# An Improved Moth-Flame Optimization Algorithm for Engineering Problems

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## Abstract

**:**

## 1. Introduction

## 2. Moth-Flame Optimization Algorithm (MFO)

#### 2.1. Biological Background of Moth-Flame Optimization Algorithm

#### 2.2. Basic Model of Moth-Flame Optimization Algorithm

- (1)
- The initial point of the helical function is selected from the initial space position of the moth;
- (2)
- The end point of the spiral is the space position corresponding to the contemporary flame;
- (3)
- The fluctuation range of the spiral should not exceed its search space.

- (1)
- By randomly selecting parameters $t$, a moth can converge to any field of flame;
- (2)
- The smaller the value of $t$, the closer the moth is to the flame;
- (3)
- As the moth gets closer and closer to the flame, its position around the flame is updated more and more rapidly.

## 3. An Improved Moth-Flame Optimization Algorithm (IMFO)

#### 3.1. Lévy Flight

_{i}is the ith moth, F

_{j}is the jth flame, and D

_{i}is the distance between the ith moth and the jth flame. When the moth spiral flight updates its position, the addition of the Lévy flight mechanism can expand the search range of the moth and prevent it from falling into local optimization. The formula of Lévy flight is as follows [28]:

_{1}and r

_{2}are random numbers between [0,1], $\phi $ is a constant 1.5, and the $\delta $ formula is as follows:

#### 3.2. Dimension-By-Dimension Evaluation Strategy

Algorithm 1 IMFO Algorithm Pseudo-Code |

1: Set population size N, maximum number of iterations ${T}_{max}$, and dimension of objective function Dim |

2: Initializes the moth position |

3: while (t < ${T}_{max}$) do |

4: Update the flame position according to Equation (9) |

5: Transboundary treatment of moths |

6: if $i=1$ |

7: find the moth and update the position of the flame according to the moth |

8: else |

9: find the moth and update the position of the flame according to the moth |

10: end |

11: Update according to Equation (8) |

12: for $i=1:size\left(Moth\_pos,1\right)$ |

13: for $j=1:size\left(Moth\_pos,2\right)$ |

14: if $i\le flame\_no$ |

15: Update Di according to Equation (9) |

16: Update S (Mi, Fj) according to Equation (11) |

17: end |

18: if $i>flame\_no$ |

19: Update Di according to Equation (9) |

20: Update S (Mi, Fj) according to Equation (6) |

21: end |

22: end |

23: end |

24: Update moth position and fitness value through dimension-by-dimension evaluation |

25: t = t +1 |

26: end while |

27: Output the best search location and its fitness value |

## 4. Experimental Studies and Comparisons

#### 4.1. Test Function and Experimental Parameter Setting

_{min}represents the theoretical optimal value.

#### 4.2. Comparison of Algorithm Parameter Settings

#### 4.3. Comparison with Other Algorithms

#### 4.4. Convergence Test

#### 4.5. Statistical Analysis

## 5. IMFO for Engineering Problems

#### 5.1. Pressure Vessel Design Problem

#### 5.2. Compression/Tension Spring Design Problem

#### 5.3. Welded Beam Design Problem

#### 5.4. Three-Bar Truss Design Problem

## 6. Conclusions and Future Work

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Two hundred successive Lévy flight paths. (

**a**) The first Lévy flight path; (

**b**) The second Lévy flight path; (

**c**) The third Lévy flight path.

**Figure 3.**Two algorithms in function F2: convergence trend comparison. (

**a**) The image of the function F2; (

**b**) Convergence trend comparsion of two algorithms.

ID | Equation | Picture | L | U | D | f_{min} |
---|---|---|---|---|---|---|

F1 | ${f}_{1}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{2}$ | −100 | 100 | 10 | 0 | |

F2 | ${f}_{2}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left|{x}_{i}\right|+{\displaystyle {\displaystyle \prod}_{i=1}^{n}}\left|{x}_{i}\right|$ | −10 | 10 | 10 | 0 | |

F3 | ${f}_{3}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}{({\displaystyle {\displaystyle \sum}_{j=1}^{i}}{x}_{j})}^{2}$ | −100 | 100 | 10 | 0 | |

F4 | ${f}_{4}\left(x\right)=ma{x}_{i}\left\{\left|{x}_{i}\right|,1\le i\le n\right\}$ | −100 | 100 | 10 | 0 | |

F5 | ${f}_{5}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n-1}}[100{\left({x}_{i+1}-{x}_{i}^{2}\right)}^{2}+{\left({x}_{i}-1\right)}^{2}]$ | −30 | 30 | 10 | 0 | |

F6 | ${f}_{6}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}{(\left[{x}_{i}+0.5\right])}^{2}$ | −100 | 100 | 10 | 0 | |

F7 | ${f}_{7}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}i{x}_{i}^{4}+random\left[0,1\right)$ | −1.28 | 1.28 | 10 | 0 | |

F8 | ${f}_{8}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}-{x}_{i}\mathrm{sin}(\sqrt{\left|{x}_{i}\right|}$ | −500 | 500 | 10 | −418.9829*D | |

F9 | ${f}_{9}\left(x\right)={\displaystyle {\displaystyle \sum}_{i=1}^{n}}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]$ | 5.12 | 5.12 | 10 | 0 | |

F10 | ${f}_{10}\left(x\right)=-20exp\left(-0.2\sqrt{\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{2}}\right)-\mathrm{exp}\left(\frac{1}{n}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+e$ | −32 | 30 | 10 | 0 | |

F11 | ${f}_{11}\left(x\right)=\frac{1}{4000}{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{x}_{i}^{2}-{\displaystyle {\displaystyle \prod}_{i=1}^{n}}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | −600 | 600 | 10 | 0 | |

F12. | $\begin{array}{l}{f}_{12}\left(x\right)=\frac{\pi}{n}\left\{10sin\left(\pi {y}_{1}\right)+{\displaystyle {\displaystyle \sum}_{i=1}^{n-1}}{\left({y}_{i}-1\right)}^{2}\left[1+10si{n}^{2}\left(\pi {y}_{i+1}\right)\right]+{\left({y}_{n}-1\right)}^{2}\right\}+{\displaystyle {\displaystyle \sum}_{i=1}^{n}}u\left({x}_{i},10,100,4\right)\\ {y}_{i}=1+\frac{{x}_{i}+1}{4},u\left({x}_{i},a,k.m\right)=\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{m},{x}_{i}a\\ 0,-a{x}_{i}a\\ k{\left(-{x}_{i}-a\right)}^{m},{x}_{i}-a\end{array}\end{array}$ | −50 | 50 | 10 | 0 | |

F13 | ${f}_{13}\left(x\right)=0.1\left\{si{n}^{2}\left(3\pi {x}_{1}\right)+{\displaystyle {\displaystyle \sum}_{i=1}^{n}}{\left({x}_{i}-1\right)}^{2}\left[1+si{n}^{2}\left(3\pi {x}_{i}+1\right)\right]+{\left({x}_{n}-1\right)}^{2}\left[1+si{n}^{2}\left(2\pi {x}_{n}\right)\right]\right\}+{\displaystyle {\displaystyle \sum}_{i=1}^{n}}u\left({x}_{i},5,100,4\right)$ | −50 | 50 | 10 | 0 |

