# Multi-Strategy Enhanced Dung Beetle Optimizer and Its Application in Three-Dimensional UAV Path Planning

^{*}

## Abstract

**:**

## 1. Introduction

- Adding a reflective learning strategy and using Beta distribution function mapping to generate reflective solutions, improving the algorithm’s search ability;
- For particles that exceeded the search space range, Levy distribution mapping was used to handle particle boundaries, enhancing the probability of global search reaching the optimal position;
- Individual crossover mechanism and dimension crossover mechanism were used to update the position of individual thief beetles, increasing the population diversity and avoiding falling into local optima;
- Applying the improved MDBO to solve the three-dimensional UAV path-planning problem, and design sets of scene experiments to verify the efficiency of the MDBO.

## 2. Dung Beetle Optimizer (DBO)

- (1)
- Ball-rolling dung beetle

- (2)
- Brood ball

- (3)
- Small dung beetle

- (4)
- Thief

## 3. The Proposed Method

#### 3.1. Dynamic Reflective Learning Strategy Based on Beta Distribution

#### 3.2. Cross Boundary Limits Method Based on Levy Distribution

#### 3.3. Cross Operators for Updating the Location of Thieves

- (1)
- Horizontal crossover search (HCS)

- (2)
- Vertical crossover search (VCS)

#### 3.4. The Detailed Process of the MDBO

Algorithm 1: The pseudo code of MDBO |

Initialize the particle’s population N; the maximum iterations T; the dimensions D. |

Initialize the positions of the dung beetlesWhile t ≤ T doCalculate the current best position and its fitness Obtain N reflective solutions by Equations (11)–(15) Update the positions of N individuals For i = 1:N doif i == ball-rolling dung beetle thenGenerate a random number $p\in (0,1)$ if p < 0.9 thenUpdate search position by Equation (1) ElseUpdate search position by Equation (3) end ifif i == brood ball thenUpdate search position by Equatio n (5)end ifif i == small dung beetle thenUpdate search position by Equation (7) end ifif i == thief thenUpdate search position by Equation (8) while t ≤ T/4 doPerform HCS using Equations (17)–(18) Perform VCS using Equation (19) end whileend ifend forUpdate the best position and its fitness t = t + 1 end whileReturn the optimal solution X^{b} and its fitness f_{b}. |

#### 3.5. Computational Complexity Analysis

## 4. Analysis of Simulation Experiments

#### 4.1. Experimental Design

#### 4.2. Sensitivity Analysis of MDBO’s Parameters

#### 4.3. Comparison of Performance on 12 Benchmark Functions

#### 4.4. Convergence Curve Analysis

^{−300}. For F4, the function curve of MDBO showed an inflection point because crossover operators can help MDBO re-exploit the optimization precision. For function F7, although the progress of convergence to 500 generations was not as good as that of ISSA, the average number of iterations of function curve convergence to the optimal value was the least and converged to the optimal value in about 100 generations. In F9 and F11, MDBO obtained the optimal global value of 0. The curve broke during iterations because the figure showed an average best value in logarithmic. For functions F8, F10, and F12, MDBO exhibited a more competitive performance than the other comparison algorithms. In summary, convergence analysis proved that MDBO had a higher success ratio than the other optimization algorithms.

#### 4.5. Wilcoxon Rank-Sum Test

#### 4.6. MDBO’s Performance on CEC2021 Suite

## 5. UAV Path-Planning Model

#### 5.1. Environment Model

#### 5.2. Path Representation

#### 5.3. Cost Function and Performance Constraints

- (1)
- Length cost

- (2)
- Flight altitude cost

- (3)
- Smooth cost

## 6. Simulation Experiments and Discussions on UAV Path Planning

#### 6.1. Scenario Setup

#### 6.2. Effect of the Cost Function Parameters

_{3}is 0.1 or 0.2. When w

_{3}is 0.1, the combination of w

_{1}and w

_{2}is {0.7,0.2} or {0.2,0.7}, and when w

_{3}is 0.2, the combination of w

_{1}and w

_{3}is {0.4,0.4} or {0.5,0.3} or {0.3,0.5}. Thus, there are a total of five combinations of design.

_{1}= 0.4, w

_{2}= 0.4, w

_{3}= 0.2, and w

_{1}= 0.3, w

_{2}= 0.5, w

_{3}= 0.2, and w

_{1}= 0.5, w

_{2}= 0.3, w

_{3}= 0.2, the performance of MDBO was second only to DBO. Although DBO had better standard deviations than MDBO under these three weight combinations, its optimal convergence solution and mean value were not as good as MDBO. In the case of the shortest path length, we believe that the performance of MDBO was still better than DBO. It is worth noting that when w

_{1}= 0.7, w

_{2}= 0.2, and w

_{3}= 0.1, MDBO ranked second only to GBO in terms of the mean value. In particular, when w

_{1}= 0.5, w

_{2}= 0.3, and w

_{3}= 0.2, the performance of MDBO was optimal. Overall, MDBO had good searchability and robustness in all testing scenarios.

