# Ranking Approach to Scheduling Repairs of a Water Distribution System for the Post-Disaster Response and Restoration Service

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## Abstract

**:**

## 1. Introduction

- C_01—length of time hospitals and firefighting flows were without supply,
- C_02—rapidity of recovery (length of time until the system’s functionalities were permanently recovered by 95%),
- C_03—resilience loss (resilience in percent of water lost over seven days that followed the event),
- C_04—average length that time nodes were without service, in minutes,
- C_05—number of nodes that awaited servicing for more than 8 consecutive hours,
- C_06—volume of water lost over seven days that followed the event.

## 2. Methodology

#### 2.1. Survey Results

- There were 8 breakdowns,
- There were 2 repair crews available.

#### 2.2. Developing the Ranking

^{3}of water. However, since pressure-deficient conditions were met in the network during most of the restoration time, the demand could not be fully supplied. To model partial supply, a pressure-driven analysis (PDA) must be used.

_{Di}), and the pressure head required (p

_{req}) to fulfill that demand in node i (p

_{req}= 20 m). Results of each of five scenarios were evaluated using provided EPANET2 [24] models. In order to solve the PDA model of a WDN with regards to the flow–pressure relationship, Wagner’s power relationship (1) was used.

- Rankings had to be developed on the provided water distribution network model that had been built under normal working conditions (BPM-EPS).
- Rankings had to be created as a part of a strategic document (i.e., the water safety plan). The prioritization scheme had to address all types of possible accidents (earthquakes, floods, wars, etc.) that could cause numerous breaks in water networks. All elements of water distribution networks had to be taken into account (water intake, water treatment stations, and water distribution networks).
- Development of the ranking had to be ‘as easy as only possible’. The key issue water utilities face, especially those from Central and Eastern Europe, is a lack of detailed data related to their networks. Only selected utilities possess reliable data on their networks and also calibrated models. Thus, the root cause behind the development of any ranking is that, in case of any damage, the strategy for network operators to follow had to be clear and intuitive. Events as extreme as earthquakes give no time to compute various scenarios, search for lacking data, research for operational hydrants, and so on. In our opinion, such events required ‘ready to do’ lists set up beforehand that covered the most critical pipes that, in turn, determined the sequence of next actions to be performed.
- During development of the rankings it was possible to include the application of compensational tanks, yet only under the condition that the volume of water collected for emergency purposes had been clearly defined.

#### 2.3. Calculation Methods

#### 2.4. Evaluation of the Ranking Solutions

- Step 1—designation of preference function values for all variants and all criteria. This function needs to be separately defined for each criterion, and each needs to receive a value from the range between 0 and 1. The smaller the function the greater the indifference of the decisionmaker; the closer to 1 the stronger their preference.
- Step 2—designation of equivalence thresholds for all variants and all criteria.
- Step 3—designation of preference thresholds for all variants, individually for each criterion.
- Step 4—determination of the multicriteria preference index and, finally, the matrix of indicators, based on the preference degree:$$\pi \left(a,b\right)=\frac{1}{k}{\displaystyle \sum}_{h=1}^{k}{P}_{h}\left(a,b\right),$$
_{h}(a, b) expresses the preference function between a and b alternatives. - Step 5—settlement of the ranking of variants based on net dominance flows, defined as:$${\mathsf{\Phi}}^{+}\left(a\right)={\displaystyle \sum}_{x\u03f5K}\pi \left(a,x\right),$$$${\mathsf{\Phi}}^{-}\left(a\right)={\displaystyle \sum}_{x\u03f5K}\pi \left(x,a\right).$$

- (a)
- Preference measure (evaluation matrix),
- (b)
- Weights,
- (c)
- Preference function.

## 3. Results

- the weight of criterion_04 (average time each node is without service) increases;
- while the score of ranking C (diameter and velocity), B (diameter and distance from the source), and ranking D (flow) increases.

- actions are represented by points;
- criteria are represented by vector;
- weights of the criteria are represented by red vector (called the decision axis).

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**The upper diagram presents the flowchart of the proposed methods. The lower one—matching tools per each step.

