# A Chaotic Hybrid Butterfly Optimization Algorithm with Particle Swarm Optimization for High-Dimensional Optimization Problems

^{1}

^{2}

^{3}

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## Abstract

**:**

## 1. Introduction

^{2}emission mitigation strategies derived from BOA [11]. Tan et al. [12] proposed an improved BOA to solve the wavelet neural networks problem based on solutions for elliptic partial differential equations. Malisetti and Pamula [13] proposed a novel BOA based on quasi opposition for the problem of cluster head selection in wireless sensor network (WSNs). Sharma et al. [14] proposed a bidirectional butterfly optimization algorithm for solving the engineering optimization problems. Above the studies of BOA, which are improvement research or applied research, there is only one paper for a hybrid algorithm with ABC and BOA.

^{−1}(x), to obtain one-, two-, and three-directional multiscroll integer and fractional order chaotic attractors, and they analyzed stabilization of the chaotic system with the application of chaos theory in the improvement of swarm intelligent optimization algorithms [48,49], and it has been recognized by scholars in the field.

## 2. The Basic Butterfly Optimization Algorithm (BOA)

## 3. The Basic Particle Swarm Optimization (PSO) Model

_{1}= c

_{2}= 2, $ran{d}_{1,}$ and $ran{d}_{2}$ are the random numbers in (0, 1). The $w$ can be calculated as:

## 4. The Proposed Algorithm

#### 4.1. Cubic Map

#### 4.2. Nonlinear Parameter Control Strategy

#### 4.3. Hybrid BOA with PSO

Algorithm 1. Pseudo-code of hybrid PSO with BOA (PSOBOA) |

1. Generate the initialize population of the butterflies X_{i} (i = 1, 2, …, n) randomly |

2. Initialize the parameter r_{1}, r_{2}, C_{1} and C_{2} |

3. Define senser modality c, power exponent a and switch probability p |

4. Calculate the fitness value of each butterflies |

5. While t = 1: the max iterations |

6. For each search agent |

7. Update the fragrance of current search agent by Equation (1) |

8. End for9. Find the best f |

10. For each search agent |

11. Set a random number r in [0,1] |

12. If r < p then |

13. Move towards best position by Equation (13) |

14. Else |

15. Move randomly by Equation (14) |

16. End if |

17. End for |

18. Update the velocity using Equation (11) |

19. Calculate the new fitness value of each butterflies |

20. If ${f}_{new}$ < best f |

21. Update the position of best f using Equation (12) |

22. End if |

23. Update the value of power exponent a |

24. t = t + 1 |

25. End while |

26. Return the best solution and its fitness value |

#### 4.4. The Proposed HPSOBOA

Algorithm 2. Pseudo-code of novel HPSOBOA |

1. Generate the initialize population of the butterflies X_{i} (i = 1, 2, …, n) using cubic map |

2. Initialize the parameter r_{1}, r_{2}, C_{1} and C_{2} and switch probability p |

3. Define senser modality c and the initial value of power exponent a |

4. Calculate the fitness value of each butterflies |

5. While t = 1: the max iterations |

6. For each search agent |

7. Update the fragrance of current search agent by Equation (1) |

8. End for 9. Find the best f |

10. For each search agent |

11. Set a random number r in [0,1] |

12. If r < p then |

13. Move towards best position by Equation (13) |

14. Else |

15. Move randomly by Equation (14) |

16. End if |

17. End for |

18. Update the velocity using Equation (11) |

19. Calculate the new fitness value of each butterflies |

20. If ${f}_{new}$ < best f |

21. Update the position of best f using Equation (12) |

22. End if |

23. Update the value of power exponent a using Equation (10) |

24. t = t + 1 |

25. End while |

26. Output the best solution |

## 5. Experiments and Comparison Results

#### 5.1. Numerical Optimization Funtions and Experiments

#### 5.1.1. The 26 Test Functions

#### 5.1.2. Experiment 1: Comparison with BOA, CBOA, PSOBOA, HPSOBOA, LBOA, and IBOA

#### 5.1.3. Experiment 2: Comparison with Other Swarm Algorithms

#### 5.1.4. Performance Measures

#### 5.2. Comparison of the Parameter Settings of Ten Algorithms

#### 5.3. Results of Experiment 1

#### 5.4. Results of Experiment 2

_{6}, F

_{7}, F

_{10}, F

_{12}, F

_{13}, F

_{14}, F

_{17}, F

_{20}, F

_{21}, F

_{23}, and F

_{25}. For functions F

_{6}, F

_{7}, F

_{10}, F

_{12}, and F

_{23}, the hybrid HPSOBOA can obtain the optimal fitness value, which is close to other algorithms but slightly worse. However, for F

_{13}, F

_{14}, F

_{17}, F

_{20}, F

_{21}, and F

_{25}, the best solutions of these functions are searched by the other algorithms, such as GWO, PSO, MPA, and IBOA, and MPA obtains the best solution twice. Additionally, the IBOA also obtains the best solution twice, which is improved by the logistic map for the control parameters. Combining the comparison results in Table 5 and Table 6, we can see that the IBOA is better than others in the SR rank, which is set to $\epsilon <{10}^{-15},$ and is called the specified value, and the order of ten algorithms is IBOA > HPSOBOA > MPA > GWO > CABOA = LBOA > PSOBOA > PSO = SCA > BOA. The order of HPSOBOA and IBOA is only different once on the function F

_{17}, and the SR of HPSOBOA is 93.33%, but the SR of IBOA is 100% for searching the global optimization value, which is set to $\epsilon <{10}^{-15},$ and is accepted in this paper. Therefore, the performance of the proposed algorithm needs to be improved in future work.

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Visualization of implemented cubic map with $\rho $ in (1.5, 3) and $\rho =2.595,$ respectively.

**Figure 2.**Variation curve of different intensity coefficients and convergence curve of test function. (

**a**) Two control parameter strategies, (

**b**) Convergence curve of Schwefel 1.2, (

**c**) Convergence curve of Schwefel 1.2 with Dim = 100 for different parameter values setting.

**Figure 3.**Convergence curve for six algorithms with Dim = 100; the six test functions’ names are Schwefel 1.2, Sumsquare, Zakharov, Rastrigin, Ackley, and Alpine, respectively.

**Figure 4.**Convergence curve for six algorithms with Dim = 300; the six test functions’ names are Schwefel 1.2, Sumsquare, Zakharov, Rastrigin, Ackley, and Alpine, respectively. (

**a**) Schwefel 1.2, (

**b**) Sumsquare, (

**c**) Zakharov, (

**d**) Rastrigin, (

**e**) Ackley, (

**f**) Alpine.

**Figure 5.**Boxplot for the 30 times fitness of six test functions with Dim = 100 and Dim = 300. (

**a**) the three functions’ names are Schwefel 1.2, Sumsquare and Zakharov with Dim = 100; (

**b**) the three functions’ names are Rastrigin, Ackley, and Alpine Dim = 100; (

**c**) the three functions’ names are Schwefel 1.2, Sumsquare, and Zakharov with Dim = 300; (

**d**) the three functions’ names are Rastrigin, Ackley, and Alpine Dim = 300.

**Figure 6.**Boxplot for the algorithms run 30 times for the fitness of 26 test functions with Dim = 30.

Name | Formula of Functions | Dim | Range | Type | f_{min} |
---|---|---|---|---|---|

Sphere | ${F}_{1}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}{x}_{i}^{2}$ | 30 | [−100,100] | U | 0 |

Schwefel 2.22 | ${F}_{2}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}\left|{x}_{i}\right|+{\displaystyle {\displaystyle \prod}_{i=1}^{Dim}}\left|{x}_{i}\right|$ | 30 | [−10,10] | U | 0 |

Schwefel 1.2 | ${F}_{3}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}{\left({\displaystyle {\displaystyle \sum}_{j=i}^{i}}{x}_{j}\right)}^{2}$ | 30 | [−100,100] | U | 0 |

Schwefel 2.21 | ${F}_{4}=\mathrm{max}\left\{\left|{x}_{i}\right|,1\le i\le Dim\right\}$ | 30 | [−10,10] | U | 0 |