#### 6.3. Impact of the Count and Position of Tasks

#### 6.4. Influence of the Number and Arrangement of Obstacles

## 7. Conclusions

## 8. Discussion

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

No. | Function Name | Search Space | Dim | f_{min} |
---|---|---|---|---|

F1 | Sphere | [−100,100] | 30/50/100 | 0 |

F2 | Schwefel 2.22 | [−10,10] | 30/50/100 | 0 |

F3 | Schwefel 1.2 | [−100,100] | 30/50/100 | 0 |

F4 | Schwefel 2.21 | [−100,100] | 30/50/100 | 0 |

F5 | Zakharov | [−5,10] | 30/50/100 | 0 |

F6 | Step | [−100,100] | 30/50/100 | 0 |

F7 | Quartic | [−1.28,1.28] | 30/50/100 | 0 |

F8 | Qing | [−500,500] | 30/50/100 | 0 |

F9 | Rastrigin | [−5.12,5.12] | 30/50/100 | 0 |

F10 | Ackley 1 | [−32,32] | 30/50/100 | 0 |

F11 | Griewank | [−600,600] | 30/50/100 | 0 |

F12 | Penalized 1 | [−50,50] | 30/50/100 | 0 |

**Table A2.**Summary of the CEC2021 test suite [32].

No. | Functions | F_{i}^{*} | |
---|---|---|---|

Unimodal Function | CEC-1 | Shifted and Rotated Bent Cigar Function | 100 |

Basic Functions | CEC-2 | Shifted and Rotated Schwefel’s Function | 1100 |

CEC-3 | Shifted and Rotated Lunacek bi-Rastrigin Function | 700 | |

CEC-4 | Expand Rosenbrock’s plus Griewangk’s Function | 1900 | |

Hybrid Functions | CEC-5 | Hybrid Function 1 (N = 3) | 1700 |

CEC-6 | Hybrid Function 2 (N = 4) | 1600 | |

CEC-7 | Hybrid Function 3 (N = 5) | 2100 | |

Composition Functions | CEC-8 | Composition Function 1 (N = 3) | 2200 |

CEC-9 | Composition Function 2 (N = 4) | 2400 | |

CEC-10 | Composition Function 3 (N = 5) | 2500 | |

Search range: [−100,100]^{D} |

No. | X | Y | Z | L | W | H |
---|---|---|---|---|---|---|

1 | 550 | 100 | 0 | 50 | 100 | 10 |

2 | 0 | 400 | 0 | 50 | 200 | 10 |

3 | 300 | 320 | 0 | 50 | 380 | 15 |

4 | 800 | 150 | 0 | 50 | 100 | 15 |

5 | 500 | 350 | 0 | 50 | 100 | 10 |

6 | 50 | 800 | 0 | 50 | 100 | 10 |

No. | X | Y | Z | L | W | H |
---|---|---|---|---|---|---|

1 | 40 | 100 | 0 | 100 | 150 | 5 |

2 | 450 | 350 | 0 | 50 | 100 | 10 |

3 | 850 | 100 | 0 | 100 | 100 | 20 |

4 | 0 | 400 | 0 | 50 | 200 | 10 |

5 | 100 | 400 | 0 | 50 | 200 | 10 |

6 | 260 | 430 | 0 | 100 | 180 | 15 |

7 | 600 | 320 | 0 | 50 | 380 | 15 |

8 | 800 | 500 | 0 | 50 | 100 | 15 |

9 | 430 | 650 | 0 | 50 | 100 | 10 |

10 | 20 | 900 | 0 | 50 | 100 | 10 |

11 | 500 | 800 | 0 | 50 | 100 | 10 |

12 | 450 | 200 | 0 | 50 | 100 | 10 |

13 | 750 | 200 | 0 | 50 | 100 | 10 |

No. | X | Y | Z | L | W | H |
---|---|---|---|---|---|---|

1 | 40 | 100 | 0 | 100 | 150 | 5 |

2 | 400 | 150 | 0 | 50 | 100 | 10 |

3 | 550 | 100 | 0 | 50 | 100 | 10 |

4 | 850 | 100 | 0 | 100 | 100 | 20 |

5 | 0 | 400 | 0 | 50 | 200 | 10 |

6 | 100 | 400 | 0 | 50 | 200 | 10 |

7 | 260 | 430 | 0 | 100 | 180 | 15 |

8 | 500 | 320 | 0 | 50 | 100 | 10 |

9 | 600 | 320 | 0 | 50 | 380 | 15 |

10 | 700 | 300 | 0 | 100 | 100 | 10 |

11 | 800 | 500 | 0 | 50 | 100 | 15 |

12 | 300 | 700 | 0 | 50 | 100 | 10 |

13 | 430 | 650 | 0 | 50 | 100 | 10 |

14 | 20 | 900 | 0 | 50 | 100 | 10 |

15 | 100 | 800 | 0 | 50 | 100 | 10 |

16 | 200 | 800 | 0 | 50 | 100 | 10 |

17 | 500 | 800 | 0 | 50 | 100 | 10 |

18 | 750 | 750 | 0 | 50 | 100 | 10 |

19 | 900 | 900 | 0 | 50 | 100 | 10 |

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**Figure 4.**Sensitivity analysis of the MDBO’s parameters. (

**a**) CEC-1; (

**b**) CEC-3; (

**c**) CEC-7; (

**d**) CEC-10.

**Figure 8.**The UAV paths of all algorithms under a different number of targets. (

**a**) The generated paths when the number of tasks was 2. (

**b**) The generated paths when the number of tasks was 3. (

**c**) The generated paths when the number of tasks was 4.

Algorithm | Parameters |
---|---|

MDBO | k = 0.1, b = 0.3, α = β = 0.5 |

POA | I = 1 or 2, R = 0.2 |

SCSO | S = 2 |

HBA | β = 6, C = 2 |

DBO | k = 0.1, b = 0.3 |

ISSA | w = 0.7 |

MCFOA | M = 4, c_{1}∈(0,1) |

GCHHO | E_{0}∈[−1,1], θ = 0 |

Fun No. | Name | 30 dim | 50 dim | 100 dim | |||
---|---|---|---|---|---|---|---|

Mean | Std | Mean | Std | Mean | Std | ||

F1 | DBO | 7.74 × 10^{−114} | 3.88 × 10^{−113} | 3.93 × 10^{−97} | 2.15 × 10^{−96} | 1.07 × 10^{−111} | 5.84 × 10^{−111} |

POA | 8.32 × 10^{−103} | 4.44 × 10^{−102} | 1.64 × 10^{−99} | 8.31 × 10^{−99} | 9.87 × 10^{−100} | 5.39 × 10^{−99} | |

HBA | 2.58 × 10^{−134} | 1.39 × 10^{−133} | 1.56 × 10^{−128} | 6.28 × 10^{−128} | 7.48 × 10^{−122} | 2.34 × 10^{−121} | |

