# A Novel Multi-Objective Five-Elements Cycle Optimization Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Basic Concepts of MOO Problem and Related Research on MOEAs

#### 2.1. Description of Multi-Objective Problems

**Definition**

**1.**

**Definition**

**2.**

**Definition**

**3.**

#### 2.2. Research on Multi-Objective Evolutionary Algorithms

## 3. Related Work and the Motivation of This Paper

#### 3.1. Related Work of Five-Elements Cycle Optimization (FECO)

#### 3.2. Motivation of This Paper

## 4. The Proposed Multi-Objective Five-Elements Cycle Optimization Algorithm

#### 4.1. General Framework of MOFECO

#### 4.2. Expression of Solutions and Population Initialization

#### 4.3. Update of Individuals

#### 4.4. Mutation of Individuals

## 5. Simulation Experiment and Result Analysis

#### 5.1. Test Problems

#### 5.2. Performance Metrics

#### 5.3. Parameter Analysis and Comparison Experiments of MOFECO

#### 5.3.1. Comparison of L and q

#### 5.3.2. Comparison of Conditions for Judging Whether to Update

#### 5.3.3. Comparison of Local-Global Update Probability ${P}_{s}$

#### 5.3.4. Comparison of Mutation Methods

#### 5.4. Comparison with Other Optimization Algorithms

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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Test Problem | Dimension D | Number of Objects m | Feature |
---|---|---|---|

ZDT1 | 30 | 2 | Convex function |

ZDT2 | 30 | 2 | Concave function |

ZDT4 | 10 | 2 | Convex function |

ZDT6 | 10 | 2 | Concave function |

DTLZ2 | 12 | 3 | Concave function |

DTLZ4 | 12 | 3 | Concave function |

DTZL5 | 12 | 3 | Concave function |

DTLZ6 | 12 | 3 | Concave function |

DTLZ7 | 22 | 3 | Mixed function |

WFG2 | 13 | 4 | convex function |

WFG3 | 13 | 4 | linear function |

WFG4 | 13 | 4 | concave function |

MaF1 | 13 | 4 | Linear function |

MaF2 | 13 | 4 | Concave function |

MaF12 | 13 | 4 | Concave function |

Parameters | Values |
---|---|

N | 100 |

T | 1000 |

L | 5 |

q | 20 |

${\mathit{r}}_{1}$ = ${\mathit{r}}_{2}$ | 1 |

$\mathit{\omega}$ | 0.5 (for bi-objective MOPs); 0.4 (others) |

${\mathit{\sigma}}_{1}$ | 0.1 |

${\mathit{\sigma}}_{2}$ | 1 |

${\mathit{\sigma}}_{3}$ | 1 |

${\mathit{P}}_{\mathit{m}}$ | 0.01 |

**Table 3.**$GD$ result of comparison of L and q in MOFECO algorithm on test functions. (The best result is bolded for each function).

Functions | $\mathit{L}=4/\mathit{q}=25$ | $\mathit{L}=5/\mathit{q}=20$ | $\mathit{L}=10/\mathit{q}=10$ | $\mathit{L}=20/\mathit{q}=5$ |
---|---|---|---|---|

ZDT1 | $2.22\times {10}^{-4}$ | $\mathbf{1.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $2.39\times {10}^{-4}$ | $2.53\times {10}^{-4}$ |

ZDT2 | $9.73\times {10}^{-6}$ | $4.02\times {10}^{-6}$ | $\mathbf{4.46}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$ | $4.50\times {10}^{-7}$ |

ZDT4 | $1.50\times {10}^{-4}$ | $\mathbf{2.39}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ | $4.60\times {10}^{-3}$ | $4.10\times {10}^{-3}$ |

ZDT6 | $3.62\times {10}^{-7}$ | $3.50\times {10}^{-7}$ | $\mathbf{1.56}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$ | $3.32\times {10}^{-7}$ |

DTLZ2 | $6.69\times {10}^{-4}$ | $\mathbf{4.67}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $1.20\times {10}^{-3}$ | $7.02\times {10}^{-4}$ |

DTLZ4 | $7.30\times {10}^{-4}$ | $1.43\times {10}^{-3}$ | $5.53\times {10}^{-4}$ | $\mathbf{4.39}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ |

DTLZ5 | $5.94\times {10}^{-5}$ | $\mathbf{3.56}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ | $7.15\times {10}^{-5}$ | $6.76\times {10}^{-5}$ |

DTLZ6 | $4.60\times {10}^{-7}$ | $4.82\times {10}^{-7}$ | $7.05\times {10}^{-7}$ | $\mathbf{4.53}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$ |

DTLZ7 | $8.80\times {10}^{-3}$ | $9.60\times {10}^{-3}$ | $8.90\times {10}^{-3}$ | $\mathbf{6.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ |

WFG2 | $6.90\times {10}^{-2}$ | $\mathbf{6.25}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $6.66\times {10}^{-2}$ | $6.79\times {10}^{-2}$ |

WFG3 | $1.74\times {10}^{-1}$ | $\mathbf{5.29}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $1.87\times {10}^{-1}$ | $2.09\times {10}^{-1}$ |

WFG4 | $2.11\times {10}^{-2}$ | $1.57\times {10}^{-2}$ | $1.70\times {10}^{-2}$ | $\mathbf{1.20}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ |

MaF1 | $8.80\times {10}^{-3}$ | $7.10\times {10}^{-3}$ | $\mathbf{6.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $9.00\times {10}^{-3}$ |

MaF2 | $4.90\times {10}^{-3}$ | $\mathbf{3.67}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $8.00\times {10}^{-3}$ | $7.20\times {10}^{-3}$ |

MaF12 | $1.88\times {10}^{-2}$ | $\mathbf{1.88}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $2.60\times {10}^{-2}$ | $2.18\times {10}^{-2}$ |

**Table 4.**$IGD$ result of comparison of L and q in MOFECO algorithms on test functions. (The best result is bolded for each function).