Step | ${F}_{5}={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}{\left({x}_{i}+0.5\right)}^{2}$ | 30 | [−10,10] | U | 0 |

Quartic | ${F}_{6}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}Dim\xb7{x}_{i}^{2}+rand\left(0,1\right)$ | 30 | [−1.28,1.28] | U | 0 |

Exponential | ${F}_{7}=exp\left(0.5{\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}{x}_{i}\right)$ | 30 | [−10,10] | U | 0 |

Sum power | ${F}_{8}={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}{\left|{x}_{i}\right|}^{\left(i+1\right)}$ | 30 | [−1,1] | U | 0 |

Sum square | ${F}_{9}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}\left(Dim\xb7{x}_{i}^{2}\right)$ | 30 | [−10,10] | U | 0 |

Rosenbrock | ${F}_{10}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}\left(100\left({x}_{i+1}-{x}_{i}^{2}\right)+{\left({x}_{i}-1\right)}^{2}\right)$ | 30 | [−5,10] | U | 0 |

Zakharov | ${F}_{11}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}{x}_{i}^{2}+{\left({\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}0.5i{x}_{i}\right)}^{2}+{\left({\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}0.5i{x}_{i}\right)}^{4}$ | 30 | [−5,10] | U | 0 |

Trid | ${F}_{12}\left(x\right)={\left({x}_{i}-1\right)}^{2}+{\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}i\xb7{\left(2{x}_{i}^{2}-{x}_{i-1}\right)}^{2}$ | 30 | [−10,10] | U | 0 |

Elliptic | ${F}_{13}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}{\left({10}^{6}\right)}^{\left(i-1\right)/\left(Dim-1\right)}\xb7{x}_{i}^{2}$ | 30 | [−100,100] | U | 0 |

Cigar | ${F}_{14}\left(x\right)={x}_{1}^{2}+{10}^{6}{\displaystyle {\displaystyle \sum}_{i=2}^{Dim}}{x}_{i}^{2}$ | 30 | [−100,100] | U | 0 |

Rastrigin | ${F}_{15}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]$ | 30 | [−5.12,5.12] | M | 0 |

NCRastrigin | ${F}_{16}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}\left[{y}_{i}^{2}-10\mathrm{cos}\left(2\pi {y}_{i}\right)+10\right],$ ${y}_{i}=\{\begin{array}{c}{x}_{i},,\left|{x}_{i}\right|<0.5\\ round\left(2{x}_{i}\right)/2,\left|{x}_{i}\right|>0.5\end{array}$ | 30 | [−5.12,5.12] | M | 0 |

Ackley | ${F}_{17}\left(x\right)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{Dim}{\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}{x}_{i}^{2}}\right)+exp\left(\frac{1}{Dim}{\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+exp\left(1\right)$ | 30 | [−50,50] | M | 0 |

Griewank | ${F}_{18}\left(x\right)=\frac{1}{4000}{\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}{x}_{i}^{2}-{\displaystyle {\displaystyle \prod}_{i=1}^{Dim}}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | 30 | [−600,600] | M | 0 |

Alpine | ${F}_{19}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}\left|{x}_{i}\xb7\mathrm{sin}\left({x}_{i}\right)+0.1{x}_{i}\right|$ | 30 | [−10,10] | M | 0 |

Penalized 1 | ${F}_{20}\left(x\right)=\frac{\pi}{Dim}\left\{{\displaystyle {\displaystyle \sum}_{i=1}^{Dim-1}}{\left({y}_{i}-1\right)}^{2}\left[1+10si{n}^{2}\left(\pi {y}_{i+1}\right)\right]+{\left({y}_{Dim-1}\right)}^{2}+10si{n}^{2}\left(\pi {y}_{1}\right)\right\}+{\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}u\left({x}_{i},10,100,4\right)$ ${y}_{i}=1+\left({x}_{i}+1\right)/4,{u}_{{y}_{i},a,k,m}=\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{m},{x}_{i}>a\\ 0,-a\le {x}_{i}\le a\\ k{\left(-{x}_{i}-a\right)}^{m},{x}_{i}<a\end{array}$ | 30 | [−100,100] | M | 0 |

Penalized 2 | ${F}_{21}\left(x\right)=\frac{1}{10}\left\{si{n}^{2}\left(\pi {x}_{1}\right)+{\displaystyle {\displaystyle \sum}_{i=1}^{Dim-1}}{\left({x}_{i}-1\right)}^{2}\left[1+si{n}^{2}\left(3\pi {x}_{i+1}\right)\right]+{\left({x}_{Dim-1}\right)}^{2}\left(1+si{n}^{2}\left(2\pi {x}_{i+1}\right)\right)\right\}+{\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}u\left({x}_{i},5,100,4\right)$ | 30 | [−100,100] | M | 0 |

Schwefel | ${F}_{22}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}\left|{x}_{i}\xb7\mathrm{sin}\left(\sqrt{\left|{x}_{i}\right|}\right)\right|$ | 30 | [−100,100] | M | 0 |

Levy | ${F}_{23}\left(x\right)=si{n}^{2}\left(3\pi {x}_{i}\right)+{\displaystyle {\displaystyle \sum}_{i=1}^{Dim-1}}{\left({x}_{i}-1\right)}^{2}\left[1+si{n}^{2}\left(3\pi {x}_{i+1}\right)\right]+\left|{x}_{Dim}-1\right|\xb7\left[1+si{n}^{2}\left(2\pi {x}_{Dim}\right)\right]$ | 30 | [−10,10] | M | 0 |

Weierstrass | ${F}_{24}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}\left({\displaystyle {\displaystyle \sum}_{k=0}^{{k}_{max}}}\left[{a}^{k}\mathrm{cos}\left(2\pi {b}^{k}\left({x}_{i}+0.5\right)\right)\right]\right)-Dim\xb7{\displaystyle {\displaystyle \sum}_{k=0}^{{k}_{max}}}\left[{a}^{k}\mathrm{cos}\left(2\pi {b}^{k}\xb70.5\right)\right],a=0.5,b=3,{k}_{max}=20$ | 30 | [−1,1] | M | 0 |

Solomon | ${F}_{25}\left(x\right)=1-cos\left(2\pi \sqrt{{\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}{x}_{i}^{2}}\right)+0.1\sqrt{{\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}{x}_{i}^{2}}$ | 30 | [−100,100] | M | 0 |

Bohachevsky | ${F}_{26}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{Dim}}\left[{x}_{i}^{2}+2{x}_{i+1}^{2}-0.3\xb7cos\left(3\pi {x}_{i}\right)\right]$ | 30 | [−10,10] | M | 0 |

NO. | Algorithms | Population Size | Parameter Settings |
---|---|---|---|

1 | Butterfly Optimization Algorithm (BOA) | 30 | a = 0.1, c(0) = 0.01, p = 0.6 |

2 | Butterfly Optimization Algorithm with Cubic map (CBOA) | 30 | a_{first} = 0.1, a_{final} = 0.3, c(0) = 0.01, p = 0.6, x(0) = 0.315, ρ = 0.295 |

3 | PSOBOA | 30 | a = 0.1, c(0) = 0.01, p = 0.6, c_{1} = c_{2} = 0.5 |

4 | Hybrid PSO with BOA and Cubic map (HPSOBOA) | 30 | a_{first} = 0.1, a_{final} = 0.3, c(0) = 0.01, p = 0.6, x(0) = 0.315, ρ = 0.295, c_{1} = c_{2} = 0.5 |

5 | Butterfly Optimization Algorithm with Lévy flights (LBOA) | 30 | a = 0.1, c(0) = 0.01, p = 0.6, λ = 1.5 |

6 | Improved Butterfly Optimization Algorithm (IBOA) | 30 | a(0) = 0.1, c(0) = 0.01, p = 0.6, r(0) = 0.33, μ = 4 |

7 | Particle Swarm Optimization (PSO) | 30 | c_{1} = c_{2} = 2, V_{max} = 1, V_{min} = −1, ω_{max} = 0.9, ω_{min} = 0.2 |