SCSO | 9.44 × 10^{−111} | 4.99 × 10^{−110} | 2.70 × 10^{−109} | 1.22 × 10^{−108} | 1.44 × 10^{−103} | 5.79 × 10^{−103} | |

GCHHO | 2.28 × 10^{−91} | 8.84 × 10^{−91} | 3.95 × 10^{−92} | 2.16 × 10^{−91} | 2.28 × 10^{−94} | 1.15 × 10^{−93} | |

ISSA | 4.45 × 10^{−14} | 1.08 × 10^{−14} | 7.07 × 10^{−14} | 1.46 × 10^{−14} | 1.40 × 10^{−13} | 1.77 × 10^{−14} | |

MCFOA | 6.53 × 10^{−11} | 1.14 × 10^{−10} | 1.90 × 10^{−10} | 3.69 × 10^{−10} | 7.94 × 10^{−10} | 1.22 × 10^{−9} | |

MDBO | 0 | 0 | 0 | 0 | 0 | 0 | |

F2 | DBO | 9.65 × 10^{−57} | 5.28 × 10^{−56} | 3.97 × 10^{−57} | 2.18 × 10^{−56} | 5.75 × 10^{−59} | 2.40 × 10^{−58} |

POA | 1.11 × 10^{−49} | 6.05 × 10^{−49} | 1.92 × 10^{−51} | 8.90 × 10^{−51} | 4.73 × 10^{−51} | 1.71 × 10^{−50} | |

HBA | 7.02 × 10^{−72} | 3.11 × 10^{−71} | 2.61 × 10^{−69} | 5.42 × 10^{−69} | 8.24 × 10^{−65} | 2.58 × 10^{−64} | |

SCSO | 3.31 × 10^{−60} | 1.13 × 10^{−59} | 6.72 × 10^{−59} | 1.24 × 10^{−58} | 5.81 × 10^{−55} | 2.40 × 10^{−54} | |

GCHHO | 2.03 × 10^{−47} | 1.11 × 10^{−46} | 3.83 × 10^{−49} | 1.65 × 10^{−48} | 6.52 × 10^{−48} | 2.78 × 10^{−47} | |

ISSA | 8.83 × 10^{−8} | 1.34 × 10^{−8} | 1.46 × 10^{−7} | 1.64 × 10^{−8} | 2.97 × 10^{−7} | 2.15 × 10^{−8} | |

MCFOA | 3.16 × 10^{−4} | 2.91 × 10^{−4} | 5.40 × 10^{−4} | 4.63 × 10^{−4} | 1.22 × 10^{−3} | 1.19 × 10^{−3} | |

MDBO | 0 | 0 | 0 | 0 | 0 | 0 | |

F3 | DBO | 2.70 × 10^{−29} | 1.48 × 10^{−28} | 1.93 × 10^{−39} | 1.06 × 10^{−38} | 4.75 × 10^{−11} | 2.60 × 10^{−10} |

POA | 6.02 × 10^{−99} | 3.30 × 10^{−98} | 3.62 × 10^{−100} | 1.81 × 10^{−99} | 6.01 × 10^{−98} | 2.24 × 10^{−97} | |

HBA | 7.39 × 10^{−96} | 3.12 × 10^{−95} | 4.41 × 10^{−87} | 2.36 × 10^{−86} | 4.22 × 10^{−74} | 2.29 × 10^{−73} | |

SCSO | 8.22 × 10^{−99} | 2.38 × 10^{−98} | 2.65 × 10^{−94} | 7.68 × 10^{−94} | 5.04 × 10^{−89} | 2.20 × 10^{−88} | |

GCHHO | 6.90 × 10^{−58} | 3.35 × 10^{−57} | 6.52 × 10^{−49} | 3.55 × 10^{−48} | 7.71 × 10^{−38} | 4.22 × 10^{−37} | |

ISSA | 4.81 × 10^{−13} | 5.24 × 10^{−13} | 1.16 × 10^{−12} | 1.47 × 10^{−12} | 4.20 × 10^{−12} | 3.91 × 10^{−12} | |

MCFOA | 1.62 × 10^{−8} | 2.07 × 10^{−8} | 9.54 × 10^{−8} | 1.32 × 10^{−7} | 5.32 × 10^{−7} | 8.46 × 10^{−7} | |

MDBO | 0 | 0 | 0 | 0 | 0 | 0 | |

F4 | DBO | 7.33 × 10^{−58} | 3.94 × 10^{−57} | 2.15 × 10^{−50} | 1.18 × 10^{−49} | 6.11 × 10^{−50} | 1.86 × 10^{−49} |

POA | 5.98 × 10^{−52} | 2.82 × 10^{−51} | 1.61 × 10^{−51} | 7.11 × 10^{−51} | 6.17 × 10^{−50} | 2.84 × 10^{−49} | |

HBA | 1.53 × 10^{−56} | 7.64 × 10^{−56} | 7.69 × 10^{−50} | 1.86 × 10^{−49} | 4.23 × 10^{−39} | 1.71 × 10^{−38} | |

SCSO | 3.79 × 10^{−51} | 1.45 × 10^{−50} | 4.92 × 10^{−49} | 1.58 × 10^{−48} | 1.24 × 10^{−47} | 5.76 × 10^{−47} | |

GCHHO | 1.60 × 10^{−46} | 7.28 × 10^{−46} | 8.34 × 10^{−45} | 3.60 × 10^{−44} | 6.73 × 10^{−45} | 2.56 × 10^{−44} | |

ISSA | 8.34 × 10^{−8} | 1.33 × 10^{−8} | 9.09 × 10^{−8} | 1.47 × 10^{−8} | 1.02 × 10^{−7} | 9.84 × 10^{−9} | |

MCFOA | 3.17 × 10^{−6} | 3.01 × 10^{−6} | 6.04 × 10^{−6} | 7.66 × 10^{−6} | 8.58 × 10^{−6} | 6.83 × 10^{−6} | |

MDBO | 0 | 0 | 0 | 0 | 0 | 0 | |