Functions | $\mathit{L}=4/\mathit{q}=25$ | $\mathit{L}=5/\mathit{q}=20$ | $\mathit{L}=10/\mathit{q}=10$ | $\mathit{L}=20/\mathit{q}=5$ |
---|---|---|---|---|

ZDT1 | $5.90\times {10}^{-3}$ | $\mathbf{5.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $5.50\times {10}^{-3}$ | $6.20\times {10}^{-3}$ |

ZDT2 | $\mathbf{5.30}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $5.40\times {10}^{-3}$ | $5.80\times {10}^{-3}$ | $6.60\times {10}^{-3}$ |

ZDT4 | $5.80\times {10}^{-3}$ | $\mathbf{5.76}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $7.04\times {10}^{-2}$ | $5.15\times {10}^{-2}$ |

ZDT6 | $4.90\times {10}^{-3}$ | $\mathbf{4.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $4.43\times {10}^{-1}$ | $3.37\times {10}^{-2}$ |

DTLZ2 | $7.55\times {10}^{-2}$ | $\mathbf{7.49}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $7.65\times {10}^{-2}$ | $8.06\times {10}^{-2}$ |

DTLZ4 | $\mathbf{2.15}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $2.21\times {10}^{-1}$ | $2.96\times {10}^{-1}$ | $6.84\times {10}^{-1}$ |

DTLZ5 | $\mathbf{6.30}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $6.70\times {10}^{-3}$ | $6.70\times {10}^{-3}$ | $6.60\times {10}^{-3}$ |

DTLZ6 | $6.60\times {10}^{-3}$ | $\mathbf{6.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $6.80\times {10}^{-3}$ | $7.30\times {10}^{-3}$ |

DTLZ7 | $2.03\times {10}^{-1}$ | $2.88\times {10}^{-1}$ | $2.17\times {10}^{-1}$ | $\mathbf{1.45}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ |

WFG2 | $5.31\times {10}^{-1}$ | $\mathbf{5.11}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $5.81\times {10}^{-1}$ | $5.61\times {10}^{-1}$ |

WFG3 | $2.15\times {10}^{-1}$ | $\mathbf{1.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $2.41\times {10}^{-1}$ | $2.81\times {10}^{-1}$ |

WFG4 | $8.17\times {10}^{-1}$ | $\mathbf{7.89}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $8.33\times {10}^{-1}$ | $7.90\times {10}^{-1}$ |

MaF1 | $\mathbf{1.19}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $1.21\times {10}^{-1}$ | $1.23\times {10}^{-1}$ | $1.29\times {10}^{-1}$ |

MaF2 | $9.66\times {10}^{-2}$ | $\mathbf{3.67}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $8.00\times {10}^{-3}$ | $7.20\times {10}^{-3}$ |

MaF12 | $1.88\times {10}^{-2}$ | $\mathbf{9.58}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $1.06\times {10}^{-1}$ | $1.11\times {10}^{-1}$ |

**Table 5.**$SI$ result of comparison of L and q in MOFECO algorithms on test functions. (The best result is bolded for each function).

Functions | $\mathit{L}=4/\mathit{q}=25$ | $\mathit{L}=5/\mathit{q}=20$ | $\mathit{L}=10/\mathit{q}=10$ | $\mathit{L}=20/\mathit{q}=5$ |
---|---|---|---|---|

ZDT1 | $4.87\times {10}^{-1}$ | $\mathbf{4.59}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $5.11\times {10}^{-1}$ | $4.66\times {10}^{-1}$ |

ZDT2 | $5.30\times {10}^{-1}$ | $\mathbf{4.54}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $5.12\times {10}^{-1}$ | $5.96\times {10}^{-1}$ |

ZDT4 | $\mathbf{4.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $5.40\times {10}^{-1}$ | $1.15\times {10}^{0}$ | $1.04\times {10}^{0}$ |

ZDT6 | $\mathbf{4.58}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $5.08\times {10}^{-1}$ | $1.08\times {10}^{0}$ | $1.18\times {10}^{0}$ |

DTLZ2 | $4.90\times {10}^{-1}$ | $\mathbf{4.86}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $5.07\times {10}^{-1}$ | $6.07\times {10}^{-1}$ |

DTLZ4 | $7.04\times {10}^{-1}$ | $\mathbf{5.65}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $1.08\times {10}^{0}$ | $1.05\times {10}^{0}$ |

DTLZ5 | $\mathbf{4.26}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.95\times {10}^{-1}$ | $5.51\times {10}^{-1}$ | $5.33\times {10}^{-1}$ |

DTLZ6 | $\mathbf{5.10}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $5.18\times {10}^{-1}$ | $5.26\times {10}^{-1}$ | $6.52\times {10}^{-1}$ |

DTLZ7 | $\mathbf{6.18}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $6.50\times {10}^{-1}$ | $7.31\times {10}^{-1}$ | $8.49\times {10}^{-1}$ |

WFG2 | $5.11\times {10}^{-1}$ | $\mathbf{4.41}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $5.11\times {10}^{-1}$ | $7.17\times {10}^{-1}$ |

WFG3 | $\mathbf{5.48}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $6.83\times {10}^{-1}$ | $5.57\times {10}^{-1}$ | $9.54\times {10}^{-1}$ |

WFG4 | $4.20\times {10}^{-1}$ | $\mathbf{4.13}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.39\times {10}^{-1}$ | $5.16\times {10}^{-1}$ |

MaF1 | $4.59\times {10}^{-1}$ | $\mathbf{4.16}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.63\times {10}^{-1}$ | $4.99\times {10}^{-1}$ |

MaF2 | $4.88\times {10}^{-1}$ | $\mathbf{4.44}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $5.30\times {10}^{-1}$ | $5.92\times {10}^{-1}$ |

MaF12 | $4.25\times {10}^{-1}$ | $\mathbf{4.06}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.35\times {10}^{-1}$ | $5.30\times {10}^{-1}$ |

**Table 6.**$GD$ result of the three update conditions of MOFECO algorithms on test functions. (The best result is bolded for each function).

Functions | Method 1 | Method 2 | Method 3 |
---|---|---|---|

ZDT1 | $\mathbf{1.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $1.40\times {10}^{-4}$ | $2.61\times {10}^{-4}$ |

ZDT2 | $\mathbf{4.02}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{6}}$ | $4.02\times {10}^{-6}$ | $4.34\times {10}^{-6}$ |

ZDT4 | $\mathbf{2.39}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ | $2.39\times {10}^{-5}$ | $1.26\times {10}^{-2}$ |

ZDT6 | $\mathbf{3.50}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$ | $3.50\times {10}^{-7}$ | $2.40\times {10}^{-3}$ |

DTLZ2 | $\mathbf{7.43}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $1.10\times {10}^{-3}$ | $1.00\times {10}^{-3}$ |

DTLZ4 | $1.43\times {10}^{-3}$ | $1.80\times {10}^{-3}$ | $\mathbf{9.39}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ |

DTLZ5 | $3.56\times {10}^{-5}$ | $3.94\times {10}^{-5}$ | $\mathbf{3.48}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ |

DTLZ6 | $\mathbf{4.82}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$ | $9.11\times {10}^{-5}$ | $2.90\times {10}^{-5}$ |

DTLZ7 | $9.60\times {10}^{-3}$ | $\mathbf{7.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $1.44\times {10}^{-2}$ |

WFG2 | $\mathbf{6.25}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $9.63\times {10}^{-2}$ | $6.68\times {10}^{-2}$ |

WFG3 | $\mathbf{5.29}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $1.94\times {10}^{-1}$ | $1.80\times {10}^{-1}$ |

WFG4 | $\mathbf{1.57}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $2.04\times {10}^{-2}$ | $1.94\times {10}^{-2}$ |

MaF1 | $\mathbf{7.10}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $8.00\times {10}^{-3}$ | $7.90\times {10}^{-3}$ |

MaF2 | $\mathbf{3.67}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $6.60\times {10}^{-3}$ | $5.40\times {10}^{-3}$ |

MaF12 | $1.88\times {10}^{-2}$ | $\mathbf{1.85}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $2.05\times {10}^{-2}$ |

**Table 7.**$IGD$ result of the three update conditions of MOFECO algorithms on test functions. (The best result is bolded for each function).