8 | Grey Wolf Optimizer (GWO) | 30 | a_{first} = 2, a_{final} = 0 |

9 | Sine Cosine Algorithm (SCA) | 30 | a = 2, r_{1}(0) = 2 |

10 | Marine Predators Algorithm (MPA) | 30 | a = 0.1, c(0) = 0.01, p = 0.6 |

Functions | BOA | CABOA | PSOBOA | HBOAPSO | LBOA | IBOA | BOA | CABOA | PSOBOA | HBOAPSO | LBOA | IBOA | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Dim = 100 | Dim = 300 | ||||||||||||

Schwefel 1.2 | Worst | 8.23 × 10^{−11} | 3.16 × 10^{−18} | 6.40 × 10^{−9} | 2.89 × 10^{−207} | 1.67 × 10^{−11} | 9.22 × 10^{−29} | 9.21 × 10^{−11} | 3.95 × 10^{−27} | 4.06 × 10^{−9} | 2.65 × 10^{−76} | 2.56 × 10^{−11} | 4.53 × 10^{−29} |

Best | 5.68 × 10^{−11} | 1.48 × 10^{−30} | 4.03 × 10^{−287} | 7.12 × 10^{−218} | 6.56 × 10^{−14} | 9.39 × 10^{−34} | 6.14 × 10^{−11} | 6.44 × 10^{−41} | 4.93 × 10^{−285} | 4.57 × 10^{−274} | 3.27 × 10^{−13} | 2.82 × 10^{−32} | |

Avg | 6.95 × 10^{−11} | 1.12 × 10^{−19} | 2.13 × 10^{−10} | 2.32 × 10^{−207} | 4.43 × 10^{−12} | 4.34 × 10^{−30} | 7.49 × 10^{−11} | 1.32 × 10^{−28} | 1.35 × 10^{−10} | 8.85 × 10^{−78} | 3.46 × 10^{−12} | 2.73 × 10^{−30} | |

Std | 6.15 × 10^{−12} | 5.76 × 10^{−19} | 1.17 × 10^{−9} | 0.00 × 10^{0} | 4.29 × 10^{−12} | 1.68 × 10^{−29} | 7.44 × 10^{−12} | 7.20 × 10^{−28} | 7.41 × 10^{−10} | 4.85 × 10^{−77} | 4.86 × 10^{−12} | 8.16 × 10^{−30} | |

rank | 5.97 | 3.97 | 1.97 | 1.17 | 4.97 | 2.97 | 5.97 | 2.80 | 1.87 | 1.63 | 4.97 | 3.77 | |

SR/% | 0.00 | 100.00 | 96.67 | 100.00 | 0.00 | 100.00 | 0.00 | 100.00 | 96.67 | 100.00 | 0.00 | 100.00 | |

Sumsquare | Worst | 1.07 × 10^{−10} | 2.33 × 10^{−12} | 2.98 × 10^{−9} | 5.82 × 10^{−20} | 1.16 × 10^{−11} | 1.35 × 10^{−29} | 1.06 × 10^{−10} | 2.50 × 10^{−12} | 5.21 × 10^{−9} | 8.92 × 10^{−24} | 1.18 × 10^{−11} | 4.98 × 10^{−30} |

Best | 6.71 × 10^{−11} | 4.45 × 10^{−19} | 3.42 × 10^{−294} | 3.47 × 10^{−294} | 1.34 × 10^{−14} | 9.55 × 10^{−34} | 7.33 × 10^{−11} | 1.00 × 10^{−16} | 9.50 × 10^{−272} | 1.35 × 10^{−292} | 2.87 × 10^{−15} | 5.72 × 10^{−33} | |

Avg | 8.63 × 10^{−11} | 2.14 × 10^{−13} | 1.01 × 10^{−10} | 1.94 × 10^{−21} | 3.20 × 10^{−12} | 1.35 × 10^{−30} | 8.95 × 10^{−11} | 1.93 × 10^{−13} | 1.98 × 10^{−10} | 2.97 × 10^{−25} | 3.03 × 10^{−12} | 1.01 × 10^{−30} | |

Std | 8.78 × 10^{−12} | 4.62 × 10^{−13} | 5.43 × 10^{−10} | 1.06 × 10^{−20} | 2.80 × 10^{−12} | 2.71 × 10^{−30} | 8.84 × 10^{−12} | 4.82 × 10^{−13} | 9.56 × 10^{−10} | 1.63 × 10^{−24} | 3.64 × 10^{−12} | 1.20 × 10^{−30} | |

rank | 5.97 | 3.93 | 1.90 | 1.43 | 4.93 | 2.83 | 5.93 | 3.90 | 1.70 | 1.77 | 4.90 | 2.80 | |

SR/% | 0.00 | 43.33 | 93.33 | 100.00 | 0.00 | 100.00 | 0.00 | 46.67 | 90.00 | 100.00 | 3.33 | 100.00 | |

Zakharov | Worst | 1.11 × 10^{−10} | 5.43 × 10^{−12} | 2.38 × 10^{−5} | 2.28 × 10^{−71} | 1.95 × 10^{−11} | 2.64 × 10^{−29} | 1.03 × 10^{−10} | 6.45 × 10^{−13} | 2.45 × 10^{−7} | 2.74 × 10^{−72} | 2.04 × 10^{−11} | 2.84 × 10^{−29} |

Best | 5.70 × 10^{−11} | 3.06 × 10^{−17} | 2.14 × 10^{−294} | 4.25 × 10^{−289} | 7.01 × 10^{−15} | 4.38 × 10^{−33} | 6.88 × 10^{−11} | 4.94 × 10^{−16} | 7.09 × 10^{−293} | 4.43 × 10^{−287} | 4.80 × 10^{−14} | 8.50 × 10^{−33} | |

Avg | 8.18 × 10^{−11} | 5.01 × 10^{−13} | 7.95 × 10^{−7} | 7.59 × 10^{−73} | 4.42 × 10^{−12} | 1.57 × 10^{−30} | 8.41 × 10^{−11} | 9.96 × 10^{−14} | 1.60 × 10^{−8} | 9.12 × 10^{−74} | 4.75 × 10^{−12} | 3.43 × 10^{−30} | |

Std | 1.13 × 10^{−11} | 1.21 × 10^{−12} | 4.35 × 10^{−6} | 4.16 × 10^{−72} | 4.89 × 10^{−12} | 4.85 × 10^{−30} | 8.57 × 10^{−12} | 1.61 × 10^{−13} | 6.11 × 10^{−8} | 5.00 × 10^{−73} | 5.45 × 10^{−12} | 6.09 × 10^{−30} | |

rank | 5.97 | 3.93 | 2.30 | 1.07 | 4.93 | 2.80 | 5.93 | 3.93 | 2.07 | 1.47 | 4.93 | 2.67 | |

SR/% | 0.00 | 43.33 | 86.67 | 100.00 | 3.33 | 100.00 | 0.00 | 36.67 | 86.67 | 100.00 | 0.00 | 100.00 | |

Rastrigin | Worst | 4.44 × 10^{−7} | 0.00 × 10^{0} | 1.39 × 10^{−9} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 2.36 × 10^{−7} | 0.00 × 10^{0} | 3.65 × 10^{−9} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} |

Best | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | |

Avg | 1.48 × 10^{−8} | 0.00 × 10^{0} | 4.65 × 10^{−11} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 7.88 × 10^{−9} | 0.00 × 10^{0} | 1.22 × 10^{−10} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | |

Std | 8.11 × 10^{−8} | 0.00 × 10^{0} | 2.55 × 10^{−10} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 4.32 × 10^{−8} | 0.00 × 10^{0} | 6.67 × 10^{−10} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | |

rank | 3.92 | 3.40 | 3.48 | 3.40 | 3.40 | 3.40 | 3.58 | 3.47 | 3.55 | 3.47 | 3.47 | 3.47 | |

SR/% | 83.33 | 100.00 | 96.67 | 100.00 | 100.00 | 100.00 | 96.67 | 100.00 | 96.67 | 100.00 | 100.00 | 100.00 | |

Ackley | Worst | 3.23 × 10^{−8} | 1.62 × 10^{−8} | 1.58 × 10^{−6} | 1.85 × 10^{−8} | 5.46 × 10^{−10} | 8.88 × 10^{−16} | 4.86 × 10^{−8} | 6.44 × 10^{−9} | 2.67 × 10^{−9} | 2.51 × 10^{−12} | 7.15 × 10^{−9} | 8.88 × 10^{−16} |

