# An Enhanced Multi-Objective Particle Swarm Optimization in Water Distribution Systems Design

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## Abstract

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## 1. Introduction

## 2. Original MOPSO Algorithm

#### 2.1. PSO Concept

#### 2.2. Formulation of WDS Design Problem

#### 2.3. Constraints Handling

## 3. Performance Metrics

## 4. Modified MOPSO Algorithm

#### 4.1. Self-Adaptive PSO Parameters Strategy

#### 4.2. Selecting Repository Members Strategy

#### 4.3. Regeneration-On-Collision Strategy

#### 4.4. Adaptive Population Size Strategy

## 5. Local Search Application

## 6. Benchmark Problems

## 7. Results and Discussion

#### 7.1. A Comparison between the Proposed Strategies

#### 7.2. Results of the Different Proposed MOPSO Algorithms

- A reduced effort is needed for parameterization of PSO since the self-adaptive PSO parameters strategy nearly optimizes their values during the search process.
- Implementing the regeneration-on-collision strategy prevents the population from being trapped in local minima areas and this strategy is more evident in medium-sized networks.
- More leaping ability is provided to MMOPSO by depending on the regeneration-on-collision and the adaptive population size strategies. The random changes of some or all particles’ positions during the search process substantially help in discovering new solutions areas within the search space.
- The real dilemma of whether to use a relatively large or relatively small population size is alleviated by depending on the adaptive population size strategy. This strategy strengthens the global search at the beginning of the search and the local search at its end.
- The local search strengthens the local exploration ability of MMOPSO in medium sized problems since it investigates nearly all the local solutions around the gained non-dominated solutions of MMOPSO final PF, providing a good opportunity to increase the available design choices for decision makers upon the real implementation of networks.

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Pseudo-Code for OMOPSO Algorithm

## Appendix B. Pseudo-Code for MMOPSO Algorithm

## References

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**Figure 3.**Box and whiskers plot for ${C}_{m}$ and $NH{V}_{m}$ values of 15 runs for OMOPSO and MMOPSO in each network.

**Figure 4.**Average curves of runtime ${C}_{m}$ and $NH{V}_{m}$ for 15 runs using the OMOPSO and MMOPSO algorithms in BIN network.

Network | ${\mathit{N}}_{\mathit{R}}$ | ${\mathit{P}}_{\mathit{min}}$ | ${\mathit{P}}_{\mathit{max}}$ | ${\mathit{NCC}}_{\mathit{min}}$ | ${\mathit{NCC}}_{\mathit{max}}$ | ${\mathit{N}}_{\mathit{av}}$ | Population Sizes | TNFE | Search Space | ||
---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{nPop}}_{\mathit{min}}$ | ${\mathit{nPop}}_{\mathit{int}}$ | ${\mathit{nPop}}_{\mathit{max}}$ | |||||||||

HAN | 1 | 30 | 100 | 1.80 | 10.97 | 6 | 60 | 120 | 240 | ≈2.02 × 10${}^{6}$ | 2.87×10${}^{26}$ |

BIN | 4 | 20 | 127 | 0.72 | 21.64 | 10 | 100 | 400 | 800 | ≈9.34 × 10${}^{6}$ | 1.00×10${}^{445}$ |

**Table 2.**Average values of ${C}_{m}$, $NH{V}_{m}$, and ${N}_{nds}$ for each strategy/algorithm (relative improvements with respect to OMOPSO are in brackets).

Strategy/Algorithm | ${\mathit{C}}_{\mathit{m}}$ | ${\mathit{NHV}}_{\mathit{m}}$ | ${\mathit{N}}_{\mathit{nds}}$ |
---|---|---|---|

OMOPSO | 0.0821 | 0.8342 | 101 |

Self-adaptive PSO Parameters | 0.0394 (52.01%) | 0.8900 (6.69%) | 93 (−7.92%) |

Selecting Repository Members | 0.0589 (28.26%) | 0.8860 (6.21%) | 122 (20.79%) |

Regeneration-on-collision | 0.0507 (38.25%) | 0.8737 (4.74%) | 113 (11.88%) |

Adaptive Population Size | 0.0270 (67.11%) | 0.9271 (11.14%) | 165 (63.37%) |

MMOPSO | 0.0259 (68.45%) | 0.9447 (13.25%) | 164 (62.38%) |

**Table 3.**Values of ${C}_{m}$, $NH{V}_{m}$, and ${N}_{nds}$ for the final PFs in HAN (relative improvements in ${C}_{m}$ and $NH{V}_{m}$ with respect to the main MOPSO search stage of OMOPSO are in brackets.)