Functions | Method 1 | Method 2 | Method 3 |
---|---|---|---|

ZDT1 | $\mathbf{5.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $5.40\times {10}^{-3}$ | $6.60\times {10}^{-3}$ |

ZDT2 | $\mathbf{5.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $5.40\times {10}^{-3}$ | $7.00\times {10}^{-3}$ |

ZDT4 | $\mathbf{5.76}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $5.76\times {10}^{-3}$ | $9.47\times {10}^{-2}$ |

ZDT6 | $\mathbf{4.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $4.70\times {10}^{-3}$ | $4.90\times {10}^{-3}$ |

DTLZ2 | $\mathbf{7.49}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $7.64\times {10}^{-2}$ | $7.55\times {10}^{-2}$ |

DTLZ4 | $\mathbf{1.21}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $1.69\times {10}^{-1}$ | $2.68\times {10}^{-1}$ |

DTLZ5 | $6.70\times {10}^{-3}$ | $\mathbf{6.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $8.30\times {10}^{-3}$ |

DTLZ6 | $\mathbf{6.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $7.10\times {10}^{-3}$ | $7.80\times {10}^{-3}$ |

DTLZ7 | $\mathbf{2.88}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $3.77\times {10}^{-1}$ | $3.76\times {10}^{-1}$ |

WFG2 | $\mathbf{5.11}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $7.21\times {10}^{-1}$ | $1.21\times {10}^{0}$ |

WFG3 | $\mathbf{1.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $2.79\times {10}^{-1}$ | $2.30\times {10}^{-1}$ |

WFG4 | $7.89\times {10}^{-1}$ | $7.96\times {10}^{-1}$ | $\mathbf{7.86}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ |

MaF1 | $\mathbf{1.21}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $1.22\times {10}^{-1}$ | $1.21\times {10}^{-1}$ |

MaF2 | $9.58\times {10}^{-2}$ | $9.78\times {10}^{-2}$ | $\mathbf{9.11}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ |

MaF12 | $\mathbf{7.43}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $7.48\times {10}^{-1}$ | $7.65\times {10}^{-1}$ |

**Table 8.**$SP$ result of the three update conditions of MOFECO algorithms on test functions. (The best result is bolded for each function).

Functions | Method 1 | Method 2 | Method 3 |
---|---|---|---|

ZDT1 | $\mathbf{8.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $8.70\times {10}^{-3}$ | $1.03\times {10}^{-2}$ |

ZDT2 | $\mathbf{7.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $7.60\times {10}^{-3}$ | $8.80\times {10}^{-3}$ |

ZDT4 | $\mathbf{7.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $7.40\times {10}^{-3}$ | $1.40\times {10}^{-2}$ |

ZDT6 | $\mathbf{6.80}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $6.80\times {10}^{-3}$ | $3.06\times {10}^{-2}$ |

DTLZ2 | $\mathbf{5.57}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $5.86\times {10}^{-2}$ | $5.73\times {10}^{-2}$ |

DTLZ4 | $\mathbf{4.33}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $5.01\times {10}^{-2}$ | $5.67\times {10}^{-2}$ |

DTLZ5 | $\mathbf{9.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $9.90\times {10}^{-3}$ | $1.18\times {10}^{-2}$ |

DTLZ6 | $\mathbf{9.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $9.70\times {10}^{-3}$ | $9.90\times {10}^{-3}$ |

DTLZ7 | $6.50\times {10}^{-2}$ | $\mathbf{5.90}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $6.29\times {10}^{-2}$ |

WFG2 | $\mathbf{3.37}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.21\times {10}^{-1}$ | $3.63\times {10}^{-1}$ |

WFG3 | $\mathbf{1.42}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $3.12\times {10}^{-1}$ | $3.01\times {10}^{-1}$ |

WFG4 | $\mathbf{4.26}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.80\times {10}^{-1}$ | $4.99\times {10}^{-1}$ |

MaF1 | $\mathbf{7.31}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $8.49\times {10}^{-2}$ | $9.19\times {10}^{-2}$ |

MaF2 | $\mathbf{6.27}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $7.01\times {10}^{-2}$ | $6.60\times {10}^{-2}$ |

MaF12 | $\mathbf{4.31}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.52\times {10}^{-2}$ | $4.81\times {10}^{-2}$ |

**Table 9.**$GD$ result of the local-global update probability ${P}_{s}$ of MOFECO algorithms on test functions. (The best result is bolded for each function).

Functions | ${\mathit{P}}_{\mathit{s}}$ Is Nonlinear Variation | ${\mathit{P}}_{\mathit{s}}$ Is Linear Variation | ${\mathit{P}}_{\mathit{s}}=0.5$ |
---|---|---|---|

ZDT1 | $\mathbf{1.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $1.72\times {10}^{-4}$ | $1.54\times {10}^{-4}$ |

ZDT2 | $4.02\times {10}^{-6}$ | $4.05\times {10}^{-6}$ | $\mathbf{3.67}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{6}}$ |

ZDT4 | $\mathbf{2.39}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ | $2.70\times {10}^{-3}$ | $5.78\times {10}^{-5}$ |

ZDT6 | $\mathbf{3.50}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$ | $3.56\times {10}^{-7}$ | $3.53\times {10}^{-7}$ |

DTLZ2 | $\mathbf{7.43}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $1.40\times {10}^{-3}$ | $1.20\times {10}^{-3}$ |

DTLZ4 | $1.43\times {10}^{-3}$ | $\mathbf{2.86}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $3.87\times {10}^{-4}$ |

DTLZ5 | $3.56\times {10}^{-5}$ | $1.18\times {10}^{-2}$ | $\mathbf{2.18}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ |

DTLZ6 | $\mathbf{4.82}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$ | $2.60\times {10}^{-6}$ | $1.73\times {10}^{-6}$ |

DTLZ7 | $\mathbf{9.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $1.03\times {10}^{-2}$ | $1.50\times {10}^{-2}$ |

WFG2 | $\mathbf{6.25}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $8.45\times {10}^{-2}$ | $9.51\times {10}^{-2}$ |

WFG3 | $\mathbf{5.29}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $1.84\times {10}^{-1}$ | $1.36\times {10}^{-1}$ |

WFG4 | $\mathbf{1.57}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $1.91\times {10}^{-2}$ | $1.83\times {10}^{-2}$ |

MaF1 | $\mathbf{7.10}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $8.40\times {10}^{-3}$ | $1.28\times {10}^{-2}$ |

MaF2 | $3.67\times {10}^{-3}$ | $5.30\times {10}^{-3}$ | $\mathbf{2.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ |

MaF12 | $1.88\times {10}^{-2}$ | $2.00\times {10}^{-2}$ | $\mathbf{1.81}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ |

**Table 10.**$IGD$ result of the local-global update probability ${P}_{s}$ of MOFECO algorithms on test functions. (The best result is bolded for each function).