Best | 1.59 × 10^{−9} | 3.02 × 10^{−10} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 4.44 × 10^{−15} | 8.88 × 10^{−16} | 2.30 × 10^{−8} | 1.80 × 10^{−10} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 4.49 × 10^{−13} | 8.88 × 10^{−16} | |

Avg | 1.34 × 10^{−8} | 3.17 × 10^{−9} | 5.26 × 10^{−8} | 6.38 × 10^{−10} | 4.02 × 10^{−11} | 8.88 × 10^{−16} | 3.38 × 10^{−8} | 2.09 × 10^{−9} | 1.75 × 10^{−10} | 1.33 × 10^{−13} | 5.19 × 10^{−10} | 8.88 × 10^{−16} | |

Std | 8.14 × 10^{−9} | 3.96 × 10^{−9} | 2.88 × 10^{−7} | 3.38 × 10^{−9} | 1.20 × 10^{−10} | 0.00 × 10^{0} | 5.63 × 10^{−9} | 1.97 × 10^{−9} | 6.66 × 10^{−10} | 5.21 × 10^{−13} | 1.33 × 10^{−9} | 0.00 × 10^{0} | |

rank | 5.97 | 4.93 | 2.10 | 2.20 | 3.90 | 1.90 | 6.00 | 4.97 | 2.08 | 2.05 | 4.00 | 1.90 | |

SR/% | 0.00 | 0.00 | 96.67 | 86.67 | 13.33 | 100.00 | 0.00 | 0.00 | 90.00 | 93.33 | 0.00 | 100.00 | |

Alpine | Worst | 1.30 × 10^{−10} | 6.56 × 10^{−7} | 1.80 × 10^{−12} | 9.13 × 10^{−41} | 9.60 × 10^{−12} | 4.41 × 10^{−19} | 7.31 × 10^{−10} | 1.80 × 10^{−6} | 1.04 × 10^{−11} | 8.54 × 10^{−23} | 6.14 × 10^{−11} | 4.97 × 10^{−19} |

Best | 2.31 × 10^{−11} | 3.36 × 10^{−11} | 3.11 × 10^{−146} | 9.85 × 10^{−136} | 2.17 × 10^{−17} | 3.81 × 10^{−21} | 5.21 × 10^{−11} | 5.68 × 10^{−12} | 8.46 × 10^{−147} | 3.37 × 10^{−131} | 9.80 × 10^{−18} | 4.96 × 10^{−21} | |

Avg | 6.94 × 10^{−11} | 2.57 × 10^{−8} | 6.01 × 10^{−14} | 3.18 × 10^{−42} | 9.82 × 10^{−13} | 7.90 × 10^{−20} | 2.67 × 10^{−10} | 6.34 × 10^{−8} | 3.46 × 10^{−13} | 3.00 × 10^{−24} | 7.53 × 10^{−12} | 1.33 × 10^{−19} | |

Std | 3.01 × 10^{−11} | 1.20 × 10^{−7} | 3.29 × 10^{−13} | 1.67 × 10^{−41} | 2.21 × 10^{−12} | 9.09 × 10^{−20} | 1.76 × 10^{−10} | 3.28 × 10^{−7} | 1.89 × 10^{−12} | 1.56 × 10^{−23} | 1.22 × 10^{−11} | 1.48 × 10^{−19} | |

rank | 5.00 | 6.00 | 1.73 | 1.37 | 4.00 | 2.90 | 5.77 | 5.23 | 1.43 | 1.73 | 4.00 | 2.83 | |

SR/% | 0.00 | 0.00 | 96.67 | 100.00 | 43.33 | 100.00 | 0.00 | 0.00 | 96.67 | 100.00 | 10.00 | 100.00 | |

Avg.rank | 5.464 | 4.361 | 2.247 | 1.772 | 4.356 | 2.800 | 5.531 | 4.050 | 2.117 | 2.019 | 4.378 | 2.906 | |

Final rank | 6 | 5 | 2 | 1 | 4 | 3 | 6 | 4 | 2 | 1 | 5 | 3 |

Functions | BOA | CABOA | PSOBOA | HBOAPSO | LBOA | IBOA | PSO | GWO | SCA | MPA | |
---|---|---|---|---|---|---|---|---|---|---|---|

F_{1} | Avg | 7.78 × 10^{−11} | 1.01 × 10^{−13} | 1.68 × 10^{−10} | 3.74 × 10^{−104} | 3.92 × 10^{−12} | 1.61 × 10^{−30} | 1.11 × 10^{−5} | 6.20 × 10^{−28} | 1.39 × 10^{1} | 4.93 × 10^{−23} |

Std | 7.67 × 10^{−12} | 2.11 × 10^{−13} | 9.17 × 10^{−10} | 2.05 × 10^{−103} | 4.46 × 10^{−12} | 3.90 × 10^{−30} | 2.12 × 10^{−5} | 7.68 × 10^{−28} | 2.88 × 10^{1} | 7.29 × 10^{−23} | |

F_{2} | Avg | 2.23 × 10^{−8} | 1.25 × 10^{−14} | 4.14 × 10^{−10} | 2.63 × 10^{−22} | 1.38 × 10^{−9} | 5.11 × 10^{−19} | 3.35 × 10^{−3} | 1.04 × 10^{−16} | 1.87 × 10^{−2} | 2.99 × 10^{−13} |

Std | 7.12 × 10^{−9} | 2.15 × 10^{−14} | 2.27 × 10^{−9} | 1.44 × 10^{−21} | 2.08 × 10^{−9} | 1.73 × 10^{−18} | 2.18 × 10^{−3} | 8.66 × 10^{−17} | 3.66 × 10^{−2} | 2.56 × 10^{−13} | |

F_{3} | Avg | 6.34 × 10^{−11} | 6.30 × 10^{−13} | 8.05 × 10^{−17} | 3.04 × 10^{−71} | 2.74 × 10^{−12} | 6.15 × 10^{−31} | 1.23 × 10^{2} | 7.24 × 10^{−6} | 8.03 × 10^{3} | 1.52 × 10^{−4} |

Std | 5.70 × 10^{−12} | 1.37 × 10^{−12} | 4.41 × 10^{−16} | 1.67 × 10^{−70} | 2.44 × 10^{−12} | 1.16 × 10^{−30} | 5.98 × 10^{2} | 1.51 × 10^{−5} | 6.30 × 10^{3} | 3.15 × 10^{−4} | |

F_{4} | Avg | 2.59 × 10^{−8} | 2.77 × 10^{−10} | 9.39 × 10^{−8} | 3.61 × 10^{−46} | 2.30 × 10^{−9} | 1.36 × 10^{−19} | 1.85 × 10^{−1} | 8.57 × 10^{−8} | 3.77 × 10^{0} | 3.29 × 10^{−10} |

Std | 2.58 × 10^{−9} | 2.96 × 10^{−10} | 5.14 × 10^{−7} | 1.97 × 10^{−45} | 2.36 × 10^{−9} | 1.97 × 10^{−19} | 4.62 × 10^{−2} | 8.56 × 10^{−8} | 1.30 × 10^{0} | 2.23 × 10^{−10} | |

F_{5} | Avg | 5.17 × 10^{0} | 8.50 × 10^{−6} | 6.47 × 10^{0} | 4.17 × 10^{−2} | 3.52 × 10^{0} | 4.44 × 10^{0} | 3.69 × 10^{−6} | 6.84 × 10^{−1} | 4.85 × 10^{0} | 1.25 × 10^{−7} |

Std | 6.09 × 10^{−1} | 1.06 × 10^{−5} | 3.90 × 10^{−1} | 6.41 × 10^{−2} | 8.50 × 10^{−1} | 8.70 × 10^{−1} | 4.74 × 10^{−6} | 4.38 × 10^{−1} | 7.32 × 10^{−1} | 4.78 × 10^{−7} | |

F_{6} | Avg | 2.03 × 10^{−3} | 2.00 × 10^{−3} | 2.53 × 10^{−4} | 2.55 × 10^{−4} | 2.10 × 10^{−3} | 1.22 × 10^{−4} | 7.98 × 10^{−2} | 1.69 × 10^{−3} | 1.19 × 10^{−1} | 1.31 × 10^{−3} |