F5 | DBO | 3.13 × 10^{−23} | 1.70 × 10^{−22} | 8.97 × 10^{−2} | 4.91 × 10^{−1} | 4.91 × 10^{1} | 1.74 × 10^{2} |

POA | 6.38 × 10^{−103} | 3.49 × 10^{−102} | 2.68 × 10^{−103} | 1.44 × 10^{−102} | 3.59 × 10^{−97} | 1.97 × 10^{−96} | |

HBA | 8.52 × 10^{−61} | 3.29 × 10^{−60} | 1.33 × 10^{−24} | 7.27 × 10^{−24} | 3.42 × 10^{−4} | 1.52 × 10^{−3} | |

SCSO | 1.67 × 10^{−92} | 6.08 × 10^{−92} | 1.05 × 10^{−86} | 5.00 × 10^{−86} | 1.09 × 10^{−72} | 5.90 × 10^{−72} | |

GCHHO | 1.26 × 10^{−33} | 6.88 × 10^{−33} | 5.00 × 10^{−29} | 2.69 × 10^{−28} | 1.92 × 10^{−5} | 1.05 × 10^{−4} | |

ISSA | 5.35 × 10^{−15} | 1.64 × 10^{−14} | 1.32 × 10^{−14} | 3.03 × 10^{−14} | 4.18 × 10^{−14} | 1.10 × 10^{−13} | |

MCFOA | 9.13 × 10^{−6} | 1.01 × 10^{−5} | 8.72 × 10^{−5} | 9.45 × 10^{−5} | 1.09 × 10^{−3} | 1.71 × 10^{−3} | |

MDBO | 0 | 0 | 0 | 0 | 0 | 0 | |

F6 | DBO | 9.15 × 10^{−3} | 4.53 × 10^{−2} | 2.97 × 10^{−1} | 2.69 × 10^{−1} | 4.68 × 10^{0} | 7.99 × 10^{−1} |

POA | 2.78 × 10^{0} | 5.92 × 10^{−1} | 5.59 × 10^{0} | 8.21 × 10^{−1} | 1.46 × 10^{1} | 1.08 × 10^{0} | |

HBA | 8.62 × 10^{−3} | 4.56 × 10^{−2} | 8.89 × 10^{−1} | 3.71 × 10^{−1} | 8.27 × 10^{0} | 9.39 × 10^{−1} | |

SCSO | 2.06 × 10^{0} | 5.98 × 10^{−1} | 4.93 × 10^{0} | 7.74 × 10^{−1} | 1.43 × 10^{1} | 1.34 × 10^{0} | |

GCHHO | 7.08 × 10^{−7} | 6.46 × 10^{−7} | 1.65 × 10^{−5} | 1.12 × 10^{−5} | 2.50 × 10^{−4} | 1.61 × 10^{−4} | |

ISSA | 3.03 × 10^{0} | 4.35 × 10^{−1} | 7.19 × 10^{0} | 6.75 × 10^{−1} | 1.87 × 10^{1} | 8.98 × 10^{−1} | |

MCFOA | 6.75 × 10^{0} | 9.29 × 10^{−2} | 1.12 × 10^{1} | 1.66 × 10^{−1} | 2.26 × 10^{1} | 2.59 × 10^{−1} | |

MDBO | 0 | 0 | 0 | 0 | 0 | 0 | |

F7 | DBO | 1.04 × 10^{−3} | 6.94 × 10^{−4} | 1.21 × 10^{−3} | 1.01 × 10^{−3} | 1.59 × 10^{−3} | 1.02 × 10^{−3} |

POA | 2.26 × 10^{−4} | 1.62 × 10^{−4} | 1.97 × 10^{−4} | 1.42 × 10^{−4} | 1.56 × 10^{−4} | 8.67 × 10^{−5} | |

HBA | 3.02 × 10^{−4} | 1.99 × 10^{−4} | 3.91 × 10^{−4} | 3.25 × 10^{−4} | 5.31 × 10^{−4} | 4.21 × 10^{−4} | |

SCSO | 8.98 × 10^{−5} | 8.64 × 10^{−5} | 1.79 × 10^{−4} | 3.71 × 10^{−4} | 2.29 × 10^{−4} | 2.99 × 10^{−4} | |

GCHHO | 2.90 × 10^{−4} | 2.73 × 10^{−4} | 2.49 × 10^{−4} | 2.28 × 10^{−4} | 4.19 × 10^{−4} | 3.78 × 10^{−4} | |

ISSA | 9.43 × 10^{−5} | 7.38 × 10^{−5} | 1.08 × 10^{−4} | 9.09 × 10^{−5} | 1.04 × 10^{−4} | 1.46 × 10^{−4} | |

MCFOA | 2.15 × 10^{−3} | 1.54 × 10^{−3} | 2.61 × 10^{−3} | 2.80 × 10^{−3} | 3.73 × 10^{−3} | 3.52 × 10^{−3} | |

MDBO | 2.81 × 10^{−5} | 2.13 × 10^{−5} | 3.04 × 10^{−5} | 2.18 × 10^{−5} | 3.50 × 10^{−5} | 2.40 × 10^{−5} | |

F8 | DBO | 2.85 × 10^{1} | 1.02 × 10^{2} | 3.44 × 10^{3} | 3.74 × 10^{3} | 8.97 × 10^{4} | 2.61 × 10^{4} |

POA | 8.65 × 10^{2} | 4.93 × 10^{2} | 6.20 × 10^{3} | 1.63 × 10^{3} | 8.63 × 10^{4} | 1.57 × 10^{4} | |

HBA | 2.84 × 10^{2} | 3.21 × 10^{2} | 5.67 × 10^{3} | 2.29 × 10^{3} | 1.16 × 10^{5} | 2.35 × 10^{4} | |

SCSO | 2.17 × 10^{3} | 1.05 × 10^{3} | 1.19 × 10^{4} | 3.20 × 10^{3} | 1.41 × 10^{5} | 3.39 × 10^{4} | |

GCHHO | 1.74 × 10^{0} | 4.61 × 10^{0} | 2.61 × 10^{1} | 2.57 × 10^{1} | 1.87 × 10^{3} | 5.35 × 10^{2} | |

