The second alternative of the second optimization phase was based on the flows in the sub-optimal networks from the first phase. The minimum and maximum flow in each pipe was determined, and the minimum and maximum diameter for each pipe (or gene) was designated according to formulas (5) and (6). Based on this principle, the reduction of the search space from 10^{454} to 10^{138} was reached. For this alternative with ten optimization runs (with various settings of crossover and mutation), the best obtained result for the Balerma network was 1,927,758€ and the worst 1,940,799€. Although this alternative gives slightly worse results than the previous one, the authors also ran a variant of the second alternative, where all the networks obtained in the first stage were considered *individually*. The main idea of this computational experiment is that the presented methodology can also be applied with fewer optimization runs in the first phase, i.e., even only with one run, where the minimum and maximum flow is the same, and the minimal and maximal diameters are based only on the minimal and maximal velocities (0.3 and 2.5 m s^{−1}) and this flow. This reduction was determined with the help of the assumption of a significant closeness between the suboptimal and global flows. This alternative was run with each of the ten suboptimal solutions from the first phase individually with the best obtained result of 1,929,863€ and the worst of 1,942,019€. All the results are summarized in Table 1. In the works of the authors cited in the table, the number of necessary iterations needed to find the penalty parameter and other settings of the algorithm was not published, so the number of iterations is not evaluated in the table. When one is reporting only the number of iterations which are necessary for the final optimization, many iterations remain hidden, i.e., those which were necessary to find the parameters of the optimization solver, e.g., the penalty parameter. This tuning is not necessary in the work presented due to using a multi-objective approach, which is one of the greatest advantages of this approach. This means that although more computational runs are necessary because of the two-step character of the proposed methodology, this does not mean that the proposed methodology is more computationally demanding.

Table 1

. | Cost of the optimized network (€) . | . | . | |
---|---|---|---|---|

Algorithm . | Minimum/Maximum . | Average . | Number of evaluations . | Source . |

NSGA-II/First phase | 1,965,341/1,997,940 | 1,988,887 | 10,000,500 | This study |

NSGA-II/Second phase alt.1 | 1,921,400^{a}/1,925,082 | 1,923,399 | 10,640,000 | This study |

NSGA-II/Second phase alt.2a | 1,933,550/1,940,799 | 1,933,771 | 9,487,750 | This study |

NSGA-II/Second phase alt.2b | 1,929,863/1,942,019 | 1,935,059 | 9,750,536 | This study |

Harmony search | 2,018,000 | – | 10,000,000 | Geem (2006) |

HD-DDS | 1,940,923 | 2,165,000 | 30,000,000 | Tolson et al. (2009) |

NLP-DE | 1,923,000 | 1,927,000 | 1,427,850 | Zheng et al. (2011) |

PEDPSO | 1,921,428 | 1,942,231 | 217,400 | Xuewei et al. (2015) |

. | Cost of the optimized network (€) . | . | . | |
---|---|---|---|---|

Algorithm . | Minimum/Maximum . | Average . | Number of evaluations . | Source . |

NSGA-II/First phase | 1,965,341/1,997,940 | 1,988,887 | 10,000,500 | This study |

NSGA-II/Second phase alt.1 | 1,921,400^{a}/1,925,082 | 1,923,399 | 10,640,000 | This study |

NSGA-II/Second phase alt.2a | 1,933,550/1,940,799 | 1,933,771 | 9,487,750 | This study |

NSGA-II/Second phase alt.2b | 1,929,863/1,942,019 | 1,935,059 | 9,750,536 | This study |

Harmony search | 2,018,000 | – | 10,000,000 | Geem (2006) |

HD-DDS | 1,940,923 | 2,165,000 | 30,000,000 | Tolson et al. (2009) |

NLP-DE | 1,923,000 | 1,927,000 | 1,427,850 | Zheng et al. (2011) |

PEDPSO | 1,921,428 | 1,942,231 | 217,400 | Xuewei et al. (2015) |

^{a}Best result.

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