Std | 8.70 × 10^{−4} | 7.89 × 10^{−4} | 3.21 × 10^{−4} | 4.00 × 10^{−4} | 9.63 × 10^{−4} | 8.06 × 10^{−5} | 3.14 × 10^{−2} | 8.21 × 10^{−4} | 1.04 × 10^{−1} | 5.47 × 10^{−4} | |

F_{7} | Avg | 1.05 × 10^{−11} | 1.48 × 10^{−62} | 8.41 × 10^{−11} | 1.48 × 10^{−62} | 5.23 × 10^{−20} | 1.19 × 10^{−19} | 0.00 × 10^{0} | 5.10 × 10^{−58} | 1.38 × 10^{−40} | 7.18 × 10^{−66} |

Std | 4.21 × 10^{−11} | 6.67 × 10^{−63} | 2.94 × 10^{−10} | 6.70 × 10^{−63} | 1.41 × 10^{−19} | 5.94 × 10^{−19} | 0.00 × 10^{0} | 1.71 × 10^{−57} | 7.24 × 10^{−40} | 7.74 × 10^{−70} | |

F_{8} | Avg | 6.33 × 10^{−14} | 6.58 × 10^{−15} | 1.42 × 10^{−17} | 3.19 × 10^{−118} | 7.51 × 10^{−16} | 1.32 × 10^{−36} | 1.37 × 10^{−14} | 2.21 × 10^{−95} | 7.27 × 10^{−5} | 1.41 × 10^{−60} |

Std | 3.60 × 10^{−14} | 1.19 × 10^{−14} | 7.78 × 10^{−17} | 1.68 × 10^{−117} | 9.49 × 10^{−16} | 4.59 × 10^{−36} | 4.69 × 10^{−14} | 1.20 × 10^{−94} | 2.25 × 10^{−4} | 5.28 × 10^{−60} | |

F_{9} | Avg | 7.01 × 10^{−11} | 2.91 × 10^{−13} | 1.87 × 10^{−16} | 2.72 × 10^{−99} | 2.36 × 10^{−12} | 5.60 × 10^{−31} | 1.67 × 10^{−4} | 1.50 × 10^{−28} | 7.67 × 10^{−1} | 1.07 × 10^{−23} |

Std | 7.91 × 10^{−12} | 7.08 × 10^{−13} | 1.02 × 10^{−15} | 1.31 × 10^{−98} | 2.76 × 10^{−12} | 1.87 × 10^{−30} | 3.97 × 10^{−4} | 1.96 × 10^{−28} | 1.13 × 10^{0} | 1.36 × 10^{−23} | |

F_{10} | Avg | 2.89 × 10^{1} | 2.87 × 10^{1} | 2.90 × 10^{1} | 2.89 × 10^{1} | 2.88 × 10^{1} | 2.89 × 10^{1} | 2.67 × 10^{1} | 2.68 × 10^{1} | 4.19 × 10^{1} | 2.53 × 10^{1} |

Std | 2.54 × 10^{−2} | 1.39 × 10^{−5} | 2.16 × 10^{−2} | 8.18 × 10^{−2} | 3.18 × 10^{−2} | 3.40 × 10^{−2} | 1.34 × 10^{0} | 7.02 × 10^{−1} | 4.32 × 10^{1} | 3.86 × 10^{−1} | |

F_{11} | Avg | 6.72 × 10^{−11} | 2.37 × 10^{−14} | 1.32 × 10^{−8} | 3.64 × 10^{−78} | 2.78 × 10^{−12} | 1.10 × 10^{−30} | 9.02 × 10^{−5} | 2.24 × 10^{−28} | 8.79 × 10^{0} | 1.09 × 10^{−23} |

Std | 6.90 × 10^{−12} | 4.24 × 10^{−14} | 6.84 × 10^{−8} | 1.99 × 10^{−77} | 2.63 × 10^{−12} | 2.90 × 10^{−30} | 1.05 × 10^{−4} | 3.10 × 10^{−28} | 1.74 × 10^{1} | 2.38 × 10^{−23} | |

F_{12} | Avg | 9.72 × 10^{−1} | 4.77 × 10^{−1} | 9.91 × 10^{−1} | 9.75 × 10^{−1} | 9.34 × 10^{−1} | 9.71 × 10^{−1} | 7.66 × 10^{−1} | 6.67 × 10^{−1} | 5.86 × 10^{2} | 6.67 × 10^{−1} |

Std | 1.14 × 10^{−2} | 3.45 × 10^{−1} | 5.24 × 10^{−3} | 8.45 × 10^{−2} | 2.11 × 10^{−2} | 7.36 × 10^{−3} | 3.39 × 10^{−1} | 2.62 × 10^{−6} | 2.29 × 10^{3} | 5.38 × 10^{−8} | |

F_{13} | Avg | 1.16 × 10^{−20} | 4.17 × 10^{−30} | 6.44 × 10^{−24} | 5.73 × 10^{−92} | 1.20 × 10^{−24} | 3.03 × 10^{−35} | 7.70 × 10^{−77} | 0.00 × 10^{0} | 2.43 × 10^{−96} | 1.97 × 10^{−162} |

Std | 6.14 × 10^{−20} | 2.18 × 10^{−29} | 3.00 × 10^{−23} | 3.14 × 10^{−91} | 5.29 × 10^{−24} | 6.42 × 10^{−35} | 3.27 × 10^{−76} | 0.00 × 10^{0} | 1.31 × 10^{−95} | 1.09 × 10^{−161} | |

F_{14} | Avg | 6.51 × 10^{−17} | 2.24 × 10^{−23} | 7.15 × 10^{−15} | 1.28 × 10^{−63} | 4.71 × 10^{−18} | 8.45 × 10^{−31} | 7.38 × 10^{−61} | 8.59 × 10^{−201} | 6.47 × 10^{−67} | 1.07 × 10^{−63} |

Std | 1.39 × 10^{−16} | 7.51 × 10^{−23} | 3.92 × 10^{−14} | 7.04 × 10^{−63} | 7.50 × 10^{−18} | 2.52 × 10^{−30} | 3.81 × 10^{−60} | 0.00 × 10^{0} | 3.37 × 10^{−66} | 5.84 × 10^{−63} | |

F_{15} | Avg | 2.51 × 10^{1} | 0.00 × 10^{0} | 1.10 × 10^{1} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 4.56 × 10^{1} | 3.36 × 10^{0} | 4.42 × 10^{1} | 0.00 × 10^{0} |

Std | 6.52 × 10^{1} | 0.00 × 10^{0} | 4.22 × 10^{1} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 1.11 × 10^{1} | 4.42 × 10^{0} | 3.71 × 10^{1} | 0.00 × 10^{0} | |

F_{16} | Avg | 9.36 × 10^{1} | 0.00 × 10^{0} | 2.00 × 10^{1} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 4.48 × 10^{1} | 8.20 × 10^{0} | 7.08 × 10^{1} | 1.01 × 10^{−8} |

Std | 8.04 × 10^{1} | 0.00 × 10^{0} | 5.23 × 10^{1} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 9.04 × 10^{0} | 5.18 × 10^{0} | 4.45 × 10^{1} | 4.72 × 10^{−8} | |

F_{17} | Avg | 1.09 × 10^{−9} | 1.84 × 10^{−9} | 5.63 × 10^{−8} | 8.96 × 10^{−11} | 2.34 × 10^{−12} | 8.88 × 10^{−16} | 1.69 × 10^{−3} | 2.79 × 10^{0} | 2.03 × 10^{1} | 1.06 × 10^{−3} |

Std | 8.16 × 10^{−10} | 1.76 × 10^{−9} | 3.06 × 10^{−7} | 4.73 × 10^{−10} | 7.87 × 10^{−12} | 0.00 × 10^{0} | 1.32 × 10^{−3} | 7.22 × 10^{0} | 5.27 × 10^{−2} | 5.83 × 10^{−3} | |

