# Improving Multi-Objective Optimization Methods of Water Distribution Networks

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

^{2}), where M is the number of objective functions and N is the population size. The fast non-dominated sorting method used in NSGA-II reduces computational complexity, making it efficient and powerful in exploring the decision space of MOPs. NSGA-II has been widely implemented and applied to various MOPs, demonstrating its effectiveness and reliability.

## 2. Materials and Methods

#### 2.1. Problem Definition of the Multi-Objective Problems

_{j}, Q

_{j}, H

_{j}, and H

_{j}

^{req}= uniformity, demand, actual head, and minimum head of node j; nr = number of reservoirs; Q

_{k}and H

_{k}= discharge and actual head of reservoir k; npu = number of pumps; P

_{i}= power of pump i; γ = specific weight of water; np

_{j}= number of pipes connected to node j; D

_{i}= diameter of pipe i connected to demand node j.

#### 2.2. Improved NSGA II

#### 2.3. Generation Methods

**G1 method**: This is the original method used in NSGA II. It involves selecting N (population size) number of parent populations and N offspring populations, evaluating their objective values, and sorting them based on non-dominated sorting and crowd distancing. The parent populations are then randomly paired to create child populations through crossover and mutation.**G2 method**: This method saves the new offspring population generated in each iteration to the archive. The archive contains all the populations generated from the start of the iteration. Parent populations are randomly selected from the archive to create new offspring populations through crossover and mutation.**G3 method**: This method focuses on specific areas of the Pareto front, such as the extreme and uncrowded areas shown in Figure 1. At each iteration, a number of points are selected from the required region, and N (population number) offspring populations are generated from these points.

- 4.
**G4 method**: This method generates offspring populations using the knee area of the Pareto front as the parent population. The knee is found by calculating the Euclidean distance of all Pareto front points to the corner, shown in Figure 1 and selecting points with the least distance. After selecting the parent population, the offspring population is generated using Equation (5).

#### 2.4. Pseudocode

#### 2.5. Parameters

- ITm1—iteration number where the G1 starts, which is equal to zero.
- ITm2—iteration number where the G2 starts.
- ITm3—iteration number where the G3 starts.
- ITm4—iteration number where the G4 starts.

- P2—probability of G2 being selected starting ITm2.
- P3—probability of G3 being selected starting ITm3.
- P4—probability of G4 being selected starting ITm4.

- sp3max—number of selected points in the maximum region as percentage of N.
- sp3min—number of selected points in the minimum region as percentage of N.
- sp3uc—number of selected points in the uncrowded region as percentage of N.
- sp4—number of selected points in the knee area as percentage of N.