Functions | ${\mathit{P}}_{\mathit{s}}$ Is Nonlinear Variation | ${\mathit{P}}_{\mathit{s}}$ Is Linear Variation | ${\mathit{P}}_{\mathit{s}}=0.5$ |
---|---|---|---|

ZDT1 | $\mathbf{5.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $5.50\times {10}^{-3}$ | $5.60\times {10}^{-3}$ |

ZDT2 | $\mathbf{5.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $5.60\times {10}^{-3}$ | $5.60\times {10}^{-3}$ |

ZDT4 | $\mathbf{5.76}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $3.01\times {10}^{-2}$ | $5.80\times {10}^{-3}$ |

ZDT6 | $4.70\times {10}^{-3}$ | $4.60\times {10}^{-3}$ | $\mathbf{4.30}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ |

DTLZ2 | $7.49\times {10}^{-2}$ | $7.66\times {10}^{-2}$ | $\mathbf{7.10}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ |

DTLZ4 | $\mathbf{1.21}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.29\times {10}^{-1}$ | $3.56\times {10}^{-1}$ |

DTLZ5 | $\mathbf{6.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $7.00\times {10}^{-3}$ | $6.90\times {10}^{-3}$ |

DTLZ6 | $\mathbf{6.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $6.70\times {10}^{-3}$ | $6.60\times {10}^{-3}$ |

DTLZ7 | $\mathbf{2.88}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $3.60\times {10}^{-1}$ | $2.92\times {10}^{-1}$ |

WFG2 | $\mathbf{5.11}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $6.26\times {10}^{-1}$ | $7.04\times {10}^{-1}$ |

WFG3 | $\mathbf{1.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $2.61\times {10}^{-1}$ | $2.96\times {10}^{-1}$ |

WFG4 | $7.89\times {10}^{-1}$ | $7.86\times {10}^{-1}$ | $\mathbf{7.74}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ |

MaF1 | $\mathbf{1.21}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $1.30\times {10}^{-1}$ | $1.62\times {10}^{-1}$ |

MaF2 | $9.58\times {10}^{-2}$ | $9.49\times {10}^{-2}$ | $\mathbf{8.34}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ |

MaF12 | $\mathbf{7.43}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $7.83\times {10}^{-1}$ | $7.43\times {10}^{-1}$ |

**Table 11.**$SP$ result of the local-global update probability ${P}_{s}$ of MOFECO algorithms on test functions. (The best result is bolded for each function).

Functions | ${\mathit{P}}_{\mathit{s}}$ Is Nonlinear Variation | ${\mathit{P}}_{\mathit{s}}$ Is Linear Variation | ${\mathit{P}}_{\mathit{s}}=0.5$ |
---|---|---|---|

ZDT1 | $\mathbf{8.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $9.30\times {10}^{-3}$ | $1.11\times {10}^{-2}$ |

ZDT2 | $\mathbf{7.50}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $7.50\times {10}^{-3}$ | $7.90\times {10}^{-3}$ |

ZDT4 | $\mathbf{7.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $9.10\times {10}^{-3}$ | $1.07\times {10}^{-2}$ |

ZDT6 | $6.80\times {10}^{-3}$ | $6.60\times {10}^{-3}$ | $\mathbf{6.50}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ |

DTLZ2 | $5.57\times {10}^{-2}$ | $5.71\times {10}^{-2}$ | $\mathbf{5.25}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ |

DTLZ4 | $\mathbf{4.33}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $5.74\times {10}^{-2}$ | $4.88\times {10}^{-2}$ |

DTLZ5 | $\mathbf{9.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $1.04\times {10}^{-2}$ | $1.01\times {10}^{-2}$ |

DTLZ6 | $\mathbf{9.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $9.60\times {10}^{-3}$ | $9.50\times {10}^{-3}$ |

DTLZ7 | $6.50\times {10}^{-2}$ | $\mathbf{5.83}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $8.18\times {10}^{-2}$ |

WFG2 | $\mathbf{3.37}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $3.94\times {10}^{-1}$ | $4.06\times {10}^{-1}$ |

WFG3 | $\mathbf{1.42}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $2.92\times {10}^{-1}$ | $2.05\times {10}^{-1}$ |

WFG4 | $\mathbf{4.26}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.88\times {10}^{-1}$ | $4.54\times {10}^{-1}$ |

MaF1 | $\mathbf{7.31}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $8.14\times {10}^{-2}$ | $5.71\times {10}^{-2}$ |

MaF12 | $\mathbf{4.31}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.67\times {10}^{-1}$ | $4.42\times {10}^{-1}$ |

**Table 12.**$GD$ result of the mutation methods of MOFECO algorithms on test functions. (The best result is bolded for each function).

Functions | Combined Mutation | Cauchy Mutation |
---|---|---|

ZDT1 | $\mathbf{1.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $1.50\times {10}^{-4}$ |

ZDT2 | $4.02\times {10}^{-6}$ | $\mathbf{2.46}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{6}}$ |

ZDT4 | $\mathbf{2.39}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{5}}$ | $1.50\times {10}^{-3}$ |

ZDT6 | $\mathbf{3.50}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$ | $1.80\times {10}^{-3}$ |

DTLZ2 | $\mathbf{7.43}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ | $1.10\times {10}^{-3}$ |

DTLZ4 | $1.43\times {10}^{-3}$ | $\mathbf{7.65}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{4}}$ |

DTLZ5 | $3.56\times {10}^{-5}$ | $\mathbf{3.36}\times {\mathbf{10}}^{-\mathbf{5}}$ |

DTLZ6 | $\mathbf{4.82}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{7}}$ | $4.89\times {10}^{-7}$ |

DTLZ7 | $9.60\times {10}^{-3}$ | $\mathbf{7.30}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ |

WFG2 | $\mathbf{6.25}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $7.82\times {10}^{-2}$ |

WFG3 | $\mathbf{5.29}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $1.87\times {10}^{-1}$ |

WFG4 | $\mathbf{1.57}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $2.03\times {10}^{-2}$ |

MaF1 | $\mathbf{7.10}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $7.40\times {10}^{-3}$ |

MaF2 | $3.67\times {10}^{-3}$ | $\mathbf{2.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ |

MaF12 | $\mathbf{1.88}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $2.01\times {10}^{-2}$ |

**Table 13.**$IGD$ result of the mutation methods of MOFECO algorithms on test functions. (The best result is bolded for each function).

