# A Comprehensive Multi-Strategy Enhanced Biogeography-Based Optimization Algorithm for High-Dimensional Optimization and Engineering Design Problems

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## Abstract

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## 1. Introduction

- (1)
- A novel framework of BBO is proposed, which is simpler and more efficient than the original BBO algorithm. Meanwhile, MSBBO has lower computational complexity than BBO.
- (2)
- MSBBO uses the example chasing strategy to eliminate the misguidance of bad information in the population. Then, the heuristic crossover and prey search–attack strategies are used to balance the exploration and exploitation of the population. MSBBO makes BBO suitable for high-dimensional optimization environments.
- (3)
- MSBBO successfully challenges the 10,000-dimensional numerical optimization problem. Compared with other meta-heuristic algorithms, its convergence performance is basically not affected by dimensions and has good ductility.

## 2. Standard BBO

Algorithm 1 Pseudo-code of the BBO. |

initialize parameters: ${S}_{max}$, I, E, N, and ${m}_{max}$ initialize the population by Equation (1) for t = 1 to T do calculate the $HSI$ and sort from best to worst calculate the ${S}_{i}$ by Equation (2), the ${\lambda}_{i}$ and ${\mu}_{i}$ by Equation (3) calculate the ${P}_{i}$ by Equation (4), the ${m}_{i}$ by Equation (5) for i = 1 to N do % Migration for j = 1 to D do if rand(0,1) < ${\lambda}_{i}$ do select the ${x}_{k}$ according to ${\left\{{\mu}_{k}\right\}}_{k=1}^{N}$ ${x}_{ij}$ = ${x}_{kj}$ end if end for % Mutation if rand(0,1) < ${m}_{i}$ do for j = 1 to D do ${x}_{ij}=l{b}_{j}+\mathit{rand}(0,1)\xb7\left(u{b}_{j}-l{b}_{j}\right)$ end for end if end for end for output the optimal solution |

## 3. Proposed Algorithm: MSBBO

#### 3.1. Motivation

#### 3.2. Example Chasing Strategy

#### 3.3. Heuristic Crossover Strategy

#### 3.4. Prey Search–Attack Operator

Algorithm 2 Pseudo-code of the MSBBO. |

initialize parameters: ${S}_{max},I,N,freq$ initialize the population by Equation (1) calculate the ${S}_{i}$ by Equation (2), the ${\lambda}_{i}$ by Equation (3) calculate the HSI and sort from best to worst for t = 1 to T do for i = 1 to N do if rand(0,1) < ${\lambda}_{i}$ do select the ${x}_{k}$ according to Equation (6) heuristic crossover of ${x}_{i}$ by Equations (7) and (8) end if calculate the ${\omega}_{1}$ by Equation (11), the ${\omega}_{2}$ by Equation (12) search the prey by Equation (9) attack the prey by Equation (10) end for calculate the HSI and sort from best to worst end for output the optimal solution |

## 4. Complexity Analysis

## 5. Experimental Results and Analysis

#### 5.1. Experiment Preparation

#### 5.2. Comparison between MSBBO and Standard BBO

#### 5.3. Comparison between MSBBO and BBO Variants

#### 5.4. Comparison between MSBBO and Other Meta-Heuristic Algorithms

#### 5.5. Comparison of MSBBO on Different High Dimensions

#### 5.6. Application on Engineering Design Problems

#### 5.6.1. Pressure Vessel Design

#### 5.6.2. Tension/Compression Spring Design

#### 5.6.3. Welded Beam Design

#### 5.6.4. Speed Reducer Design

#### 5.6.5. Step-Cone Pulley Problem

#### 5.6.6. Robot Gripper Problem

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 5.**Convergence curves and boxplots of MSBBO and BBO on different benchmark functions ($D=30,50,100$).

**Figure 7.**Convergence curves and boxplots of MSBBO and BBO variants on different benchmark functions ($D=200$).

**Figure 8.**Convergence curves and boxplots of MSBBO and other meta-heuristic algorithms on different benchmark functions ($D=500$).

**Figure 9.**Convergence curves of MSBBO on $f1,f3,f7,f10,f16$ and $f24$ ($D=500,1000,2000,5000$ and 10,000).

Biogeography Theory | Biogeography-Based Optimization Algorithm |
---|---|

Habitats (Islands) | Individuals (candidate solutions) |

Habitat suitability index ($HSI$) | Objective function value (fitness) |

Suitability index variables ($SIVs$) | Characteristic variables of solutions |

Catastrophic events destroyed the habitat | Mutation |

The number of habitats | Population size (the number of solutions) |

Habitats with low HSI immigrate species | Inferior solutions accept variables |

Habitats with high HSI emigrate species | Superior solutions share variables |

Function | Search Space | $\mathit{f}\left({\mathit{x}}^{*}\right)$ |
---|---|---|

$f1\left(x\right)={\sum}_{i=1}^{D}i{x}_{i}^{2}$ | ${[-10,10]}^{D}$ | 0 |

$f2\left(x\right)={\sum}_{i=1}^{D}\left|{x}_{i}\right|+{\prod}_{i=1}^{D}\left|{x}_{i}\right|$ | ${[-10,10]}^{D}$ | 0 |

$f3\left(x\right)={\sum}_{i=1}^{D}{x}_{i}^{2}+{\left({\sum}_{i=1}^{D}0.5i{x}_{i}\right)}^{2}+{\left({\sum}_{i=1}^{D}0.5i{x}_{i}\right)}^{4}$ | ${[-5,10]}^{D}$ | 0 |

$f4\left(x\right)={max}_{i=1}^{D}\left\{\left|{x}_{i}\right|\right\}$ | ${[-100,100]}^{D}$ | 0 |

$f5\left(x\right)={\sum}_{i=1}^{D}{\left({\sum}_{j=1}^{i}{x}_{j}\right)}^{2}\times (1+0.4|N(0,1)\left|\right)$ | ${[-100,100]}^{D}$ | 0 |