F_{18} | Avg | 7.64 × 10^{−12} | 1.70 × 10^{−14} | 2.61 × 10^{−8} | 0.00 × 10^{0} | 3.48 × 10^{−13} | 0.00 × 10^{0} | 5.33 × 10^{−3} | 1.31 × 10^{−3} | 2.17 × 10^{−1} | 0.00 × 10^{0} |

Std | 6.94 × 10^{−12} | 1.82 × 10^{−14} | 1.35 × 10^{−7} | 0.00 × 10^{0} | 8.78 × 10^{−13} | 0.00 × 10^{0} | 7.48 × 10^{−3} | 4.99 × 10^{−3} | 2.13 × 10^{−1} | 0.00 × 10^{0} | |

F_{19} | Avg | 1.90 × 10^{−10} | 6.76 × 10^{−6} | 4.77 × 10^{−7} | 2.54 × 10^{−45} | 6.32 × 10^{−14} | 8.93 × 10^{−20} | 1.15 × 10^{−3} | 5.15 × 10^{−4} | 3.02 × 10^{−1} | 2.12 × 10^{−14} |

Std | 1.00 × 10^{−10} | 3.13 × 10^{−5} | 1.78 × 10^{−6} | 1.39 × 10^{−44} | 1.73 × 10^{−13} | 1.19 × 10^{−19} | 9.36 × 10^{−4} | 7.29 × 10^{−4} | 5.38 × 10^{−1} | 1.52 × 10^{−14} | |

F_{20} | Avg | 5.56 × 10^{−1} | 1.90 × 10^{−4} | 8.75 × 10^{−1} | 2.84 × 10^{−3} | 3.06 × 10^{−1} | 4.97 × 10^{−1} | 4.44 × 10^{0} | 4.74 × 10^{−2} | 1.17 × 10^{6} | 5.79 × 10^{−5} |

Std | 1.40 × 10^{−1} | 4.90 × 10^{−4} | 2.11 × 10^{−1} | 3.79 × 10^{−3} | 9.96 × 10^{−2} | 1.37 × 10^{−1} | 2.62 × 10^{0} | 2.27 × 10^{−2} | 2.83 × 10^{6} | 3.17 × 10^{−4} | |

F_{21} | Avg | 3.52 × 10^{0} | 1.36 × 10^{−2} | 4.42 × 10^{0} | 3.93 × 10^{−2} | 2.41 × 10^{0} | 3.15 × 10^{0} | 1.89 × 10^{−6} | 9.27 × 10^{−1} | 3.48 × 10^{6} | 1.38 × 10^{−2} |

Std | 5.92 × 10^{−1} | 4.57 × 10^{−2} | 7.32 × 10^{−1} | 4.57 × 10^{−2} | 5.15 × 10^{−1} | 4.40 × 10^{−1} | 3.64 × 10^{−6} | 2.80 × 10^{−1} | 6.37 × 10^{6} | 3.69 × 10^{−2} | |

F_{22} | Avg | 9.76 × 10^{0} | 5.52 × 10^{−16} | 3.42 × 10^{−7} | 8.38 × 10^{−77} | 1.56 × 10^{−3} | 3.56 × 10^{−26} | 6.71 × 10^{−4} | 5.34 × 10^{−1} | 1.42 × 10^{1} | 1.11 × 10^{−1} |

Std | 7.84 × 10^{0} | 4.56 × 10^{−16} | 1.88 × 10^{−6} | 4.30 × 10^{−76} | 8.54 × 10^{−3} | 7.84 × 10^{−26} | 2.74 × 10^{−3} | 4.25 × 10^{−1} | 1.87 × 10^{0} | 1.72 × 10^{−1} | |

F_{23} | Avg | 1.17 × 10^{1} | 4.35 × 10^{−4} | 1.75 × 10^{1} | 7.28 × 10^{−2} | 8.42 × 10^{0} | 9.83 × 10^{0} | 4.77 × 10^{−2} | 1.42 × 10^{0} | 1.74 × 10^{1} | 1.35 × 10^{−1} |

Std | 2.66 × 10^{0} | 4.66 × 10^{−4} | 3.79 × 10^{0} | 1.87 × 10^{−1} | 2.56 × 10^{0} | 2.47 × 10^{0} | 6.58 × 10^{−2} | 1.13 × 10^{0} | 3.58 × 10^{0} | 1.13 × 10^{−1} | |

F_{24} | Avg | 6.21 × 10^{−1} | 0.00 × 10^{0} | 3.36 × 10^{−11} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 9.15 × 10^{−1} | 5.05 × 10^{0} | 9.91 × 10^{0} | 0.00 × 10^{0} |

Std | 1.95 × 10^{0} | 0.00 × 10^{0} | 1.84 × 10^{−10} | 0.00 × 10^{0} | 0.00 × 10^{0} | 0.00 × 10^{0} | 1.40 × 10^{0} | 2.35 × 10^{0} | 2.00 × 10^{0} | 0.00 × 10^{0} | |

F_{25} | Avg | 7.65 × 10^{−1} | 7.30 × 10^{−2} | 1.41 × 10^{−1} | 2.53 × 10^{−8} | 3.65 × 10^{−2} | 2.25 × 10^{−32} | 1.15 × 10^{0} | 3.48 × 10^{−1} | 1.66 × 10^{0} | 9.95 × 10^{−2} |

Std | 2.21 × 10^{−1} | 4.47 × 10^{−2} | 1.21 × 10^{−1} | 1.38 × 10^{−7} | 4.88 × 10^{−2} | 5.88 × 10^{−32} | 3.41 × 10^{−1} | 1.13 × 10^{−1} | 2.15 × 10^{0} | 8.20 × 10^{−17} | |

F_{26} | Avg | 7.96 × 10^{−11} | 6.54 × 10^{−15} | 6.50 × 10^{−12} | 0.00 × 10^{0} | 3.22 × 10^{−12} | 0.00 × 10^{0} | 1.00 × 10^{−1} | 0.00 × 10^{0} | 7.60 × 10^{−1} | 0.00 × 10^{0} |

Std | 8.77 × 10^{−12} | 1.52 × 10^{−14} | 3.54 × 10^{−11} | 0.00 × 10^{0} | 2.68 × 10^{−12} | 0.00 × 10^{0} | 3.29 × 10^{−1} | 0.00 × 10^{0} | 1.27 × 10^{0} | 0.00 × 10^{0} |

Functions | BOA | CABOA | PSOBOA | HBOAPSO | LBOA | IBOA | PSO | GWO | SCA | MPA |
---|---|---|---|---|---|---|---|---|---|---|

F_{1} | 0.00 | 43.33 | 93.33 | 100.00 | 0.00 | 100.00 | 0.00 | 100.00 | 0.00 | 100.00 |

F_{2} | 0.00 | 76.67 | 86.67 | 100.00 | 0.00 | 100.00 | 0.00 | 100.00 | 0.00 | 0.00 |

F_{3} | 0.00 | 30.00 | 100.00 | 100.00 | 3.33 | 100.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F_{4} | 0.00 | 0.00 | 90.00 | 100.00 | 0.00 | 100.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F_{5} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F_{6} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F_{7} | 0.00 | 100.00 | 56.67 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 |

F_{8} | 0.00 | 80.00 | 100.00 | 100.00 | 100.00 | 100.00 | 80.00 | 100.00 | 0.00 | 100.00 |

F_{9} | 0.00 | 56.67 | 100.00 | 100.00 | 3.33 | 100.00 | 0.00 | 100.00 | 0.00 | 100.00 |

F_{10} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F_{11} | 0.00 | 63.33 | 90.00 | 100.00 | 0.00 | 100.00 | 0.00 | 100.00 | 0.00 | 100.00 |

F_{12} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F_{13} | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 |

F_{14} | 100.00 | 100.00 | 96.67 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 | 100.00 |

F_{15} | 50.00 | 100.00 | 86.67 | 100.00 | 100.00 | 100.00 | 0.00 | 0.00 | 0.00 | 100.00 |

F_{16} | 3.33 | 100.00 | 90.00 | 100.00 | 100.00 | 100.00 | 0.00 | 0.00 | 0.00 | 50.00 |

F_{17} | 0.00 | 0.00 | 83.33 | 93.33 | 30.00 | 100.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F_{18} | 0.00 | 46.67 | 93.33 | 100.00 | 23.33 | 100.00 | 0.00 | 93.33 | 0.00 | 100.00 |