Functions | Combined Mutation | Cauchy Mutation |
---|---|---|

ZDT1 | $\mathbf{5.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $5.60\times {10}^{-3}$ |

ZDT2 | $\mathbf{5.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $6.40\times {10}^{-3}$ |

ZDT4 | $\mathbf{5.76}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $1.80\times {10}^{-2}$ |

ZDT6 | $4.70\times {10}^{-3}$ | $\mathbf{4.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ |

DTLZ2 | $\mathbf{7.49}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $7.54\times {10}^{-2}$ |

DTLZ4 | $\mathbf{1.21}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $3.83\times {10}^{-1}$ |

DTLZ5 | $\mathbf{6.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $7.00\times {10}^{-3}$ |

DTLZ6 | $\mathbf{6.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $6.60\times {10}^{-3}$ |

DTLZ7 | $\mathbf{2.88}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.30\times {10}^{-1}$ |

WFG2 | $5.11\times {10}^{-1}$ | $\mathbf{4.95}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ |

WFG3 | $\mathbf{1.60}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $3.01\times {10}^{-1}$ |

WFG4 | $\mathbf{7.89}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $7.97\times {10}^{-1}$ |

MaF1 | $\mathbf{1.21}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $1.21\times {10}^{-1}$ |

MaF2 | $9.58\times {10}^{-2}$ | $\mathbf{8.34}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ |

MaF12 | $\mathbf{7.43}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $7.63\times {10}^{-1}$ |

**Table 14.**$SP$ results of the mutation methods of MOFECO algorithms on test functions. (The best result is bolded for each function).

Functions | Combined Mutation | Cauchy Mutation |
---|---|---|

ZDT1 | $\mathbf{8.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $9.10\times {10}^{-3}$ |

ZDT2 | $\mathbf{7.50}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $8.10\times {10}^{-3}$ |

ZDT4 | $\mathbf{7.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $8.80\times {10}^{-3}$ |

ZDT6 | $\mathbf{6.80}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $1.74\times {10}^{-2}$ |

DTLZ2 | $\mathbf{5.57}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $5.63\times {10}^{-2}$ |

DTLZ4 | $4.33\times {10}^{-2}$ | $\mathbf{3.76}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ |

DTLZ5 | $\mathbf{9.70}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $1.13\times {10}^{-2}$ |

DTLZ6 | $\mathbf{9.40}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{3}}$ | $9.60\times {10}^{-3}$ |

DTLZ7 | $6.50\times {10}^{-2}$ | $\mathbf{5.34}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ |

WFG2 | $\mathbf{3.37}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.33\times {10}^{-1}$ |

WFG3 | $\mathbf{1.42}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $3.02\times {10}^{-1}$ |

WFG4 | $\mathbf{4.26}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.84\times {10}^{-1}$ |

MaF1 | $\mathbf{7.31}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ | $8.42\times {10}^{-2}$ |

MaF2 | $6.27\times {10}^{-2}$ | $\mathbf{5.71}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{2}}$ |

MaF12 | $\mathbf{4.31}\mathbf{\times}{\mathbf{10}}^{\mathbf{-}\mathbf{1}}$ | $4.52\times {10}^{-1}$ |

Algorithms | EAreal | PSO | NSLS Operator | ||||||
---|---|---|---|---|---|---|---|---|---|

$\mathit{proC}$ | $\mathit{proM}$ | $\mathit{disC}$ | $\mathit{disM}$ | $\mathit{div}$ | $\mathit{rate}$ | $\mathit{\omega}$ | $\mathit{mu}$ | $\mathit{delta}$ | |

NSGA-II | 1 | 1 | 20 | 20 | / | / | / | / | / |

PESA-II | 1 | 1 | 20 | 20 | 10 | / | / | / | / |

KnEA | 1 | 1 | 20 | 20 | / | 0.4 | / | / | / |

MOPSO | / | / | / | / | / | / | 0.4 | / | / |

NSLS | / | / | / | / | / | / | / | 0.5 | 0.1 |

**Table 16.**$GD$ result of the six compared algorithms on test functions. (“+”, “∼” and “-” respectively represent that MOFECO is better than, similar to and inferior to the other five algorithms).

Problems | MOFECO | NSGA-II | MOPSO | PESA-II | KnEA | NSLS |
---|---|---|---|---|---|---|

ZDT1 | $1.40\times {10}^{-4}$ | $1.54\times {10}^{-4}$ (+) | $6.94\times {10}^{-4}$ (+) | $1.72\times {10}^{-4}$ (+) | $5.36\times {10}^{-5}$ (-) | $5.04\times {10}^{-4}$ (+) |

ZDT2 | $4.02\times {10}^{-6}$ | $2.36\times {10}^{-5}$ (+) | $2.00\times {10}^{-3}$ (+) | $5.75\times {10}^{-4}$ (+) | $5.59\times {10}^{-6}$ (+) | $3.80\times {10}^{-3}$ (+) |

ZDT4 | $2.39\times {10}^{-5}$ | $2.81\times {10}^{-5}$ (+) | $9.30\times {10}^{-3}$ (+) | $2.09\times {10}^{-2}$ (+) | $1.42\times {10}^{-2}$ (+) | $1.81\times {10}^{-2}$ (+) |

ZDT6 | $3.50\times {10}^{-7}$ | $4.02\times {10}^{-6}$ (+) | $9.13\times {10}^{-2}$ (+) | $6.10\times {10}^{-3}$ (+) | $4.36\times {10}^{-6}$ (+) | $7.30\times {10}^{-3}$ (+) |

DTLZ2 | $4.67\times {10}^{-4}$ | $1.20\times {10}^{-3}$ (+) | $4.70\times {10}^{-3}$ (+) | $1.40\times {10}^{-3}$ (+) | $5.03\times {10}^{-4}$ (+) | $5.56\times {10}^{-4}$ (+) |

DTLZ4 | $1.43\times {10}^{-3}$ | $1.10\times {10}^{-3}$ (-) | $4.60\times {10}^{-3}$ (+) | $1.42\times {10}^{-3}$ (∼) | $4.24\times {10}^{-4}$ (-) | $6.05\times {10}^{-4}$ (-) |

DTLZ5 | $3.56\times {10}^{-5}$ | $2.30\times {10}^{-4}$ (+) | $1.40\times {10}^{-3}$ (+) | $2.19\times {10}^{-4}$ (+) | $5.00\times {10}^{-4}$ (+) | $1.46\times {10}^{-5}$ (-) |

DTLZ6 | $4.82\times {10}^{-7}$ | $4.82\times {10}^{-6}$ (+) | $5.20\times {10}^{-3}$ (+) | $1.80\times {10}^{-3}$ (+) | $4.73\times {10}^{-6}$ (+) | $4.85\times {10}^{-6}$ (+) |