$f6\left(x\right)={\sum}_{i=1}^{D}i{x}_{i}^{4}+rand$ | ${[-100,100]}^{D}$ | 0 |

$f7\left(x\right)={\sum}_{i=1}^{D}{z}_{i}^{2}-450,z=x-o$ | ${[-100,100]}^{D}$ | −450 |

$f8\left(x\right)={\sum}_{i=1}^{D}{\left|{x}_{i}\right|}^{i+1}$ | ${[-1,1]}^{D}$ | 0 |

$f9\left(x\right)=exp\left(0.5{\sum}_{i=1}^{D}\left|{x}_{i}\right|\right)-1$ | ${[-1.28,1.28]}^{D}$ | 0 |

$f10\left(x\right)={\sum}_{i=1}^{D}{\left({10}^{6}\right)}^{\frac{i-1}{D-1}}{x}_{i}^{2}$ | ${[-100,100]}^{D}$ | 0 |

$f11\left(x\right)={\sum}_{i=1}^{D}{\u230a{x}_{i}+0.5\u230b}^{2}$ | ${[-100,100]}^{D}$ | 0 |

$f12\left(x\right)={\sum}_{i=1}^{D}{\left({\sum}_{j=1}^{i}{x}_{j}\right)}^{2}$ | ${[-100,100]}^{D}$ | 0 |

$f13\left(x\right)={\sum}_{i=1}^{D}\left[{z}_{i}^{2}-10cos\left(2\pi {z}_{i}\right)+10\right]-330,z=x-o$ | ${[-5.12,5.12]}^{D}$ | −330 |

$f14\left(x\right)={\sum}_{i=1}^{D}\left[{z}_{i}^{2}-10cos\left(2\pi {z}_{i}\right)+10\right],$ ${z}_{i}=\left\{\begin{array}{c}{x}_{i},\left|{x}_{i}\right|<0.5\hfill \\ \phantom{\rule{4.pt}{0ex}}\mathrm{round}\phantom{\rule{4.pt}{0ex}}\left(2{x}_{i}\right)/2,\phantom{\rule{4.pt}{0ex}}\mathrm{else}\phantom{\rule{4.pt}{0ex}}\hfill \end{array}\right.$ | ${[-5.12,5.12]}^{D}$ | 0 |

$f15\left(x\right)=-20exp\left(-0.2\sqrt{{\sum}_{i=1}^{D}\frac{{z}_{i}^{2}}{D}}\right)-exp\left[\frac{1}{D}{\sum}_{i=1}^{D}cos\left(2\pi {z}_{i}\right)\right]$ $+e-120,z=x-o$ | ${[-32,32]}^{D}$ | −140 |

$f16\left(x\right)={\sum}_{i=1}^{D}\left|{x}_{i}sin\left({x}_{i}\right)+0.1{x}_{i}\right|$ | ${[-10,10]}^{D}$ | 0 |

$f17\left(x\right)=\frac{1}{4000}\left[{\sum}_{i=1}^{D}{\left({z}_{i}-100\right)}^{2}\right]-\left[{\prod}_{i=1}^{D}cos\left(\frac{{z}_{i}-100}{\sqrt{i}}\right)\right]$ $-179,z=x-o$ | ${[-600,600]}^{D}$ | −180 |

$f18\left(x\right)=-cos\left(2\pi \sqrt{{\sum}_{i}^{D}{x}_{i}^{2}}\right)+0.1\times \sqrt{{\sum}_{i}^{D}{x}_{i}^{2}}+1$ | ${[-100,100]}^{D}$ | 0 |

$f19\left(x\right)={\sum}_{i=1}^{D}\left\{{\sum}_{k=0}^{{k}_{max}}\left[{a}^{k}cos\left(2\pi {b}^{k}\left({x}_{i}+0.5\right)\right)\right]\right\}-$ $D{\sum}_{k=0}^{{k}_{max}}\left[{a}^{k}cos\left(2\pi {b}^{k}\times 0.5\right)\right],$ $a=0.5,b=3,{k}_{max}=20$ | ${[-0.5,0.5]}^{D}$ | 0 |

$f20\left(x\right)=\frac{\pi}{D}\{10{sin}^{2}\left(\pi {y}_{i}\right)+{\sum}_{i=1}^{D-1}{\left({y}_{i}-1\right)}^{2}\left[1+10{sin}^{2}\left(\pi {y}_{i+1}\right)\right]$ $+{\left({y}_{D}-1\right)}^{2}\}+{\sum}_{i=1}^{D}u\left({x}_{i},10,100,4\right)$ ${y}_{i}=0.25({x}_{i}+1)+1$, $u\left({x}_{i},a,k,m\right)=\left\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{m},{x}_{i}>a\hfill \\ 0,=-a\le {x}_{i}\le a\hfill \\ k{\left(-{x}_{i}-a\right)}^{m},{x}_{i}<-a\hfill \end{array}\right.$ | ${[-50,50]}^{D}$ | 0 |

$f21\left(x\right)=0.1\{{sin}^{2}\left(3\pi {x}_{1}\right)+{\sum}_{i=1}^{D-1}{\left({x}_{i}-1\right)}^{2}\left[1+{sin}^{2}\left(3\pi {x}_{i+1}\right)\right]$ $+\left({x}_{D}-1\right)\left[1+{sin}^{2}\left(2\pi {x}_{D}\right)\right]\}+{\sum}_{i=1}^{D}u\left({x}_{i},5,100,4\right)$ $u\left({x}_{i},a,k,m\right)=\left\{\begin{array}{c}k{\left({x}_{i}-a\right)}^{m},{x}_{i}>a\hfill \\ 0,-a\le {x}_{i}\le a\hfill \\ k{\left(-{x}_{i}-a\right)}^{m},{x}_{i}<-a\hfill \end{array}\right.$ | ${[-50,50]}^{D}$ | 0 |