ID | Equation | Picture | L | U | D | f_{min} |
---|---|---|---|---|---|---|

F14 | ${f}_{14}\left(x\right)={[0.002+{{\displaystyle \sum}}_{j=1}^{25}\frac{1}{j+{{\displaystyle \sum}}_{i=1}^{2}{\left({x}_{i}-{x}_{ij}\right)}^{6}}]}^{-1}$ | −65.536 | 65.536 | 2 | 1 | |

F15 | ${f}_{15}\left(x\right)={{\displaystyle \sum}}_{i=1}^{11}{\left[{a}_{i}-\frac{{x}_{1}\left({b}_{i}^{2}+{b}_{i}{x}_{2}\right)}{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}\right]}^{2}$ | −5 | 5 | 4 | 0.0003 | |

F16 | ${f}_{16}\left(x\right)=4{x}_{1}^{2}-2.1{x}_{1}^{4}+\frac{1}{3}{x}_{1}^{6}+{x}_{1}{x}_{2}-4{x}_{2}^{2}+4{x}_{2}^{4}$ | −5 | 5 | 2 | −1.0316 | |

F17 | ${f}_{17}\left(x\right)=10+10\times \left(1-\frac{0.125}{\pi}\right)\mathrm{cos}\left({x}_{1}\right)+{\left({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}-6\right)}^{2}$ | [5, 0] | [10, 15] | 2 | 0.398 | |

F18 | ${f}_{18}\left(x\right)=[1+{\left(1+{x}_{1}+{x}_{2}\right)}^{2}\left(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}\right)]\times [30+{\left(2{x}_{1}-3{x}_{2}\right)}^{2}\times (18-32{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2}$)] | −2 | 2 | 2 | 3 | |

F19 | ${f}_{19}\left(x\right)=-{{\displaystyle \sum}}_{i=1}^{4}{c}_{i}\mathrm{exp}(-{\displaystyle {\displaystyle \sum}_{j=1}^{3}}{a}_{ij}{\left({x}_{j}-{p}_{ij}\right)}^{2})$ | 0 | 1 | 3 | −3.86 | |

F20 | ${F}_{20}=-{\displaystyle {\sum}_{i=1}^{4}{c}_{i}\mathrm{exp}(-{\displaystyle {\sum}_{j=1}^{6}{a}_{ij}{({x}_{j}-{p}_{ij})}^{2}})}$ | 0 | 1 | 6 | −3.32 | |

F21 | ${F}_{21}=-{{\displaystyle {\sum}_{i=1}^{5}[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}]}}^{-1}$ | 0 | 10 | 4 | −10.1532 | |

F22 | ${F}_{22}=-{{\displaystyle {\sum}_{i=1}^{7}[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}]}}^{-1}$ | 0 | 10 | 4 | −10.4028 | |

F23 | ${F}_{23}=-{{\displaystyle {\sum}_{i=1}^{10}[(X-{a}_{i}){(X-{a}_{i})}^{T}+{c}_{i}]}}^{-1}$ | 0 | 10 | 4 | −10.5363 |

Algorithm | The Parameters | The Value |
---|---|---|

Moth–Flame Optimization (MFO) | b | 1 |

Going Cosine Algorithm (SCA) | a. | 2 |

Bat Algorithm (BA) | Frequency minimum (Q_{min}) | 0 |

Frequency maximum (Q_{max}) | 2 | |

Loudness (A) | 0.5 | |

Pulse rate | 0.5 | |

Spotted Hyena Optimizer (SHO) | The Control Parameter ($\overrightarrow{h}$) | (5, 0] |

$\overrightarrow{M}$ Constant | [0.5, 1] | |

Particle Swarm Optimization (PSO) | Maximum Inertia weight (W_{max}) | 0.9 |

Minimum Inertia weight (W_{min}) | 0.2 | |

Maximum Velocity (V_{max}) | 6 | |

Cognitive coefficient (C_{1}) | 2 | |

Cognitive coefficient (C_{2}) | 2 | |

Whale Optimization Algorithm (WOA) | a | (2, 0) |

a_{2} | (2, 1) | |

b | 1 | |

Grey Wolf Optimizer (GWO) | The Control Parameter ($\overrightarrow{a}$) | (2, 0) |

Salp Swarm Algorithm (SSA) | l | 2 |

F | IMFO | MFO | SCA | BA | SHO | PSO | WOA | GWO | SSA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | |

F1 | $\mathbf{6.09}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{181}}$ | 0 | $3.05\times {10}^{-29}$ | $1.28\times {10}^{-28}$ | $5.59\times {10}^{-27}$ | $1.65\times {10}^{-26}$ | $8.97\times {10}^{-7}$ | $3.93\times {10}^{-14}$ | $5.03\times {10}^{-8}$ | $2.72\times {10}^{-7}$ | $1.21\times {10}^{-39}$ | $6.63\times {10}^{-39}$ | $7.22\times {10}^{-148}$ | $3.93\times {10}^{-147}$ | $2.15\times {10}^{-58}$ | $5.40\times {10}^{-58}$ | $6.62\times {10}^{-10}$ | $2.44\times {10}^{-10}$ |

F2 | $5.50\times {10}^{-98}$ | $2.83\times {10}^{-97}$ | $1.27\times {10}^{-18}$ | $2.83\times {10}^{-18}$ | $4.51\times {10}^{-19}$ | $1.56\times {10}^{-18}$ | $3.62\times {10}^{-2}$ | $6.17\times {10}^{-3}$ | 0 | 0 | $6.87\times {10}^{-20}$ | $2.52\times {10}^{-19}$ | $2.31\times {10}^{-101}$ | $9.16\times {10}^{-101}$ | $1.39\times {10}^{-34}$ | $2.11\times {10}^{-34}$ | $2.80\times {10}^{-2}$ | $1.53\times {10}^{-1}$ |

F3 | $\mathbf{4.02}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{147}}$ | $\mathbf{2.19}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{146}}$ | $3.33\times {10}^{2}$ | $1.27\times {10}^{3}$ | $2.36\times {10}^{-7}$ | $1.25\times {10}^{-6}$ | $1.50\times {10}^{-6}$ | $3.56\times {10}^{-13}$ | $2.64\times {10}^{4}$ | $4.45\times {10}^{4}$ | $3.16\times {10}^{-12}$ | $7.49\times {10}^{-12}$ | $2.39\times {10}^{4}$ | $1.03\times {10}^{4}$ | $2.02\times {10}^{-14}$ | $1.08\times {10}^{-13}$ | $1.76\times {10}^{-9}$ | $9.69\times {10}^{-10}$ |