ISSA | 3.18 × 10^{3} | 4.46 × 10^{2} | 1.80 × 10^{4} | 1.19 × 10^{3} | 1.73 × 10^{5} | 1.05 × 10^{4} | |

MCFOA | 9.36 × 10^{3} | 1.59 × 10^{2} | 4.27 × 10^{4} | 2.68 × 10^{2} | 3.37 × 10^{5} | 1.79 × 10^{3} | |

MDBO | 2.29 × 10^{−7} | 1.99 × 10^{−7} | 1.29 × 10^{−5} | 1.07 × 10^{−5} | 3.07 × 10^{−3} | 4.26 × 10^{−3} | |

F9 | DBO | 9.96 × 10^{−2} | 5.45 × 10^{−1} | 0 | 0 | 2.32 × 10^{0} | 1.27 × 10^{1} |

POA | 0 | 0 | 0 | 0 | 0 | 0 | |

HBA | 0 | 0 | 0 | 0 | 0 | 0 | |

SCSO | 0 | 0 | 0 | 0 | 0 | 0 | |

GCHHO | 0 | 0 | 0 | 0 | 0 | 0 | |

ISSA | 0 | 0 | 0 | 0 | 0 | 0 | |

MCFOA | 4.68 × 10^{−6} | 8.78 × 10^{−6} | 8.24 × 10^{−6} | 1.61 × 10^{−5} | 2.47 × 10^{−5} | 3.87 × 10^{−5} | |

MDBO | 0 | 0 | 0 | 0 | 0 | 0 | |

F10 | DBO | 1.01 × 10^{−15} | 6.49 × 10^{−16} | 8.88 × 10^{−16} | 0 | 1.01 × 10^{−15} | 6.49 × 10^{−16} |

POA | 3.61 × 10^{−15} | 1.53 × 10^{−15} | 3.97 × 10^{−15} | 1.23 × 10^{−15} | 3.85 × 10^{−15} | 1.35 × 10^{−15} | |

HBA | 6.64 × 10^{−1} | 3.64 × 10^{0} | 2.66 × 10^{0} | 6.89 × 10^{0} | 3.32 × 10^{0} | 7.55 × 10^{0} | |

SCSO | 8.88 × 10^{−16} | 0 | 8.88 × 10^{−16} | 0 | 8.88 × 10^{−16} | 0 | |

GCHHO | 8.88 × 10^{−16} | 0 | 8.88 × 10^{−16} | 0 | 8.88 × 10^{−16} | 0 | |

ISSA | 4.84 × 10^{−8} | 4.18 × 10^{−9} | 4.69 × 10^{−8} | 3.79 × 10^{−9} | 4.75 × 10^{−8} | 2.78 × 10^{−9} | |

MCFOA | 1.74 × 10^{−5} | 1.45 × 10^{−5} | 1.74 × 10^{−5} | 1.88 × 10^{−5} | 1.50 × 10^{−5} | 1.58 × 10^{−5} | |

MDBO | 8.88 × 10^{−16} | 0 | 8.88 × 10^{−16} | 0 | 8.88 × 10^{−16} | 0 | |

F11 | DBO | 1.80 × 10^{−3} | 9.87 × 10^{−3} | 0 | 0 | 0 | 0 |

POA | 0 | 0 | 0 | 0 | 0 | 0 | |

HBA | 0 | 0 | 0 | 0 | 0 | 0 | |

SCSO | 0 | 0 | 0 | 0 | 0 | 0 | |

GCHHO | 0 | 0 | 0 | 0 | 0 | 0 | |

ISSA | 9.89 × 10^{−14} | 4.25 × 10^{−14} | 1.01 × 10^{−13} | 4.09 × 10^{−14} | 1.24 × 10^{−13} | 3.34 × 10^{−14} | |

MCFOA | 1.74 × 10^{−13} | 3.89 × 10^{−13} | 1.74 × 10^{−13} | 3.92 × 10^{−13} | 2.69 × 10^{−13} | 4.75 × 10^{−13} | |

MDBO | 0 | 0 | 0 | 0 | 0 | 0 | |

F12 | DBO | 5.13 × 10^{−4} | 1.64 × 10^{−3} | 4.77 × 10^{−3} | 6.03 × 10^{−3} | 6.45 × 10^{−2} | 2.34 × 10^{−2} |

POA | 1.61 × 10^{−1} | 8.04 × 10^{−2} | 2.81 × 10^{−1} | 8.59 × 10^{−2} | 4.74 × 10^{−1} | 8.76 × 10^{−2} | |

HBA | 4.44 × 10^{−4} | 1.64 × 10^{−3} | 1.88 × 10^{−2} | 8.56 × 10^{−3} | 1.43 × 10^{−1} | 5.33 × 10^{−2} | |

SCSO | 9.95 × 10^{−2} | 4.05 × 10^{−2} | 2.06 × 10^{−1} | 5.90 × 10^{−2} | 3.77 × 10^{−1} | 7.31 × 10^{−2} | |

GCHHO | 1.34 × 10^{−7} | 1.49 × 10^{−7} | 5.88 × 10^{−7} | 6.09 × 10^{−7} | 1.66 × 10^{−6} | 1.10 × 10^{−6} | |

ISSA | 2.35 × 10^{−1} | 4.33 × 10^{−2} | 4.13 × 10^{−1} | 6.52 × 10^{−2} | 6.38 × 10^{−1} | 6.17 × 10^{−2} | |

MCFOA | 1.33 × 10^{0} | 1.81 × 10^{−1} | 1.23 × 10^{0} | 7.90 × 10^{−2} | 1.13 × 10^{0} | 2.51 × 10^{−2} | |

MDBO | 1.57 × 10^{−32} | 5.57 × 10^{−48} | 9.42 × 10^{−33} | 2.78 × 10^{−48} | 4.71 × 10^{−33} | 1.39 × 10^{−48} |

Fun No. | DBO | POA | HBA | SCSO | GCHHO | ISSA | MCFOA |
---|---|---|---|---|---|---|---|

p Value R | p Value R | p Value R | p Value R | p Value R | p Value R | p Value R | |

