# An Interactive Estimation of the Distribution Algorithm Integrated with Surrogate-Assisted Fitness

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## Abstract

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## 1. Introduction

- (1)
- Introducing a probabilistic model to EDA. Unlike machine learning methods, the proposed method is particularly well-suited for small training data and boosts the efficiency of EDA significantly;
- (2)
- Presenting a fitness estimation approach that markedly improves the precision of fitness prediction;
- (3)
- Proposing a novel interactive distribution estimation algorithm that enhances the quality of the interactive evolutionary algorithm.

## 2. The Proposed Method

#### 2.1. Preference Probability Model

#### 2.2. Surrogate-Assisted Fitness Evaluation

#### 2.3. Interactive Estimation of the Distribution Algorithm

## 3. Experimental Study

#### 3.1. Experimental Setup

- (1)
- The traditional interactive genetic algorithm (IGA);
- (2)
- The Kano-integrated interactive genetic algorithm (Kano-IGA), proposed in [23];
- (3)
- The interactive genetic algorithm with BP neural network-based user cognitive surrogate model (BP-IGA), proposed in [24];
- (4)
- An interactive estimation of the distribution algorithm with RBF neural network-based fitness evaluation (RBF-IEDA), proposed in [16].

#### 3.2. Results and Analysis

#### 3.2.1. The Parameter $\lambda $

#### 3.2.2. The Result Analyses of SAF-IEDA with the Comparison Methods

#### 3.2.3. Application to the Indoor Lighting Optimization

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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References | Algorithms | Problems |
---|---|---|

Zhou, et al. [13] | DE/AEDA | Differential evolution strategy and adaptive learning rate mechanism are integrated into EDA |

Pérez-Rodríguez, et al. [14] | HMEDA | Mallows distribution probability model |

Pérez-Rodríguez [15] | QCEDA | Probability model uses a distance-based ranking model and the moth–flame algorithm. |

Chen Yang, et al. [16] | RBF-IEDA | Surrogate-assisted fitness evaluation-based RBF |

Wang Leuohong, et al. [17] | APL-IEC | Algorithmic probability IEC |

Chen Yang, et al. [18] | LMIEDA | Language model-based IEDA |

Bao Lin, et al. [19] | RBM-EDA | Surrogate-assisted fitness evaluation-based RBM |

Dewancker, I. et al. [20] | Active Utility Function Preference Learning | Utility function learning |

Amalia Utamima [21] | AEDA | EDA plus a lottery procedure, an elitism strategy, and a neighborhood search |

$N$ | Population size |

${X}_{i}$ | Individual i in the population |

${x}_{i}$ | The i-th decision variable of individual $X$ |

c | Number of decision variables |

${x}_{i,{i}_{m}}$ | The ${i}_{m}$-th attribute value of decision variable ${x}_{i}$ |

m | Number of attribute values |

$P$ | Set of individual attribute values |

${X}_{*}$ | Optimal individual (specific individual) |

$U(p)$ | Utility function of attribute p |

Nc | Number of user evaluations of individuals |

$\tilde{f}({X}_{i})$ | Fitness value of individual ${X}_{i}$ |

$t({X}_{j})$ | Evaluation time for individual ${X}_{j}$ |

$\mu ({x}_{i}={x}_{i,{i}_{h}})$ | Phenotypic similarity when the value of decision variable ${x}_{i}$ is ${x}_{i,{i}_{h}}$ |

Users | Evolutionary Generation | Fitness of Top-Nc | ${\mathit{D}}_{\mathbf{NBS}}$ | Runtime (s) | NOS | SR (%) |
---|---|---|---|---|---|---|

Use 1 | 7.7 (1.033) | 95.70 (3.561) | 4.33 (0.687) | 181.30 (20.548) | 167 (11.253) | 96.24 (0.405) |

Use 2 | 7.3 (0.816) | 94.56 (2.351) | 5.71 (0.592) | 188.56 (22.562) | 162 (13.052) | 95.62 (0.561) |

