# PSO, a Swarm Intelligence-Based Evolutionary Algorithm as a Decision-Making Strategy: A Review

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## Abstract

**:**

## 1. Introduction

## 2. Methodology of the Literature Review

#### 2.1. Scientific Articles

#### 2.2. Patents Employing the PSO Algorithm

#### 2.3. Copyright with the PSO Algorithm

## 3. Results of the Literature Review

#### 3.1. Analysis and Decision-Making

#### 3.2. Multi-Criteria Methods

#### 3.3. Metaheuristics

#### 3.4. Particle Swarm Algorithm (PSO)

#### 3.4.1. PSO Applications

#### 3.4.2. Structure of the Classic PSO Algorithm

**Steps****1.**- Initialization

- 1.
- Set the control parameters: N, ω, ${c}_{1}$, ${c}_{2}$, ${r}_{1}$, ${r}_{2}$, and T (definition in Table 9), the number current number of iterations with a value of 1 ($t=1$) because it is the first iteration, and the fitness function to initialize the swarm.The variable ω makes convergence happen in fewer iterations and maintains the balance between local and global searches. A smaller value of ω leads to a local search and a larger value than a global search. If ω has a large value, the algorithm starts the search globally and ends with a local search [108,109].
- 2.
- Determine the first local position of the particles randomly, considering $i=\{1,2,\cdots ,N\}$:$$C{P}_{N}^{i}\left(t\right)=(C{P}_{1}^{i}\left(t\right),C{P}_{2}^{i}\left(t\right),\cdots ,C{P}_{N}^{i}\left(t\right))$$
- 3.
- Randomly set the first local velocity of the particles (from 1 to N):$${V}_{N}^{i}\left(t\right)=({V}_{1}^{i}\left(t\right),{V}_{2}^{i}\left(t\right),\cdots ,{V}_{N}^{i}\left(t\right))$$
- 4.
- Evaluate the fitness function with the first position (Equation (1)), to obtain the best current optimum:$$C{F}_{N}^{i}\left(t\right)=f\left(x\right)=f\left(C{P}_{N}^{i}\left(t\right)\right)$$
- 5.
- To establish the best local position (LBP), we use the first local position (Equation (1)). While, for the first best local optimum (LBF), we use the current best optimum (Equation (3)):$$LB{P}_{N}^{i}\left(t\right)=C{P}_{N}^{i}\left(t\right)$$$$LB{F}_{N}^{i}\left(t\right)=C{F}_{N}^{i}\left(t\right)$$
- 6.
- To obtain the best global optimum with the maximum value of the best local optimum (Equation (5)); that is, the maximum value of the dataset of the best local optimum (LBF):$$GBF\left(t\right)=max\left(LB{F}_{N}^{i}\left(t\right)\right)$$
- 7.
- To obtain the best global position, we extract the particle position in i from the best global optimum (Equation (6)). Position (z) provides the value of the best local position and becomes the best global position:z = particle position in i of GBF(t)$$GBP\left(t\right)=LBF\left(z\right)$$To continue with step 2, we increment the current number of iterations ($t=t+1$).

**Steps****2.**- Position updating and local optimal

- 1.
- Update speed and position of the particle.From this step, ($t-1$) indicates the value of the previous iteration, while (t) is the current iteration.To update the velocity, we use the inertial weight coefficient (ω), the learning factors (${c}_{1}$ and ${c}_{2}$), and the particle circulation values (${r}_{1}$ and ${r}_{2}$), as well as the values of the previous iteration of the speed, local position, and best local and global positions.Meanwhile, for the current position, we add the previous current position and the new speed:$${V}_{N}^{i}\left(t\right)=\omega ({V}_{N}^{i}(t-1)+{c}_{1}{r}_{1}(LB{P}_{N}^{i}(t-1)-C{P}_{N}^{i}(t-1))+{c}_{2}{r}_{2}(GBP(t-1)-C{P}_{N}^{i}(t-1))$$$$C{P}_{N}^{i}\left(t\right)=C{P}_{N}^{i}(t-1)+{V}_{N}^{i}\left(t\right)$$
- 2.
- To obtain the best current optimum, evaluating the fitness function with the current position (Equation (9)):$$C{F}_{N}^{i}\left(t\right)=f\left(x\right)=f\left(C{P}_{N}^{i}\left(t\right)\right)$$
- 3.
- Update best local position with current position (Equation (9)):$$LB{P}_{N}^{i}\left(t\right)=CP\left(t\right)$$
- 4.
- 5.
- Obtain the best global optimum with the maximum value of the best local optimum (Equation (12)). That is, the maximum value of the dataset of the best local optimum (LBF):$$GBF\left(t\right)=max\left(LB{F}_{N}^{i}\left(t\right)\right)$$
- 6.
- To obtain the best global position, we extract the particle position in i from the best global optimum (Equation (11)). That position (z) provides the value of the best local position and becomes the best global position:z = particle position in i of GBF(t)$$GBP\left(t\right)=LBP\left(z\right)$$