F4 | $\mathbf{3.36}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{82}}$ | $\mathbf{1.83}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{81}}$ | 1.40 | 3.47 | $3.38\times {10}^{-7}$ | $1.69\times {10}^{-6}$ | $4.95\times {10}^{-2}$ | $1.06\times {10}^{-2}$ | $3.13\times {10}^{-6}$ | $8.14\times {10}^{-6}$ | $6.41\times {10}^{-10}$ | $1.61\times {10}^{-9}$ | $4.11\times {10}^{1}$ | $3.15\times {10}^{1}$ | $1.61\times {10}^{-14}$ | $2.14\times {10}^{-14}$ | $1.54\times {10}^{-5}$ | $4.64\times {10}^{-6}$ |

F5 | 6.45 | $\mathbf{1.22}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $1.73\times {10}^{2}$ | $5.54\times {10}^{2}$ | 7.06 | $4.04\times {10}^{-1}$ | 2.32 | 3.13 | $2.89\times {10}^{1}$ | $1.39\times {10}^{-1}$ | 4.00 | 1.94 | $2.73\times {10}^{1}$ | $5.79\times {10}^{-1}$ | $2.69\times {10}^{1}$ | $6.69\times {10}^{-1}$ | $6.56\times {10}^{1}$ | $1.53\times {10}^{2}$ |

F6 | $1.18\times {10}^{-12}$ | $1.02\times {10}^{-12}$ | $3.07\times {10}^{-30}$ | $6.24\times {10}^{-30}$ | $3.24\times {10}^{-1}$ | $1.36\times {10}^{-1}$ | $8.96\times {10}^{-7}$ | $7.14\times {10}^{-14}$ | 4.32 | 3.31 | $\mathbf{1.03}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{33}}$ | $\mathbf{4.53}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{33}}$ | $1.67\times {10}^{-1}$ | $1.58\times {10}^{-1}$ | $4.87\times {10}^{-1}$ | $2.68\times {10}^{-1}$ | $6.25\times {10}^{-10}$ | $1.79\times {10}^{-10}$ |

F7 | $\mathbf{7.12}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ | $\mathbf{5.64}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ | $5.90\times {10}^{-3}$ | $4.10\times {10}^{-3}$ | $1.30\times {10}^{-3}$ | $1.40\times {10}^{-3}$ | 2.81 | 3.57 | 3.57 | $1.46\times {10}^{1}$ | $3.81\times {10}^{-3}$ | $2.21\times {10}^{-3}$ | $1.50\times {10}^{-3}$ | $1.80\times {10}^{-3}$ | $8.79\times {10}^{-4}$ | $5.58\times {10}^{-4}$ | $5.20\times {10}^{-3}$ | $2.00\times {10}^{-3}$ |

F8 | $\mathbf{-}\mathbf{3.35}\mathbf{\times}{\mathbf{10}}^{\mathbf{3}}$ | $2.53\times {10}^{2}$ | $-3.08\times {10}^{-3}$ | $3.25\times {10}^{2}$ | $-2.25\times {10}^{3}$ | $\mathbf{1.83}\mathbf{\times}{\mathbf{10}}^{\mathbf{2}}$ | N/A | N/A | $-2.17\times {10}^{3}$ | $5.37\times {10}^{2}$ | $-2.37\times {10}^{3}$ | $4.71\times {10}^{2}$ | $-1.13\times {10}^{4}$ | $1.57\times {10}^{3}$ | $-6.28\times {10}^{3}$ | $9.40\times {10}^{2}$ | $-2.76\times {10}^{3}$ | $3.37\times {10}^{2}$ |

F9 | 0 | 0 | $2.39\times {10}^{1}$ | $1.26\times {10}^{1}$ | $2.04\times {10}^{-1}$ | 1.12 | $1.42\times {10}^{1}$ | $6.35\times {10}^{1}$ | $9.53\times {10}^{1}$ | $9.42\times {10}^{1}$ | 4.08 | 1.80 | 0 | 0 | $8.09\times {10}^{-1}$ | 2.26 | $1.76\times {10}^{1}$ | 7.91 |

F10 | $\mathbf{8.88}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{16}}$ | 0 | $4.80\times {10}^{-15}$ | $1.08\times {10}^{-15}$ | $9.06\times {10}^{-15}$ | $9.34\times {10}^{-15}$ | 1.66 | 1.00 | $4.77\times {10}^{-3}$ | $2.60\times {10}^{-2}$ | $4.80\times {10}^{-15}$ | $1.08\times {10}^{-15}$ | $3.61\times {10}^{-15}$ | $2.22\times {10}^{-15}$ | $1.57\times {10}^{-14}$ | $2.96\times {10}^{-15}$ | $6.66\times {10}^{-1}$ | $9.62\times {10}^{-1}$ |

F11 | 0 | 0 | $1.60\times {10}^{-1}$ | $9.62\times {10}^{-2}$ | $5.27\times {10}^{-2}$ | $1.61\times {10}^{-1}$ | $1.08\times {10}^{-7}$ | $1.03\times {10}^{-15}$ | $1.12\times {10}^{-15}$ | $4.44\times {10}^{-15}$ | $1.84\times {10}^{-1}$ | $1.57\times {10}^{-1}$ | 0 | 0 | $1.90\times {10}^{-3}$ | $5.90\times {10}^{-3}$ | $2.17\times {10}^{-1}$ | $1.12\times {10}^{-1}$ |

F12. | $2.62\times {10}^{-13}$ | $4.83\times {10}^{-13}$ | $4.55\times {10}^{-1}$ | $8.38\times {10}^{-1}$ | $6.53\times {10}^{-2}$ | $2.28\times {10}^{-2}$ | $2.38\times {10}^{-4}$ | $1.70\times {10}^{-6}$ | 1.19 | $7.51\times {10}^{-1}$ | $\mathbf{4.74}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{32}}$ | $\mathbf{6.09}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{34}}$ | $6.10\times {10}^{-3}$ | $5.70\times {10}^{-3}$ | $3.75\times {10}^{-2}$ | $1.87\times {10}^{-2}$ | $2.14\times {10}^{-1}$ | $4.90\times {10}^{-1}$ |

F13 | $3.80\times {10}^{-12}$ | $6.06\times {10}^{-12}$ | $3.70\times {10}^{-3}$ | $5.30\times {10}^{-3}$ | $2.37\times {10}^{-1}$ | $7.46\times {10}^{-2}$ | $2.21\times {10}^{-1}$ | $3.21\times {10}^{-2}$ | 2.98 | $3.14\times {10}^{-2}$ | $\mathbf{2.05}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{32}}$ | $\mathbf{3.47}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{32}}$ | $2.12\times {10}^{-1}$ | $1.45\times {10}^{-1}$ | $5.51\times {10}^{-1}$ | $2.22\times {10}^{-1}$ | $1.80\times {10}^{-3}$ | $4.20\times {10}^{-3}$ |

