# Pelican Optimization Algorithm: A Novel Nature-Inspired Algorithm for Engineering Applications

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Motivation

#### 1.2. Research Gap

#### 1.3. Contribiution

#### 1.4. Paper Organization

## 2. Background

## 3. Pelican Optimization Algorithm

#### 3.1. Inspiration and Behavior of Pelican during Hunting

#### 3.2. Mathematical Model of the Proposed POA

- (i)
- Moving towards prey (exploration phase).
- (ii)
- Winging on the water surface (exploitation phase).

#### 3.2.1. Phase 1: Moving towards Prey (Exploration Phase)

#### 3.2.2. Phase 2: Winging on the Water Surface (Exploitation Phase)

#### 3.2.3. Steps Repetition, Pseudo-Code, and Flowchart of the Proposed POA

Algorithm 1. Pseudo-code of POA. | ||||

Start POA. | ||||

1. | Input the optimization problem information. | |||

2. | Determine the POA population size (N) and the number of iterations (T). | |||

3. | Initialization of the position of pelicans and calculate the objective function. | |||

4. | For t = 1:T | |||

5. | Generate the position of the prey at random. | |||

6. | For I = 1:N | |||

7. | Phase 1: Moving towards prey (exploration phase). | |||

8. | For j = 1:m | |||

9. | Calculate new status of the jth dimension using Equation (4). | |||

10. | End. | |||

11. | Update the ith population member using Equation (5). | |||

12. | Phase 2: Winging on the water surface (exploitation phase). | |||

13. | For j = 1:m. | |||

14. | Calculate new status of the jth dimension using Equation (6). | |||

15. | End. | |||

16. | Update the ith population member using Equation (7). | |||

17. | End. | |||

18. | Update best candidate solution. | |||

19. | End. | |||

20. | Output best candidate solution obtained by POA. | |||

End POA. |

#### 3.3. Computational Complexity of the Proposed POA

## 4. Simulation Studies and Results

#### 4.1. Evaluation of Unimodal Functions

#### 4.2. Evaluation of High-Dimensional Multimodal Functions

#### 4.3. Evaluation of Fixed-Dimensional Multimodal Functions

#### 4.4. Statistical Analysis

#### 4.5. Sensitivity Analysis

## 5. Discussion

## 6. POA for Real-World Applications

#### 6.1. Pressure Vessel Design

#### 6.2. Speed Reducer Design Problem

#### 6.3. Welded Beam Design

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

#### 6.5. The POA’s Applicability in Image Processing and Sensor Networks

## 7. Conclusions and Future Works

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

Objective Function | Range | Dimensions | ${\mathit{F}}_{\mathit{m}\mathit{i}\mathit{n}}$ |
---|---|---|---|

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

${F}_{2}\left(x\right)={\displaystyle \sum}_{i=1}^{m}\left|{x}_{i}\right|+{\displaystyle \prod}_{i=1}^{m}\left|{x}_{i}\right|$ | $\left[-10,10\right]$ | 30 | 0 |

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

${F}_{4}\left(x\right)=max\left\{\left|{x}_{i}\right|,1\le i\le m\right\}$ | $\left[-100,100\right]$ | 30 | 0 |

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

${F}_{6}\left(x\right)={\displaystyle \sum}_{i=1}^{m}{\left(\left[{x}_{i}+0.5\right]\right)}^{2}$ | $\left[-100,100\right]$ | 30 | 0 |

${F}_{7}\left(x\right)={\displaystyle \sum}_{i=1}^{m}i{x}_{i}^{4}+random\left(0,1\right)$ | $\left[-1.28,1.28\right]$ | 30 | 0 |

Objective Function | Range | Dimensions | F_{min} |
---|---|---|---|

${F}_{8}\left(x\right)={\displaystyle \sum}_{i=1}^{m}-{x}_{i}\mathrm{sin}\left(\sqrt{\left|{x}_{i}\right|}\right)$ | $\left[-500,500\right]$ | 30 | −12,569 |

${F}_{9}\left(x\right)={\displaystyle \sum}_{i=1}^{m}\left[{x}_{i}^{2}-10\mathrm{cos}\left(2\pi {x}_{i}\right)+10\right]$ | $\left[-5.12,5.12\right]$ | 30 | 0 |

${F}_{10}\left(x\right)=-20\mathrm{exp}\left(-0.2\sqrt{\frac{1}{m}{\displaystyle \sum}_{i=1}^{m}{x}_{i}^{2}}\right)-\mathrm{exp}\left(\frac{1}{m}{\displaystyle \sum}_{i=1}^{m}\mathrm{cos}\left(2\pi {x}_{i}\right)\right)+20+e$ | $\left[-32,32\right]$ | 30 | 0 |

${F}_{11}\left(x\right)=\frac{1}{4000}{\displaystyle \sum}_{i=1}^{m}{x}_{i}^{2}-{\displaystyle \prod}_{i=1}^{m}\mathrm{cos}\left(\frac{{x}_{i}}{\sqrt{i}}\right)+1$ | $\left[-600,600\right]$ | 30 | 0 |

${F}_{12}\left(x\right)=\frac{\pi}{m}\left\{10\mathrm{sin}\left(\pi {y}_{1}\right)+{\displaystyle \sum}_{i=1}^{m}{\left({y}_{i}-1\right)}^{2}\left[1+10{\mathrm{sin}}^{2}\left(\pi {y}_{i+1}\right)\right]+{\left({y}_{n}-1\right)}^{2}\right\}+{\displaystyle \sum}_{i=1}^{m}u\left({x}_{i},10,100,4\right),$ where $u\left({x}_{i},a,i,n\right)=\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{n},{x}_{i}-a;\\ 0,-a\le {x}_{i}\le a\\ k{\left(-{x}_{i}-a\right)}^{n},{x}_{i}-a.\end{array};$ | $\left[-50,50\right]$ | 30 | 0 |

${F}_{13}\left(x\right)=0.1\left\{{\mathrm{sin}}^{2}\left(3\pi {x}_{1}\right)+{\displaystyle \sum}_{i=1}^{m}{\left({x}_{i}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(3\pi {x}_{i}+1\right)\right]+{\left({x}_{n}-1\right)}^{2}\left[1+{\mathrm{sin}}^{2}\left(2\pi {x}_{m}\right)\right]\right\}+{\displaystyle \sum}_{i=1}^{m}u\left({x}_{i},5,100,4\right),$ | $\left[-50,50\right]$ | 30 | 0 |

