# An Improved Multiobjective Particle Swarm Optimization Based on Culture Algorithms

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Preliminary

#### 2.1. Cultural Algorithms

#### 2.2. Multiobjective Optimization Problems

_{1}(x), f

_{2}(x), …, f

_{n}(x)]

s.t.g(x) ≤ 0, h(x) = 0

**Definition**

**1.**

_{i}(x) ≤ f

_{i}(y), $\exists $i, f

_{i}(x) < f

_{i}(y).

**Definition**

**4.**

_{ε}y, if f(x) · (1 + ε) $\prec $ f(y).

#### 2.3. Particle Swam Optimization

_{1}and c

_{2}are the acceleration coefficients. Parameters r

_{1}and r

_{2}are uniformly distributed random numbers within the range [0, 1]. The parameters p${}_{id}^{(k)}$ and p${}_{gd}^{(k)}$ indicate the best position that the i-th particle has found so far, and the best position that the global swarm has found so far. The procedure of PSO is shown in Algorithm 1.

Algorithm 1. The Procedure of Particle Swarm Optimization (PSO) |

(1) Calculate the fitness function of each particle. |

(2) Update p${}_{id}^{(k)}$ and p${}_{gd}^{(k)}$. |

(3) Update the velocity and position of each particle. |

(4) If the stop criterion is not met, go to (1), else p${}_{g}^{(k)}$ is the best position |

## 3. The Proposed Algorithm

#### 3.1. Three Kinds of Knowledge Based on Population

_{1}, U

_{1}, L

_{2}, U

_{2}to L

_{1}’, U

_{1}’, L

_{2}’, U

_{2}’.

_{i}= min f

_{i}(x), U

_{i}= max f

_{i}(x)

#### 3.2. Personal Guide

Algorithm 2. The Detailed Procedure of A Personal Guide |

For the d-th dimension of the i-th particle, |

(1) b_{n+1} = max({pbest_{id}|i = 1, 2, …, n}); |

b_{1} = min({pbest_{id}|i = 1, 2, …, n}); |

b_{i} = b_{1} + (b_{n+1} – b_{1})·(i – 1)/n, i = 2, 3, …, n |

(2) s_{i} = rand(b_{i}, b_{i}_{+1}); |

(3) Replace x${}_{i}^{d}$ with s_{i} and get a new particle |

(4) If the newly generated particle dominates x_{i}, then it replaces x_{i}; break; |

if not, the newly generated particle is discarded |

#### 3.3. Global Guide

#### 3.4. Dominate Criteria Based on the Hypercube

#### 3.5. Mutation Operator

_{m}= s + α·β

_{j}

_{j}– l

_{j}, p is a positive real number, v is a uniformly distributed random number in [0, 1], and β

_{j}is the range of j-th dimension.

#### 3.6. Updating the Repository

Algorithm 3. The Update Function of the Dominance Archive |

Suppose element f’ is a new element that is to be judged whether to enter the repository (rep). |

(1) Input rep, f |

(2) If ¬$\exists $f’$\in $ rep:hypercube(f)$\prec $hypercube(f’) then rep = rep∪{f}E |

(3) If $\exists $f’:hypercube(f’) = hypercube(f)∩f$\prec $ε + f’ then rep = rep∪{f}f’ |

(4) If ¬$\exists $f’:hypercube(f’) = hypercube(f)∪hypercube(f’)$\prec $hypercube(f), then rep = rep∪{f} |

(5) Output rep |

## 4. Experiments

_{1}= 2.5 – (ita/max_ita) × (2.5 – 0.5)

_{2}= 0.5 – (ita/max_ita) × (0.9 – 0.4)

#### 4.1. Quantitative Performance Evaluations

#### 4.2. Comparison of the Results Using Hypercube-Based $\epsilon $-Domination and Normal Domination Hypervolume

#### 4.3. Comparison of the Result with and without the Mutation Operator

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**Figure 6.**Pareto fronts for DTLZ1: (