F_{19} | 0.00 | 0.00 | 86.67 | 100.00 | 56.67 | 100.00 | 0.00 | 33.33 | 0.00 | 23.33 |

F_{20} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F_{21} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F_{22} | 16.67 | 100.00 | 90.00 | 100.00 | 96.67 | 100.00 | 0.00 | 0.00 | 0.00 | 13.33 |

F_{23} | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F_{24} | 23.33 | 100.00 | 96.67 | 100.00 | 100.00 | 100.00 | 0.00 | 0.00 | 0.00 | 100.00 |

F_{25} | 0.00 | 0.00 | 10.00 | 100.00 | 30.00 | 100.00 | 0.00 | 0.00 | 0.00 | 0.00 |

F_{26} | 0.00 | 86.67 | 93.33 | 100.00 | 3.33 | 100.00 | 0.00 | 100.00 | 0.00 | 100.00 |

times | 2 | 7 | 4 | 18 | 7 | 19 | 3 | 9 | 3 | 11 |

SR rank | 8 | 5 | 6 | 2 | 5 | 1 | 7 | 4 | 7 | 3 |

Rank | BOA | CABOA | PSOBOA | HBOAPSO | LBOA | IBOA | PSO | GWO | SCA | MPA |
---|---|---|---|---|---|---|---|---|---|---|

F_{1} | 7.97 | 5.93 | 2.10 | 1.53 | 6.97 | 2.83 | 9.00 | 3.83 | 10.00 | 4.83 |

F_{2} | 8.00 | 4.87 | 1.93 | 1.63 | 6.97 | 2.83 | 9.03 | 3.87 | 9.97 | 5.90 |

F_{3} | 6.00 | 4.00 | 1.70 | 1.33 | 5.00 | 2.97 | 9.00 | 7.03 | 10.00 | 7.97 |

F_{4} | 7.17 | 4.07 | 1.83 | 1.47 | 5.97 | 2.87 | 9.00 | 7.77 | 10.00 | 4.87 |

F_{5} | 8.83 | 2.93 | 10.00 | 4.07 | 6.00 | 7.13 | 2.07 | 4.93 | 8.03 | 1.00 |

F_{6} | 6.87 | 6.67 | 2.43 | 1.77 | 7.07 | 1.80 | 9.47 | 5.27 | 9.53 | 4.13 |

F_{7} | 9.53 | 3.68 | 9.43 | 3.65 | 8.00 | 7.03 | 1.00 | 4.53 | 6.00 | 2.13 |

F_{8} | 8.97 | 6.97 | 1.90 | 1.60 | 6.20 | 4.93 | 7.87 | 2.73 | 10.00 | 3.83 |

F_{9} | 8.00 | 6.00 | 1.90 | 1.30 | 7.00 | 2.93 | 9.00 | 3.93 | 10.00 | 4.93 |

F_{10} | 7.70 | 3.90 | 9.13 | 6.97 | 5.30 | 6.30 | 2.97 | 2.63 | 9.07 | 1.03 |

F_{11} | 7.90 | 5.90 | 2.67 | 1.03 | 6.90 | 2.87 | 9.00 | 3.87 | 10.00 | 4.87 |

F_{12} | 6.23 | 2.97 | 7.83 | 8.60 | 4.70 | 6.17 | 4.00 | 2.77 | 10.00 | 1.73 |

F_{13} | 10.00 | 7.20 | 5.03 | 3.13 | 8.93 | 7.40 | 5.70 | 1.00 | 4.57 | 2.03 |

F_{14} | 9.97 | 7.57 | 3.83 | 2.07 | 8.97 | 7.30 | 5.67 | 1.50 | 4.33 | 3.80 |

F_{15} | 5.90 | 3.68 | 4.28 | 3.68 | 3.68 | 3.68 | 9.47 | 7.80 | 9.13 | 3.68 |

F_{16} | 8.90 | 3.18 | 4.02 | 3.18 | 3.18 | 3.18 | 8.53 | 7.13 | 9.07 | 4.62 |

F_{17} | 6.60 | 7.57 | 2.47 | 2.12 | 3.57 | 1.98 | 8.83 | 6.10 | 9.87 | 5.90 |

F_{18} | 7.87 | 5.82 | 3.30 | 2.97 | 6.75 | 2.97 | 9.00 | 3.37 | 10.00 | 2.97 |

F_{19} | 6.33 | 7.33 | 2.23 | 1.37 | 4.17 | 2.80 | 9.00 | 6.77 | 10.00 | 5.00 |

F_{20} | 7.03 | 2.00 | 8.03 | 3.00 | 5.03 | 6.03 | 8.87 | 4.00 | 10.00 | 1.00 |

F_{21} | 8.00 | 2.83 | 9.00 | 3.97 | 6.00 | 7.00 | 1.50 | 5.00 | 10.00 | 1.70 |

F_{22} | 8.40 | 4.93 | 2.30 | 1.03 | 4.03 | 2.87 | 6.53 | 8.33 | 9.73 | 6.83 |

F_{23} | 8.00 | 1.10 | 9.70 | 2.27 | 6.07 | 6.93 | 2.80 | 4.97 | 9.30 | 3.87 |

F_{24} | 6.43 | 3.58 | 3.68 | 3.58 | 3.58 | 3.58 | 7.97 | 9.00 | 10.00 | 3.58 |

F_{25} | 8.77 | 4.80 | 5.90 | 1.07 | 3.47 | 2.00 | 9.03 | 6.77 | 9.20 | 4.00 |

F_{26} | 7.97 | 4.60 | 3.53 | 3.23 | 6.97 | 3.23 | 9.00 | 3.23 | 10.00 | 3.23 |

Avg-rank | 7.82 | 4.77 | 4.62 | 2.75 | 5.79 | 4.29 | 7.05 | 4.93 | 9.15 | 3.83 |

Final rank | 9 | 5 | 4 | 1 | 7 | 3 | 8 | 6 | 10 | 2 |

Ranksum | BOA | CABOA | PSOBOA | LBOA | IBOA | PSO | GWO | SCA | MPA |
---|---|---|---|---|---|---|---|---|---|

F_{1} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 0.035137 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F_{2} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 0.325527 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F_{3} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 0.001597 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F_{4} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 0.014412 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F_{5} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 1.31 × 10^{−8} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F_{6} | 4.20 × 10^{−10} | 2.15 × 10^{−10} | 0.200949 | 1.33 × 10^{−10} | 0.520145 | 3.02 × 10^{−11} | 7.38 × 10^{−10} | 3.02 × 10^{−11} | 2.44 × 10^{−9} |

F_{7} | 5.18 × 10^{−12} | 1.00 × 10^{0} | 5.18 × 10^{−12} | 5.18 × 10^{−12} | 5.16 × 10^{−12} | 1.19 × 10^{−13} | 0.009689 | 5.18 × 10^{−12} | 9.85 × 10^{−11} |

F_{8} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 0.122353 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F_{9} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 0.001302 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F_{10} | 0.340288 | 3.02 × 10^{−11} | 3.16 × 10^{−5} | 0.003671 | 0.200949 | 2.20 × 10^{−7} | 4.50 × 10^{−11} | 1.34 × 10^{−5} | 3.02 × 10^{−11} |

F_{11} | 0.340288 | 3.02 × 10^{−11} | 3.16 × 10^{−5} | 0.003671 | 0.200949 | 2.20 × 10^{−7} | 4.50 × 10^{−11} | 1.34 × 10^{−5} | 3.02 × 10^{−11} |

F_{12} | 8.48 × 10^{−9} | 4.69 × 10^{−8} | 4.12 × 10^{−6} | 8.48 × 10^{−9} | 8.48 × 10^{−9} | 1.43 × 10^{−8} | 5.57 × 10^{−10} | 3.02 × 10^{−11} | 5.57 × 10^{−10} |

F_{13} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 0.00557 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 1.21 × 10^{−12} | 1.29 × 10^{−9} | 2.53 × 10^{−4} |

F_{14} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 0.001953 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 4.62 × 10^{−10} | 0.09049 | 7.69 × 10^{−8} | 9.06 × 10^{−8} |