$f22\left(x\right)={\sum}_{i=1}^{n-1}F\left({x}_{i},{x}_{i+1}\right)+F\left({x}_{n},{x}_{1}\right),$ | ${[-100,100]}^{D}$ | 0 |

$F(x,y)={\left({x}^{2}+{y}^{2}\right)}^{0.25}\xb7\left[{sin}^{2}\left(50{\left({x}^{2}+{y}^{2}\right)}^{0.1}\right)+1\right]$ | ||

$f23\left(x\right)={\sum}_{i=1}^{D-1}\left[{x}_{i}^{2}+2{x}_{i+1}^{2}-0.3cos\left(3\pi {x}_{i}\right)cos\left(3\pi {x}_{i+1}\right)+0.3\right]$ | ${[-100,100]}^{D}$ | 0 |

$f24\left(x\right)={\sum}_{i=1}^{D/4}[{\left({x}_{4i-3}+10{x}_{4i-2}\right)}^{2}+5{\left({x}_{4i-1}-{x}_{4i}\right)}^{2}$ $+{\left({x}_{4i-2}-2{x}_{4i-1}\right)}^{4}+10{\left({x}_{4i-3}-{x}_{4i}\right)}^{4}]$ | ${[-4,5]}^{D}$ | 0 |

F | BBO (D = 30) | MSBBO (D = 30) | BBO (D = 50) | MSBBO (D = 50) | BBO (D = 100) | MSBBO (D = 100) | ||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

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

$f1$ | 1.74E+00 | 4.83E-01 | 0.00E+00 | 0.00E+00 | 1.47E+01 | 1.80E+01 | 0.00E+00 | 0.00E+00 | 3.56E+02 | 3.33E+03 | 0.00E+00 | 0.00E+00 |

$f2$ | 1.24E+00 | 6.40E-02 | 0.00E+00 | 0.00E+00 | 3.49E+00 | 2.53E-01 | 0.00E+00 | 0.00E+00 | 1.65E+01 | 2.82E+00 | 0.00E+00 | 0.00E+00 |

$f3$ | 6.89E+01 | 2.99E+02 | 0.00E+00 | 0.00E+00 | 2.12E+02 | 1.55E+03 | 0.00E+00 | 0.00E+00 | 7.91E+02 | 7.05E+03 | 0.00E+00 | 0.00E+00 |

$f4$ | 9.75E+00 | 3.53E+00 | 0.00E+00 | 0.00E+00 | 2.01E+01 | 5.39E+00 | 0.00E+00 | 0.00E+00 | 4.12E+01 | 6.70E+00 | 0.00E+00 | 0.00E+00 |

$f5$ | 1.88E+04 | 2.09E+07 | 0.00E+00 | 0.00E+00 | 5.51E+04 | 1.37E+08 | 0.00E+00 | 0.00E+00 | 2.11E+05 | 9.67E+08 | 0.00E+00 | 0.00E+00 |

$f6$ | 4.61E+02 | 1.88E+05 | 2.42E-05 | 2.47E-05 | 1.40E+04 | 1.73E+08 | 3.03E-05 | 4.05E-05 | 2.25E+06 | 8.81E+11 | 4.52E-05 | 4.41E-05 |

$f7$ | 1.29E+01 | 2.25E+01 | 0.00E+00 | 0.00E+00 | 6.61E+01 | 2.46E+02 | 0.00E+00 | 0.00E+00 | 8.55E+02 | 1.73E+04 | 0.00E+00 | 0.00E+00 |

$f8$ | 8.86E-06 | 2.13E-10 | 0.00E+00 | 0.00E+00 | 1.60E-05 | 5.24E-10 | 0.00E+00 | 0.00E+00 | 2.26E-05 | 1.56E-09 | 0.00E+00 | 0.00E+00 |

$f9$ | 8.93E-02 | 2.46E-04 | 0.00E+00 | 0.00E+00 | 2.78E-01 | 1.72E-03 | 0.00E+00 | 0.00E+00 | 2.30E+00 | 1.07E-01 | 0.00E+00 | 0.00E+00 |

$f10$ | 3.70E+05 | 8.62E+10 | 0.00E+00 | 0.00E+00 | 1.36E+06 | 6.38E+11 | 0.00E+00 | 0.00E+00 | 9.03E+06 | 9.35E+12 | 0.00E+00 | 0.00E+00 |

$f11$ | 1.29E+01 | 2.50E+01 | 0.00E+00 | 0.00E+00 | 7.63E+01 | 4.23E+02 | 0.00E+00 | 0.00E+00 | 8.49E+02 | 2.05E+04 | 0.00E+00 | 0.00E+00 |

$f12$ | 1.77E+03 | 5.33E+05 | 0.00E+00 | 0.00E+00 | 2.40E+04 | 6.00E+07 | 0.00E+00 | 0.00E+00 | 9.86E+05 | 4.18E+10 | 0.00E+00 | 0.00E+00 |

$f13$ | 4.76E+00 | 1.70E+00 | 0.00E+00 | 0.00E+00 | 1.73E+01 | 7.91E+00 | 0.00E+00 | 0.00E+00 | 8.17E+01 | 5.71E+01 | 0.00E+00 | 0.00E+00 |

$f14$ | 4.66E+00 | 2.02E+00 | 0.00E+00 | 0.00E+00 | 1.62E+01 | 6.40E+00 | 0.00E+00 | 0.00E+00 | 6.10E+01 | 1.37E+01 | 0.00E+00 | 0.00E+00 |

$f15$ | 1.85E+00 | 9.78E-02 | 4.44E-16 | 0.00E+00 | 2.93E+00 | 5.39E-02 | 4.44E-16 | 0.00E+00 | 4.98E+00 | 8.09E-02 | 4.44E-16 | 0.00E+00 |

$f16$ | 8.14E-02 | 9.15E-04 | 0.00E+00 | 0.00E+00 | 4.06E-01 | 8.68E-03 | 0.00E+00 | 0.00E+00 | 3.63E+00 | 2.76E-01 | 0.00E+00 | 0.00E+00 |