Objective Function | Range | Dimensions | F_{min} |
---|---|---|---|

${F}_{14}\left(x\right)={\left(\frac{1}{500}+{\displaystyle \sum}_{j=1}^{25}\frac{1}{j+{{\displaystyle \sum}}_{i=1}^{2}{\left({x}_{i}-{a}_{ij}\right)}^{6}}\right)}^{-1}$ | $\left[-65.53,65.53\right]$ | 2 | 0.998 |

${F}_{15}\left(x\right)={\displaystyle \sum}_{i=1}^{11}{\left[{a}_{i}-\frac{{x}_{1}\left({b}_{i}^{2}+{b}_{i}{x}_{2}\right)}{{b}_{i}^{2}+{b}_{i}{x}_{3}+{x}_{4}}\right]}^{2}$ | $\left[-5,5\right]$ | 4 | 0.00030 |

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

${F}_{17}\left(x\right)={\left({x}_{2}-\frac{5.1}{4{\pi}^{2}}{x}_{1}^{2}+\frac{5}{\pi}{x}_{1}-6\right)}^{2}+10\left(1-\frac{1}{8\pi}\right)\mathrm{cos}{x}_{1}+10$ | [-5, 10] $\times $ [0, 15] | 2 | 0.398 |

${F}_{18}\left(x\right)=\left[1+{\left({x}_{1}+{x}_{2}+1\right)}^{2}\left(19-14{x}_{1}+3{x}_{1}^{2}-14{x}_{2}+6{x}_{1}{x}_{2}+3{x}_{2}^{2}\right)\right]\xb7\left[30+{\left(2{x}_{1}-3{x}_{2}\right)}^{2}\xb7\left(18-32{x}_{1}+12{x}_{1}^{2}+48{x}_{2}-36{x}_{1}{x}_{2}+27{x}_{2}^{2}\right)\right]$ | $\left[-5,5\right]$ | 2 | 3 |

${F}_{19}\left(x\right)=-{\displaystyle \sum}_{i=1}^{4}{c}_{i}\mathrm{exp}\left(-{\displaystyle \sum}_{j=1}^{3}{a}_{ij}{\left({x}_{j}-{P}_{ij}\right)}^{2}\right)$ | $\left[0,1\right]$ | 3 | −3.86 |

${F}_{20}\left(x\right)=-{\displaystyle \sum}_{i=1}^{4}{c}_{i}\mathrm{exp}\left(-{\displaystyle \sum}_{j=1}^{6}{a}_{ij}{\left({x}_{j}-{P}_{ij}\right)}^{2}\right)$ | $\left[0,1\right]$ | 6 | −3.22 |

${F}_{21}\left(x\right)=-{\displaystyle \sum}_{i=1}^{5}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+6{c}_{i}\right]}^{-1}$ | $\left[0,10\right]$ | 4 | −10.1532 |

${F}_{22}\left(x\right)=-{\displaystyle \sum}_{i=1}^{7}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+6{c}_{i}\right]}^{-1}$ | $\left[0,10\right]$ | 4 | −10.4029 |

${F}_{23}\left(x\right)=-{\displaystyle \sum}_{i=1}^{10}{\left[\left(X-{a}_{i}\right){\left(X-{a}_{i}\right)}^{T}+6{c}_{i}\right]}^{-1}$ | $\left[0,10\right]$ | 4 | −10.5364 |

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Algorithm | Parameter | Value |
---|---|---|

MPA | Binary vector | U = 0 or 1 |

Random vector | $R\mathrm{is}\mathrm{a}\mathrm{vector}\mathrm{of}\mathrm{uniform}\mathrm{random}\mathrm{numbers}\mathrm{in}\left[0,1\right].$ | |

Constant number | p = 0.5 | |

Fish Aggregating Devices (FADs) | FADs = 0.2 | |

TSA | c1, c2, c3 | random numbers lie in the interval [0, 1]. |

Pmin | 1 | |

Pmax | 4 | |

WOA | l is a random number in [−1, 1]. | |

$r\mathrm{is}\mathrm{a}\mathrm{random}\mathrm{vector}\mathrm{in}\left[0,1\right].$ | ||

Convergence parameter (a) | a: Linear reduction from 2 to 0. | |

GWO | Convergence parameter (a) | a: Linear reduction from 2 to 0. |

TLBO | random number | rand is a random number from interval [0, 1]. |

T_{F}: teaching factor | ${T}_{F}=\mathrm{round}\left[\left(1+rand\right)\right]$ | |

GSA | Alpha | 20 |

G_{0} | 100 | |

Rnorm | 2 | |

Rnorm | 1 | |

PSO | Velocity limit | 10% of dimension range |

Topology | Fully connected | |

Inertia weight | Linear reduction from 0.9 to 0.1 | |

Cognitive and social constant | $\left({C}_{1},{C}_{2}\right)=\left(2,2\right)$ | |

GA | Type | Real coded |

Mutation | Gaussian (Probability = 0.05) | |

Crossover | Whole arithmetic (Probability = 0.8) | |

Selection | Roulette wheel (Proportionate) |

GA | PSO | GSA | TLBO | GWO | WOA | TSA | MPA | POA | ||
---|---|---|---|---|---|---|---|---|---|---|

F_{1} | avg | 11.6208 | 4.1728 × 10^{−4} | 2.0259 × 10^{−16} | 3.8324 × 10^{−59} | 1.0896 × 10^{−57} | 5.37 × 10^{−62} | 5.7463 × 10^{−37} | 3.2612 × 10^{−20} | 2.87 × 10^{−258} |

std | 2.6142 × 10^{−11} | 3.6142 × 10^{−21} | 6.9113 × 10^{−30} | 9.6318 × 10^{−72} | 5.1462 × 10^{−73} | 5.78 × 10^{−78} | 6.3279 × 10^{−20} | 1.5264 × 10^{−19} | 4.51 × 10^{−514} | |

bsf | 5.593489 | 2 × 10^{−10} | 8.2 × 10^{−18} | 9.36 × 10^{−61} | 7.73 × 10^{−61} | 1.61 × 10^{−65} | 1.14 × 10^{−62} | 3.41 × 10^{−28} | 7.62 × 10^{−264} | |

med | 11.04546 | 9.92 × 10^{−7} | 1.78 × 10^{−17} | 4.69 × 10^{−60} | 1.08 × 10^{−59} | 8.42 × 10^{−54} | 3.89 × 10^{−38} | 1.27 × 10^{−19} | 8.2 × 10^{−248} | |