F_{15} | 1.27 × 10^{−5} | NaN | 0.041926 | NaN | NaN | 1.21 × 10^{−12} | 1.19 × 10^{−12} | 1.21 × 10^{−12} | NaN |

F_{16} | 1.21 × 10^{−12} | NaN | 0.041926 | NaN | NaN | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.27 × 10^{−5} |

F_{17} | 6.03 × 10^{−11} | 2.89 × 10^{−11} | 0.248673 | 5.93 × 10^{−7} | 0.160802 | 2.37 × 10^{−12} | 2.80 × 10^{−10} | 2.37 × 10^{−12} | 6.24 × 10^{−10} |

F_{18} | 1.21 × 10^{−12} | 4.57 × 10^{−12} | 0.160802 | 4.57 × 10^{−12} | NaN | 1.21 × 10^{−12} | 0.160802 | 1.21 × 10^{−12} | NaN |

F_{19} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 0.003671 | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F_{20} | 3.02 × 10^{−11} | 3.08 × 10^{−8} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.34 × 10^{−11} | 3.02 × 10^{−11} | 1.09 × 10^{−10} |

F_{21} | 3.02 × 10^{−11} | 9.51 × 10^{−6} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 4.12 × 10^{−6} |

F_{22} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 1.58 × 10^{−4} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} |

F_{23} | 3.02 × 10^{−11} | 1.39 × 10^{−6} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 1.11 × 10^{−4} | 2.37 × 10^{−10} | 3.02 × 10^{−11} | 1.86 × 10^{−6} |

F_{24} | 1.95 × 10^{−9} | NaN | 0.333711 | NaN | NaN | 1.21 × 10^{−12} | 1.21 × 10^{−12} | 1.21 × 10^{−12} | NaN |

F_{25} | 3.02 × 10^{−11} | 5.49 × 10^{−11} | 1.09 × 10^{−10} | 8.89 × 10^{−10} | 8.48 × 10^{−9} | 1.90 × 10^{−11} | 3.02 × 10^{−11} | 3.02 × 10^{−11} | 1.55 × 10^{−11} |

F_{26} | 1.21 × 10^{−12} | 2.93 × 10^{−5} | 0.160802 | 1.21 × 10^{−12} | NaN | 1.21 × 10^{−12} | NaN | 1.21 × 10^{−12} | NaN |

H | BOA | CABOA | PSOBOA | LBOA | IBOA | PSO | GWO | SCA | MPA |
---|---|---|---|---|---|---|---|---|---|

F_{1} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{2} | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{3} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{4} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{5} | 1 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |

F_{6} | 1 | 0 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{7} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{8} | 1 | 1 | 0 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{9} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{10} | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |

F_{11} | 0 | 1 | 1 | 1 | 0 | 1 | 1 | 1 | 1 |

F_{12} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{13} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{14} | 1 | 1 | 1 | 1 | 1 | 1 | 0 | 1 | 1 |

F_{15} | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 0 |

F_{16} | 1 | 0 | 1 | 0 | 0 | 1 | 1 | 1 | 1 |

F_{17} | 1 | 1 | 0 | 1 | 0 | 1 | 1 | 1 | 1 |

F_{18} | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

F_{19} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{20} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{21} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{22} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{23} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{24} | 1 | 0 | 0 | 0 | 0 | 1 | 1 | 1 | 0 |

F_{25} | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 | 1 |

F_{26} | 1 | 1 | 0 | 1 | 0 | 1 | 0 | 1 | 0 |

t-tset | BOA | CABOA | PSOBOA | LBOA | IBOA | PSO | GWO | SCA | MPA |
---|---|---|---|---|---|---|---|---|---|

F_{1} | 6.6456 | 6.6456 | 2.1068 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 |

F_{2} | 6.6456 | 6.6456 | 0.9832 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 |

F_{3} | 6.6456 | 6.6456 | 3.1565 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 |

F_{4} | 6.6456 | 6.6456 | 2.4468 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 |

F_{5} | 6.6456 | −6.6456 | 6.6456 | 6.6456 | 6.6456 | −6.6456 | 5.6846 | 6.6456 | −6.6456 |

F_{6} | 6.2464 | 6.3499 | 1.2789 | 6.4238 | 0.6431 | 6.6456 | 6.1577 | 6.6456 | 5.9655 |

F_{7} | 6.9005 | 0.0000 | 6.9005 | 6.9005 | 6.9010 | −7.4180 | 2.5867 | 6.9005 | −6.4692 |

F_{8} | 6.6456 | 6.6456 | 1.5450 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 |

F_{9} | 6.6456 | 6.6456 | 3.2156 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 |

F_{10} | 0.9536 | −6.6456 | 4.1618 | −2.9051 | −1.2789 | −5.1819 | −6.5865 | 4.3540 | −6.6456 |

F_{11} | 6.6456 | 6.6456 | 3.7183 | 6.6456 | 6.6456 | 6.6456 | −6.5865 | 6.6456 | 6.6456 |

F_{12} | −5.7585 | −5.4628 | −4.6053 | −5.7585 | −5.7585 | −5.6698 | −6.2021 | 6.6456 | −6.2021 |

F_{13} | 6.6456 | 6.6456 | 2.7721 | 6.6456 | 6.6456 | 6.6456 | −7.1040 | 6.0690 | −3.6591 |

F_{14} | 6.6456 | 6.6456 | 3.0973 | 6.6456 | 6.6456 | 6.2316 | −1.6928 | 5.3741 | 5.3446 |

F_{15} | 4.3649 | NaN | 2.0343 | NaN | NaN | 7.1040 | 7.1063 | 7.1040 | NaN |

F_{16} | 7.1040 | NaN | 2.0343 | NaN | NaN | 7.1040 | 7.1040 | 7.1040 | 4.3650 |

F_{17} | 6.5431 | 6.6523 | 1.1536 | 4.9936 | −1.4024 | 7.0110 | 6.3094 | 7.0110 | 6.1844 |

F_{18} | 7.1040 | 6.9183 | 1.4024 | 6.9182 | NaN | 7.1040 | 1.4024 | 7.1040 | NaN |

F_{19} | 6.6456 | 6.6456 | 2.9051 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 |

F_{20} | 6.6456 | −5.5368 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6308 | 6.6456 | −6.4534 |

F_{21} | 6.6456 | −4.4279 | 6.6456 | 6.6456 | 6.6456 | −6.6456 | 6.6456 | 6.6456 | −4.6053 |

F_{22} | 6.6456 | 6.6456 | 3.7774 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 | 6.6456 |

F_{23} | 6.6456 | −4.8271 | 6.6456 | 6.6456 | 6.6456 | 3.8661 | 6.3351 | 6.6456 | 4.7680 |

F_{24} | 6.0023 | NaN | 0.9667 | NaN | NaN | 7.1040 | 7.1040 | 7.1040 | NaN |

F_{25} | 6.6456 | 6.5569 | 6.4534 | 6.1281 | 5.7585 | 6.7136 | 6.6456 | 6.6456 | 6.7434 |

F_{26} | 7.1040 | 4.1785 | 1.4024 | 7.1040 | NaN | 7.1040 | NaN | 7.1040 | NaN |

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## Share and Cite

**MDPI and ACS Style**

Zhang, M.; Long, D.; Qin, T.; Yang, J.
A Chaotic Hybrid Butterfly Optimization Algorithm with Particle Swarm Optimization for High-Dimensional Optimization Problems. *Symmetry* **2020**, *12*, 1800.
https://doi.org/10.3390/sym12111800

**AMA Style**

Zhang M, Long D, Qin T, Yang J.
A Chaotic Hybrid Butterfly Optimization Algorithm with Particle Swarm Optimization for High-Dimensional Optimization Problems. *Symmetry*. 2020; 12(11):1800.
https://doi.org/10.3390/sym12111800

**Chicago/Turabian Style**

Zhang, Mengjian, Daoyin Long, Tao Qin, and Jing Yang.
2020. "A Chaotic Hybrid Butterfly Optimization Algorithm with Particle Swarm Optimization for High-Dimensional Optimization Problems" *Symmetry* 12, no. 11: 1800.
https://doi.org/10.3390/sym12111800