**Steps****3.**- Obtaining the best global-optimal position

- 1.
- If the current iteration is less than the total iterations, it is convergent; therefore, we increment the iteration and continue from step 2:$$t<T\to (t=t+1)$$
- 2.
- The process ends when the total iteration is equal to or greater than the current iteration. We obtain the best position and the best optimum from the last values of the best global position and the best global optimum:$$t\ge T=\left\{\begin{array}{c}pbest\left(T\right)=GBP\left(t\right)\hfill \\ gbest\left(T\right)=GBF\left(t\right)\hfill \end{array}\right.$$

## 4. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

PSO | Particle Swarm Optimization |

EA | Evolutionary Algorithms |

GA | Genetic Algorithms |

SA | Simulated Annealed Algorithm |

TS | Taboo |

ACO | Colony of Ants Algorithm |

BA | Bat Algorithm |

ABC | Artificial Bee Colony Algorithm |

USPTO | United States Patent and Trademark Office |

SW | Programs, Systems, or Software |

MCDM | Multi-Criteria Decision Making |

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Period | IEEExplore | ProQuest | SAGE | JSTOR | ScienceDirect | Google Scholar |
---|---|---|---|---|---|---|

1997–2001 | 1 | 88 | 2240 | 9446 | 3427 | 1455 |

2002–2006 | 2 | 235 | 3324 | 11,499 | 6962 | 3575 |

2007–2011 | 57 | 665 | 4645 | 13,831 | 13,077 | 9720 |

2012–2016 | 43 | 2066 | 5955 | 14,946 | 24,062 | 19,700 |

2017–2021 | 53 | 3366 | 7935 | 6126 | 37,977 | 30,360 |

Period | ScienceDirect | IEEExplore | ProQuest | JSTOR | SAGE | Google Scholar |
---|---|---|---|---|---|---|