F14 | 2.65 | $7.05\times {10}^{-1}$ | 2.02 | 1.61 | 1.53 | $8.92\times {10}^{-1}$ | $1.23\times {10}^{1}$ | 4.54 | $1.02\times {10}^{1}$ | 3.16 | 3.56 | 2.79 | 2.11 | 2.49 | 4.13 | 3.99 | $\mathbf{9.98}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $\mathbf{2.66}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{16}}$ |

F15 | $\mathbf{3.36}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $\mathbf{7.90}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ | $2.30\times {10}^{-3}$ | $3.90\times {10}^{-3}$ | $9.75\times {10}^{-4}$ | $4.17\times {10}^{-4}$ | $2.34\times {10}^{-3}$ | $6.11\times {10}^{-5}$ | $1.42\times {10}^{-2}$ | $2.00\times {10}^{-2}$ | $8.31\times {10}^{-4}$ | $2.06\times {10}^{-4}$ | $6.11\times {10}^{-4}$ | $3.58\times {10}^{-4}$ | $3.70\times {10}^{-3}$ | $7.60\times {10}^{-3}$ | $1.50\times {10}^{-3}$ | $3.60\times {10}^{-3}$ |

F16 | −1.03 | $\mathbf{6.78}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{16}}$ | −1.03 | $6.78\times {10}^{-16}$ | −1.03 | $2.11\times {10}^{-5}$ | −1.00 | $2.22\times {10}^{-2}$ | $-6.87\times {10}^{-1}$ | $3.43\times {10}^{-1}$ | −1.03 | $6.71\times {10}^{-16}$ | −1.03 | $4.14\times {10}^{-10}$ | −1.03 | $6.49\times {10}^{-9}$ | −1.03 | $5.77\times {10}^{-15}$ |

F17 | $\mathbf{3.98}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | 0 | $3.98\times {10}^{-1}$ | 0 | $3.99\times {10}^{-1}$ | $1.60\times {10}^{-3}$ | $4.50\times {10}^{-1}$ | $6.13\times {10}^{-2}$ | $5.72\times {10}^{-1}$ | $2.13\times {10}^{-1}$ | $3.98\times {10}^{-1}$ | 0 | $3.98\times {10}^{-1}$ | $5.84\times {10}^{-6}$ | $3.98\times {10}^{-1}$ | $2.28\times {10}^{-5}$ | N/A | N/A |

F18 | 3.00 | $\mathbf{1.20}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{15}}$ | 3.00 | $1.89\times {10}^{-15}$ | 3.00 | $2.40\times {10}^{-5}$ | 6.60 | $2.38\times {10}^{2}$ | 9.58 | 9.68 | 3.00 | $1.23\times {10}^{-15}$ | 3.00 | $1.61\times {10}^{-5}$ | 3.00 | $8.86\times {10}^{-6}$ | 3.00 | $1.02\times {10}^{-13}$ |

F19 | −3.86 | $\mathbf{2.71}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{15}}$ | −3.86 | $2.71\times {10}^{-15}$ | −3.86 | $2.60\times {10}^{-3}$ | −3.76 | $7.14\times {10}^{-2}$ | −3.24 | $3.74\times {10}^{-1}$ | −3.86 | $2.71\times {10}^{-15}$ | −3.86 | $2.70\times {10}^{-3}$ | −3.86 | $2.50\times {10}^{-3}$ | −3.86 | $2.18\times {10}^{-14}$ |

F20 | −3.31 | $4.10\times {10}^{-2}$ | −3.22 | $6.20\times {10}^{-2}$ | −2.79 | $4.48\times {10}^{-1}$ | −3.24 | $\mathbf{2.77}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | −1.48 | $4.28\times {10}^{-1}$ | −3.26 | $6.03\times {10}^{-2}$ | −3.21 | $1.20\times {10}^{-1}$ | −3.28 | $5.95\times {10}^{-2}$ | −3.24 | $5.97\times {10}^{-2}$ |

F21 | −5.06 | $\mathbf{9.47}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{16}}$ | −6.63 | 3.27 | −2.62 | 1.96 | −5.23 | $8.66\times {10}^{-1}$ | $-5.60\times {10}^{-1}$ | $3.47\times {10}^{-1}$ | −8.21 | 2.62 | −8.71 | 2.46 | −9.14 | 2.06 | −8.14 | 2.98 |

F22 | −5.26 | $\mathbf{9.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | −7.98 | 3.30 | −3.53 | 2.29 | −5.26 | $9.42\times {10}^{-1}$ | $-7.38\times {10}^{-1}$ | $2.88\times {10}^{-1}$ | −8.91 | 2.54 | −8.99 | 2.63 | $\mathbf{-}\mathbf{1.01}\mathbf{\times}{\mathbf{10}}^{\mathbf{1}}$ | 1.34 | −9.01 | 2.62 |

F23 | −5.31 | $\mathbf{9.87}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | −8.66 | 3.21 | −4.32 | 1.86 | −5.27 | 1.05 | $-8.26\times {10}^{-1}$ | $2.46\times {10}^{-1}$ | −9.92 | 1.91 | −8.10 | 3.1 | $\mathbf{-}\mathbf{1.03}\mathbf{\times}{\mathbf{10}}^{\mathbf{1}}$ | 1.48 | −8.16 | 3.48 |

F | IMFO | MFO | SCA | BA | SHO | PSO | WOA | GWO | SSA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | |

F1 | $1.68\times {10}^{-109}$ | $5.34\times {10}^{-109}$ | $3.00\times {10}^{3}$ | $4.66\times {10}^{3}$ | $3.55\times {10}^{-2}$ | $1.06\times {10}^{-1}$ | $5.60\times {10}^{2}$ | $7.29\times {10}^{-12}$ | $6.04\times {10}^{-2}$ | $3.31\times {10}^{-1}$ | $3.59\times {10}^{-9}$ | $\mathbf{9.38}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{151}}$ | $\mathbf{3.90}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{150}}$ | $7.75\times {10}^{-59}$ | $2.75\times {10}^{-58}$ | $1.20\times {10}^{-8}$ | $2.58\times {10}^{-9}$ | |

F2 | $1.37\times {10}^{-66}$ | $4.46\times {10}^{-66}$ | $2.93\times {10}^{1}$ | $2.26\times {10}^{1}$ | $3.23\times {10}^{-5}$ | $8.57\times {10}^{-5}$ | 1.17 | $1.89\times {10}^{-1}$ | 0 | 0 | $6.63\times {10}^{-4}$ | $1.89\times {10}^{-3}$ | $3.67\times {10}^{-104}$ | $1.76\times {10}^{-103}$ | $9.90\times {10}^{-35}$ | $1.22\times {10}^{-34}$ | 1.44 | 1.73 |