F_{2} | avg | 4.6942 | 0.3114 | 7.0605 × 10^{−7} | 4.6237 × 10^{−34} | 2.0509 × 10^{−33} | 2.51 × 10^{−55} | 4.5261 × 10^{−38} | 6.3214 × 10^{−11} | 1.43× 10^{−128} |

std | 5.4318 × 10^{−14} | 4.4667 × 10^{−16} | 8.5637 × 10^{−23} | 9.3719 × 10^{−49} | 6.3195 × 10^{−29} | 5.60 × 10^{−58} | 2.6591 × 10^{−40} | 3.6249 × 10^{−11} | 2.90× 10^{−129} | |

bsf | 1.591137 | 0.001741 | 1.59 × 10^{−8} | 1.32 × 10^{−35} | 1.55 × 10^{−35} | 3.42 × 10^{−63} | 8.26 × 10^{−43} | 4.25 × 10^{−18} | 2.61 × 10^{−131} | |

med | 2.463873 | 0.130114 | 2.33 × 10^{−8} | 4.37 × 10^{−35} | 6.38 × 10^{−35} | 1.59 × 10^{−51} | 8.26 × 10^{−41} | 3.18 × 10^{−11} | 7.1 × 10^{−123} | |

F_{3} | avg | 1361.2743 | 588.3012 | 280.6014 | 7.0772 × 10^{−14} | 4.7206 × 10^{−14} | 7.5621 × 10^{−9} | 5.6230 × 10^{−20} | 0.0819 | 1.88× 10^{−256} |

std | 6.6096 × 10^{−12} | 9.7117 × 10^{−12} | 5.2497 × 10^{−12} | 8.9637 × 10^{−30} | 6.5225 × 10^{−28} | 1.02 × 10^{−18} | 7.0925 × 10^{−19} | 0.1370 | 5.16× 10^{−614} | |

bsf | 1014.689 | 1.614937 | 81.91242 | 1.21 × 10^{−16} | 4.75 × 10^{−20} | 1.9738 × 10^{−11} | 7.29 × 10^{−30} | 0.032038 | 7.36 × 10^{−262} | |

med | 1510.715 | 54.15445 | 291.4308 | 1.86 × 10^{−15} | 1.59 × 10^{−16} | 17085.2 | 9.81 × 10^{−21} | 0.378658 | 8.2 × 10^{−244} | |

F_{4} | avg | 2.0396 | 4.3693 | 2.6319 × 10^{−8} | 8.9196 × 10^{−14} | 1.9925 × 10^{−13} | 0.0013 | 3.1162 × 10^{−22} | 6.3149 × 10^{−8} | 2.36× 10^{−133} |

std | 4.3321× 10^{−14} | 4.2019 × 10^{−15} | 5.3017 × 10^{−23} | 1.7962 × 10^{−29} | 1.8305 × 10^{−28} | 0.0877 | 6.3129 × 10^{−21} | 2.3687 × 10^{−9} | 8.37× 10^{−134} | |

bsf | 1.389849 | 1.60441 | 2.09 × 10^{−09} | 6.41 × 10^{−16} | 3.43 × 10^{−16} | 0.0001 | 1.87 × 10^{−52} | 3.42 × 10^{−17} | 6.08 × 10^{−138} | |

med | 2.09854 | 3.260672 | 3.34 × 10^{−09} | 1.54 × 10^{−15} | 7.3 × 10^{−15} | 0.0010 | 3.13 × 10^{−27} | 3.03 × 10^{−08} | 2.8 × 10^{−123} | |

F_{5} | avg | 308.4196 | 50.5412 | 36.01528 | 147.6214 | 27.1786 | 27.17543 | 28.8592 | 46.0408 | 27.1253 |

std | 3.0412 × 10^{−12} | 1.8529 × 10^{−13} | 2.6091 × 10^{−13} | 6.3017 × 10^{−13} | 8.7029 × 10^{−14} | 0.393959 | 4.3219 × 10^{−3} | 0.4199 | 1.91× 10^{−15} | |

bsf | 160.5013 | 3.647051 | 25.83811 | 120.7932 | 25.21201 | 26.43249 | 28.53831 | 41.58682 | 26.2052 | |

med | 279.5174 | 28.69298 | 26.07475 | 142.8936 | 26.70874 | 26.93542 | 28.53913 | 42.49068 | 28.707 | |

F_{6} | avg | 15.6231 | 20.2691 | 0 | 0.5531 | 0.6518 | 0.071527 | 5.7268 × 10^{−20} | 0.3894 | 0 |

std | 7.3160 × 10^{−14} | 2.6314 | 0 | 3.1971 × 10^{−15} | 5.3096 × 10^{−16} | 0.006113 | 2.1163 × 10^{−24} | 0.2001 | 0 | |

bsf | 6 | 5 | 0 | 0 | 1.57 × 10^{−05} | 0.014645 | 6.74 × 10^{−26} | 0.274582 | 0 | |

med | 13.5 | 19 | 0 | 0 | 0.621487 | 0.029296 | 6.74 × 10^{−21} | 0.406648 | 0 | |

F_{7} | avg | 8.6517 × 10^{−2} | 0.3218 | 0.0234 | 0.0011 | 0.0077 | 0.00103 | 8.2196 × 10^{−4} | 1.2561× 10^{−3} | 9.37× 10^{−6} |

std | 8.9206 × 10^{−17} | 3.4333 × 10^{−16} | 7.1526 × 10^{−17} | 3.2610 × 10^{−18} | 7.2307 × 10^{−19} | 1.12 × 10^{−5} | 9.6304 × 10^{−5} | 9.6802× 10^{−3} | 8.03× 10^{−20} | |

bsf | 0.002111 | 0.029593 | 0.01006 | 0.001362 | 0.000248 | 4.24 × 10^{−5} | 0.000104 | 0.001429 | 7.05 × 10^{−07} | |

med | 0.005365 | 0.107872 | 0.016995 | 0.002912 | 0.000629 | 0.00215 | 0.000367 | 0.00218 | 4.86 × 10^{−05} |