$f17$ | 1.10E+00 | 1.07E-03 | 0.00E+00 | 0.00E+00 | 1.60E+00 | 2.64E-02 | 0.00E+00 | 0.00E+00 | 8.29E+00 | 1.44E+00 | 0.00E+00 | 0.00E+00 |

$f18$ | 2.07E+00 | 5.86E-02 | 0.00E+00 | 0.00E+00 | 3.79E+00 | 1.73E-01 | 0.00E+00 | 0.00E+00 | 8.91E+00 | 4.75E-01 | 0.00E+00 | 0.00E+00 |

$f19$ | 3.22E+00 | 1.41E-01 | 0.00E+00 | 0.00E+00 | 7.51E+00 | 3.37E-01 | 0.00E+00 | 0.00E+00 | 2.56E+01 | 3.23E+00 | 0.00E+00 | 0.00E+00 |

$f20$ | 5.18E-01 | 1.38E-01 | 7.07E-02 | 2.90E-03 | 6.54E-01 | 1.32E-01 | 2.93E-01 | 2.07E-02 | 3.04E+00 | 4.73E-01 | 8.21E-01 | 9.38E-02 |

$f21$ | 3.54E+00 | 6.57E-01 | 5.95E-01 | 6.07E-02 | 6.62E+00 | 2.17E+00 | 3.54E+00 | 7.36E-01 | 3.81E+02 | 4.84E+05 | 1.53E+01 | 1.37E+00 |

$f22$ | 4.11E+01 | 2.09E+01 | 0.00E+00 | 0.00E+00 | 8.52E+01 | 6.65E+01 | 0.00E+00 | 0.00E+00 | 2.66E+02 | 1.65E+02 | 0.00E+00 | 0.00E+00 |

$f23$ | 4.43E+01 | 2.40E+02 | 0.00E+00 | 0.00E+00 | 2.23E+02 | 3.63E+03 | 0.00E+00 | 0.00E+00 | 2.53E+03 | 1.93E+05 | 0.00E+00 | 0.00E+00 |

$f24$ | 3.86E+00 | 6.75E+00 | 0.00E+00 | 0.00E+00 | 1.61E+01 | 7.63E+01 | 0.00E+00 | 0.00E+00 | 1.55E+02 | 2.71E+03 | 0.00E+00 | 0.00E+00 |

w/t/l | 0/0/24 | - | 0/0/24 | - | 0/0/24 | - |

F | PRBBO (2017) | BBOSB (2018) | HGBBO (2020) | FABBO (2021) | BLEHO (2022) | MPBBO (2022) | BBOIMAM (2022) | MSBBO | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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

$f1$ | 1.43E-02 | 2.09E-05 | 2.01E+03 | 6.06E+04 | 2.13E-15 | 1.23E-30 | 1.47E-13 | 1.95E-27 | 1.80E+02 | 2.65E+03 | 7.93E-04 | 1.63E-07 | 1.29E+02 | 4.84E+02 | 0.00E+00 | 0.00E+00 |

$f2$ | 2.98E-02 | 3.18E-05 | 3.42E+01 | 1.25E+01 | 4.19E-11 | 1.56E-22 | 3.21E-07 | 6.91E-16 | 2.66E+01 | 2.54E+01 | 3.81E+02 | 7.19E+04 | 1.11E+01 | 5.55E-01 | 0.00E+00 | 0.00E+00 |

$f3$ | 2.08E+03 | 5.83E+04 | 2.97E+03 | 6.08E+04 | 3.78E+03 | 3.27E+04 | 2.54E+03 | 6.29E+04 | 1.46E+05 | 2.34E+10 | 2.61E+02 | 5.32E+03 | 1.55E+03 | 2.19E+04 | 0.00E+00 | 0.00E+00 |

$f4$ | 3.01E+01 | 1.99E+01 | 2.98E+01 | 7.53E+00 | 2.39E-02 | 7.06E-05 | 6.89E+01 | 1.20E+02 | 2.09E-02 | 1.67E-04 | 1.18E+01 | 3.09E+00 | 2.12E+01 | 3.17E+00 | 0.00E+00 | 0.00E+00 |

$f5$ | 4.05E+05 | 5.84E+09 | 5.09E+05 | 2.89E+09 | 6.95E+05 | 7.22E+10 | 4.68E+05 | 4.62E+09 | 1.87E+05 | 2.02E+09 | 2.18E+05 | 4.41E+09 | 1.78E+05 | 6.67E+08 | 0.00E+00 | 0.00E+00 |

$f6$ | 4.61E+01 | 1.68E+03 | 4.07E+03 | 1.46E+06 | 5.15E-02 | 1.35E-04 | 2.50E-01 | 3.86E-03 | 7.06E+03 | 4.18E+06 | 8.82E-01 | 3.27E-01 | 3.86E+04 | 8.48E+07 | 4.81E-05 | 4.41E-05 |

$f7$ | 1.98E-02 | 5.78E-05 | 2.20E+01 | 4.68E+00 | 1.28E-16 | 6.78E-33 | 1.02E-13 | 3.38E-28 | 3.04E+01 | 1.59E+01 | 3.51E-04 | 8.51E-08 | 1.41E+02 | 2.83E+02 | 0.00E+00 | 0.00E+00 |

$f8$ | 2.35E-21 | 1.17E-41 | 2.29E-05 | 6.36E-10 | 1.24E-109 | 2.68E-218 | 8.64E-01 | 3.33E-01 | 1.91E-23 | 7.38E-46 | 1.86E-19 | 8.30E-38 | 1.60E-06 | 4.63E-12 | 0.00E+00 | 0.00E+00 |

$f9$ | 1.40E-03 | 7.94E-08 | 2.31E+05 | 2.39E+10 | 4.94E-12 | 3.90E-24 | 1.99E-08 | 2.94E-18 | 4.69E+00 | 3.14E+00 | 1.65E-04 | 1.47E-09 | 1.74E+00 | 3.74E-02 | 0.00E+00 | 0.00E+00 |

