# Comparison of Searching Behaviour of Three Evolutionary Algorithms Applied to Water Distribution System Design Optimization

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Evolutionary Algorithms and Their Parameterization

#### 2.2. Case Studies

## 3. Searching Measure Metrics

#### 3.1. Search Quality Measure Metrics

#### 3.2. Convergence Measure Metrics

## 4. Case Study Results and Discussion

#### 4.1. Search Quality Results

#### 4.2. Convergence Measure Results

#### 4.3. Solution Comparisons

## 5. Conclusions

- The three EAs have exhibited significantly different searching behaviours in both the solution space and decision space. Such differences can be related to their underlying model structures and the searching mechanisms. This demonstrates that the proposed run-time metrics are effective in revealing the searching properties of the EAs.
- From the obtained real-time metric results, it can be concluded that the DE has an overall best ability to locate feasible solutions as well as high quality solutions for the WDS design problems. The ACO performed overall the worst in identifying the optimal solutions as observed in this study. This provides a guideline for the selection of the algorithms for WDS design optimization.
- If the computational budget is rather limited, the GA can identify better solutions than the DE as shown in this study. This suggests that the GA is promising for the case when the optimization solution needs to be identified in a short time such as real-time WDS operation and management.

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Solution quality metric ${f}_{best}(G)$ of the three different types of EAs applied to the five case studies.

**Figure 3.**Solution quality metric ${f}_{avg}(G)$ of the three different types of EAs applied to the five case studies.

**Figure 4.**Solution quality metric $PF\%$ of the three different types of EAs applied to the five case studies.

**Figure 5.**Solution quality metric ${D}_{mean}(G)$ of the three different types of EAs applied to the five case studies.

**Figure 6.**Solution quality metric $PB\%$ of the three different types of EAs applied to the five case studies.

**Table 1.**Details of the parameterization strategies of the EAs applied to the five case studies. Parameters are as follows: where $N$ is the population size, ${P}_{c}$ and $CR$ are the crossover probabilities for the GA and DE respectively, ${P}_{m}$ and $F$ are their mutation probabilities; $\alpha $ = pheromone weighting factor; $\beta $ = visibility weighting factor; $\rho $ = pheromone decay factor; Q = pheromone update coefficient; N = number of ants; $\sigma $ = number of elitist ants; Maximum number of allowed generations for each algorithm is 10,000.

The Algorithm Name | Parameters of Each Algorithm | Parameter Values for Five Different Case Studies | ||||
---|---|---|---|---|---|---|

Hanoi (HP) | Extend Hanoi (EHP) | ZJ | Balerma (BN) | Rural Network (RN) | ||

ACO | $N$ | 100 | 100 | 500 | 500 | 500 |

$Q$ | $1.1\times {10}^{7}$ | $1.1\times {10}^{7}$ | $5.3\times {10}^{7}$ | $2.2\times {10}^{7}$ | $5.3\times {10}^{8}$ | |

$\alpha $ | 1 | 1 | 1 | 1 | 1 | |

$\beta $ | 0.98 | 0.98 | 0.98 | 0.98 | 0.98 | |

$\sigma $ | 5 | 5 | 5 | 5 | 5 | |

DE | $N$ | 100 | 100 | 500 | 1000 | 1000 |

$F$ | 0.5 | 0.5 | 0.3 | 0.3 | 0.3 | |

${C}_{R}$ | 0.5 | 0.5 | 0.5 | 0.5 | 0.5 | |

GA | $N$ | 100 | 100 | 500 | 1000 | 1000 |

${P}_{c}$ | 0.9 | 0.9 | 0.9 | 0.9 | 0.9 | |

${P}_{m}$ | 0.02 | 0.02 | 0.06 | 0.02 | 0.02 |

Case Study | No. of Decision Variables | No. of Diameter Options | Size of Total Search Space | Pressure Head Constraint | Reference |
---|---|---|---|---|---|

Hanoi (HP) | 34 | 6 | $2.86\times {10}^{26}$ | $\ge 30\text{}\mathrm{m}$ | [8] |

Extend Hanoi (EHP) | 34 | 10 | $1\times {10}^{34}$ | $\ge 30\text{}\mathrm{m}$ | [34] |

ZJ | 164 | 14 | $9.23\times {10}^{187}$ | $\ge 22\text{}\mathrm{m}$ | [36] |

Balerma (BN) | 454 | 10 | $1\times {10}^{454}$ | $\ge 20\text{}\mathrm{m}$ | [17] |

Rural network (RN) | 476 | 15 | $6.58\times {10}^{559}$ | $\ge 0\text{}\mathrm{m}$ | [37] |

**Table 3.**Solution statistics of the three algorithms, where the percentage indicates the difference between the best solutions found by the three algorithms in this study and the current best known solutions in literature.

Case Studies | Current Best Known Solution | GA | DE | ACO | ||||||
---|---|---|---|---|---|---|---|---|---|---|

Best Solution | Worst Solution | Average Cost | Best Solution | Worst Solution | Average Cost | Best Solution | Worst Solution | Average Cost | ||

Hanoi (HP) ($M) | 6.081 [17] | 6.174 (1.5%) | 6.560 | 6.395 | 6.081 (0%) | 6.329 | 6.155 | 7.198 (18.4%) | 7.761 | 7.440 |

Extend Hanoi (EHP) ($M) | 5.346 [34] | 5.357 (0.2%) | 5.616 | 5.460 | 5.338 (−0.2%) | 5.364 | 5.341 | 5.432 (1.6%) | 5.718 | 5.582 |

ZJ ($M) | 7.082 [36] | 7.248 (2.3%) | 7.509 | 7.345 | 7.125 (0.6%) | 7.176 | 7.140 | 7.579 (7.0%) | 8.0362 | 7.787 |

Balerma (BN) (€M) | 1.923 [36] | 2.045 (6.3%) | 2.152 | 2.103 | 2.002 (4.1%) | 2.075 | 2.040 | 2.332 (21.3%) | 2.846 | 2.618 |

Rural Network (RN) ($M) | 31.220 [37] | 33.010 (5.7%) | 35.821 | 34.332 | 30.989 (−0.7%) | 31.776 | 31.396 | 36.737 (17.7%) | 39.963 | 38.332 |

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**MDPI and ACS Style**

Bi, W.; Xu, Y.; Wang, H. Comparison of Searching Behaviour of Three Evolutionary Algorithms Applied to Water Distribution System Design Optimization. *Water* **2020**, *12*, 695.
https://doi.org/10.3390/w12030695

**AMA Style**

Bi W, Xu Y, Wang H. Comparison of Searching Behaviour of Three Evolutionary Algorithms Applied to Water Distribution System Design Optimization. *Water*. 2020; 12(3):695.
https://doi.org/10.3390/w12030695

**Chicago/Turabian Style**

Bi, Weiwei, Yihui Xu, and Hongyu Wang. 2020. "Comparison of Searching Behaviour of Three Evolutionary Algorithms Applied to Water Distribution System Design Optimization" *Water* 12, no. 3: 695.
https://doi.org/10.3390/w12030695