GA | PSO | GSA | TLBO | GWO | WOA | TSA | MPA | POA | ||
---|---|---|---|---|---|---|---|---|---|---|

F_{8} | avg | −8210.3415 | −6899.9556 | −2854.5207 | −7410.8016 | −5903.3711 | −7239.1 | −5737.7822 | −3611.2271 | −9336.7304 |

std | 833.5126 | 625.4286 | 2641576 | 513.4752 | 467.8216 | 261.0117 | 39.5203 | 811.1459 | 2.64× 10^{−12} | |

bsf | −9717.68 | −8501.44 | −3969.23 | −9103.77 | −7227.05 | −7568.9 | −5706.3 | −4419.9 | −9850.21 | |

med | −8117.66 | −7098.95 | −2671.33 | −7735.22 | −5774.63 | −7124.8 | −5669.63 | −3632.84 | −8505.55 | |

F_{9} | avg | 62.1441 | 57.0503 | 16.5714 | 10.1379 | 8.1036 × 10^{−14} | 0 | 6.0311 × 10^{−3} | 139.9806 | 0 |

std | 2.1637 × 10^{−13} | 6.0013 × 10^{−14} | 6.1972 × 10^{−14} | 4.9631 × 10^{−14} | 4.6537 × 10^{−29} | 0 | 5.6146 × 10^{−3} | 25.9024 | 0 | |

bsf | 36.86623 | 27.85883 | 4.974795 | 9.873963 | 0 | 0 | 0.004776 | 128.2306 | 0 | |

med | 61.67858 | 55.22468 | 15.42187 | 10.88657 | 0 | 0 | 0.005871 | 154.6214 | 0 | |

F_{10} | avg | 3.8134 | 2.6304 | 3.5438 × 10^{−9} | 0.2691 | 8.6234 × 10^{−13} | 3.91 × 10^{−15} | 8.6247 × 10^{−13} | 8.6291 × 10^{−11} | 8.88× 10^{−16} |

std | 6.8972 × 10^{−15} | 6.9631 × 10^{−15} | 2.7054 × 10^{−24} | 6.4129 × 10^{−14} | 5.6719 × 10^{−28} | 7.01 × 10^{−30} | 1.6240 × 10^{−12} | 5.3014 × 10^{−11} | 0 | |

bsf | 2.757203 | 1.155151 | 2.64 × 10^{−09} | 0.156305 | 1.51 × 10^{−14} | 8.88 × 10^{−16} | 8.14 × 10^{−15} | 1.68 × 10^{−18} | 8.88 × 10^{−16} | |

med | 3.120322 | 2.170083 | 3.64 × 10^{−09} | 0.261541 | 1.51 × 10^{−14} | 4.44 × 10^{−15} | 1.1 × 10^{−13} | 1.05 × 10^{−11} | 8.88 × 10^{−16} | |

F_{11} | avg | 1.1973 | 0.0364 | 3.9123 | 0.5912 | 0.0013 | 2.03 × 10^{−4} | 5.3614 × 10^{−7} | 0 | 0 |

std | 4.8521 × 10^{−15} | 2.6398 × 10^{−17} | 4.0306 × 10^{−14} | 6.2914 × 10^{−15} | 6.1294 × 10^{−17} | 1.82 × 10^{−17} | 6.3195 × 10^{−7} | 0 | 0 | |

bsf | 1.140471 | 7.29 × 10^{−09} | 1.519288 | 0.310117 | 0 | 0 | 4.23 × 10^{−15} | 0 | 0 | |

med | 1.227231 | 0.029473 | 3.424268 | 0.582026 | 0 | 0 | 8.77 × 10^{−07} | 0 | 0 | |

F_{12} | avg | 0.0469 | 0.4792 | 0.0341 | 0.0219 | 0.0364 | 0.007728 | 0.0372 | 0.0815 | 0.0583 |

std | 1.7456 × 10^{−14} | 9.3071 × 10^{−15} | 2.0918 × 10^{−16} | 2.6195 × 10^{−14} | 1.3604 × 10^{−13} | 8.07E-05 | 8.6391 × 10^{−2} | 0.0162 | 2.73 × 10^{−16} | |

bsf | 0.018364 | 0.000145 | 5.57 × 10^{−20} | 0.002031 | 0.019294 | 0.001142 | 0.035428 | 0.077912 | 0.0452 | |

med | 0.04179 | 0.1556 | 1.48 × 10^{−19} | 0.015181 | 0.032991 | 0.003887 | 0.050935 | 0.082108 | 0.1464 | |

F_{13} | avg | 1.2106 | 0.5156 | 0.0017 | 0.3306 | 0.5561 | 0.193293 | 2.8041 | 0.4875 | 1.42866 |

std | 3.5630 × 10^{−15} | 4.1427 × 10^{−16} | 1.9741 × 10^{−13} | 5.6084 × 10^{−15} | 5.6219 × 10^{−15} | 0.022767 | 3.9514 × 10^{−11} | 0.1041 | 2.83× 10^{−15} | |

bsf | 0.49809 | 9.99 × 10^{−07} | 1.18 × 10^{−18} | 0.038266 | 0.297822 | 0.029662 | 2.63175 | 0.280295 | 1.428663 | |

med | 1.218053 | 0.043997 | 2.14 × 10^{−18} | 0.282764 | 0.578323 | 0.146503 | 2.66175 | 0.579854 | 2.976773 |

GA | PSO | GSA | TLBO | GWO | WOA | TSA | MPA | POA | ||
---|---|---|---|---|---|---|---|---|---|---|

F_{14} | avg | 0.9969 | 2.3909 | 3.9505 | 2.4998 | 4.1140 | 1.106143 | 2.061 | 0.9980 | 0.9980 |

std | 6.3124 × 10^{−14} | 8.0126 × 10^{−15} | 8.9631 × 10^{−15} | 6.3014 × 10^{−15} | 1.3679 × 10^{−14} | 0.48689 | 5.6213 × 10^{−7} | 1.9082 × 10^{−15} | 0 | |

bsf | 0.998004 | 0.998004 | 0.999508 | 0.998391 | 0.998004 | 0.998004 | 0.9979 | 0.9980 | 0.9980 | |

med | 0.998018 | 0.998004 | 2.986658 | 2.275231 | 2.982105 | 0.998004 | 1.912608 | 0.9980 | 0.9980 | |

F_{15} | avg | 0.0042 | 0.0528 | 0.0027 | 0.0031 | 0.0059 | 0.000463 | 0.0005 | 0.0028 | 0.0003 |

std | 1.6317 × 10^{−17} | 2.6159 × 10^{−18} | 3.6051 × 10^{−18} | 6.3195 × 10^{−16} | 3.0598 × 10^{−17} | 1.22 × 10^{−7} | 1.6230 × 10^{−5} | 1.2901 × 10^{−14} | 1.21× 10^{−19} | |

bsf | 0.000775 | 0.000307 | 0.000805 | 0.002206 | 0.000307 | 0.000313 | 0.000264 | 0.00027 | 0.0003 | |

med | 0.002074 | 0.000307 | 0.002311 | 0.003185 | 0.000308 | 0.000492 | 0.00039 | 0.0027 | 0.0003 | |