F3 | $\mathbf{1.53}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{151}}$ | $\mathbf{8.39}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{151}}$ | $1.91\times {10}^{4}$ | $1.21\times {10}^{4}$ | $4.91\times {10}^{3}$ | $3.89\times {10}^{3}$ | $1.13\times {10}^{-1}$ | $7.13\times {10}^{-2}$ | $2.07\times {10}^{4}$ | $4.36\times {10}^{4}$ | $1.56\times {10}^{1}$ | 6.49 | $1.68\times {10}^{4}$ | $8.91\times {10}^{3}$ | $1.90\times {10}^{-13}$ | $9.85\times {10}^{-13}$ | $3.21\times {10}^{2}$ | $1.88\times {10}^{2}$ |

F4 | $\mathbf{4.15}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{43}}$ | $\mathbf{2.27}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{42}}$ | $6.60\times {10}^{1}$ | 7.45 | $1.87\times {10}^{1}$ | 8.21 | $4.65\times {10}^{-1}$ | $2.57\times {10}^{-2}$ | $6.41\times {10}^{-6}$ | $2.35\times {10}^{-5}$ | $5.71\times {10}^{-1}$ | $1.63\times {10}^{-1}$ | $3.99\times {10}^{1}$ | $3.24\times {10}^{1}$ | $1.47\times {10}^{-14}$ | $1.39\times {10}^{-14}$ | 7.81 | 2.68 |

F5 | $2.81\times {10}^{1}$ | $\mathbf{1.22}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $2.75\times {10}^{4}$ | $4.17\times {10}^{4}$ | $7.37\times {10}^{2}$ | $1.98\times {10}^{3}$ | $\mathbf{2.63}\mathbf{\times}{\mathbf{10}}^{\mathbf{1}}$ | 3.81 | $2.89\times {10}^{1}$ | $1.27\times {10}^{-1}$ | $4.80\times {10}^{1}$ | $3.03\times {10}^{1}$ | $2.71\times {10}^{1}$ | $5.50\times {10}^{-1}$ | $2.70\times {10}^{1}$ | $7.05\times {10}^{-1}$ | $1.32\times {10}^{2}$ | $2.15\times {10}^{2}$ |

F6 | 5.75 | $2.82\times {10}^{-1}$ | $1.67\times {10}^{3}$ | $3.80\times {10}^{3}$ | 4.88 | $9.75\times {10}^{-1}$ | $1.71\times {10}^{-5}$ | $\mathbf{4.95}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{12}}$ | 6.14 | 2.12 | $\mathbf{8.04}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{9}}$ | $2.12\times {10}^{-8}$ | $7.42\times {10}^{2}$ | $8.12\times {10}^{-2}$ | $5.77\times {10}^{-1}$ | $3.30\times {10}^{-1}$ | $1.11\times {10}^{-8}$ | $2.60\times {10}^{-9}$ |

F7 | 5.79 | $2.13\times {10}^{-1}$ | 4.02 | 6.59 | $3.88\times {10}^{-2}$ | $5.79\times {10}^{-2}$ | $2.17\times {10}^{1}$ | $7.86\times {10}^{1}$ | 3.08 | $1.32\times {10}^{1}$ | $6.92\times {10}^{-2}$ | $2.50\times {10}^{-2}$ | $1.77\times {10}^{-3}$ | $1.99\times {10}^{-3}$ | $\mathbf{8.66}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $\mathbf{4.32}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $1.04\times {10}^{-1}$ | $4.95\times {10}^{-2}$ |

F8 | $-9.11\times {10}^{-3}$ | $7.15\times {10}^{2}$ | $-8.57\times {10}^{3}$ | $9.58\times {10}^{2}$ | $-3.81\times {10}^{3}$ | $\mathbf{2.52}\mathbf{\times}{\mathbf{10}}^{\mathbf{2}}$ | N/A | N/A | $-2.14\times {10}^{3}$ | $5.15\times {10}^{2}$ | $-6.32\times {10}^{3}$ | $1.14\times {10}^{3}$ | $\mathbf{-}\mathbf{1.07}\mathbf{\times}{\mathbf{10}}^{\mathbf{4}}$ | $1.82\times {10}^{3}$ | $-5.99\times {10}^{3}$ | $7.70\times {10}^{2}$ | $-7.55\times {10}^{3}$ | $8.39\times {10}^{2}$ |

F9 | 0 | 0 | $1.62\times {10}^{2}$ | $3.92\times {10}^{1}$ | $2.23\times {10}^{1}$ | $3.25\times {10}^{1}$ | $3.02\times {10}^{1}$ | $1.93\times {10}^{2}$ | $8.13\times {10}^{1}$ | $8.70\times {10}^{1}$ | $4.60\times {10}^{1}$ | $1.05\times {10}^{1}$ | $1.89\times {10}^{-15}$ | $1.04\times {10}^{-14}$ | $9.85\times {10}^{-1}$ | 2.11 | $5.87\times {10}^{1}$ | $2.20\times {10}^{1}$ |

F10 | 0 | 0 | $1.57\times {10}^{1}$ | 6.41 | $1.55\times {10}^{1}$ | 8.11 | 2.09 | $1.77\times {10}^{-1}$ | $9.01\times {10}^{-11}$ | $4.51\times {10}^{-10}$ | $8.29\times {10}^{-5}$ | $1.74\times {10}^{-4}$ | $4.20\times {10}^{-15}$ | $2.63\times {10}^{-15}$ | $1.64\times {10}^{-14}$ | $3.16\times {10}^{-15}$ | 2.12 | $9.42\times {10}^{-1}$ |

F11 | $\mathbf{8.88}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{16}}$ | 0 | $1.51\times {10}^{1}$ | $3.42\times {10}^{1}$ | $3.01\times {10}^{-1}$ | $2.89\times {10}^{-1}$ | $1.23\times {10}^{-6}$ | $1.37\times {10}^{-12}$ | $1.85\times {10}^{-14}$ | $1.01\times {10}^{-13}$ | $9.76\times {10}^{-3}$ | $1.00\times {10}^{-2}$ | $4.36\times {10}^{-3}$ | $1.67\times {10}^{-2}$ | $2.87\times {10}^{-3}$ | $6.95\times {10}^{-3}$ | $7.14\times {10}^{-3}$ | $8.74\times {10}^{-3}$ |

F12. | 0 | 0 | $6.41\times {10}^{-1}$ | $8.95\times {10}^{-1}$ | $5.21\times {10}^{1}$ | $2.47\times {10}^{2}$ | $1.46\times {10}^{-2}$ | $6.44\times {10}^{-4}$ | $9.58\times {10}^{-1}$ | $8.27\times {10}^{-1}$ | $1.75\times {10}^{-10}$ | $3.92\times {10}^{-10}$ | $7.11\times {10}^{-3}$ | $5.93\times {10}^{-3}$ | $4.03\times {10}^{-2}$ | $1.64\times {10}^{-2}$ | 5.36 | 2.70 |

F13 | $7.85\times {10}^{-1}$ | $4.25\times {10}^{-2}$ | $7.93\times {10}^{-1}$ | 1.87 | $2.81\times {10}^{2}$ | $8.63\times {10}^{2}$ | 1.17 | $3.10\times {10}^{-1}$ | 2.98 | $2.81\times {10}^{-2}$ | $\mathbf{2.93}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $\mathbf{8.62}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $2.11\times {10}^{-1}$ | $1.76\times {10}^{-1}$ | $5.57\times {10}^{-1}$ | $2.06\times {10}^{-1}$ | $7.05\times {10}^{-2}$ | $2.67\times {10}^{-1}$ |