DTLZ7 | $9.60\times {10}^{-3}$ | $3.00\times {10}^{-3}$ (-) | $1.01\times {10}^{-2}$ (+) | $2.70\times {10}^{-3}$ (-) | $1.50\times {10}^{-3}$ (-) | $2.90\times {10}^{-3}$ (-) |

WFG2 | $6.25\times {10}^{-2}$ | $8.77\times {10}^{-2}$ (+) | $7.12\times {10}^{-2}$ (+) | $6.44\times {10}^{-2}$ (+) | $6.51\times {10}^{-2}$ (+) | $6.66\times {10}^{-2}$ (+) |

WFG3 | $5.29\times {10}^{-2}$ | $1.85\times {10}^{-1}$ (+) | $1.46\times {10}^{-1}$ (+) | $1.63\times {10}^{-1}$ (+) | $1.21\times {10}^{-1}$ (+) | $1.62\times {10}^{-1}$ (+) |

WFG4 | $1.57\times {10}^{-2}$ | $1.68\times {10}^{-2}$ (+) | $3.61\times {10}^{-2}$ (+) | $2.58\times {10}^{-2}$ (+) | $1.59\times {10}^{-2}$ (∼) | $4.39\times {10}^{-2}$ (+) |

MaF1 | $7.10\times {10}^{-3}$ | $3.97\times {10}^{-3}$ (-) | $9.72\times {10}^{-3}$ (+) | $4.29\times {10}^{-3}$ (-) | $1.75\times {10}^{-3}$ (-) | $6.14\times {10}^{-3}$ (-) |

MaF2 | $3.67\times {10}^{-3}$ | $8.13\times {10}^{-3}$ (+) | $4.88\times {10}^{-3}$ (+) | $3.79\times {10}^{-3}$ (+) | $4.35\times {10}^{-3}$ (+) | $6.19\times {10}^{-3}$ (+) |

MaF12 | $1.88\times {10}^{-2}$ | $2.01\times {10}^{-2}$ (+) | $4.11\times {10}^{-2}$ (+) | $2.20\times {10}^{-2}$ (+) | $2.01\times {10}^{-2}$ (+) | $5.83\times {10}^{-2}$ (+) |

+ | / | 12 | 15 | 12 | 10 | 11 |

- | / | 3 | 0 | 2 | 4 | 4 |

∼ | / | 0 | 0 | 1 | 1 | 0 |

**Table 17.**$PD$ result of the six compared algorithms on test functions. (“+”, “∼” and “-” respectively represent that MOFECO is better than, similar to and inferior to the other five algorithms).

Problems | MOFECO | NSGA-II | MOPSO | PESA-II | KnEA | NSLS |
---|---|---|---|---|---|---|

ZDT1 | $1.73\times {10}^{3}$ | $1.71\times {10}^{3}$ (+) | $1.47\times {10}^{3}$ (+) | $1.45\times {10}^{3}$ (+) | $1.06\times {10}^{3}$ (+) | $1.52\times {10}^{3}$ (+) |

ZDT2 | $1.74\times {10}^{3}$ | $1.73\times {10}^{3}$ (∼) | $1.53\times {10}^{3}$ (+) | $1.44\times {10}^{3}$ (+) | $1.30\times {10}^{3}$ (+) | $1.36\times {10}^{3}$ (+) |

ZDT4 | $1.69\times {10}^{3}$ | $1.63\times {10}^{3}$ (+) | $8.40\times {10}^{2}$ (+) | $1.57\times {10}^{3}$ (+) | $7.27\times {10}^{2}$ (+) | $1.98\times {10}^{3}$ (+) |

ZDT6 | $1.62\times {10}^{3}$ | $1.45\times {10}^{3}$ (+) | $1.30\times {10}^{3}$ (+) | $1.27\times {10}^{3}$ (+) | $1.53\times {10}^{3}$ (+) | $1.35\times {10}^{3}$ (+) |

DTLZ2 | $1.73\times {10}^{5}$ | $1.87\times {10}^{5}$ (-) | $1.99\times {10}^{5}$ (-) | $1.85\times {10}^{5}$ (-) | $1.00\times {10}^{5}$ (+) | $1.56\times {10}^{5}$ (+) |

DTLZ4 | $1.32\times {10}^{5}$ | $1.88\times {10}^{5}$ (-) | $1.69\times {10}^{5}$ (-) | $1.70\times {10}^{5}$ (-) | $8.39\times {10}^{4}$ (+) | $8.41\times {10}^{4}$ (+) |

DTLZ5 | $7.30\times {10}^{4}$ | $7.10\times {10}^{4}$ (+) | $7.27\times {10}^{4}$ (+) | $6.29\times {10}^{4}$ (+) | $7.36\times {10}^{4}$ (-) | $7.20\times {10}^{4}$ (+) |

DTLZ6 | $7.97\times {10}^{4}$ | $7.20\times {10}^{4}$ (+) | $6.85\times {10}^{4}$ (+) | $6.66\times {10}^{4}$ (+) | $7.57\times {10}^{4}$ (+) | $6.94\times {10}^{4}$ (+) |

DTLZ7 | $1.42\times {10}^{5}$ | $2.00\times {10}^{5}$ (-) | $1.72\times {10}^{5}$ (-) | $1.35\times {10}^{5}$ (+) | $1.42\times {10}^{5}$ (∼) | $1.75\times {10}^{5}$ (-) |

WFG2 | $9.78\times {10}^{6}$ | $8.80\times {10}^{6}$ (+) | $9.78\times {10}^{6}$ (∼) | $9.55\times {10}^{6}$ (+) | $5.3\times {10}^{6}$ (+) | $8.57\times {10}^{6}$ (+) |

WFG3 | $8.38\times {10}^{6}$ | $1.20\times {10}^{7}$ (-) | $8.17\times {10}^{6}$ (+) | $7.13\times {10}^{6}$ (+) | $1.02\times {10}^{7}$ (-) | $1.38\times {10}^{7}$ (-) |

WFG4 | $1.92\times {10}^{7}$ | $1.90\times {10}^{7}$ (+) | $1.84\times {10}^{7}$ (+) | $1.68\times {10}^{7}$ (+) | $7.06\times {10}^{6}$ (+) | $1.69\times {10}^{7}$ (+) |

MaF1 | $3.54\times {10}^{6}$ | $3.26\times {10}^{6}$ (+) | $3.49\times {10}^{6}$ (+) | $3.09\times {10}^{6}$ (+) | $2.74\times {10}^{6}$ (+) | $3.06\times {10}^{6}$ (+) |

MaF2 | $2.74\times {10}^{6}$ | $2.57\times {10}^{6}$ (+) | $2.30\times {10}^{6}$ (+) | $2.14\times {10}^{6}$ (+) | $1.52\times {10}^{6}$ (+) | $2.67\times {10}^{6}$ (+) |

MaF12 | $1.96\times {10}^{7}$ | $1.85\times {10}^{7}$ (+) | $1.90\times {10}^{7}$ (+) | $1.93\times {10}^{7}$ (+) | $9.25\times {10}^{6}$ (+) | $1.55\times {10}^{7}$ (+) |

+ | / | 10 | 11 | 13 | 12 | 12 |

- | / | 4 | 3 | 2 | 2 | 3 |

∼ | / | 1 | 1 | 0 | 1 | 0 |

**Table 18.**$HV$ result of the six compared algorithms on test functions. (“+”, “∼” and “-” respectively represent that MOFECO is better than, similar to and inferior to the other five algorithms).