F1 | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} |

F2 | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} |

F3 | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} |

F4 | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} |

F5 | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} |

F6 | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} |

F7 | 3.02 × 10^{−11} | 4.08 × 10^{−11} | 5.07 × 10^{−10} | 8.88 × 10^{−6} | 2.83 × 10^{−8} | 6.55 × 10^{−4} | 3.34 × 10^{−11} |

F8 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F9 | 8.15 × 10^{−2} | NaN | NaN | NaN | NaN | NaN | 1.21 × 10^{−12} |

F10 | NaN | 8.99 × 10^{−11} | 1.61 × 10^{−1} | NaN | NaN | 1.21 × 10^{−12} | 1.21 × 10^{−12} |

F11 | NaN | NaN | NaN | NaN | NaN | 1.21 × 10^{−12} | 1.66 × 10^{−11} |

F12 | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} |

+/−/= | 9/1/2 | 10/0/2 | 10/0/2 | 9/0/3 | 9/0/3 | 11/0/1 | 12/0/0 |

Fun No. | DBO | POA | HBA | SCSO | GCHHO | ISSA | MCFOA | MDBO | |
---|---|---|---|---|---|---|---|---|---|

CEC-1 | Mean | 3.62 × 10^{7} | 6.57 × 10^{9} | 5.71 × 10^{3} | 2.49 × 10^{9} | 4.20 × 10^{3} | 1.04 × 10^{10} | 5.05 × 10^{10} | 8.99 × 10^{2} |

Std. | 3.47 × 10^{7} | 3.63 × 10^{9} | 4.24 × 10^{3} | 2.08 × 10^{9} | 3.42 × 10^{3} | 2.34 × 10^{9} | 6.21 × 10^{8} | 1.54 × 10^{3} | |

p-value | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.08 × 10^{8} | 3.02 × 10^{−11} | 1.47 × 10^{−7} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | N/A | |

Signed-rank test | + | + | + | + | + | + | + | ||

CEC-2 | Mean | 3.50 × 10^{3} | 3.12 × 10^{3} | 2.86 × 10^{3} | 3.67 × 10^{3} | 3.02 × 10^{3} | 5.79 × 10^{3} | 9.27 × 10^{3} | 1.78 × 10^{3} |

Std. | 5.78 × 10^{2} | 4.21 × 10^{2} | 7.74 × 10^{2} | 4.92 × 10^{2} | 5.90 × 10^{2} | 2.99 × 10^{2} | 1.86 × 10^{2} | 2.59 × 10^{2} | |

p-value | 3.02 × 10^{−11} | 3.69 × 10^{−11} | 3.96 × 10^{−8} | 3.02 × 10^{−11} | 5.07 × 10^{−10} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | N/A | |

Signed-rank test | + | + | + | + | + | + | + | ||

CEC-3 | Mean | 8.40 × 10^{2} | 9.39 × 10^{2} | 8.00 × 10^{2} | 9.06 × 10^{2} | 8.72 × 10^{2} | 9.72 × 10^{2} | 1.18 × 10^{3} | 7.58 × 10^{2} |

Std. | 4.23 × 10^{1} | 2.99 × 10^{1} | 2.89 × 10^{1} | 3.52 × 10^{1} | 3.53 × 10^{1} | 3.14 × 10^{1} | 5.25 × 10^{0} | 1.34 × 10^{1} | |

p-value | 6.70 × 10^{−11} | 3.02 × 10^{−11} | 1.10 × 10^{−8} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | N/A | |

Signed-rank test | + | + | + | + | + | + | + | ||

CEC-4 | Mean | 1.94 × 10^{3} | 3.51 × 10^{3} | 1.91 × 10^{3} | 3.47 × 10^{3} | 1.91 × 10^{3} | 2.15 × 10^{4} | 3.86 × 10^{7} | 1.91 × 10^{3} |

Std. | 6.81 × 10^{1} | 1.78 × 10^{3} | 4.80 × 10^{0} | 3.37 × 10^{3} | 5.67 × 10^{0} | 1.48 × 10^{4} | 2.19 × 10^{6} | 4.73 × 10^{0} | |

p-value | 9.83 × 10^{−8} | 3.02 × 10^{−11} | 6.20 × 10^{−1} | 3.02 × 10^{−11} | 1.24 × 10^{−3} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | N/A | |

Signed-rank test | + | + | − | + | + | + | + | ||

CEC-5 | Mean | 1.22 × 10^{6} | 1.35 × 10^{5} | 1.64 × 10^{5} | 7.79 × 10^{5} | 4.75 × 10^{5} | 2.04 × 10^{6} | 4.81 × 10^{7} | 3.21 × 10^{5} |

Std. | 9.62 × 10^{5} | 9.98 × 10^{4} | 1.21 × 10^{5} | 5.78 × 10^{5} | 2.66 × 10^{5} | 6.98 × 10^{5} | 5.83 × 10^{6} | 1.72 × 10^{5} | |

p-value | 3.26 × 10^{−7} | 5.27 × 10^{−5} | 4.98 × 10^{−4} | 3.18 × 10^{−4} | 1.99 × 10^{−2} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | N/A | |

Signed-rank test | + | + | + | + | + | + | + | ||

CEC-6 | Mean | 2.24 × 10^{3} | 2.24 × 10^{3} | 2.10 × 10^{3} | 2.18 × 10^{3} | 2.00 × 10^{3} | 2.92 × 10^{3} | 7.66 × 10^{3} | 1.67 × 10^{3} |