F_{16} | avg | −1.0307 | −1.0312 | −1.0309 | −1.0310 | −1.0316 | −1.0316 | −1.0314 | −1.0315 | −1.0316 |

std | 9.1449 × 10^{−15} | 3.2496 × 10^{−15} | 5.4162 × 10^{−15} | 1.3061 × 10^{−14} | 3.0816 × 10^{−15} | 2.38 × 10^{−20} | 6.0397 × 10^{−15} | 2.1679 × 10^{−15} | 1.93× 10^{−18} | |

bsf | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.0316 | −1.03161 | −1.0316 | −1.03163 | |

med | −1.0309 | −1.0311 | −1.0310 | −1.0308 | −1.0316 | −1.0316 | −1.0311 | −1.0312 | −1.03163 | |

F_{17} | avg | 0.4401 | 0.7951 | 0.3980 | 0.3978 | 0.3981 | 0.39788 | 0.3987 | 0.3991 | 0.3978 |

std | 1.4109 × 10^{−16} | 3.9801 × 10^{−5} | 1.0291 × 10^{−16} | 2.1021 × 10^{−15} | 6.0391 × 10^{−16} | 1.42 × 10^{−12} | 6.1472 × 10^{−15} | 5.9317 × 10^{−14} | 0 | |

bsf | 0.3978 | 0.3978 | 0.3978 | 0.3978 | 0.3978 | 0.397887 | 0.3980 | 0.3982 | 0.3978 | |

med | 0.4016 | 0.6521 | 0.3979 | 0.3978 | 0.3979 | 0.397887 | 0.3990 | 0.3977 | 0.3978 | |

F_{18} | avg | 4.3601 | 3.0010 | 3.0016 | 3.0010 | 3.0009 | 3.000009 | 3 | 3.0013 | 3 |

std | 2.6108 × 10^{−15} | 1.1041 × 10^{−14} | 3.7159 × 10^{−15} | 7.6013 × 10^{−14} | 5.0014 × 10^{−14} | 2.42 × 10^{−15} | 5.6148 × 10^{−14} | 2.3017 × 10^{−14} | 1.09× 10^{−16} | |

bsf | 3.0002 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | 3 | |

med | 3.7581 | 3.0005 | 3.0008 | 3.0006 | 3.0006 | 3.000001 | 3 | 3.0009 | 3 | |

F_{19} | avg | −3.8519 | −3.8627 | −3.8627 | −3.8615 | −3.8617 | −3.86068 | −3.8205 | −3.8627 | −3.86278 |

std | 3.6015 × 10^{−14} | 7.0114 × 10^{−14} | 5.3419 × 10^{−14} | 1.0314 × 10^{−14} | 9.6041 × 10^{−14} | 6.55 × 10^{−6} | 6.7514 × 10^{−14} | 2.6197 × 10^{−14} | 6.45× 10^{−16} | |

bsf | −3.86278 | −3.8627 | −3.8627 | −3.8625 | −3.8627 | −3.86278 | −3.8366 | −3.8627 | −3.86278 | |

med | −3.8413 | −3.8560 | −3.8627 | −3.8620 | −3.8612 | −3.86216 | −3.8066 | −3.8627 | −3.86278 | |

F_{20} | avg | −2.8301 | −3.2626 | −3.0402 | −3.1927 | −3.2481 | −3.22298 | −3.3201 | −3.3195 | −3.3220 |

std | 3.7124 × 10^{−15} | 3.4567 × 10^{−15} | 5.2179 × 10^{−13} | 5.3140 × 10^{−14} | 3.3017 × 10^{−14} | 0.008173 | 6.5203 × 10^{−14} | 9.8160 × 10^{−10} | 1.97× 10^{−16} | |

bsf | −3.31342 | −3.322 | −3.322 | −3.26174 | −3.32199 | −3.32198 | −3.3212 | −3.3213 | −3.322 | |

med | −2.96828 | −3.2160 | −2.9014 | −3.2076 | −3.26248 | −3.19935 | −3.3206 | −3.3211 | −3.322 | |

F_{21} | avg | −4.2593 | −5.4236 | −5.2014 | −9.2049 | −9.6602 | −8.87635 | −5.1477 | −9.9561 | −10.1532 |

std | 2.3631 × 10^{−8} | 6.3014 × 10^{−9} | 5.8961 × 10^{−8} | 3.8715 × 10^{−14} | 5.3391 × 10^{−14} | 5.123359 | 6.1974 × 10^{−12} | 8.7195 × 10^{−10} | 1.93× 10^{−16} | |

bsf | −7.82781 | −8.0267 | −7.3506 | −9.6638 | −10.1532 | −10.1531 | −7.5020 | −10.1532 | −10.1532 | |

med | −4.16238 | −5.10077 | −3.64802 | −9.1532 | −10.1526 | −10.1518 | −5.5020 | −10.1531 | −10.1532 | |

F_{22} | avg | −5.1183 | −7.6351 | −9.0241 | −10.0399 | −10.4199 | −9.33732 | −5.0597 | −10.2859 | −10.4029 |

std | 6.1697 × 10^{−14} | 5.0610 × 10^{−14} | 5.0231 × 10^{−11} | 6.7925 × 10^{−13} | 6.1496 × 10^{−14} | 4.752577 | 3.1673 × 10^{−14} | 7.3596 × 10^{−10} | 3.57× 10^{−16} | |

bsf | −9.1106 | −10.4024 | −10.4026 | −10.4023 | −10.4021 | −10.4028 | −9.06249 | −10.4029 | −10.4029 | |

med | −5.0296 | −10.4020 | −10.4017 | −10.1836 | −10.4015 | −10.4013 | −5.06249 | −10.4027 | −10.4029 | |