#### 2.6. Case Studies

## 3. Results

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

- Filion, Y.R.; MacLean, H.L.; Karney, B.W. Life-Cycle Energy Analysis of a Water Distribution System. J. Infrastruct. Syst.
**2004**, 10, 120–130. [Google Scholar] [CrossRef] [Green Version] - Kleiner, Y.; Adams, B.J.; Rogers, J.S. Water Distribution Network Renewal Planning. J. Comput. Civ. Eng.
**2001**, 15, 15–26. [Google Scholar] [CrossRef] [Green Version] - Cunha, M.D.C.; Sousa, J. Water Distribution Network Design Optimization: Simulated Annealing Approach. J. Water Resour. Plan. Manag.
**1999**, 125, 215–221. [Google Scholar] - Zecchin, A.C.; Simpson, A.R.; Maier, H.R.; Leonard, M.; Roberts, A.J.; Berrisford, M.J. Application of two ant colony optimisation algorithms to water distribution system optimisation. Math. Comput. Model.
**2006**, 44, 451–468. [Google Scholar] [CrossRef] [Green Version] - Bi, W.; Chen, M.; Shen, S.; Huang, Z.; Chen, J. A Many-Objective Analysis Framework for Large Real-World Water Distribution System Design Problems. Water
**2022**, 14, 557. [Google Scholar] - Wu, W.; Maier, H.R.; Simpson, A.R. Multiobjective optimization of water distribution systems accounting for economic cost, hydraulic reliability, and greenhouse gas emissions. Water Resour. Res.
**2013**, 49, 1211–1225. [Google Scholar] [CrossRef] [Green Version] - Awe, O.M.; Okolie, S.T.A.; Fayomi, O.S.I. Optimization of Water Distribution Systems: A Review. J. Phys. Conf. Ser.
**2019**, 1378, 022068. [Google Scholar] [CrossRef] - Farmani, R.; Savic, D.A.; Walters, G.A. Evolutionary multi-objective optimization in water distribution network design. Eng. Optim.
**2005**, 37, 167–183. [Google Scholar] [CrossRef] - Prasad, T.D.; Park, N.-S. Multiobjective genetic algorithms for design of water distribution networks. J. Water Resour. Plan. Manag.
**2004**, 130, 73–82. [Google Scholar] - Prasad, T.D.; Hong, S.-H.; Park, N. Reliability based design of water distribution networks using multi-objective genetic algorithms. KSCE J. Civ. Eng.
**2003**, 7, 351–361. [Google Scholar] [CrossRef] - Saldarriaga, J.; Takahashi, S.; Hernández, F.; Escovar, M. Multi-objective water distribution system design using an expert algorithm. In Proceedings of the Urban Water Management Challenges Oppurtunities—11th International Conference on Computing and Control for the Water Industry, CCWI 2011, Exeter, UK, 5–7 September 2011; Volume 3. [Google Scholar]
- Todini, E. Looped water distribution networks design using a resilience index based heuristic approach. Urban Water
**2000**, 2, 115–122. [Google Scholar] [CrossRef] - Prasad, T.D.; Tanyimboh, T.T. Entry Bassed Design of “Anytown” Water Distribution Network; Kruger: Montreal, QC, Canada, 2008. [Google Scholar]
- Maier, H.R.; Kapelan, Z.; Kasprzyk, J.; Kollat, J.; Matott, L.S.; Cunha, M.C.; Dandy, G.C.; Gibbs, M.S.; Keedwell, E.; Marchi, A.; et al. Evolutionary algorithms and other metaheuristics in water resources: Current status, research challenges and future directions. Environ. Model. Softw.
**2014**, 62, 271–299. [Google Scholar] [CrossRef] [Green Version] - Nicklow, J.; Asce, F.; Reed, P.; Asce, M.; Savic, D.; Dessalegne, T.; Harrell, L.; Chan-Hilton, A.; Karamouz, M.; Minsker, B.; et al. State of the Art for Genetic Algorithms and Beyond in Water Resources Planning and Management ASCE Task Committee on Evolutionary Computation in Environmental and Water Resources Engineering. J. Water Resour. Plan. Manag.
**2010**, 136, 412–432. [Google Scholar] - Zhou, A.; Qu, B.Y.; Li, H.; Zhao, S.Z.; Suganthan, P.N.; Zhang, Q. Multiobjective evolutionary algorithms: A survey of the state of the art. Swarm Evol. Comput.
**2011**, 1, 32–49. [Google Scholar] [CrossRef] - Wang, M.; Dai, G.; Hu, H. Improved NSGA-II algorithm for Optimization of Constrained Functions. In Proceedings of the 2010 International Conference on Machine Vision and Human-Machine Interface, Kaifeng, China, 24–25 April 2010; pp. 673–675. [Google Scholar] [CrossRef]
- D’Souza, R.G.L.; Sekaran, K.C.; Kandasamy, A. Improved NSGA-II Based on a Novel Ranking Scheme. J. Comput. Civ. Eng.
**2010**, 2, 91–95. [Google Scholar] - Verma, S.; Pant, M.; Snasel, V. A Comprehensive Review on NSGA-II for Multi-Objective Combinatorial Optimization Problems. IEEE Access
**2021**, 9, 57757–57791. [Google Scholar] [CrossRef] - Liu, T.; Gao, X.; Wang, L. Multi-objective optimization method using an improved NSGA-II algorithm for oil-gas production process. J. Taiwan Inst. Chem. Eng.
**2015**, 57, 42–53. [Google Scholar] [CrossRef] - Wang, Q.; Wang, L.; Huang, W.; Wang, Z.; Liu, S.; Savić, D.A. Parameterization of NSGA-II for the optimal design of water distribution systems. Water
**2019**, 11, 971. [Google Scholar] [CrossRef] [Green Version] - Deb, K.; Pratap, A.; Agarwal, S.; Meyarivan, T. A fast and elitist multiobjective genetic algorithm: NSGA-II. IEEE Trans. Evol. Comput.
**2002**, 6, 182–197. [Google Scholar] [CrossRef] [Green Version] - Kapelan, Z.S.; Savic, D.A.; Walters, G.A. Multiobjective design of water distribution systems under uncertainty. Water Resour. Res.
**2005**, 41, 1–15. [Google Scholar] [CrossRef] - Deb, K.; Jain, H. An evolutionary many-objective optimization algorithm using reference-point-based nondominated sorting approach, Part I: Solving problems with box constraints. IEEE Trans. Evol. Comput.
**2014**, 18, 577–601. [Google Scholar] [CrossRef] - Mohamad Shirajuddin, T.; Muhammad, N.S.; Abdullah, J. Optimization problems in water distribution systems using Non-dominated Sorting Genetic Algorithm II: An overview. Ain Shams Eng. J.
**2022**, 14, 101932. [Google Scholar] [CrossRef] - Wang, Q.; Guidolin, M.; Dragan, S.; Zoran, K. Two-Objective Design of Benchmark Problems of a Water Distribution System via MOEAs: Towards the Best-Known Approximation of the True Pareto Front. J. Water Resour. Plan. Manag.
**2015**, 141, 04014060. [Google Scholar] [CrossRef] [Green Version] - Cunha, M.; Marques, J. A New Multiobjective Simulated Annealing Algorithm—MOSA-GR: Application to the Optimal Design of Water Distribution Networks. Water Resour. Res.
**2020**, 56, e2019WR025852. [Google Scholar] [CrossRef] - Bin Mahmoud, A.A.; Piratla, K.R. Comparative evaluation of resilience metrics for water distribution systems using a pressure driven demand-based reliability approach. J. Water Supply Res. Technol.
**2018**, 67, 517–530. [Google Scholar] [CrossRef] - Zhan, X.; Meng, F.; Liu, S.; Fu, G. Comparing Performance Indicators for Assessing and Building Resilient Water Distribution Systems. J. Water Resour. Plan. Manag.
**2020**, 146, 06020012. [Google Scholar] [CrossRef] - Jayaram, N.; Srinivasan, K. Performance-based optimal design and rehabilitation of water distribution networks using life cycle costing. Water Resour. Res.
**2008**, 44, 1417. [Google Scholar] [CrossRef]