Use 3 | 7.0 (0.876) | 95.21 (2.803) | 4.04 (0.754) | 179.63 (19.554) | 165 (12.455) | 96.53 (0.684) |

Use 4 | 7.4 (0.948) | 96.89 (3.149) | 4.31 (0.653) | 168.37 (18.615) | 182 (14.027) | 96.37 (0.515) |

Use 5 | 7.0 (0.994) | 96.43 (3.153) | 3.65 (0.626) | 175.48 (18.327) | 172 (13.587) | 95.75 (0.632) |

Use 6 | 7.6 (0.949) | 95.82 (2.362) | 4.28 (0.654) | 187.56 (22.355) | 166 (14.396) | 95.83 (0.426) |

Use 7 | 7.4 (0.843) | 96.87 (3.257) | 5.23 (0.735) | 194.25 (22.452) | 189 (12.473) | 95.85 (0.654) |

Use 8 | 7.5 (0.738) | 97.02 (1.758) | 3.36 (0.593) | 186.56 (20.354) | 192 (12.951) | 96.58 (0.488) |

Use 9 | 7.7 (0.865) | 96.26 (2.354) | 4.56 (0.827) | 179.19 (24.228) | 176 (11.558) | 95.51 (0.584) |

Use 10 | 7.3 (0.952) | 97.28 (12.528) | 4.27 (0.774) | 167.63 (15.736) | 187 (13.825) | 96.46 (0.752) |

Algorithms | Evolutionary Generation | Fitness of Top-Nc | ${\mathit{D}}_{\mathbf{NBS}}$ | Runtime (s) | NOS | SR (%) |
---|---|---|---|---|---|---|

IGA vs. SAF-IEDA | + | + | + | + | + | + |

Kano-IGA vs. SAF-IEDA | + | + | + | + | + | + |

BP-IGA vs. SAF-IEDA | + | + | + | + | + | + |

RBF-IEDA vs. SAF-IEDA | + | + | + | + | + | + |

Independent Sample Test p-Value (Two-Sided) | Evolutionary Generation | Fitness of Top-Nc | ${\mathit{D}}_{\mathbf{NBS}}$ | Runtime (s) | NOS |
---|---|---|---|---|---|

SAF-IEDA | SAF-IEDA | SAF-IEDA | SAF-IEDA | SAF-IEDA | |

Kano-IGA | 0.018 | 0.023 | 0.028 | 0.017 | 0.001 |

BP-IGA | 0.037 | 0.033 | 0.020 | 0.003 | 0.012 |

RBF-IEDA | 0.028 | 0.032 | 0.042 | 0.035 | 0.014 |

IGA | 0.013 | 0.028 | 0.039 | 0.016 | 0.028 |

Levene test for equality of variances | {0.352, 0.637, 0.256, 382} | {0.385, 0.417, 0.228, 0.673} | {0.753, 0.525, 0.257, 0.702} | {0259, 0.450, 0.441, 0.335} | {0.204, 0.198, 0.657, 0.284} |

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**MDPI and ACS Style**

Qiao, Z.; Guo, G.; Zhang, Y.
An Interactive Estimation of the Distribution Algorithm Integrated with Surrogate-Assisted Fitness. *Symmetry* **2023**, *15*, 1852.
https://doi.org/10.3390/sym15101852

**AMA Style**

Qiao Z, Guo G, Zhang Y.
An Interactive Estimation of the Distribution Algorithm Integrated with Surrogate-Assisted Fitness. *Symmetry*. 2023; 15(10):1852.
https://doi.org/10.3390/sym15101852

**Chicago/Turabian Style**

Qiao, Zhanzhou, Guangsong Guo, and Yong Zhang.
2023. "An Interactive Estimation of the Distribution Algorithm Integrated with Surrogate-Assisted Fitness" *Symmetry* 15, no. 10: 1852.
https://doi.org/10.3390/sym15101852