F_{23} | avg | −6.5675 | −6.1653 | −8.9091 | −9.2916 | −10.1319 | −9.45231 | −10.3675 | −10.1409 | −10.5364 |

std | 5.6014 × 10^{−14} | 5.3917 × 10^{−15} | 8.0051 × 10^{−14} | 5.2673 × 10^{−14} | 2.6912 × 10^{−15} | 9.47 × 10^{−9} | 2.9637 × 10^{−12} | 5.0981 × 10^{−10} | 3.97× 10^{−16} | |

bsf | −10.2227 | −10.5364 | −10.5364 | −10.5340 | −10.5363 | −10.5363 | −10.3683 | −10.5364 | −10.5364 | |

med | −6.5629 | −4.50554 | −10.5360 | −9.6717 | −10.5361 | −10.5349 | −10.3613 | −10.2159 | −10.5364 |

Functions Type | Compared Algorithms | |||||||
---|---|---|---|---|---|---|---|---|

POA and MPA | POA and TSA | POA and WOA | POA and GWO | POA and TLBO | POA and GSA | POA and PSO | POA and GA | |

Unimodal | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0156 | 0.0312 | 0.0156 | 0.0156 |

High-dimensional multimodal | 0.3125 | 0.2187 | 0.1562 | 0.8437 | 0.3125 | 0.3125 | 0.1562 | 0.1562 |

Fixed-dimensional multimodal | 0.0195 | 0.0039 | 0.0078 | 0.0117 | 0.0058 | 0.0195 | 0.0039 | 0.0019 |

Objective Function | Number of Population Members | |||
---|---|---|---|---|

20 | 30 | 50 | 80 | |

F_{1} | 9.3343 × 10^{−212} | 1.6451 × 10^{−235} | 2.87 × 10^{−258} | 7.3038 × 10^{−260} |

F_{2} | 1.5489 × 10^{−98} | 2.303 × 10^{−119} | 1.42 × 10^{−128} | 2.0842 × 10^{−132} |

F_{3} | 1.6656 × 10^{−206} | 9.9891 × 10^{−249} | 1.879 × 10^{−256} | 2.1553 × 10^{−259} |

F_{4} | 6.0489 × 10^{−112} | 1.4332 × 10^{−127} | 2.36 × 10^{−133} | 3.6451 × 10^{−136} |

F_{5} | 28.4440 | 27.1418 | 27.1253 | 25.4195 |

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

F_{7} | 0.0001 | 8.8865 × 10^{−6} | 9.37 × 10^{−6} | 1.3305 × 10^{−6} |

F_{8} | −7727.8678 | −8924.3072 | −9336.7304 | −9385.8725 |

F_{9} | 0 | 0 | 0 | 0 |

F_{10} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} |

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

F_{12} | 0.2944 | 0.0369 | 0.0583 | 0.0142 |

F_{13} | 2.9548 | 2.0214 | 1.4286 | 2.0471 |

F_{14} | 1.6403 | 1.0120 | 0.9980 | 0.9980 |

F_{15} | 0.0024 | 0.0003 | 0.0003 | 0.0003 |

F_{16} | −1.0311 | −1.0314 | −1.0316 | −1.03163 |

F_{17} | 0.3987 | 0.3983 | 0.3978 | 0.3978 |

F_{18} | 3.0003 | 3.0001 | 3.0000 | 3.0000 |

F_{19} | −3.8615 | −3.8625 | −3.8628 | −3.8628 |

F_{20} | −3.3041 | −3.3120 | −3.322 | −3.322 |

F_{21} | −7.3492 | −10.1529 | −10.1532 | −10.1532 |

F_{22} | −8.0110 | −10.4023 | −10.4029 | −10.4029 |

F_{23} | −8.6436 | −10.5357 | −10.5364 | −10.5364 |

Objective Function | Maximum Number of Iterations | |||
---|---|---|---|---|

100 | 500 | 800 | 1000 | |

F_{1} | 2.7725 × 10^{−19} | 6.2604 × 10^{−115} | 4.3539 × 10^{−185} | 2.87 × 10^{−258} |

F_{2} | 1.1541 × 10^{−9} | 3.5658 × 10^{−57} | 1.61505 × 10^{−94} | 1.42 × 10^{−128} |

F_{3} | 2.1172 × 10^{−19} | 5.0884 × 10^{−117} | 6.461 × 10^{−180} | 1.879 × 10^{−256} |

F_{4} | 5.9252 × 10^{−10} | 1.8962 × 10^{−56} | 3.1178 × 10^{−92} | 2.36 × 10^{−133} |

F_{5} | 28.9350 | 28.5274 | 28.3259 | 27.1253 |

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

F_{7} | 0.0007 | 0.0001 | 9.0872 × 10^{−5} | 9.37 × 10^{−6} |

F_{8} | −6753.5658 | −8063.7455 | −8208.3044 | −9336.7304 |

F_{9} | 0 | 0 | 0 | 0 |

F_{10} | 1.1932 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} |

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

F_{12} | 0.5768 | 0.2211 | 0.1673 | 0.0583 |

F_{13} | 2.8999 | 2.7595 | 2.7286 | 1.4286 |

F_{14} | 1.0012 | 0.9996 | 0.9980 | 0.9980 |

F_{15} | 0.0013 | 0.0007 | 0.0004 | 0.0003 |

F_{16} | −1.0310 | −1.0314 | −1.0316 | −1.03163 |

F_{17} | 0.3983 | 0.3972 | 0.3978 | 0.3978 |

F_{18} | 3.0172 | 3.0120 | 3.0001 | 3.0000 |

F_{19} | −3.7928 | −3.8598 | −3.8628 | −3.8628 |

F_{20} | −3.2810 | −3.3160 | −3.3041 | −3.322 |

F_{21} | −9.8968 | −9.6433 | −9.8982 | −10.1532 |

F_{22} | −10.4002 | −10.4018 | −10.4022 | −10.4029 |

F_{23} | −10.5358 | −10.5361 | −10.5363 | −10.5364 |

OF | R Value | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

0.1 | 0.2 | 0.3 | 0.4 | 0.5 | 0.6 | 0.7 | 0.8 | 0.9 | 1 | |

F_{1} | 4.84 × 10^{−244} | 2.87 × 10^{−258} | 7.98 × 10^{−246} | 3.79 × 10^{−244} | 6.25 × 10^{−240} | 6.31 × 10^{−235} | 2.32 × 10^{−231} | 4.98 × 10^{−227} | 6.44 × 10^{−224} | 1.04 × 10^{−221} |

F_{2} | 1.50 × 10^{−126} | 1.42 × 10^{−128} | 2.72 × 10^{−125} | 7.70 × 10^{−125} | 2.01 × 10^{−123} | 3.85 × 10^{−122} | 1.89 × 10^{−121} | 2.56 × 10^{−120} | 4.69 × 10^{−119} | 6.50 × 10^{−115} |

