# Generation of Benchmark Problems for Optimal Design of Water Distribution Systems

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## Abstract

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## 1. Introduction

## 2. Methodology

_{c}(D

_{i}) is the construction cost according to pipe diameter per unit length, L

_{i}is the pipe length, D

_{i}is the pipe diameter, P

_{j}is the penalty function for ensuring that the pressure constraints are satisfied, N is the number of pipes, and M is the number of nodes. If a design solution does not meet the nodal pressure requirements, a penalty function is added to the objective function as given follows [24]:

_{j}is the nodal pressure at node j, h

_{min}is the minimum pressure requirement at node j, and α and β are penalty function constants. Note that other hydraulic or water quality requirements, such as allowable flow velocity, water age, and residual chlorine concentration, can also be considered in water distribution system design [25].

## 3. Applications and Results

^{34}. The Hazen–Williams roughness coefficient for calculating friction head loss is assumed to be 130 for all pipes. In addition, the minimum required pressure head at each demand node is 30 m for the Hanoi network. Default problem characteristic factors are printed in bold given in Table 4. Four values were considered for each factor and 20 benchmark problems were generated in this study.

## 4. Conclusions and Future Research

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 3.**Performance comparison results (mean of average improvement ratio in percentage). (

**a**) Genetic algorithms (GAs); (

**b**) simulated annealing (SA); (

**c**) harmony search algorithm (HSA); and (

**d**) water cycle algorithm (WCA).

**Figure 4.**Performance comparison results (standard deviation of average improvement ratio). (

**a**) GAs; (

**b**) SA; (

**c**) HSA; and (

**d**) WCA.

**Figure 5.**Performance comparison results (mean of best improvement ratio in percentage). (

**a**) GAs; (

**b**) SA; (

**c**) HSA; and (

**d**) WCA.

**Figure 6.**Performance comparison results (standard deviation of best improvement ratio). (

**a**) GAs; (

**b**) SA; (

**c**) HSA; and (

**d**) WCA.

Factor | Definition |
---|---|

n | Number of pipes |

m | Number of candidate pipe diameter options |

p | Pressure constraint |

c | Roughness coefficient |

d | Nodal demand multiplier |

**Table 2.**Factors applied in engineering benchmarks modified from the two-loop network design problem.

Figure | Values Used |
---|---|

n | [8, 16, 24] |

m | [5, 6, 7] |

p | [30, 35, 40] |

C | [130, 100, 70] |

D | [1.0, 1.5, 2.0] |

Modified factor | n | ||

Used value | 8 | 16 | 24 |

Global optimum ($) | 882,000 | 1,764,000 | 2,646,000 |

No. of candidate designs | 390,625 | 1.53E + 11 | 5.96E + 16 |

No. of feasible designs | 9956 | 9.91E + 07 | 9.87E + 11 |

Ratio of feasible designs (%) | 2.5487 | 0.0650 | 0.0017 |

Modified factor | m | ||

Used value | 5 | 6 | 7 |

Global optimum ($) | 882,000 | 841,000 | 474,000 |

No. of candidate designs | 390,625 | 1,679,616 | 5,764,801 |

No. of feasible designs | 9956 | 38,721 | 389,145 |

Ratio of feasible designs (%) | 2.5487 | 2.3053 | 6.7504 |

Modified factor | p | ||

Used value | 30 | 35 | 40 |

Global optimum ($) | 882,000 | 949,000 | 1,354,000 |

No. of candidate designs | 390,625 | 390,625 | 390,625 |

No. of feasible designs | 9956 | 4725 | 2294 |

Ratio of feasible designs (%) | 2.5487 | 1.2096 | 0.5873 |

Modified factor | c | ||

Used value | 130 | 100 | 70 |

Global optimum ($) | 882,000 | 961,000 | 1,403,000 |

No. of candidate designs | 390,625 | 390,625 | 390,625 |

No. of feasible designs | 9956 | 4101 | 1895 |

Ratio of feasible designs (%) | 2.5487 | 1.0499 | 0.4851 |

Modified factor | d | ||

Used value | 1.0 | 1.5 | 2.0 |

Global optimum ($) | 882,000 | 1,010,000 | 1,476,000 |

No. of candidate designs | 390,625 | 390,625 | 390,625 |

No. of feasible designs | 9956 | 3346 | 1326 |

Ratio of feasible designs (%) | 2.5487 | 0.8566 | 0.3395 |

Factor | Values Used |
---|---|

n | [34, 68, 102, 136] |

m | [6, 8, 10, 12] |

p | [30, 35, 40, 45] |

c | [130, 125, 120, 115] |

d | [1.0, 1.05, 1.1, 1.15] |

Factors | Success Rate (%) | ||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

RS | GAs | SA | HSA | WCA | |||||||||||

10,000 FEs* | 30,000 FEs | 50,000 FEs | 10,000 FEs | 30,000 FEs | 50,000 FEs | 10,000 FEs | 30,000 FEs | 50,000 FEs | 10,000 FEs | 30,000 FEs | 50,000 FEs | 10,000 FEs | 30,000 FEs | 50,000 FEs | |

n | 0 | 0 | 0 | 23.75 | 68.75 | 100 | 6.25 | 8.75 | 15 | 100 | 100 | 100 | 100 | 100 | 100 |

m | 0 | 0 | 0 | 55 | 100 | 100 | 15 | 17.5 | 27.5 | 100 | 100 | 100 | 100 | 100 | 100 |

p | 0 | 0 | 0 | 63.75 | 100 | 100 | 7.5 | 10 | 16.25 | 100 | 100 | 100 | 100 | 100 | 100 |

c | 0 | 0 | 0 | 46.25 | 100 | 100 | 10 | 12.5 | 18.75 | 100 | 100 | 100 | 100 | 100 | 100 |

d | 0 | 0 | 0 | 51.25 | 98.75 | 100 | 6.25 | 8.75 | 15 | 100 | 100 | 100 | 100 | 100 | 100 |

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**MDPI and ACS Style**

Lee, H.M.; Jung, D.; Sadollah, A.; Yoo, D.G.; Kim, J.H. Generation of Benchmark Problems for Optimal Design of Water Distribution Systems. *Water* **2019**, *11*, 1637.
https://doi.org/10.3390/w11081637

**AMA Style**

Lee HM, Jung D, Sadollah A, Yoo DG, Kim JH. Generation of Benchmark Problems for Optimal Design of Water Distribution Systems. *Water*. 2019; 11(8):1637.
https://doi.org/10.3390/w11081637

**Chicago/Turabian Style**

Lee, Ho Min, Donghwi Jung, Ali Sadollah, Do Guen Yoo, and Joong Hoon Kim. 2019. "Generation of Benchmark Problems for Optimal Design of Water Distribution Systems" *Water* 11, no. 8: 1637.
https://doi.org/10.3390/w11081637