Problems | MOFECO | NSGA-II | MOPSO | PESA-II | KnEA | NSLS |
---|---|---|---|---|---|---|

ZDT1 | $8.70\times {10}^{-1}$ | $8.70\times {10}^{-1}$ (∼) | $8.58\times {10}^{-1}$ (+) | $8.62\times {10}^{-1}$ (+) | $7.62\times {10}^{-1}$ (+) | $8.66\times {10}^{-1}$ (+) |

ZDT2 | $5.37\times {10}^{-1}$ | $5.35\times {10}^{-1}$ (+) | $5.28\times {10}^{-1}$ (+) | $5.25\times {10}^{-1}$ (+) | $3.90\times {10}^{-1}$ (+) | $4.95\times {10}^{-1}$ (+) |

ZDT4 | $8.68\times {10}^{-1}$ | $8.61\times {10}^{-1}$ (+) | $6.88\times {10}^{-1}$ (+) | $8.62\times {10}^{-1}$ (+) | $6.64\times {10}^{-1}$ (+) | $6.15\times {10}^{-1}$ (+) |

ZDT6 | $4.33\times {10}^{-1}$ | $4.33\times {10}^{-1}$ (∼) | $3.95\times {10}^{-1}$ (+) | $4.28\times {10}^{-1}$ (+) | $4.30\times {10}^{-1}$ (+) | $4.33\times {10}^{-1}$ (∼) |

DTLZ2 | $6.70\times {10}^{-1}$ | $7.08\times {10}^{-1}$ (-) | $6.61\times {10}^{-1}$ (+) | $6.94\times {10}^{-1}$ (-) | $7.21\times {10}^{-1}$ (-) | $7.46\times {10}^{-1}$ (-) |

DTLZ4 | $8.08\times {10}^{-1}$ | $6.92\times {10}^{-1}$ (+) | $6.78\times {10}^{-1}$ (+) | $7.14\times {10}^{-1}$ (+) | $7.09\times {10}^{-1}$ (+) | $7.07\times {10}^{-1}$ (+) |

DTLZ5 | $1.33\times {10}^{-1}$ | $1.33\times {10}^{-1}$ (∼) | $1.27\times {10}^{-1}$ (+) | $1.28\times {10}^{-1}$ (+) | $1.29\times {10}^{-1}$ (+) | $1.33\times {10}^{-1}$ (∼) |

DTLZ6 | $1.33\times {10}^{-1}$ | $1.33\times {10}^{-1}$ (∼) | $1.30\times {10}^{-1}$ (+) | $1.27\times {10}^{-1}$ (+) | $1.33\times {10}^{-1}$ (∼) | $1.33\times {10}^{-1}$ (∼) |

DTLZ7 | $1.30\times {10}^{0}$ | $1.58\times {10}^{0}$ (-) | $1.42\times {10}^{0}$ (-) | $1.53\times {10}^{0}$ (-) | $1.61\times {10}^{0}$ (-) | $1.62\times {10}^{0}$ (-) |

WFG2 | $5.47\times {10}^{2}$ | $5.40\times {10}^{2}$ (+) | $4.73\times {10}^{2}$ (+) | $5.26\times {10}^{2}$ (+) | $5.41\times {10}^{2}$ (+) | $5.01\times {10}^{2}$ (+) |

WFG3 | $5.01\times {10}^{0}$ | $5.19\times {10}^{0}$ (-) | $0.00\times {10}^{0}$ (+) | $0.00\times {10}^{0}$ (+) | $3.93\times {10}^{0}$ (+) | $1.81\times {10}^{0}$ (+) |

WFG4 | $3.25\times {10}^{2}$ | $3.50\times {10}^{2}$ (-) | $2.43\times {10}^{2}$ (+) | $2.60\times {10}^{2}$ (+) | $3.71\times {10}^{2}$ (-) | $3.20\times {10}^{2}$ (+) |

MaF1 | $5.84\times {10}^{-2}$ | $6.19\times {10}^{-2}$ (-) | $4.75\times {10}^{-2}$ (+) | $6.16\times {10}^{-2}$ (-) | $7.51\times {10}^{-2}$ (-) | $5.03\times {10}^{-2}$ (+) |

MaF2 | $1.37\times {10}^{-1}$ | $1.32\times {10}^{-1}$ (+) | $1.18\times {10}^{-1}$ (+) | $1.15\times {10}^{-1}$ (+) | $1.33\times {10}^{-1}$ (+) | $1.36\times {10}^{-1}$ (∼) |

MaF12 | $2.97\times {10}^{2}$ | $3.10\times {10}^{2}$ (-) | $2.42\times {10}^{2}$ (+) | $2.89\times {10}^{2}$ (+) | $3.64\times {10}^{2}$ (-) | $2.26\times {10}^{2}$ (+) |

+ | / | 5 | 14 | 12 | 9 | 9 |

- | / | 6 | 1 | 3 | 5 | 2 |

∼ | / | 4 | 0 | 0 | 1 | 4 |

**Table 19.**$SP$ result of the six compared algorithms on test functions. (“+”, “∼” and “-” respectively represent that MOFECO is better than, similar to and inferior to the other five algorithms).

Problems | MOFECO | NSGA-II | MOPSO | PESA-II | KnEA | NSLS |
---|---|---|---|---|---|---|

ZDT1 | $8.70\times {10}^{-3}$ | $6.90\times {10}^{-3}$ (-) | $9.60\times {10}^{-3}$ (+) | $1.07\times {10}^{-2}$ (+) | $5.90\times {10}^{-3}$ (-) | $7.70\times {10}^{-3}$ (-) |

ZDT2 | $7.60\times {10}^{-3}$ | $7.70\times {10}^{-3}$ (+) | $2.02\times {10}^{-2}$ (+) | $1.45\times {10}^{-2}$ (+) | $4.60\times {10}^{-3}$ (-) | $3.91\times {10}^{-2}$ (+) |

ZDT4 | $7.40\times {10}^{-3}$ | $7.60\times {10}^{-3}$ (+) | $1.62\times {10}^{-2}$ (+) | $2.19\times {10}^{-1}$ (+) | $1.46\times {10}^{-1}$ (+) | $5.74\times {10}^{-3}$ (-) |