F | IMFO | MFO | SCA | BA | SHO | PSO | WOA | GWO | SSA | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | Ave | Std | |

F1 | $3.95\times {10}^{-74}$ | $2.02\times {10}^{-73}$ | $2.97\times {10}^{4}$ | $1.33\times {10}^{4}$ | $5.10\times {10}^{3}$ | $4.21\times {10}^{3}$ | $1.02\times {10}^{-1}$ | $1.15\times {10}^{-1}$ | $3.84\times {10}^{2}$ | $1.96\times {10}^{3}$ | 2.63 | 1.00 | $\mathbf{2.19}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{148}}$ | $\mathbf{9.25}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{148}}$ | $2.93\times {10}^{-29}$ | $3.39\times {10}^{-29}$ | 2.13 | 1.44 |

F2 | $2.14\times {10}^{-51}$ | $1.14\times {10}^{-50}$ | $1.87\times {10}^{2}$ | $5.35\times {10}^{1}$ | 1.65 | 1.39 | $1.06\times {10}^{1}$ | $1.02\times {10}^{1}$ | 0 | 0 | $1.18\times {10}^{1}$ | 5.15 | $\mathbf{1.51}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{100}}$ | $\mathbf{7.91}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{100}}$ | $5.40\times {10}^{-18}$ | $2.88\times {10}^{-18}$ | $2.27\times {10}^{1}$ | 3.94 |

F3 | $\mathbf{3.72}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{40}}$ | $\mathbf{2.04}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{39}}$ | $1.97\times {10}^{5}$ | $6.35\times {10}^{4}$ | $1.86\times {10}^{5}$ | $5.02\times {10}^{4}$ | $1.60\times {10}^{1}$ | $1.99\times {10}^{2}$ | $8.95\times {10}^{5}$ | $7.53\times {10}^{5}$ | $1.20\times {10}^{4}$ | $3.55\times {10}^{3}$ | $8.59\times {10}^{5}$ | $1.46\times {10}^{5}$ | $2.36\times {10}^{1}$ | $8.07\times {10}^{1}$ | $4.26\times {10}^{4}$ | $1.91\times {10}^{4}$ |

F4 | $\mathbf{1.26}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$ | $\mathbf{6.91}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$ | $9.35\times {10}^{1}$ | 1.85 | $8.61\times {10}^{1}$ | 3.17 | $8.30\times {10}^{-1}$ | $1.70\times {10}^{-2}$ | $5.75\times {10}^{-4}$ | $8.79\times {10}^{-4}$ | 8.99 | 1.41 | $7.55\times {10}^{1}$ | $2.73\times {10}^{1}$ | $4.92\times {10}^{-3}$ | $1.02\times {10}^{-2}$ | $2.75\times {10}^{1}$ | 4.45 |

F5 | $9.85\times {10}^{1}$ | $3.78\times {10}^{-1}$ | $6.85\times {10}^{7}$ | $5.06\times {10}^{7}$ | $5.97\times {10}^{7}$ | $3.23\times {10}^{7}$ | $1.76\times {10}^{2}$ | $1.09\times {10}^{4}$ | $1.82\times {10}^{2}$ | $3.32\times {10}^{2}$ | $2.15\times {10}^{3}$ | $1.07\times {10}^{3}$ | $\mathbf{9.77}\mathbf{\times}{\mathbf{10}}^{\mathbf{1}}$ | $\mathbf{3.71}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $9.77\times {10}^{1}$ | $6.82\times {10}^{-1}$ | $2.23\times {10}^{3}$ | $2.06\times {10}^{3}$ |

F6 | $2.30\times {10}^{1}$ | $\mathbf{2.12}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $3.58\times {10}^{4}$ | $1.70\times {10}^{4}$ | $5.99\times {10}^{3}$ | $5.26\times {10}^{3}$ | 1.58 | 2.00 | $5.61\times {10}^{2}$ | $2.62\times {10}^{3}$ | 3.13 | 1.54 | 1.76 | $7.26\times {10}^{-1}$ | 9.34 | $8.26\times {10}^{-1}$ | 3.02 | 2.34 |

F7 | $\mathbf{1.37}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $\mathbf{1.14}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $1.76\times {10}^{2}$ | $1.51\times {10}^{2}$ | $7.02\times {10}^{1}$ | $4.04\times {10}^{1}$ | $8.50\times {10}^{1}$ | $1.72\times {10}^{3}$ | $2.85\times {10}^{1}$ | $8.65\times {10}^{1}$ | $1.46\times {10}^{3}$ | $2.79\times {10}^{2}$ | $1.78\times {10}^{-3}$ | $1.99\times {10}^{-3}$ | $2.77\times {10}^{-3}$ | $1.26\times {10}^{-3}$ | 1.41 | $2.85\times {10}^{-1}$ |

F8 | $-4.59\times {10}^{3}$ | $\mathbf{4.53}\mathbf{\times}{\mathbf{10}}^{\mathbf{2}}$ | $-2.38\times {10}^{4}$ | $2.16\times {10}^{3}$ | $-7.18\times {10}^{3}$ | $4.69\times {10}^{2}$ | N/A | N/A | $-3.99\times {10}^{3}$ | $9.68\times {10}^{2}$ | $-1.98\times {10}^{4}$ | $4.47\times {10}^{3}$ | $\mathbf{-}\mathbf{3.79}\mathbf{\times}{\mathbf{10}}^{\mathbf{4}}$ | $5.46\times {10}^{3}$ | $-1.67\times {10}^{4}$ | $2.41\times {10}^{3}$ | $-2.43\times {10}^{4}$ | $1.96\times {10}^{3}$ |

F9 | 0 | 0 | $7.60\times {10}^{2}$ | $7.43\times {10}^{1}$ | $2.21\times {10}^{2}$ | $9.65\times {10}^{1}$ | $1.38\times {10}^{2}$ | $2.35\times {10}^{3}$ | $3.71\times {10}^{2}$ | $4.06\times {10}^{2}$ | $4.72\times {10}^{2}$ | $5.75\times {10}^{1}$ | 0 | 0 | 1.14 | 2.54 | $1.63\times {10}^{2}$ | $3.12\times {10}^{1}$ |

F10 | $\mathbf{8.88}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{16}}$ | 0 | $1.98\times {10}^{1}$ | $2.29\times {10}^{-1}$ | $1.95\times {10}^{1}$ | 4.05 | 2.13 | $1.19\times {10}^{-1}$ | $4.23\times {10}^{-1}$ | $9.44\times {10}^{-1}$ | 2.65 | $2.98\times {10}^{-1}$ | $4.20\times {10}^{-15}$ | $2.79\times {10}^{-15}$ | $1.12\times {10}^{-13}$ | $8.06\times {10}^{-15}$ | 7.16 | 1.14 |