$f10$ | 2.32E+01 | 9.02E+01 | 3.98E+05 | 6.59E+09 | 3.59E-10 | 2.85E-20 | 7.12E+01 | 6.03E+03 | 1.02E+07 | 5.47E+12 | 2.77E+04 | 1.42E+08 | 1.01E+06 | 6.47E+10 | 0.00E+00 | 0.00E+00 |

$f11$ | 0.00E+00 | 0.00E+00 | 4.75E+01 | 4.72E+01 | 0.00E+00 | 0.00E+00 | 4.54E+01 | 1.82E+02 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 1.52E+02 | 2.70E+02 | 0.00E+00 | 0.00E+00 |

$f12$ | 7.56E+01 | 7.15E+02 | 3.30E+05 | 1.82E+09 | 1.22E-11 | 8.06E-23 | 3.76E-09 | 4.10E-18 | 2.06E+06 | 6.25E+10 | 3.59E+01 | 6.74E+01 | 5.17E+05 | 2.37E+09 | 0.00E+00 | 0.00E+00 |

$f13$ | 6.63E+02 | 6.61E+02 | 7.12E+02 | 2.79E+03 | 1.31E-11 | 2.21E-22 | 1.14E+03 | 1.11E+04 | 3.34E+02 | 4.49E+03 | 3.99E+02 | 4.80E+03 | 1.02E+02 | 1.64E+02 | 0.00E+00 | 0.00E+00 |

$f14$ | 5.08E+02 | 7.19E+02 | 5.15E+02 | 2.58E+03 | 4.40E-04 | 2.96E-07 | 1.28E+03 | 1.76E+04 | 2.65E+02 | 2.27E+04 | 4.85E+02 | 3.67E+03 | 8.06E+01 | 3.05E+01 | 0.00E+00 | 0.00E+00 |

$f15$ | 1.26E-02 | 3.66E-06 | 3.11E+00 | 4.65E-02 | 1.99E+01 | 6.88E-04 | 1.60E+00 | 4.80E-01 | 3.63E+00 | 5.71E-02 | 6.84E+00 | 6.53E+01 | 2.26E+00 | 1.79E-02 | 4.44E-16 | 0.00E+00 |

$f16$ | 9.36E-01 | 6.22E-01 | 3.81E+01 | 1.74E+01 | 4.32E-02 | 3.99E-04 | 6.42E+00 | 8.08E+01 | 2.44E+01 | 2.45E+01 | 9.14E+00 | 8.52E+00 | 3.73E+00 | 5.73E-01 | 0.00E+00 | 0.00E+00 |

$f17$ | 5.28E-03 | 3.29E-06 | 1.78E-01 | 4.85E-04 | 1.11E-16 | 0.00E+00 | 3.84E-03 | 2.21E-05 | 1.26E+00 | 1.28E-03 | 2.79E-03 | 1.35E-05 | 2.31E+00 | 2.83E-02 | 0.00E+00 | 0.00E+00 |

$f18$ | 2.81E+00 | 3.55E-02 | 4.71E+00 | 8.88E-02 | 4.01E-01 | 3.98E-03 | 2.12E+00 | 2.09E-01 | 4.19E+00 | 7.28E-02 | 1.42E+00 | 2.51E-02 | 5.67E+00 | 4.96E-02 | 0.00E+00 | 0.00E+00 |

$f19$ | 9.38E-01 | 8.32E-03 | 1.88E+02 | 2.05E+01 | 6.87E-10 | 4.51E-20 | 6.35E+01 | 3.29E+01 | 1.26E+02 | 1.36E+02 | 8.27E-01 | 9.15E-03 | 4.10E+01 | 1.62E+00 | 0.00E+00 | 0.00E+00 |

$f20$ | 2.06E-02 | 3.31E-04 | 1.76E+00 | 2.49E-01 | 7.96E-06 | 3.26E-12 | 7.82E+00 | 1.06E+01 | 1.14E+01 | 4.55E+00 | 3.49E+00 | 2.35E+00 | 6.89E-02 | 1.59E-04 | 9.47E-01 | 7.06E-02 |

$f21$ | 2.73E-01 | 1.79E-02 | 1.84E+01 | 2.05E+01 | 8.71E-04 | 1.11E-07 | 1.09E+02 | 8.51E+03 | 5.11E+01 | 2.38E+02 | 2.52E-02 | 6.21E-04 | 6.59E+00 | 6.89E-01 | 3.21E+01 | 2.05E+00 |

$f22$ | 1.17E+02 | 1.02E+02 | 6.11E+02 | 2.38E+03 | 2.42E-03 | 5.70E-07 | 1.25E+03 | 7.29E+03 | 1.09E+03 | 5.35E+03 | 4.06E+02 | 7.15E+04 | 3.13E+02 | 1.75E+02 | 0.00E+00 | 0.00E+00 |

$f23$ | 7.21E-01 | 1.03E-01 | 1.22E+02 | 8.72E+01 | 4.34E-15 | 1.81E-29 | 4.16E+00 | 7.08E+00 | 1.70E+02 | 1.61E+02 | 2.11E-02 | 2.94E-04 | 4.70E+02 | 4.07E+03 | 0.00E+00 | 0.00E+00 |

$f24$ | 8.75E+00 | 8.98E+00 | 1.85E+03 | 8.66E+04 | 8.30E-01 | 5.30E-02 | 1.90E-01 | 5.15E-03 | 2.65E+01 | 4.97E+01 | 2.95E+00 | 2.06E+00 | 2.15E+01 | 3.55E+01 | 0.00E+00 | 0.00E+00 |

w/t/l | 2/1/21 | 1/0/23 | 2/1/21 | 0/0/24 | 0/1/23 | 1/1/22 | 2/0/22 | - |

F | GWO (2014) | WOA (2016) | SSA (2019) | ChOA (2019) | MPA (2020) | GJO (2022) | BWO (2022) | MSBBO | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