**a**) the Pareto fronts obtained using normal domination criterion; (

**b**) the Pareto fronts obtained using hypercube-based ε-domination.

**Figure 7.**Pareto fronts for DTLZ7: (

**a**) the Pareto fronts obtained using normal domination criterion; (

**b**) the Pareto fronts obtained using hypercube-based ε-domination.

**Figure 8.**Pareto fronts for ZDT1: (

**a**) the Pareto fronts obtained without using the mutation operator; (

**b**) the Pareto fronts obtained using the mutation operator.

**Figure 9.**Pareto fronts for ZDT3: (

**a**) the Pareto fronts obtained without using the mutation operator; (

**b**) the Pareto fronts obtained using the mutation operator.

Test Functions | MOPSO | IC MOPSO | Cultural MOQPSO | Cultural MOPSO | NSGA-II | CD MOPSO | |
---|---|---|---|---|---|---|---|

ZDT1 | mean | 5.56 × 10^{−1} | 8.73 × 10^{−1} | 8.42 × 10^{−1} | 8.66 × 10^{−1} | 8.11 × 10^{−1} | 8.74 × 10^{−1} |

standard deviation (std) | 9.03 × 10^{−2} | 5.38 × 10^{−3} | 8.01 × 10^{−2} | 7.00 × 10^{−3} | 8.46 × 10^{−3} | 4.76 × 10^{−3} | |

p-value | 8.42 × 10^{−2} | 3.72 × 10^{−2} | 8.82 × 10^{−5} | 6.65 × 10^{−40} | 6.25 × 10^{−1} | ||

ZDT2 | mean | 2.88 × 10^{−1} | 5.38 × 10^{−1} | 4.72 × 10^{−1} | 4.20 × 10^{−1} | 1.20 × 10^{−1} | 5.37 × 10^{−1} |

std | 1.19 × 10^{−1} | 7.88 × 10^{−3} | 1.07 × 10^{−1} | 1.91 × 10^{−1} | 2.22 × 10^{−2} | 6.43 × 10^{−3} | |

p-value | 1.69 × 10^{−16} | 1.40 × 10^{−3} | 1.31 × 10^{−5} | 6.53 × 10^{−66} | 8.25 × 10^{−1} | ||

ZDT3 | mean | 1.00 | 1.01 | 9.39 × 10^{−1} | 7.61 × 10^{−1} | 7.60 × 10^{−1} | 1.01 |

std | 5.17 × 10^{−3} | 4.01 × 10^{-3} | 1.43 × 10^{−1} | 4.00 × 10^{−1} | 1.31 × 10^{−2} | 4.37 × 10^{−3} | |

p-value | 1.72 × 10^{−7} | 0.0110 | 1.35 × 10^{−3} | 2.35 × 10^{−66} | 8.76 × 10^{−1} | ||

ZDT4 | mean | 0.00 | 8.71 × 10^{−1} | 8.67 × 10^{-1} | 0.00 | 3.66 × 10^{−1} | 0.00 |

std | 0.00 | 6.08 × 10^{−3} | 5.18 × 10^{−3} | 0.00 | 1.77 × 10^{−1} | 0.00 | |

p-value | 1.80 × 10^{−118} | 1.59 × 10^{−2} | 1.80 × 10^{−118} | 1.80 × 10^{−118} | 1.80 × 10^{−118} | ||

ZDT6 | mean | 4.94 × 10^{−1} | 5.04 × 10^{−1} | 4.65 × 10^{−1} | 5.03 × 10^{−1} | 0.00 | 5.05 × 10^{−1} |

std | 7.43 × 10^{−3} | 4.88 × 10^{−3} | 1.12 × 10^{−1} | 7.63 × 10^{−3} | 0.00 | 5.77 × 10^{−3} | |

p-value | 5.97 × 10^{−08} | 6.31 × 10^{−2} | 8.61 × 10^{−1} | 3.40 × 10^{−110} | 8.76 × 10^{−1} | ||

DTLZ1 | mean | 0.00 | 1.30 | 1.30 | 0.00 | 0.00 | 0.00 |

std | 0.00 | 2.02 × 10^{−3} | 3.07 × 10^{−3} | 0.00 | 0.00 | 0.00 | |

p-value | 2.90 × 10^{−156} | 6.44 × 10^{−1} | 2.90 × 10^{−156} | 2.90 × 10^{−156} | 2.90 × 10^{−156} | ||