F11 | 0 | 0 | $2.71\times {10}^{2}$ | $1.05\times {10}^{2}$ | $6.81\times {10}^{1}$ | $4.91\times {10}^{1}$ | $2.45\times {10}^{-2}$ | $1.30\times {10}^{-3}$ | 8.59 | $3.86\times {10}^{1}$ | $4.49\times {10}^{-2}$ | $2.13\times {10}^{-2}$ | $3.91\times {10}^{-3}$ | $2.14\times {10}^{-2}$ | $1.65\times {10}^{-3}$ | $6.27\times {10}^{-3}$ | $6.83\times {10}^{-1}$ | $2.14\times {10}^{-1}$ |

F12 | 1.04 | $2.75\times {10}^{-2}$ | $1.05\times {10}^{8}$ | $1.44\times {10}^{8}$ | $1.49\times {10}^{8}$ | $9.88\times {10}^{7}$ | $1.17\times {10}^{-1}$ | $\mathbf{2.56}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | 1.15 | $4.57\times {10}^{-1}$ | 2.00 | $9.86\times {10}^{-1}$ | $\mathbf{1.69}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $7.46\times {10}^{-3}$ | $2.54\times {10}^{-1}$ | $6.66\times {10}^{-2}$ | $1.76\times {10}^{1}$ | 3.65 |

F13 | 9.76 | $\mathbf{3.22}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $2.88\times {10}^{8}$ | $2.45\times {10}^{8}$ | $3.33\times {10}^{8}$ | $2.42\times {10}^{8}$ | 9.54 | 1.89 | 9.89 | $5.42\times {10}^{-1}$ | $1.34\times {10}^{1}$ | $1.03\times {10}^{1}$ | 1.73 | $7.62\times {10}^{-1}$ | 6.27 | $4.34\times {10}^{-1}$ | $1.79\times {10}^{2}$ | $1.91\times {10}^{1}$ |

F | MFO | SCA | BA | SHO | PSO | WOA | GWO | SSA |
---|---|---|---|---|---|---|---|---|

F1 | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $1.21\times {10}^{-12}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | |

F2 | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $1.21\times {10}^{-12}$ | $3.02\times {10}^{-11}$ | $4.50\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ |

F3 | $3.00\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $1.21\times {10}^{-12}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ |

F4 | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $1.24\times {10}^{-9}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.01\times {10}^{-11}$ |

F5 | $3.71\times {10}^{-1}$ | $3.20\times {10}^{-9}$ | $3.02\times {10}^{-11}$ | $2.72\times {10}^{-11}$ | $7.04\times {10}^{-7}$ | $6.77\times {10}^{-5}$ | $7.48\times {10}^{-2}$ | $7.29\times {10}^{-3}$ |

F6 | $3.01\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.01\times {10}^{-11}$ | $2.26\times {10}^{-11}$ | $2.36\times {10}^{-12}$ | $3.02\times {10}^{-11}$ | $3.01\times {10}^{-11}$ | $3.01\times {10}^{-11}$ |

F7 | $3.02\times {10}^{-11}$ | $1.09\times {10}^{-10}$ | $3.02\times {10}^{-11}$ | $2.69\times {10}^{-3}$ | $3.02\times {10}^{-11}$ | $1.60\times {10}^{-7}$ | $9.26\times {10}^{-9}$ | $3.02\times {10}^{-11}$ |

F8 | $4.96\times {10}^{-1}$ | $3.02\times {10}^{-11}$ | N/A | $3.02\times {10}^{-11}$ | $3.47\times {10}^{-10}$ | $2.84\times {10}^{-1}$ | $1.17\times {10}^{-9}$ | $3.44\times {10}^{-6}$ |

F9 | $1.21\times {10}^{-12}$ | $4.19\times {10}^{-2}$ | $1.19\times {10}^{-12}$ | $2.79\times {10}^{-3}$ | $1.13\times {10}^{-12}$ | $3.34\times {10}^{-1}$ | N/A | $1.20\times {10}^{-12}$ |

F10 | $1.19\times {10}^{-13}$ | $6.66\times {10}^{-13}$ | $1.13\times {10}^{-12}$ | N/A | $2.90\times {10}^{-13}$ | $9.16\times {10}^{-9}$ | $1.55\times {10}^{-13}$ | $1.18\times {10}^{-12}$ |

F11 | $1.21\times {10}^{-12}$ | $2.21\times {10}^{-6}$ | $1.21\times {10}^{-12}$ | N/A | $1.21\times {10}^{-12}$ | $1.37\times {10}^{-3}$ | $5.38\times {10}^{-6}$ | $1.21\times {10}^{-12}$ |

F12 | $3.98\times {10}^{-4}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $2.40\times {10}^{-11}$ | $1.88\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.00\times {10}^{-11}$ |

F13 | $7.85\times {10}^{-3}$ | $3.02\times {10}^{-11}$ | $3.01\times {10}^{-11}$ | $2.26\times {10}^{-11}$ | $3.16\times {10}^{-12}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ | $3.02\times {10}^{-11}$ |

F14 | $3.87\times {10}^{-1}$ | $6.26\times {10}^{-2}$ | $2.19\times {10}^{-11}$ | $5.42\times {10}^{-8}$ | $8.49\times {10}^{-1}$ | $1.94\times {10}^{-2}$ | 1.00 | $9.94\times {10}^{-10}$ |

F15 | $3.54\times {10}^{-11}$ | $6.07\times {10}^{-11}$ | $8.17\times {10}^{-7}$ | $3.69\times {10}^{-11}$ | $6.72\times {10}^{-10}$ | $9.83\times {10}^{-8}$ | $3.87\times {10}^{-1}$ | $3.33\times {10}^{-11}$ |

F16 | N/A | $1.21\times {10}^{-12}$ | $2.71\times {10}^{-14}$ | $5.77\times {10}^{-9}$ | N/A | N/A | N/A | |

F17 | N/A | $1.21\times {10}^{-12}$ | $4.16\times {10}^{-14}$ | $1.21\times {10}^{-12}$ | N/A | $1.30\times {10}^{-7}$ | $1.21\times {10}^{-12}$ | N/A |

F18 | N/A | $1.21\times {10}^{-12}$ | $1.61\times {10}^{-1}$ | $1.21\times {10}^{-12}$ | N/A | $4.57\times {10}^{-12}$ | $1.21\times {10}^{-12}$ | N/A |

F19 | $3.34\times {10}^{-1}$ | $1.21\times {10}^{-12}$ | $8.64\times {10}^{-14}$ | $1.21\times {10}^{-12}$ | N/A | $1.21\times {10}^{-12}$ | $1.21\times {10}^{-12}$ | N/A |

F20 | $1.18\times {10}^{-4}$ | $9.17\times {10}^{-12}$ | $1.99\times {10}^{-7}$ | $9.17\times {10}^{-12}$ | $4.65\times {10}^{-1}$ | $1.83\times {10}^{-6}$ | $5.78\times {10}^{-7}$ | $9.13\times {10}^{-6}$ |