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

$f1$ | 6.10E-56 | 5.02E-56 | 4.90E-103 | 2.03E-102 | 5.73E-05 | 1.32E-04 | 1.06E-12 | 9.06E-13 | 2.92E-125 | 7.86E-125 | 1.31E-16 | 1.49E-16 | 1.89E-04 | 1.51E-04 | 0.00E+00 | 0.00E+00 |

$f2$ | 1.00E-33 | 6.29E-34 | 1.15E-108 | 3.19E-108 | 6.69E-03 | 5.18E-03 | 2.18E-09 | 1.27E-09 | 1.63E-06 | 1.15E-05 | 4.44E-140 | 5.38E-140 | 1.83E-02 | 7.94E-03 | 0.00E+00 | 0.00E+00 |

$f3$ | 7.02E+03 | 5.32E+03 | 7.97E+03 | 3.52E+02 | 1.22E+00 | 3.08E+00 | 6.57E+02 | 4.06E+02 | 2.90E-02 | 2.01E-02 | 1.05E+04 | 1.05E+04 | 1.39E+00 | 1.26E+00 | 0.00E+00 | 0.00E+00 |

$f4$ | 9.94E+01 | 7.07E+01 | 9.89E+01 | 3.91E-01 | 1.05E-04 | 9.49E-05 | 2.63E+02 | 2.13E+02 | 6.28E-42 | 9.13E-42 | 2.13E+02 | 1.76E+02 | 3.50E-04 | 1.57E-04 | 0.00E+00 | 0.00E+00 |

$f5$ | 4.82E+05 | 2.99E+05 | 4.32E+07 | 2.83E+07 | 2.89E+00 | 2.67E+00 | 1.31E+06 | 6.79E+05 | 3.30E+02 | 6.46E+02 | 1.80E+07 | 1.12E+07 | 9.73E+00 | 7.54E+00 | 0.00E+00 | 0.00E+00 |

$f6$ | 6.84E-03 | 3.73E-03 | 2.00E+02 | 2.81E+02 | 3.52E-03 | 3.15E-03 | 6.51E-03 | 3.31E-03 | 2.59E-04 | 1.11E-04 | 2.74E+05 | 1.82E+05 | 2.31E-03 | 1.40E-03 | 4.98E-05 | 5.99E-05 |

$f7$ | 3.83E-56 | 3.73E-56 | 1.73E-102 | 1.13E-101 | 1.00E-05 | 1.75E-05 | 2.64E-13 | 1.89E-13 | 6.28E-126 | 1.02E-125 | 1.60E-15 | 2.02E-15 | 7.46E-05 | 6.95E-05 | 0.00E+00 | 0.00E+00 |

$f8$ | 5.65E-24 | 6.55E-24 | 1.40E-05 | 2.08E-05 | 2.60E-12 | 6.30E-12 | 2.61E+00 | 1.92E+00 | 0.00E+00 | 0.00E+00 | 2.15E-06 | 2.35E-06 | 1.20E-12 | 1.40E-12 | 0.00E+00 | 0.00E+00 |

$f9$ | 3.89E-15 | 3.31E-15 | 0.00E+00 | 0.00E+00 | 4.19E-04 | 3.59E-04 | 8.62E-11 | 5.33E-11 | 0.00E+00 | 0.00E+00 | 4.66E-16 | 2.54E-16 | 1.26E-03 | 4.90E-04 | 0.00E+00 | 0.00E+00 |

$f10$ | 2.56E-53 | 1.85E-53 | 2.51E-102 | 1.55E-101 | 1.05E+00 | 1.71E+00 | 6.21E-10 | 5.63E-10 | 1.01E-121 | 3.56E-121 | 4.16E-26 | 5.50E-26 | 5.85E+00 | 5.03E+00 | 0.00E+00 | 0.00E+00 |

$f11$ | 0.00E+00 | 0.00E+00 | 2.81E+01 | 3.49E+01 | 0.00E+00 | 0.00E+00 | 1.40E-01 | 3.51E-01 | 0.00E+00 | 0.00E+00 | 5.64E+02 | 4.66E+02 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f12$ | 4.23E-52 | 2.49E-52 | 1.34E-98 | 1.68E-98 | 1.39E+00 | 1.16E+00 | 8.24E-09 | 6.54E-09 | 8.73E-122 | 1.61E-121 | 9.39E-15 | 1.11E-14 | 2.95E+00 | 1.60E+00 | 0.00E+00 | 0.00E+00 |

$f13$ | 0.00E+00 | 0.00E+00 | 1.25E+02 | 5.98E+02 | 7.46E-06 | 1.48E-05 | 3.88E-06 | 4.23E-06 | 0.00E+00 | 0.00E+00 | 1.17E+03 | 1.50E+03 | 4.42E-05 | 4.06E-05 | 0.00E+00 | 0.00E+00 |

$f14$ | 1.24E+00 | 1.48E+00 | 7.29E+01 | 1.49E+02 | 1.34E-05 | 2.52E-05 | 2.82E-01 | 2.85E-01 | 0.00E+00 | 0.00E+00 | 8.99E+01 | 8.57E+01 | 3.69E-05 | 2.41E-05 | 0.00E+00 | 0.00E+00 |

$f15$ | 9.05E-14 | 7.37E-14 | 3.25E+00 | 7.11E+00 | 1.64E-04 | 1.58E-04 | 5.13E+01 | 4.48E+01 | 4.44E-15 | 0.00E+00 | 1.08E-10 | 1.21E-10 | 4.77E-04 | 2.40E-04 | 4.44E-16 | 0.00E+00 |

$f16$ | 2.01E-32 | 2.16E-32 | 3.13E+01 | 5.94E+01 | 2.32E-04 | 2.24E-04 | 1.83E-07 | 1.93E-07 | 2.77E-75 | 7.32E-75 | 2.55E+01 | 2.76E+01 | 1.85E-03 | 6.85E-04 | 0.00E+00 | 0.00E+00 |