DTLZ2 | mean | 6.74 × 10^{-1} | 7.15 × 10^{−1} | 4.27 × 10^{−1} | 2.46 × 10^{−1} | 6.93 × 10^{−1} | 7.20 × 10^{−1} |

std | 1.50 × 10^{−2} | 9.74 × 10^{−3} | 2.41 × 10^{−2} | 1.26 × 10^{−2} | 8.35 × 10^{−3} | 8.12 × 10^{−3} | |

p-value | 4.76 × 10^{−18} | 3.83 × 10^{−54} | 1.33 × 10^{−78} | 3.17 × 10^{−13} | 3.85 × 10^{−2} | ||

DTLZ3 | mean | 0.00 | 7.16 × 10^{−1} | 7.15 × 10^{−1} | 0.00 | 0.00 | 0.00 |

std | 0.00 | 7.63 × 10^{−3} | 9.61 × 10^{−3} | 0.00 | 0.00 | 0.00 | |

p-value | 8.20 × 10^{−108} | 9.47 × 10^{−1} | 8.20 × 10^{−108} | 8.20 × 10^{−108} | 8.20 × 10^{−108} | ||

DTLZ4 | mean | 6.82 × 10^{−1} | 7.29 × 10^{−1} | 5.46 × 10^{−1} | 1.67 × 10^{−1} | 6.65 × 10^{−1} | 7.22 × 10^{−1} |

std | 1.56 × 10^{−2} | 7.45 × 10^{−3} | 5.70 × 10^{−2} | 6.76 × 10^{−2} | 3.11 × 10^{−2} | 8.74 × 10^{−3} | |

p-value | 1.55 × 10^{−21} | 9.53 × 10^{−25} | 6.11 × 10^{−47} | 9.22 × 10^{−16} | 8.66 × 10^{−4} | ||

DTLZ5 | mean | 4.34 × 10^{−1} | 4.39 × 10^{−1} | 4.35 × 10^{−1} | 2.43 × 10^{−1} | 4.38 × 10^{−1} | 4.38 × 10^{−1} |

std | 6.21 × 10^{−3} | 4.94×10^{−3} | 8.00 × 10^{−3} | 4.96 × 10^{−3} | 5.74 × 10^{−3} | 5.86 × 10^{−3} | |

p-value | 1.65 × 10^{−3} | 3.72 × 10^{−2} | 2.13 × 10^{−77} | 5.64 × 10^{−1} | 3.14 × 10^{−1} | ||

DTLZ6 | mean | 4.29 × 10^{−1} | 4.40×10^{−1} | 4.21 × 10^{−1} | 2.36 × 10^{−1} | 0.00 | 4.39 × 10^{−1} |

std | 6.99 × 10^{−3} | 5.97×10^{−3} | 7.53 × 10^{−3} | 2.81 × 10^{−2} | 0.00 | 6.85 × 10^{−3} | |

p-value | 1.43 × 10^{−8} | 5.64 × 10^{−15} | 3.58 × 10^{−43} | 9.89 × 10^{−102} | 6.28 × 10^{−1} | ||

DTLZ7 | mean | 1.27 | 1.43 | 1.21 | 5.44 × 10^{−1} | 1.47 × 10^{−1} | 1.38 |

std | 2.93 × 10^{−2} | 3.20 × 10^{−2} | 4.10 × 10^{−2} | 8.36 × 10^{−2} | 4.43 × 10^{−2} | 4.25 × 10^{−2} | |

p-value | 1.19 × 10^{−28} | 2.91 × 10^{−31} | 1.98 × 10^{−51} | 5.57 × 10^{−73} | 5.54 × 10^{−6} | ||

Better/similar/worse | 12/0/0 | 9/3/0 | 11/1/0 | 11/1/0 | 5/6/1 |

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**MDPI and ACS Style**

Jia, C.; Zhu, H.
An Improved Multiobjective Particle Swarm Optimization Based on Culture Algorithms. *Algorithms* **2017**, *10*, 46.
https://doi.org/10.3390/a10020046

**AMA Style**

Jia C, Zhu H.
An Improved Multiobjective Particle Swarm Optimization Based on Culture Algorithms. *Algorithms*. 2017; 10(2):46.
https://doi.org/10.3390/a10020046

**Chicago/Turabian Style**

Jia, Chunhua, and Hong Zhu.
2017. "An Improved Multiobjective Particle Swarm Optimization Based on Culture Algorithms" *Algorithms* 10, no. 2: 46.
https://doi.org/10.3390/a10020046