F21 | $2.77\times {10}^{-2}$ | $1.91\times {10}^{-7}$ | $2.71\times {10}^{-14}$ | $1.21\times {10}^{-12}$ | $3.53\times {10}^{-7}$ | $1.33\times {10}^{-8}$ | $7.47\times {10}^{-10}$ | $2.22\times {10}^{-8}$ |

F22 | $1.83\times {10}^{-3}$ | $1.65\times {10}^{-7}$ | $1.61\times {10}^{-12}$ | $1.72\times {10}^{-12}$ | $5.47\times {10}^{-13}$ | $7.23\times {10}^{-2}$ | $1.72\times {10}^{-12}$ | $4.59\times {10}^{-12}$ |

F23 | $6.69\times {10}^{-3}$ | $1.23\times {10}^{-2}$ | $7.35\times {10}^{-11}$ | $1.72\times {10}^{-12}$ | $8.04\times {10}^{-13}$ | $1.90\times {10}^{-6}$ | $4.56\times {10}^{-11}$ | $3.93\times {10}^{-5}$ |

Algorithm | Rank Mean (d = 10) | Rank | Rank Mean (d = 30) | Rank | Rank Mean (d = 100) | Rank |
---|---|---|---|---|---|---|

IMFO | 2.71 | 1 | 2.92 | 1 | 2.63 | 1 |

MFO | 5.45 | 6 | 8.25 | 9 | 7.17 | 9 |

SCA | 5.52 | 7 | 6.83 | 8 | 6.83 | 8 |

BA | 6.52 | 8 | 4.92 | 5 | 4.00 | 4 |

SHO | 7.86 | 9 | 5.67 | 7 | 5.83 | 5 |

PSO | 3.38 | 2 | 4.17 | 4 | 2.79 | 2 |

WOA | 4.24 | 4 | 3.50 | 3 | 3.08 | 3 |

GWO | 4.19 | 3 | 3.17 | 2 | 6.25 | 6 |

SSA | 5.12 | 5 | 5.58 | 6 | 6.42 | 7 |

Algorithm | Variable | Target Cost | |||
---|---|---|---|---|---|

T_{s} | T_{h} | R | L | ||

MFO [1] | 0.8125 | 0.4375 | 42.098445 | 176.636596 | 6059.7143 |

OMFO [22] | 0.81386 | 0.4022 | 42.1689 | 175.7651 | 5949.1830 |

ES [31] | 0.812500 | 0.437500 | 42.098087 | 176.640518 | 6059.7456 |

CSDE [33] | 0.8125 | 0.4375 | 42.10 | 176.6 | 6059.7133 |

CPSO [34] | 0.812500 | 0.437500 | 42.091266 | 176.746500 | 6061.0777 |

LFD [35] | 0.8777 | 0.4339 | 45.4755 | 139.0654 | 6080 |

WOA [36] | 0.812500 | 0.437500 | 42.098209 | 176.638998 | 6059.7410 |

MOSCA [37] | 0.7781909 | 0.3830476 | 40.3207539 | 199.9841994 | 5880.71150 |

RDWOA [38] | 0.793769 | 0.39236 | 41.127973 | 189.045124 | 5912.53868 |

CCMWOA [39] | 0.779661 | 0.385611 | 40.34738 | 199.6141 | 5895.2039 |

LWOA [40] | 0.778858 | 0.385321 | 40.32609 | 200 | 5893.339 |

GA [41] | 0.8125 | 0.4345 | 40.323900 | 200.000000 | 6288.7445 |

BFGSOLMFO [42] | 0.778675 | 0.385392 | 40.342876 | 199.754805 | 5889.7080 |

IMFO [43] | 0.77455 | 0.38320 | 40.31962 | 200.00000 | 5870.12398 |

IMFO | 0.7781948 | 0.3846621 | 40.32097 | 199.9812 | 5885.3778 |

Algorithm | Variable | The Target Weight | ||
---|---|---|---|---|

d | D | N | ||

MFO [1] | 0.051994457 | 0.36410932 | 10.868421862 | 0.0126669 |

LAFBA [27] | 0.051663 | 0.356074 | 11.333400 | 0.0126720 |

ES [31] | 0.051989 | 0.363965 | 10.890522 | 0.0126810 |

CPSO [34] | 0.051728 | 0.357644 | 11.244543 | 0.0126747 |

LFD [35] | 0.0517 | 0.3575 | 11.2442 | 0.0127 |

WOA [36] | 0.051207 | 0.345215 | 12.0043032 | 0.0126763 |

CCMWOA [39] | 0.051843 | 0.360444 | 11.07410 | 0.0126660 |

LWOA [40] | 0.051124 | 0.342922 | 12.16119 | 0.0126920 |

GA [41] | 0.051480 | 0.351661 | 11.632201 | 1.0127048 |

RO [44] | 0.051370 | 0.349096 | 11.762790 | 0.0126788 |

IMFO | 0.05159 | 0.354337 | 11.4301 | 0.012666 |

Algorithm | Variable | Target Cost | |||
---|---|---|---|---|---|

h | l | t | b | ||

LMFO [26] | 0.2020 | 3.3575 | 9.0938 | 0.2061 | 1.7165 |

LAFBA [27] | 0.184706815 | 3.642655691 | 9.134897358 | 0.205254053 | 1.7287 |

CSDE [33] | 0.2057 | 3.47 | 9.037 | 0.2057 | 1.724849 |

CPSO [34] | 0.202369 | 3.544214 | 9.048210 | 0.205723 | 1.728024 |

LFD [35] | 0.1857 | 3.9070 | 9.1552 | 0.2051 | 1.77 |

WOA [36] | 0.205396 | 3.484293 | 9.037426 | 0.206276 | 1.730499 |

RO [44] | 0.203687 | 3.528467 | 9.004233 | 0.207241 | 1.735344 |

SFO [45] | 0.2038 | 3.6630 | 9.0506 | 0.2064 | 1.73231 |

IMFO | 0.20573 | 3.4702 | 9.0375 | 0.20573 | 1.7249 |

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## Share and Cite

**MDPI and ACS Style**

Li, Y.; Zhu, X.; Liu, J.
An Improved Moth-Flame Optimization Algorithm for Engineering Problems. *Symmetry* **2020**, *12*, 1234.
https://doi.org/10.3390/sym12081234

**AMA Style**

Li Y, Zhu X, Liu J.
An Improved Moth-Flame Optimization Algorithm for Engineering Problems. *Symmetry*. 2020; 12(8):1234.
https://doi.org/10.3390/sym12081234

**Chicago/Turabian Style**

Li, Yu, Xinya Zhu, and Jingsen Liu.
2020. "An Improved Moth-Flame Optimization Algorithm for Engineering Problems" *Symmetry* 12, no. 8: 1234.
https://doi.org/10.3390/sym12081234