**Figure 6.**Pareto front results of Improved NSGA II and BKPF of: (

**a**) TLN; (

**b**) HAN; (

**c**) GOY; (

**d**) FOS; (

**e**) PES.

**Figure 7.**Distribution of non-dominated solutions from each MOEA in the BKPF of PES network [26].

**Table 1.**Benchmark network characteristics, WS = water sources, DV = decision variables, PD = pipe diameter options, NFE = number of function evaluation.

Problem | Name | WS | DV | PD | Search Space | NFE |
---|---|---|---|---|---|---|

Two-Loop Network | TLN | 1 | 8 | 14 | 1.48 × 10^{9} | 100,000 |

Hanoi Network | HAN | 1 | 34 | 6 | 2.87 × 10^{26} | 600,000 |

GoYang Network | GOY | 1 | 30 | 8 | 1.24 × 10^{27} | 600,000 |

Fossolo Network | FOS | 1 | 58 | 22 | 7.25 × 10^{77} | 1,000,000 |

Pescara Network | PES | 3 | 99 | 13 | 1.91 × 10^{110} | 1,000,000 |

**Table 2.**Results of the improved NSGA II and the five MOEAS in [26] CS = case study; CWS = center for water systems; BKPF = best known pareto front; AEA = all the evolutionary algorithms; TNDS = total non-dominated solutions from improved NSGA II; ENDS = equal to BKPF; DS = dominated by BKPF solutions; NDS = solutions that are non-dominated to BKPF; BNDS = solutions that dominated by BKPF, TF = True Front.

CS | CWS BKPF | Wang et al. [26] Results | Improved NSGA II-Results | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|

NSGA-II | ɛ-MOEA | ɛ-NSGA-II | AMAGAM | Borg | AEA | TNDS | ENDS | DS | NDS | BNDS | ||

TLN | 114 ^{TF} | 77 | 64 | 64 | 76 | 65 | 77 | 114 | 114 | 0 | 0 | 0 |

HAN | 575 | 39 | 8 | 9 | 35 | 10 | 39 | 716 | 10 | 115 | 664 | 40 |

GOY | 489 | 29 | 2 | 39 | 57 | 15 | 67 | 341 | 303 | 97 | 27 | 11 |

FOS | 474 | 48 | 10 | 42 | 21 | 31 | 140 | 532 | 3 | 58 | 39 | 490 |

PES | 782 | 82 | 58 | 24 | 41 | 49 | 215 | 247 | 1 | 437 | 100 | 146 |

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**MDPI and ACS Style**

Kidanu, R.A.; Cunha, M.; Salomons, E.; Ostfeld, A.
Improving Multi-Objective Optimization Methods of Water Distribution Networks. *Water* **2023**, *15*, 2561.
https://doi.org/10.3390/w15142561

**AMA Style**

Kidanu RA, Cunha M, Salomons E, Ostfeld A.
Improving Multi-Objective Optimization Methods of Water Distribution Networks. *Water*. 2023; 15(14):2561.
https://doi.org/10.3390/w15142561

**Chicago/Turabian Style**

Kidanu, Rahel Amare, Maria Cunha, Elad Salomons, and Avi Ostfeld.
2023. "Improving Multi-Objective Optimization Methods of Water Distribution Networks" *Water* 15, no. 14: 2561.
https://doi.org/10.3390/w15142561