Stage | OMOPSO | OMOPSO^{+} | MMOPSO | MMOPSO^{+} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{C}}_{\mathit{m}}$ | ${\mathit{NHV}}_{\mathit{m}}$ | ${\mathit{N}}_{\mathit{nds}}$ | ${\mathit{C}}_{\mathit{m}}$ | ${\mathit{NHV}}_{\mathit{m}}$ | ${\mathit{N}}_{\mathit{nds}}$ | ${\mathit{C}}_{\mathit{m}}$ | ${\mathit{NHV}}_{\mathit{m}}$ | ${\mathit{N}}_{\mathit{nds}}$ | ${\mathit{C}}_{\mathit{m}}$ | ${\mathit{NHV}}_{\mathit{m}}$ | ${\mathit{N}}_{\mathit{nds}}$ | |

Main MOPSO search | 0.0177 | 0.9486 | 166 | 0.0177 | 0.9486 | 166 | 0.0132 (25.42%) | 0.9727 (2.54%) | 242 | 0.0132 (25.42%) | 0.9727 (2.54%) | 242 |

Main local Search | N/A | N/A | N/A | 0.0061 (65.54%) | 0.9718 (2.45%) | 234 | N/A | N/A | N/A | 0.0054 (69.49%) | 0.9894 (4.30%) | 298 |

Secondary MOPSO search | 0.0066 (62.71%) | 0.9860 (3.94%) | 193 | 0.0051 (71.19%) | 0.9840 (3.73%) | 276 | 0.0122 (31.07%) | 0.9730 (2.57%) | 234 | 0.0053 (70.06%) | 0.9896 (4.32%) | 319 |

Secondary local search | N/A | N/A | N/A | 0.0039 (77.97%) | 0.9903 (4.40%) | 312 | N/A | N/A | N/A | 0.0041 (76.84%) | 0.9917 (4.54%) | 363 |

**Table 4.**Values of ${C}_{m}$, $NH{V}_{m}$, and ${N}_{nds}$ for the final PFs in BIN (relative improvements in ${C}_{m}$ and $NH{V}_{m}$ with respect to the main MOPSO search stage of OMOPSO are in brackets.)

Stage | OMOPSO | OMOPSO^{+} | MMOPSO | MMOPSO^{+} | ||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

${\mathit{C}}_{\mathit{m}}$ | ${\mathit{NHV}}_{\mathit{m}}$ | ${\mathit{N}}_{\mathit{nds}}$ | ${\mathit{C}}_{\mathit{m}}$ | ${\mathit{NHV}}_{\mathit{m}}$ | ${\mathit{N}}_{\mathit{nds}}$ | ${\mathit{C}}_{\mathit{m}}$ | ${\mathit{NHV}}_{\mathit{m}}$ | ${\mathit{N}}_{\mathit{nds}}$ | ${\mathit{C}}_{\mathit{m}}$ | ${\mathit{NHV}}_{\mathit{m}}$ | ${\mathit{N}}_{\mathit{nds}}$ | |

Main MOPSO search | 0.1239 | 0.8389 | 151 | 0.1239 | 0.8389 | 151 | 0.0781 (36.97%) | 0.8931 (6.46%) | 289 | 0.0781 (36.97%) | 0.8931 (6.46%) | 289 |

Main local Search | N/A | N/A | N/A | 0.1218 (1.69%) | 0.8465 (0.91%) | 808 | N/A | N/A | N/A | 0.0782 (36.88%) | 0.8985 (7.10%) | 2655 |

Secondary MOPSO search | 0.0966 (22.03%) | 0.8783 (4.70%) | 149 | 0.0995 (19.69%) | 0.8900 (6.09%) | 365 | 0.0483 (61.02%) | 0.9321 (11.11%) | 399 | 0.0705 (43.10%) | 0.9179 (9.42%) | 911 |

Secondary Local search | N/A | N/A | N/A | 0.0956 (22.84%) | 0.8931 (6.46%) | 741 | N/A | N/A | N/A | 0.0695 (43.91%) | 0.9193 (9.58%) | 5455 |

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**MDPI and ACS Style**

Torkomany, M.R.; Hassan, H.S.; Shoukry, A.; Abdelrazek, A.M.; Elkholy, M. An Enhanced Multi-Objective Particle Swarm Optimization in Water Distribution Systems Design. *Water* **2021**, *13*, 1334.
https://doi.org/10.3390/w13101334

**AMA Style**

Torkomany MR, Hassan HS, Shoukry A, Abdelrazek AM, Elkholy M. An Enhanced Multi-Objective Particle Swarm Optimization in Water Distribution Systems Design. *Water*. 2021; 13(10):1334.
https://doi.org/10.3390/w13101334

**Chicago/Turabian Style**

Torkomany, Mohamed R., Hassan Shokry Hassan, Amin Shoukry, Ahmed M. Abdelrazek, and Mohamed Elkholy. 2021. "An Enhanced Multi-Objective Particle Swarm Optimization in Water Distribution Systems Design" *Water* 13, no. 10: 1334.
https://doi.org/10.3390/w13101334