$f17$ | 6.05E-04 | 2.99E-03 | 1.22E-03 | 5.39E-03 | 4.19E-06 | 1.12E-05 | 1.14E-02 | 1.34E-02 | 0.00E+00 | 0.00E+00 | 1.81E-02 | 2.12E-02 | 1.55E-05 | 1.48E-05 | 0.00E+00 | 0.00E+00 |

$f18$ | 2.69E-01 | 1.39E-01 | 1.22E+00 | 5.44E-01 | 6.06E-04 | 1.01E-03 | 2.20E-01 | 1.34E-01 | 1.60E-01 | 4.95E-02 | 4.27E+00 | 2.85E+00 | 1.93E-02 | 1.86E-02 | 0.00E+00 | 0.00E+00 |

$f19$ | 5.24E-14 | 7.35E-14 | 6.75E-14 | 8.46E-14 | 2.17E-01 | 1.51E-01 | 5.22E-08 | 4.40E-08 | 0.00E+00 | 0.00E+00 | 2.06E-13 | 1.69E-13 | 5.17E-01 | 2.59E-01 | 0.00E+00 | 0.00E+00 |

$f20$ | 1.22E+00 | 1.04E+00 | 2.34E+05 | 1.58E+05 | 1.19E-09 | 2.14E-09 | 2.45E+00 | 1.99E+00 | 2.66E-02 | 3.30E-03 | 4.80E+04 | 2.68E+04 | 1.86E-04 | 2.52E-04 | 1.09E+00 | 3.05E-02 |

$f21$ | 8.61E+01 | 5.98E+01 | 1.47E+05 | 9.36E+04 | 4.57E-07 | 1.31E-06 | 9.03E+01 | 4.26E+01 | 3.84E+01 | 1.08E+00 | 9.46E+04 | 6.91E+04 | 4.24E-06 | 3.10E-06 | 8.30E+01 | 4.75E+00 |

$f22$ | 2.27E-15 | 9.95E-16 | 4.51E-65 | 6.96E-65 | 6.88E+00 | 4.09E+00 | 3.72E-03 | 2.28E-03 | 1.63E-43 | 4.36E-43 | 9.25E-82 | 1.04E-81 | 1.57E+01 | 2.86E+00 | 0.00E+00 | 0.00E+00 |

$f23$ | 0.00E+00 | 0.00E+00 | 1.40E-16 | 4.57E-16 | 2.77E-04 | 5.38E-04 | 4.36E-11 | 4.60E-11 | 0.00E+00 | 0.00E+00 | 4.77E-15 | 3.19E-15 | 1.96E-03 | 1.86E-03 | 0.00E+00 | 0.00E+00 |

$f24$ | 4.10E-06 | 4.43E-06 | 1.60E-05 | 4.39E-05 | 7.55E-07 | 1.22E-06 | 1.50E-06 | 1.45E-06 | 2.27E-124 | 1.15E-123 | 1.55E-01 | 1.04E-01 | 4.66E-06 | 4.38E-06 | 0.00E+00 | 0.00E+00 |

w/t/l | 0/3/21 | 0/1/23 | 2/1/21 | 0/0/24 | 2/8/14 | 0/0/24 | 2/1/21 | - |

F | MSBBO (D = 500) | MSBBO (D = 1000) | MSBBO (D = 2000) | MSBBO (D = 5000) | MSBBO (D = 10,000) | |||||
---|---|---|---|---|---|---|---|---|---|---|

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

$f1$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f2$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f3$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f4$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f5$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f6$ | 4.98E-05 | 5.99E-05 | 5.10E-05 | 3.81E-05 | 5.37E-05 | 9.76E-05 | 5.87E-05 | 6.03E-05 | 6.02E-05 | 5.94E-05 |

$f7$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f8$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f9$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f10$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f11$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f12$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f13$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f14$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f15$ | 4.44E-16 | 0.00E+00 | 4.44E-16 | 0.00E+00 | 4.44E-16 | 0.00E+00 | 4.44E-16 | 0.00E+00 | 4.44E-16 | 0.00E+00 |

$f16$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f17$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f18$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f19$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f20$ | 1.09E+00 | 3.05E-02 | 1.13E+00 | 1.79E-02 | 1.16E+00 | 6.15E-02 | 1.16E+00 | 6.31E-02 | 1.17E+00 | 6.43E-02 |

$f21$ | 8.30E+01 | 4.75E+00 | 1.67E+02 | 9.49E+00 | 3.32E+02 | 1.97E+01 | 8.46E+02 | 3.09E+01 | 1.90E+03 | 4.12E+01 |

$f22$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f23$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

$f24$ | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 | 0.00E+00 |

Algorithm | Optimal Values for Variables | Optimal Cost | |||
---|---|---|---|---|---|

${\mathit{T}}_{\mathit{s}}$ | ${\mathit{T}}_{\mathit{h}}$ | R | L | ||

COA | 0.78425309 | 0.38785440 | 40.63463345 | 195.95592683 | 5902.85656647 |

EDO | 0.77997855 | 0.38558510 | 40.38924403 | 199.85289900 | 5909.44930611 |

OMA | 0.77827133 | 0.38470014 | 40.32491200 | 199.92634333 | 5885.51298728 |

SHO | 0.78255188 | 0.38700919 | 40.54668188 | 196.86310662 | 5893.43974421 |

SCSO | 0.79522133 | 0.38960618 | 40.66736096 | 195.21548366 | 5976.11395352 |

MSBBO | 0.77816864 | 0.38464916 | 40.31961873 | 200 | 5885.33277894 |

Algorithm | Optimal Values for Variables | Optimal Cost | ||
---|---|---|---|---|