2001–1997 | 177 | 14 | 0 | 244 | 55 | 7 |

2006–2002 | 533 | 64 | 0 | 891 | 165 | 15 |

2011–2007 | 1689 | 333 | 0 | 2283 | 269 | 78 |

2016–2012 | 4548 | 614 | 6 | 4768 | 251 | 181 |

2021–2017 | 9212 | 889 | 0 | 6361 | 46 | 322 |

Database/Period | 1995–2001 | 2002–2011 | 2012–2021 |
---|---|---|---|

ScienceDirect | 130 | 3617 | 33,520 |

IEEE | 64 | 9296 | 18,232 |

SAGE | 1 | 510 | 3701 |

ProQuest | 0 | 12 | 117 |

JSTOR | 0 | 3 | 5 |

TOTALS | 195 | 13,438 | 55,578 |

Country | 2009–2012 | 2013–2017 | 2019–2021 |
---|---|---|---|

Saudi Arabia | 1 | 0 | 0 |

Germany | 0 | 1 | 0 |

France | 0 | 3 | 0 |

United States of America | 4 | 8 | 0 |

Taiwan | 0 | 0 | 1 |

Ireland | 0 | 0 | 1 |

People’s Republic of China | 0 | 0 | 10 |

Patent | Inventors | Year of Assignment | Reference |
---|---|---|---|

8793200 | Chen Yang et al. | 2014 | [20] |

7558762 | Owechko Yuri et al. | 2009 | [21] |

7599894 | Owechko Yuri, Medasani Swarup | 2009 | [22] |

7636700 | Owechko Yuri, Medasani Swarup | 2009 | [23] |

7672911 | Owechko Yuri, Medasani Swarup | 2010 | [24] |

8645294 | Owechko Yuri et al. | 2014 | [25] |

8213709 | Medasani Swarup et al. | 2012 | [26] |

8370114 | Saisan Payam | 2013 | [27] |

8589315 | Medasani Swarup, Owechko Yuri | 2013 | [28] |

8437558 | Medasani Swarup, Owechko Yuri | 2013 | [29] |

Year | Text | Serials | Computer Files |
---|---|---|---|

2002 | 1 | 0 | 0 |

2004 | 2 | 0 | 0 |

2009 | 4 | 0 | 0 |

2010 | 1 | 0 | 0 |

2011 | 3 | 0 | 0 |

2012 | 1 | 1 | 0 |

2013 | 1 | 0 | 0 |

2014 | 3 | 0 | 0 |

2015 | 0 | 1 | 0 |

2016 | 1 | 0 | 0 |

2017 | 3 | 1 | 0 |

2018 | 1 | 0 | 0 |

2019 | 1 | 0 | 0 |

MCDM | Proposed by: | Year | References |
---|---|---|---|

ELECTRE | Bernard Roy | 1968 | [45] |

TOPSIS | Hwang Yoon | 1981 | [46,47] |

AHP | Saaty | 1981 | [36,48] |

VIKOR | Opricovic | 1988 | [48,49] |

PROMETHEE | Brans and Mareschar | 2005 | [50,51] |

MOORA | Brauers and Zavadskas | 2006 | [52,53] |

CODAS | Ghorabee | 2016 | [54,55] |

MCDM-Fuzzy | Proposed by: | Year | References |
---|---|---|---|

FS | Lotfi A. Zaden and Dieter Klaua | 1965 | [45,53] |

IFS | Krassimir Atanassov | 1986 | [56] |

BFS | Zhang Wen-Ran | 1994 | [56,57] |

Fuzzy PSO | Bo Wang, GuoQiang Liang and ChaLin, Wang | 2006 | [58] |

Fuzzy TOPSIS | Chen and Tsao | 2008 | [59] |

PFS | Zadeh and Yanger | 2013 | [41,60] |

q-ROF | Yager | 2017 | [51,52] |

TSFS | Smarandache, Florentin | 2019 | [61,62] |

Notation | Definition |
---|---|

i | Independent unit, called a particle. |

N | Number of particles in the swarm, also called search space dimension or population $N=\{{i}_{1},{i}_{2},\cdots ,{i}_{N}\}.$ |

$\omega $ | Coefficient of the weight of inertia, increases or decreases the speed, keeping the current speed and moment of the particle. |

$t,T$ | Current number of iterations (t) and the total number of iterations expected to be performed (T). |

${c}_{1},{c}_{2}$ | Non-negative acceleration factors, known as learning factors. Driving each particle towards the pbest and gbest positions. |

${r}_{1},{r}_{2}$ | Numbers between [0, 1], for the circulation of each particle. That is, they simulate the influence of nature on the particle. |

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## Share and Cite

**MDPI and ACS Style**

Ramírez-Ochoa, D.-D.; Pérez-Domínguez, L.A.; Martínez-Gómez, E.-A.; Luviano-Cruz, D.
PSO, a Swarm Intelligence-Based Evolutionary Algorithm as a Decision-Making Strategy: A Review. *Symmetry* **2022**, *14*, 455.
https://doi.org/10.3390/sym14030455

**AMA Style**

Ramírez-Ochoa D-D, Pérez-Domínguez LA, Martínez-Gómez E-A, Luviano-Cruz D.
PSO, a Swarm Intelligence-Based Evolutionary Algorithm as a Decision-Making Strategy: A Review. *Symmetry*. 2022; 14(3):455.
https://doi.org/10.3390/sym14030455

**Chicago/Turabian Style**

Ramírez-Ochoa, Dynhora-Danheyda, Luis Asunción Pérez-Domínguez, Erwin-Adán Martínez-Gómez, and David Luviano-Cruz.
2022. "PSO, a Swarm Intelligence-Based Evolutionary Algorithm as a Decision-Making Strategy: A Review" *Symmetry* 14, no. 3: 455.
https://doi.org/10.3390/sym14030455