\mathrm{max}}\left\{\left|{x}_{i}\right|,1\le i\le 30\right\}$ | 50 | [−100, 100] | 0 |

5 | ${f}_{5}\left(x\right)={\displaystyle \sum _{i=1}^{n-1}}[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}]$ | 50 | [−30, 30] | 0 |

6 | ${f}_{6}\left(x\right)={\displaystyle \sum _{i=1}^{n}}{\left(\left|{x}_{i}+0.5\right|\right)}^{2}$ | 50 | [−100, 100] | 0 |

7 | ${f}_{7}\left(x\right)={\displaystyle \sum _{i=1}^{n}}i{x}_{i}^{4}+random\left[0,1\right)$ | 50 | [−1.28, 1.28] | 0 |

8 | ${f}_{8}\left(x\right)={\displaystyle \sum _{i=1}^{n}}\left(-{x}_{i}\mathrm{sin}\left(\sqrt{\left|{x}_{i}\right|}\right)\right)$ | 50 | [−500, 500] | −12,659.5 |

9 | ${f}_{9}\left(x\right)={\displaystyle \sum _{i=1}^{n}}{\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]}^{2}$ | 50 | [−5.12, 5.12] | 0 |

10 | ${f}_{10}\left(x\right)=-20\mathrm{exp}(-0.2\sqrt{\frac{1}{n}{\displaystyle \sum _{i=1}^{n}}{x}_{i}^{2}}-\mathrm{exp}(\frac{1}{n}{\displaystyle \sum _{i=1}^{n}}\mathrm{cos}\left(2\pi {x}_{i}\right))+e+20$ | 50 | [−32, 32] | 0 |

Test Function | Index | PSO | WCA | WOA | BES | LSCBES |
---|---|---|---|---|---|---|

f_{1} | Average Value | 1.78 × 10^{1} | 7.65 × 10^{−11} | 3.46 × 10^{−14} | 2.35 × 10^{−36} | 0 |

Standard Deviation | 3.26 × 10^{0} | 4.37 × 10^{−11} | 2.15 × 10^{−14} | 1.27 × 10^{−36} | 0 | |

f_{2} | Average Value | 2.33 × 10^{1} | 6.38 × 10^{−7} | 9.26 × 10^{−13} | 4.96 × 10^{−183} | 0 |

Standard Deviation | 4.37 × 10^{0} | 4.96 × 10^{−7} | 4.13 × 10^{−12} | 1.49 × 10^{−185} | 0 | |

f_{3} | Average Value | 1.16 × 10^{2} | 2.87 × 10^{1} | 7.31 × 10^{−10} | 3.67 × 10^{−223} | 0 |

Standard Deviation | 4.46 × 10^{0} | 1.74 × 10^{0} | 2.33 × 10^{−10} | 0 | 0 | |

f_{4} | Average Value | 2.45 × 10^{0} | 5.43 × 10^{−1} | 8.39 × 10^{−13} | 3.21 × 10^{−226} | 0 |

Standard Deviation | 7.46 × 10^{1} | 2.76 × 10^{−1} | 2.91 × 10^{−13} | 1.23 × 10^{−227} | 0 | |

f_{5} | Average Value | 7.29 × 10^{3} | 7.77 × 10^{2} | 2.15 × 10^{1} | 3.45 × 10^{−6} | 1.43 × 10^{−6} |

Standard Deviation | 3.18 × 10^{3} | 6.45 × 10^{2} | 1.46 × 10^{1} | 2.94 × 10^{−6} | 4.36 × 10^{−6} | |

f_{6} | Average Value | 2.44 × 10^{1} | 8.14 × 10^{−10} | 6.45 × 10^{−9} | 3.77 × 10^{−6} | 7.92 × 10^{−22} |

Standard Deviation | 5.73 × 10^{0} | 5.37 × 10^{−10} | 2.78 × 10^{−9} | 8.73 × 10^{−7} | 6.45 × 10^{−22} | |

f_{7} | Average Value | 6.26 × 10^{1} | 9.46 × 10^{−3} | 7.26 × 10^{−3} | 2.33 × 10^{−5} | 3.92 × 10^{−6} |

Standard Deviation | 2.91 × 10^{1} | 3.42 × 10^{−2} | 5.42 × 10^{−3} | 4.26 × 10^{−5} | 2.35 × 10^{−6} | |

f_{8} | Average Value | −7.65 × 10^{3} | −5.64 × 10^{3} | −2.15 × 10^{3} | −2.13 × 10^{3} | −7.94 × 10^{2} |

Standard Deviation | 2.45 × 10^{3} | 4.94 × 10^{2} | 8.92 × 10^{1} | 7.96 × 10^{−3} | 4.32 × 10^{2} | |

f_{9} | Average Value | 1.66 × 10^{2} | 7.86 × 10^{−6} | 5.45 × 10^{−16} | 6.57 × 10^{−98} | 2.15 × 10^{−199} |