ZDT6 | $6.80\times {10}^{-3}$ | $6.84\times {10}^{-3}$ (+) | $1.97\times {10}^{-1}$ (+) | $4.53\times {10}^{-2}$ (+) | $7.90\times {10}^{-3}$ (+) | $7.66\times {10}^{-2}$ (+) |

DTLZ2 | $5.57\times {10}^{-2}$ | $5.85\times {10}^{-2}$ (+) | $5.13\times {10}^{-2}$ (-) | $5.16\times {10}^{-2}$ (-) | $7.16\times {10}^{-2}$ (+) | $3.43\times {10}^{-2}$ (-) |

DTLZ4 | $4.33\times {10}^{-2}$ | $5.48\times {10}^{-2}$ (+) | $4.89\times {10}^{-2}$ (+) | $5.23\times {10}^{-2}$ (+) | $9.10\times {10}^{-2}$ (+) | $4.84\times {10}^{-2}$ (+) |

DTLZ5 | $9.70\times {10}^{-3}$ | $9.30\times {10}^{-3}$ (-) | $1.52\times {10}^{-2}$ (+) | $1.33\times {10}^{-2}$ (+) | $1.80\times {10}^{-2}$ (+) | $7.40\times {10}^{-3}$ (-) |

DTLZ6 | $9.40\times {10}^{-3}$ | $1.12\times {10}^{-2}$ (+) | $1.69\times {10}^{-2}$ (+) | $2.04\times {10}^{-2}$ (+) | $1.03\times {10}^{-2}$ (+) | $6.40\times {10}^{-2}$ (-) |

DTLZ7 | $6.50\times {10}^{-2}$ | $7.18\times {10}^{-2}$ (+) | $6.69\times {10}^{-2}$ (+) | $6.14\times {10}^{-2}$ (-) | $4.97\times {10}^{-2}$ (-) | $4.25\times {10}^{-2}$ (-) |

WFG2 | $3.37\times {10}^{-1}$ | $4.88\times {10}^{-1}$ (+) | $3.62\times {10}^{-1}$ (+) | $3.40\times {10}^{-1}$ (+) | $4.37\times {10}^{-1}$ (+) | $3.64\times {10}^{-1}$ (+) |

WFG3 | $1.42\times {10}^{-1}$ | $3.43\times {10}^{-1}$ (+) | $2.01\times {10}^{-1}$ (+) | $1.79\times {10}^{-1}$ (+) | $4.06\times {10}^{-1}$ (+) | $1.99\times {10}^{-1}$ (+) |

WFG4 | $4.26\times {10}^{-1}$ | $5.23\times {10}^{-1}$ (+) | $4.87\times {10}^{-1}$ (+) | $4.32\times {10}^{-1}$ (+) | $7.24\times {10}^{-1}$ (+) | $4.29\times {10}^{-1}$ (+) |

MaF1 | $7.31\times {10}^{-2}$ | $8.36\times {10}^{-2}$ (+) | $7.06\times {10}^{-2}$ (-) | $6.23\times {10}^{-2}$ (-) | $1.05\times {10}^{-1}$ (+) | $7.18\times {10}^{-2}$ (-) |

MaF2 | $6.27\times {10}^{-2}$ | $6.28\times {10}^{-2}$ (∼) | $6.08\times {10}^{-2}$ (-) | $5.39\times {10}^{-2}$ (-) | $8.64\times {10}^{-2}$ (+) | $3.21\times {10}^{-2}$ (-) |

MaF12 | $4.31\times {10}^{-1}$ | $5.18\times {10}^{-1}$ (+) | $4.53\times {10}^{-1}$ (+) | $4.33\times {10}^{-1}$ (+) | $7.17\times {10}^{-1}$ (+) | $6.85\times {10}^{-1}$ (+) |

+ | / | 12 | 12 | 11 | 12 | 7 |

- | / | 2 | 3 | 4 | 3 | 8 |

∼ | / | 1 | 0 | 0 | 0 | 0 |

**Table 20.**$SI$ result of the six compared algorithms on test functions. (“+”, “∼” and “-” respectively represent that MOFECO is better than, similar to and inferior to the other five algorithms).

Problems | MOFECO | NSGA-II | MOPSO | PESA-II | KnEA | NSLS |
---|---|---|---|---|---|---|

ZDT1 | $4.59\times {10}^{-1}$ | $4.01\times {10}^{-1}$ (-) | $6.69\times {10}^{-1}$ (+) | $9.18\times {10}^{-1}$ (+) | $6.98\times {10}^{-1}$ (+) | $3.69\times {10}^{-1}$ (-) |

ZDT2 | $4.54\times {10}^{-1}$ | $4.59\times {10}^{-1}$ (+) | $7.91\times {10}^{-1}$ (+) | $1.02\times {10}^{0}$ (+) | $4.90\times {10}^{-1}$ (+) | $8.28\times {10}^{-1}$ (+) |

ZDT4 | $5.40\times {10}^{-1}$ | $4.65\times {10}^{-1}$ (-) | $8.27\times {10}^{-1}$ (+) | $1.03\times {10}^{0}$ (+) | $9.64\times {10}^{-1}$ (+) | $3.50\times {10}^{-1}$ (-) |

ZDT6 | $5.08\times {10}^{-1}$ | $5.19\times {10}^{-1}$ (+) | $1.21\times {10}^{0}$ (+) | $1.07\times {10}^{0}$ (+) | $4.44\times {10}^{-1}$ (-) | $5.75\times {10}^{-1}$ (+) |

DTLZ2 | $4.86\times {10}^{-1}$ | $5.24\times {10}^{-1}$ (+) | $3.62\times {10}^{-1}$ (-) | $4.24\times {10}^{-1}$ (-) | $4.63\times {10}^{-1}$ (-) | $1.34\times {10}^{-1}$ (-) |

DTLZ4 | $5.65\times {10}^{-1}$ | $4.99\times {10}^{-1}$ (-) | $3.41\times {10}^{-1}$ (-) | $4.04\times {10}^{-1}$ (-) | $8.17\times {10}^{-1}$ (+) | $3.40\times {10}^{-1}$ (-) |

DTLZ5 | $4.95\times {10}^{-1}$ | $4.79\times {10}^{-1}$ (-) | $6.90\times {10}^{-1}$ (+) | $9.17\times {10}^{-1}$ (+) | $8.14\times {10}^{-1}$ (+) | $2.30\times {10}^{-1}$ (-) |

DTLZ6 | $5.18\times {10}^{-1}$ | $6.89\times {10}^{-1}$ (+) | $9.97\times {10}^{-1}$ (+) | $1.23\times {10}^{0}$ (+) | $4.00\times {10}^{-1}$ (-) | $2.10\times {10}^{-1}$ (-) |

DTLZ7 | $6.50\times {10}^{-1}$ |