F_{3} | 6.84 × 10^{−256} | 1.879 × 10^{−256} | 3.92 × 10^{−251} | 4.90 × 10^{−248} | 1.83 × 10^{−244} | 4.39 × 10^{−241} | 8.56 × 10^{−236} | 2.83 × 10^{−236} | 8.20 × 10^{−235} | 1.96 × 10^{−234} |

F_{4} | 3.50 × 10^{−126} | 2.36 × 10^{−133} | 8.99 × 10^{−120} | 1.96 × 10^{−123} | 1.90 × 10^{−126} | 2.60 × 10^{−122} | 4.96 × 10^{−115} | 4.04 × 10^{−112} | 1.40 × 10^{−112} | 6.74 × 10^{−110} |

F_{5} | 27.5583 | 27.1253 | 27.5641 | 27.5912 | 27.8162 | 28.4294 | 28.5964 | 28.6237 | 28.6907 | 28.7015 |

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

F_{7} | 3.43 × 10^{−5} | 9.37 × 10^{−6} | 4.86 × 10^{−5} | 7.62 × 10^{−5} | 4.31 × 10^{−5} | 2.06 × 10^{−4} | 2.71 × 10^{−4} | 4.63 × 10^{−4} | 3.66 × 10^{−4} | 5.70 × 10^{−4} |

F_{8} | −8934.1836 | −9336.7304 | −8963.8127 | −8898.2760 | −8702.3872 | −8629.6948 | −8485.2713 | −8212.2289 | −8070.2688 | −7919.3914 |

F_{9} | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 |

F_{10} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} | 8.88 × 10^{−16} |

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

F_{12} | 0.1542 | 0.0583 | 0.0629 | 0.0701 | 0.0821 | 0.08659 | 0.08826 | 0.09184 | 0.09633 | 0.097571 |

F_{13} | 2.8516 | 1.4286 | 2.1295 | 2.5203 | 2.591 | 2.6314 | 2.4736 | 2.3871 | 2.7630 | 2.8532 |

F_{14} | 0.9980 | 0.9980 | 0.9980 | 0.9980 | 0.9980 | 0.9980 | 0.9980 | 0.9980 | 0.9980 | 0.9980 |

F_{15} | 0.0003 | 0.0003 | 0.0003 | 0.0003 | 0.0003 | 0.0003 | 0.0003 | 0.0003 | 0.0003 | 0.0003 |

F_{16} | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 | −1.03163 |

F_{17} | 0.3978 | 0.3978 | 0.3978 | 0.3978 | 0.3978 | 0.3978 | 0.3978 | 0.3978 | 0.3978 | 0.3978 |

F_{18} | 3.0000 | 3.0000 | 3.0000 | 3.0000 | 3.0000 | 3.0000 | 3.0000 | 3.0000 | 3.0000 | 3.0000 |

F_{19} | −3.8628 | −3.8628 | −3.8628 | −3.8628 | −3.8628 | −3.8628 | −3.8628 | −3.8628 | −3.8628 | −3.8628 |

F_{20} | −3.322 | −3.322 | −3.322 | −3.3219 | −3.3218 | −3.3218 | −3.1984 | −3.1821 | −3.1167 | −3.0126 |

F_{21} | −10.1532 | −10.1532 | −10.1531 | −10.1531 | −10.1529 | −10.1527 | −9.8965 | −9.9623 | −9.2196 | −9.1637 |

F_{22} | −10.4029 | −10.4029 | −10.4027 | −10.4027 | −10.3827 | −10.3561 | −10.0032 | −9.7304 | −9.1931 | −9.0157 |

F_{23} | −10.5364 | −10.5364 | −10.5363 | −10.5363 | −10.2195 | −10.0412 | −9.6318 | −9.2305 | −9.1027 | −10.0081 |

Algorithm | Optimum Variables | Optimum Cost | |||
---|---|---|---|---|---|

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

POA | 0.778035 | 0.384607 | 40.31261 | 199.9972 | 5883.0278 |

MPA | 0.782101 | 0.386813 | 40.51662 | 200 | 5915.005 |

TSA | 0.78293 | 0.386583 | 40.52943 | 200 | 5918.816 |

WOA | 0.782856 | 0.386606 | 40.52252 | 200 | 5920.845 |

GWO | 0.849948 | 0.420657 | 44.03535 | 157.1635 | 6041.572 |

TLBO | 0.821665 | 0.420022 | 41.95814 | 184.4906 | 6168.059 |

GSA | 1.091229 | 0.954362 | 49.59196 | 170.3348 | 11608.05 |

PSO | 0.756124 | 0.401538 | 40.65478 | 198.9927 | 5919.78 |

GA | 1.105021 | 0.911112 | 44.67868 | 180.5572 | 6582.773 |

Algorithm | Best | Mean | Worst | Std. Dev. | Median |
---|---|---|---|---|---|

POA | 5883.0278 | 5887.082 | 5894.256 | 24.35317 | 5886.457 |

MPA | 5915.005 | 5890.388 | 5895.267 | 2.894447 | 5889.171 |

TSA | 5918.816 | 5894.47 | 5897.571 | 13.91696 | 5893.595 |

WOA | 5920.845 | 6534.769 | 7398.285 | 534.3861 | 6419.322 |

GWO | 6041.572 | 6480.544 | 7254.542 | 327.1705 | 6400.679 |

TLBO | 6168.059 | 6329.924 | 6515.61 | 126.6723 | 6321.477 |

GSA | 11608.05 | 6843.963 | 7162.87 | 5793.52 | 6841.052 |

PSO | 5919.78 | 6267.137 | 7009.253 | 496.3761 | 6115.746 |

GA | 6582.773 | 6647.309 | 8009.442 | 657.8518 | 7589.802 |

Algorithm | Optimum Variables | Optimum Cost | ||||||
---|---|---|---|---|---|---|---|---|