Standard Deviation | 2.74 × 10^{1} | 3.82 × 10^{−6} | 4.37 × 10^{−16} | 5.42 × 10^{−99} | 4.37 × 10^{−199} | |

f_{10} | Average Value | 1.08 × 10^{1} | 7.12 × 10^{−7} | 6.64 × 10^{−12} | 8.42 × 10^{−16} | 2.41 × 10^{−19} |

Standard Deviation | 2.32 × 10^{0} | 2.34 × 10^{−8} | 4.33 × 10^{−12} | 2.11 × 10^{−18} | 0 |

Serial Number | Coordinate | Cargo Volume | Serial Number | Coordinate | Cargo Volume |
---|---|---|---|---|---|

1 | (1625, 2413) | 20 | 17 | (4027, 2106) | 90 |

2 | (3710, 924) | 90 | 18 | (4135, 2419) | 70 |

3 | (4213, 2256) | 90 | 19 | (3864, 2217) | 100 |

4 | (3694, 1403) | 60 | 20 | (3655, 2543) | 50 |

5 | (3476, 1537) | 70 | 21 | (4122, 2795) | 50 |

6 | (3319, 1558) | 70 | 22 | (4257, 2931) | 50 |

7 | (3238, 1231) | 40 | 23 | (3429, 1908) | 80 |

8 | (2793, 1546) | 90 | 24 | (3507, 2376) | 70 |

9 | (2894, 1793) | 90 | 25 | (3451, 2712) | 80 |

10 | (3154, 1425) | 70 | 26 | (3275, 3014) | 40 |

11 | (2857, 2236) | 60 | 27 | (3167, 3455) | 40 |

12 | (2346, 1498) | 40 | 28 | (3345, 3716) | 60 |

13 | (2476, 1154) | 40 | 29 | (2296, 2437) | 70 |

14 | (1819, 1479) | 40 | 30 | (3004, 3152) | 50 |

15 | (1684, 829) | 20 | 31 | (2754, 3666) | 30 |

16 | (3729, 1683) | 80 |

Algorithm | Site Selection Plan | Distance ∗ Cargo Volume | Number of Iterations |
---|---|---|---|

LSCBES | (5, 9, 12, 18, 25, 27) | 6.1069 × 10^{5} | 33 |

BES | (3, 5, 9, 12, 20, 27) | 6.1934 × 10^{5} | 28 |

WOA | (5, 11, 14, 18, 25, 27) | 6.3114 × 10^{5} | 30 |

WCA | (5, 8, 14, 18, 20, 27) | 6.2412 × 10^{5} | 52 |

PSO | (4, 6, 12, 18, 25, 27) | 6.4463 × 10^{5} | 42 |

LSCBES | BES | WOA | WCA | PSO | |||||
---|---|---|---|---|---|---|---|---|---|

Distribution Centera | Distribution Range | Distribution Centera | Distribution Range | Distribution Centera | Distribution Range | Distribution Centera | Distribution Range | Distribution Centera | Distribution Range |

5 | 2, 4, 6, 7, 10, 16, 23 | 3 | 17, 18, 21, 22 | 5 | 2, 4, 6, 7, 10, 16, 23 | 5 | 2, 4, 6, 7, 10, 16, 23 | 4 | 2, 16 |

9 | 8, 11, 29 | 5 | 2, 4, 6, 7, 16, 23 | 11 | 8, 9, 29 | 8 | 9, 11, 12, 13 | 6 | 5, 7, 8, 9, 11, 23 |

12 | 1, 13, 14, 15 | 9 | 8, 10, 11 | 14 | 1, 12, 13, 15 | 14 | 1, 15, 29 | 12 | 1, 13, 14, 15, 29 |

18 | 3, 17, 19, 21, 22 | 12 | 1, 13, 14, 15, 29 | 18 | 3, 17, 19, 21, 22 | 18 | 3, 17, 21, 22 | 18 | 3, 17, 19, 21, 22 |

25 | 20, 24, 26 | 20 | 19, 24, 25 | 25 | 20, 24, 26 | 20 | 19, 24, 25 | 25 | 20, 24, 26 |

27 | 28, 30, 31 | 27 | 26, 30, 31, 28 | 27 | 28, 30, 31 | 27 | 26, 28, 30, 31 | 27 | 28, 30, 31 |

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**MDPI and ACS Style**

Tong, Y.; Cheng, X.
Location of Logistics Distribution Center Based on Improved Bald Eagle Algorithm. *Sustainability* **2022**, *14*, 9036.
https://doi.org/10.3390/su14159036

**AMA Style**

Tong Y, Cheng X.
Location of Logistics Distribution Center Based on Improved Bald Eagle Algorithm. *Sustainability*. 2022; 14(15):9036.
https://doi.org/10.3390/su14159036

**Chicago/Turabian Style**

Tong, Yanfen, and Xianbao Cheng.
2022. "Location of Logistics Distribution Center Based on Improved Bald Eagle Algorithm" *Sustainability* 14, no. 15: 9036.
https://doi.org/10.3390/su14159036