Std. | 2.53 × 10^{2} | 1.88 × 10^{2} | 3.42 × 10^{2} | 2.22 × 10^{2} | 2.01 × 10^{2} | 2.43 × 10^{2} | 1.54 × 10^{2} | 6.62 × 10^{1} | |

p-value | 4.08 × 10^{−11} | 3.34 × 10^{−11} | 1.96 × 10^{−10} | 3.34 × 10^{−11} | 4.50 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | N/A | |

Signed-rank test | + | + | + | + | + | + | + | ||

CEC-7 | Mean | 6.19 × 10^{5} | 2.60 × 10^{4} | 9.52 × 10^{4} | 2.56 × 10^{5} | 1.37 × 10^{5} | 1.18 × 10^{6} | 7.69 × 10^{8} | 2.03 × 10^{5} |

Std. | 7.71 × 10^{5} | 3.27 × 10^{4} | 8.49 × 10^{4} | 3.08 × 10^{5} | 9.82 × 10^{4} | 5.94 × 10^{5} | 2.63 × 10^{7} | 1.32 × 10^{5} | |

p-value | 1.33 × 10^{−2} | 5.97 × 10^{−9} | 6.20 × 10^{−4} | 6.73 × 10^{−1} | 5.37 × 10^{−2} | 9.92 × 10^{−11} | 3.02 × 10^{−11} | N/A | |

Signed-rank test | + | + | + | − | − | + | + | ||

CEC-8 | Mean | 2.38 × 10^{3} | 3.12 × 10^{3} | 2.84 × 10^{3} | 2.93 × 10^{3} | 2.41 × 10^{3} | 3.60 × 10^{3} | 9.53 × 10^{3} | 2.51 × 10^{3} |

Std. | 3.15 × 10^{2} | 8.44 × 10^{2} | 1.48 × 10^{3} | 9.68 × 10^{2} | 5.57 × 10^{2} | 6.70 × 10^{2} | 1.26 × 10^{2} | 6.20 × 10^{2} | |

p-value | 1.25 × 10^{−4} | 5.09 × 10^{−8} | 9.03 × 10^{−4} | 6.01 × 10^{−8} | 2.32 × 10^{−2} | 8.35 × 10^{−8} | 3.02 × 10^{−11} | N/A | |

Signed-rank test | + | + | + | + | + | + | + | ||

CEC-9 | Mean | 3.00 × 10^{3} | 3.01 × 10^{3} | 2.96 × 10^{3} | 2.94 × 10^{3} | 2.94 × 10^{3} | 3.04 × 10^{3} | 4.55 × 10^{3} | 2.92 × 10^{3} |

Std. | 8.21 × 10^{1} | 5.69 × 10^{1} | 1.29 × 10^{2} | 4.64 × 10^{1} | 4.87 × 10^{1} | 2.59 × 10^{1} | 1.86 × 10^{1} | 9.77 × 10^{1} | |

p-value | 1.11 × 10^{−3} | 1.09 × 10^{−5} | 6.52 × 10^{−1} | 5.30 × 10^{−1} | 5.49 × 10^{−1} | 2.92 × 10^{−9} | 3.02 × 10^{−11} | N/A | |

Signed-rank test | + | + | − | − | − | + | + | ||

CEC-10 | Mean | 2.98 × 10^{3} | 3.12 × 10^{3} | 2.96 × 10^{3} | 3.07 × 10^{3} | 2.98 × 10^{3} | 3.56 × 10^{3} | 1.12 × 10^{4} | 2.99 × 10^{3} |

Std. | 4.91 × 10^{1} | 1.24 × 10^{2} | 4.07 × 10^{1} | 7.76 × 10^{1} | 3.62 × 10^{1} | 1.32 × 10^{2} | 1.95 × 10^{2} | 2.24 × 10^{1} | |

p-value | 9.93 × 10^{−2} | 1.85 × 10^{−8} | 6.91 × 10^{−4} | 2.15 × 10^{−6} | 2.23 × 10^{−1} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | N/A | |

Signed-rank test | − | + | + | + | − | = | + | ||

+/−/=/gm | 62/8/0/54 |

W | GBO | HPO | GWO | DBO | FOA | PSO | MDBO | |
---|---|---|---|---|---|---|---|---|

w_{1} = 0.7, w_{2} = 0.2, w_{3} = 0.1 | Best | 3.12 × 10^{1} | 8.01 × 10^{1} | 1.05 × 10^{2} | 3.01 × 10^{1} | 4.09 × 10^{1} | 1.81 × 10^{2} | 3.00 × 10^{1} |

Mean | 6.89× 10^{1} | 1.78 × 10^{2} | 2.56 × 10^{2} | 3.56 × 10^{1} | 5.95 × 10^{1} | 2.68 × 10^{2} | 3.37 × 10^{1} | |

Std | 4.54 × 10^{1} | 6.03 × 10^{1} | 1.15 × 10^{2} | 5.36 × 10^{0} | 1.27 × 10^{1} | 5.65 × 10^{1} | 2.92 × 10^{0} | |

w_{1} = 0.2, w_{2} = 0.7, w_{3} = 0.1 | Best | 3.20 × 10^{1} | 1.15 × 10^{2} | 1.39 × 10^{2} | 3.56 × 10^{1} | 4.70 × 10^{1} | 1.93 × 10^{2} | 3.36 × 10^{1} |

Mean | 1.08 × 10^{2} | 2.63 × 10^{2} | 2.81 × 10^{2} | 4.47 × 10^{1} | 7.91 × 10^{1} | 3.12 × 10^{2} | 4.33 × 10^{1} | |

Std | 7.53 × 10^{1} | 8.33 × 10^{1} | 1.24 × 10^{2} | 5.74 × 10^{0} | 1.38 × 10^{1} | 6.32 × 10^{1} | 5.51 × 10^{0} | |

w_{1} = 0.4, w_{2} = 0.4, w_{3} = 0.2 | Best | 3.09 × 10^{1} | 1.85 × 10^{2} | 8.88 × 10^{1} | 2.97 × 10^{1} | 4.71 × 10^{1} | 2.87 × 10^{2} | 2.80 × 10^{1} |