b | m | p | l_{1} | l_{2} | d_{1} | d_{2} | ||

POA | 3.5 | 0.7 | 17 | 7.3 | 7.8 | 3.350215 | 5.286683 | 2996.3482 |

MPA | 3.503341 | 0.7 | 17 | 7.3 | 7.8 | 3.352946 | 5.291384 | 3000.05 |

TSA | 3.508443 | 0.7 | 17 | 7.381059 | 7.815726 | 3.359526 | 5.289411 | 3002.789 |

WOA | 3.501769 | 0.7 | 17 | 8.3 | 7.8 | 3.354088 | 5.289358 | 3007.266 |

GWO | 3.510256 | 0.7 | 17 | 7.410236 | 7.816034 | 3.359752 | 5.28942 | 3004.429 |

TLBO | 3.510509 | 0.7 | 17 | 7.3 | 7.8 | 3.462751 | 5.291858 | 3032.078 |

GSA | 3.6018 | 0.7 | 17 | 8.3 | 7.8 | 3.371343 | 5.291869 | 3052.646 |

PSO | 3.512008 | 0.7 | 17 | 8.35 | 7.8 | 3.363882 | 5.290367 | 3069.095 |

GA | 3.521884 | 0.7 | 17 | 8.37 | 7.8 | 3.368653 | 5.291363 | 3030.517 |

Algorithm | Best | Mean | Worst | Std. Dev. | Median |
---|---|---|---|---|---|

POA | 2996.3482 | 2999.88 | 3001.491 | 1.782335 | 2998.715 |

MPA | 3000.05 | 3002.04 | 3006.292 | 1.933476 | 3001.586 |

TSA | 3002.789 | 3008.25 | 3011.159 | 5.84261 | 3006.923 |

WOA | 3007.266 | 3107.736 | 3213.743 | 79.70181 | 3107.736 |

GWO | 3004.429 | 3031.264 | 3063.407 | 13.02901 | 3029.453 |

TLBO | 3032.078 | 3068.37 | 3107.263 | 18.08866 | 3068.061 |

GSA | 3052.646 | 3172.87 | 3366.564 | 92.64666 | 3159.277 |

PSO | 3069.095 | 3189.072 | 3315.85 | 17.13229 | 3200.746 |

GA | 3030.517 | 3297.965 | 3622.361 | 57.06912 | 3291.288 |

Algorithm | Optimum Variables | Optimum Cost | |||
---|---|---|---|---|---|

h | l | T | b | ||

POA | 0.205719 | 3.470104 | 9.038353 | 0.205722 | 1.725021 |

MPA | 0.205604 | 3.475541 | 9.037606 | 0.205852 | 1.726006 |

TSA | 0.205719 | 3.476098 | 9.038771 | 0.20627 | 1.72734 |

WOA | 0.19745 | 3.315724 | 10.000 | 0.201435 | 1.820759 |

GWO | 0.205652 | 3.472797 | 9.042739 | 0.20575 | 1.725817 |

TLBO | 0.204736 | 3.536998 | 9.006091 | 0.210067 | 1.759525 |

GSA | 0.147127 | 5.491842 | 10.000 | 0.217769 | 2.173293 |

PSO | 0.164204 | 4.033348 | 10.000 | 0.223692 | 1.874346 |

GA | 0.206528 | 3.636599 | 10.000 | 0.20329 | 1.836617 |

Algorithm | Best | Mean | Worst | Std. Dev. | Median |
---|---|---|---|---|---|

POA | 1.724968 | 1.726504 | 1.728593 | 0.004328 | 1.725779 |

MPA | 1.726006 | 1.727209 | 1.727445 | 0.000287 | 1.727168 |

TSA | 1.72734 | 1.72851 | 1.728946 | 0.001158 | 1.728469 |

WOA | 1.820759 | 2.232094 | 3.05067 | 0.324785 | 2.246459 |

GWO | 1.725817 | 1.731064 | 1.743044 | 0.00487 | 1.728802 |

TLBO | 1.759525 | 1.819111 | 1.874907 | 0.027565 | 1.821584 |

GSA | 2.173293 | 2.546274 | 3.00606 | 0.256064 | 2.49711 |

PSO | 1.874346 | 2.120935 | 2.321981 | 0.034848 | 2.098726 |

GA | 1.836617 | 1.364618 | 2.036875 | 0.139597 | 1.937297 |

Algorithm | Optimum Variables | Optimum Cost | ||
---|---|---|---|---|

d | D | p | ||

POA | 0.051892 | 0.361608 | 11.00793 | 0.012666 |

MPA | 0.051154 | 0.34382 | 12.09792 | 0.012677 |

TSA | 0.050188 | 0.341609 | 12.0759 | 0.012681 |

WOA | 0.05001 | 0.310476 | 15.003 | 0.013195 |

GWO | 0.05001 | 0.316019 | 14.22908 | 0.012819 |

TLBO | 0.05079 | 0.334846 | 12.72523 | 0.012712 |

GSA | 0.05001 | 0.317375 | 14.23152 | 0.012876 |

PSO | 0.05011 | 0.310173 | 14.0028 | 0.013039 |

GA | 0.05026 | 0.316414 | 15.24265 | 0.012779 |

Algorithm | Best | Mean | Worst | Std. Dev. | Median |
---|---|---|---|---|---|

POA | 0.012666 | 0.012688 | 0.012677 | 0.001022 | 0.012685 |

MPA | 0.012677 | 0.012693 | 0.012724 | 0.005623 | 0.012696 |

TSA | 0.012681 | 0.012706 | 0.01273 | 0.004157 | 0.012709 |

WOA | 0.013195 | 0.014828 | 0.017875 | 0.002274 | 0.013202 |

GWO | 0.012819 | 0.014474 | 0.017852 | 0.001623 | 0.014031 |

TLBO | 0.012712 | 0.012849 | 0.013008 | 7.81E-05 | 0.012854 |

GSA | 0.012876 | 0.013448 | 0.014222 | 0.000287 | 0.013377 |

PSO | 0.013039 | 0.014046 | 0.016263 | 0.002074 | 0.013011 |

GA | 0.012779 | 0.013079 | 0.015225 | 0.000375 | 0.012961 |

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## Share and Cite

**MDPI and ACS Style**

Trojovský, P.; Dehghani, M.
Pelican Optimization Algorithm: A Novel Nature-Inspired Algorithm for Engineering Applications. *Sensors* **2022**, *22*, 855.
https://doi.org/10.3390/s22030855

**AMA Style**

Trojovský P, Dehghani M.
Pelican Optimization Algorithm: A Novel Nature-Inspired Algorithm for Engineering Applications. *Sensors*. 2022; 22(3):855.
https://doi.org/10.3390/s22030855

**Chicago/Turabian Style**

Trojovský, Pavel, and Mohammad Dehghani.
2022. "Pelican Optimization Algorithm: A Novel Nature-Inspired Algorithm for Engineering Applications" *Sensors* 22, no. 3: 855.
https://doi.org/10.3390/s22030855