$\mathit{d}$ | $\mathit{D}$ | P | ||

COA | 0.05122526 | 0.34565821 | 12.14215544 | 0.01269823 |

EDO | 0.05190445 | 0.36182344 | 11.22500450 | 0.01267212 |

OMA | 0.05356063 | 0.40339534 | 8.53973118 | 0.01272961 |

SHO | 0.05058048 | 0.33062883 | 13.22719241 | 0.01268814 |

SCSO | 0.05718457 | 0.50390409 | 5.53685327 | 0.01318243 |

MSBBO | 0.05120413 | 0.34516305 | 12.26044071 | 0.01266959 |

Algorithm | Optimal Values for Variables | Optimal Cost | |||
---|---|---|---|---|---|

h | l | t | b | ||

COA | 0.18479628 | 3.68381216 | 9.22577702 | 0.19867497 | 1.69837299 |

EDO | 0.19793629 | 3.36058378 | 9.18941514 | 0.19902434 | 1.67299413 |

OMA | 0.19883231 | 3.33736530 | 9.19202432 | 0.19883231 | 1.67021773 |

SHO | 0.17440082 | 3.86323716 | 9.20290796 | 0.19878156 | 1.70196639 |

SCSO | 0.17898507 | 3.68017384 | 9.47591891 | 0.19754191 | 1.72245954 |

MSBBO | 0.19883231 | 3.33736530 | 9.19202432 | 0.19883231 | 1.67021773 |

Algorithm | Optimal Values for Variables | Optimal Cost | ||||||
---|---|---|---|---|---|---|---|---|

b | m | z | ${\mathit{l}}_{\mathbf{1}}$ | ${\mathit{l}}_{\mathbf{2}}$ | ${\mathit{d}}_{\mathbf{1}}$ | ${\mathit{d}}_{\mathbf{2}}$ | ||

COA | 3.50000068 | 0.7 | 17 | 7.3 | 8.007338125 | 3.354940375 | 5.286998297 | 3002.176768192 |

EDO | 3.50025780 | 0.70000066 | 17 | 7.3 | 7.723941215 | 3.350658886 | 5.286670403 | 2994.758133972 |

OMA | 3.5 | 0.7 | 17 | 7.3 | 7.715319911 | 3.350540949 | 5.286654465 | 2994.424465758 |

SHO | 3.50004682 | 0.7 | 17 | 7.30178682 | 7.715591607 | 3.350558567 | 5.286654691 | 2994.469209053 |

SCSO | 3.51205842 | 0.7 | 17 | 7.3 | 7.766370701 | 3.351151796 | 5.286672013 | 3000.448065947 |

MSBBO | 3.50000001 | 0.7 | 17 | 7.30000014 | 7.715320035 | 3.350540986 | 5.286654467 | 2994.424489954 |

Algorithm | Optimal Values for Variables | Optimal Cost | ||||
---|---|---|---|---|---|---|

${\mathit{d}}_{\mathbf{1}}$ | ${\mathit{d}}_{\mathbf{2}}$ | ${\mathit{d}}_{\mathbf{3}}$ | ${\mathit{d}}_{\mathbf{4}}$ | $\mathit{w}$ | ||

COA | 17.43376313 | 29.03743308 | 50.95003894 | 89.57133620 | 89.72893135 | 8.80527504 |

EDO | 17.00069596 | 28.33533123 | 50.82602160 | 84.55984091 | 89.95618331 | 8.19564986 |

OMA | 16.96572313 | 28.25752810 | 50.79671071 | 84.49571607 | 90 | 8.18149598 |

SHO | 17.12626161 | 28.25787354 | 50.79728294 | 84.49688761 | 89.99901887 | 8.19717274 |

SCSO | 18.25834926 | 28.68767069 | 51.89516285 | 88.88993899 | 88.65129296 | 8.75660602 |

MSBBO | 16.96572695 | 28.25753037 | 50.79673307 | 84.49572374 | 89.99999337 | 8.18149598 |

Algorithm | Optimal Values for Variables | Optimal Cost | ||||||
---|---|---|---|---|---|---|---|---|

$\mathit{a}$ | $\mathit{b}$ | $\mathit{c}$ | $\mathit{d}$ | $\mathit{e}$ | $\mathit{f}$ | $\mathit{\delta}$ | ||

COA | 149.99754208 | 149.65008561 | 159.71207516 | 0.00198394 | 10.06515284 | 115.49379010 | 1.57052121 | 3.49048709 |

EDO | 149.96736172 | 98.92864085 | 199.99158377 | 49.94588624 | 150 | 124.89353758 | 2.86707789 | 3.55150434 |

OMA | 147.04217688 | 134.34472269 | 200 | 12.48032203 | 149.35062039 | 106.73574813 | 2.44307744 | 2.84065650 |

SHO | 144.95148508 | 144.77644040 | 100.00021186 | 0.05032617 | 11.72681600 | 100.42656588 | 1.36167252 | 5.26168942 |

SCSO | 149.14088858 | 148.88101175 | 148.11100725 | 0.04507554 | 60.56845351 | 108.40010554 | 1.99300044 | 3.63308379 |

MSBBO | 149.61527355 | 149.39619170 | 199.20669308 | 8.60E-16 | 149.63248094 | 108.80577404 | 2.42474600 | 2.69788610 |

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© 2024 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Gao, C.; Li, T.; Gao, Y.; Zhang, Z.
A Comprehensive Multi-Strategy Enhanced Biogeography-Based Optimization Algorithm for High-Dimensional Optimization and Engineering Design Problems. *Mathematics* **2024**, *12*, 435.
https://doi.org/10.3390/math12030435

**AMA Style**

Gao C, Li T, Gao Y, Zhang Z.
A Comprehensive Multi-Strategy Enhanced Biogeography-Based Optimization Algorithm for High-Dimensional Optimization and Engineering Design Problems. *Mathematics*. 2024; 12(3):435.
https://doi.org/10.3390/math12030435

**Chicago/Turabian Style**

Gao, Chenyang, Teng Li, Yuelin Gao, and Ziyu Zhang.
2024. "A Comprehensive Multi-Strategy Enhanced Biogeography-Based Optimization Algorithm for High-Dimensional Optimization and Engineering Design Problems" *Mathematics* 12, no. 3: 435.
https://doi.org/10.3390/math12030435