Mean | 1.11 × 10^{2} | 3.60 × 10^{2} | 4.50 × 10^{2} | 3.59 × 10^{1} | 7.96 × 10^{1} | 4.86 × 10^{2} | 3.40 × 10^{1} | |

Std | 8.11 × 10^{1} | 1.11 × 10^{2} | 1.40 × 10^{2} | 4.77 × 10^{0} | 2.08 × 10^{1} | 8.36 × 10^{1} | 7.62 × 10^{0} | |

w_{1} = 0.3, w_{2} = 0.5, w_{3} = 0.2 | Best | 3.56 × 10^{1} | 1.66 × 10^{2} | 1.82 × 10^{2} | 3.10 × 10^{1} | 5.67 × 10^{1} | 3.53 × 10^{2} | 2.89 × 10^{1} |

Mean | 1.49 × 10^{2} | 4.13 × 10^{2} | 4.41 × 10^{2} | 3.63 × 10^{1} | 9.26 × 10^{1} | 5.33 × 10^{2} | 3.47 × 10^{1} | |

Std | 8.66 × 10^{1} | 1.30 × 10^{2} | 1.64 × 10^{2} | 3.45 × 10^{0} | 2.37 × 10^{1} | 1.02 × 10^{2} | 4.39 × 10^{0} | |

w_{1} = 0.5, w_{2} = 0.3, w_{3} = 0.2 | Best | 2.99 × 10^{1} | 1.41 × 10^{2} | 1.83 × 10^{2} | 2.85 × 10^{1} | 5.86 × 10^{1} | 3.30 × 10^{2} | 2.61 × 10^{1} |

Mean | 1.13 × 10^{2} | 3.14 × 10^{2} | 4.35 × 10^{2} | 3.37 × 10^{1} | 8.72 × 10^{1} | 4.81 × 10^{2} | 3.32 × 10^{1} | |

Std | 1.11 × 10^{2} | 1.05 × 10^{2} | 1.65 × 10^{2} | 7.27 × 10^{0} | 1.96 × 10^{1} | 1.04 × 10^{2} | 8.63 × 10^{0} |

Tasks’ Numbers | Target Coordinates |
---|---|

2 | [250,650,5], [500,450,10] |

3 | [250,650,5], [300,300,7], [700,300,10] |

4 | [250,650,5], [300,300,7], [600,800,12], [900,400,2] |

Task Numbers | GBO | HPO | GWO | DBO | FOA | PSO | MDBO | |
---|---|---|---|---|---|---|---|---|

2 | Best | 5.22 × 10^{1} | 4.58 × 10^{2} | 3.22 × 10^{2} | 5.22 × 10^{1} | 5.52 × 10^{1} | 4.96 × 10^{2} | 4.58 × 10^{1} |

Mean | 1.06 × 10^{2} | 6.65 × 10^{2} | 6.00 × 10^{2} | 6.18 × 10^{1} | 6.93 × 10^{1} | 7.54 × 10^{2} | 5.50 × 10^{1} | |

Std | 7.78 × 10^{1} | 1.25 × 10^{2} | 2.10 × 10^{2} | 7.76 × 10^{0} | 7.78 × 10^{0} | 1.17 × 10^{2} | 5.02 × 10^{0} | |

3 | Best | 3.55 × 10^{1} | 3.28 × 10^{2} | 2.62 × 10^{2} | 3.61 × 10^{1} | 3.85 × 10^{1} | 4.91 × 10^{2} | 3.55 × 10^{1} |

Mean | 6.58 × 10^{1} | 5.77 × 10^{2} | 4.91 × 10^{2} | 4.10 × 10^{1} | 5.90 × 10^{1} | 7.11 × 10^{2} | 3.60 × 10^{1} | |

Std | 4.60 × 10^{1} | 1.63 × 10^{2} | 2.49 × 10^{2} | 6.34 × 10^{0} | 1.24 × 10^{1} | 1.82 × 10^{2} | 7.38 × 10^{−}^{1} | |

4 | Best | 6.23 × 10^{1} | 8.01 × 10^{2} | 4.54 × 10^{2} | 8.21 × 10^{1} | 1.16 × 10^{2} | 1.05 × 10^{3} | 6.16 × 10^{1} |

Mean | 1.62 × 10^{2} | 1.04 × 10^{3} | 8.55 × 10^{2} | 1.37 × 10^{2} | 1.50 × 10^{2} | 1.51 × 10^{3} | 9.73 × 10^{1} | |

Std | 8.28 × 10^{1} | 2.08 × 10^{2} | 2.18 × 10^{2} | 4.68 × 10^{1} | 2.16 × 10^{1} | 2.77 × 10^{2} | 3.28 × 10^{1} |

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## Share and Cite

**MDPI and ACS Style**

Shen, Q.; Zhang, D.; Xie, M.; He, Q.
Multi-Strategy Enhanced Dung Beetle Optimizer and Its Application in Three-Dimensional UAV Path Planning. *Symmetry* **2023**, *15*, 1432.
https://doi.org/10.3390/sym15071432

**AMA Style**

Shen Q, Zhang D, Xie M, He Q.
Multi-Strategy Enhanced Dung Beetle Optimizer and Its Application in Three-Dimensional UAV Path Planning. *Symmetry*. 2023; 15(7):1432.
https://doi.org/10.3390/sym15071432

**Chicago/Turabian Style**

Shen, Qianwen, Damin Zhang, Mingshan Xie, and Qing He.
2023. "Multi-Strategy Enhanced Dung Beetle Optimizer and Its Application in Three-Dimensional UAV Path Planning" *Symmetry* 15, no. 7: 1432.
https://doi.org/10.3390/sym15071432