Team | Breakdown ID | Number of Answers | |||
---|---|---|---|---|---|

Task 1 | Task 2 | Task 3 | Task 4 | ||

1 | 1 | 19 | 3 | 0 | 2 |

2 | 4 | 18 | 1 | 0 | |

3 | 1 | 0 | 13 | 9 | |

4 | 0 | 8 | 5 | 10 | |

2 | 5 | 0 | 10 | 8 | 5 |

6 | 18 | 1 | 4 | 0 | |

7 | 1 | 3 | 11 | 8 | |

8 | 3 | 3 | 4 | 12 |

Task | Breakdown ID | |
---|---|---|

Crew 1 | Crew 2 | |

1 | 1 | 6 |

2 | 2 | 5 |

3 | 3 | 7 |

4 | 4 | 8 |

Break/Leak | Diameter | Pipes to Isolate for Repair Break | Notes | Valves | Closing Time | Average Time |
---|---|---|---|---|---|---|

ID | (mm) | ID (*) | (-) | (-) | (min) | (h) |

437 | 500 | 437 (0), 599 (1), 6069 (1), 169 (1) | 3 | 45 | 14.00 | |

3602 | 200 | 3602 (2) | 2 | 30 | 7.00 | |

2112 | 200 | 2112 (1), 2106 (1), 2123 (1), 2124 (1) | 4 | 60 | 8.00 | |

3562 | 200 | 3562 (1), 3584 (1), 3721 (1), 3720 (1) | important | 4 | 60 | 8.00 |

**Table 4.**Input evaluation matrix for the preference ranking organization method for enrichment evaluation (PROMETHEE).

Scenario | Alternatives | Results from MATLAB and EPANET Matlab Toolkit | |||||
---|---|---|---|---|---|---|---|

C_01 | C_02 | C_03 | C_04 | C_05 | C_06 | ||

(min) | (min) | (min%) | (min) | (-) | (L) | ||

Scenario 1 | A | 3000 | 460 | 41,648 | 216 | 801 | 76,913,854 |

B | 3165 | 460 | 42,333 | 213 | 795 | 77,166,926 | |

C | 3045 | 460 | 41,324 | 212 | 759 | 77,640,589 | |

D | 2970 | 460 | 41,098 | 211 | 760 | 77,370,499 | |

E | 2970 | 460 | 42,051 | 217 | 793 | 78,046,971 | |

F | 3555 | 460 | 47,564 | 261 | 799 | 85,936,068 | |

Scenario 2 | A | 3510 | 276 | 19,697 | 46 | 134 | 57,045,249 |

B | 3765 | 276 | 20,129 | 48 | 143 | 56,725,233 | |

C | 3720 | 276 | 19,233 | 44 | 135 | 56,820,145 | |

D | 3795 | 276 | 19,379 | 45 | 138 | 56,480,788 | |

E | 3720 | 276 | 19,182 | 42 | 134 | 56,988,294 | |

F | 3960 | 276 | 18,608 | 37 | 120 | 59,438,741 | |

Scenario 3 | A | 3690 | 364 | 26,224 | 36 | 73 | 82,986,155 |

B | 3720 | 364 | 28,692 | 38 | 78 | 82,561,084 | |

C | 3690 | 364 | 28,597 | 38 | 77 | 84,812,665 | |

D | 3375 | 364 | 28,207 | 38 | 76 | 82,910,628 | |

E | 3480 | 364 | 28,927 | 38 | 74 | 84,615,279 | |

F | 3900 | 457 | 29,308 | 34 | 79 | 91,362,983 | |

Scenario 4 | A | 2565 | 364 | 32,555 | 80 | 166 | 74,506,055 |

B | 2430 | 364 | 33,823 | 82 | 198 | 74,246,491 | |

C | 2610 | 364 | 33,063 | 80 | 168 | 75,216,764 | |

D | 2445 | 364 | 33,470 | 81 | 168 | 75,133,769 | |

E | 2625 | 364 | 32,949 | 81 | 171 | 75,596,054 | |

F | 2790 | 364 | 36,002 | 103 | 399 | 78,722,686 | |

Scenario 5 | A | 3300 | 364 | 25,279 | 80 | 150 | 64,664,999 |

B | 3600 | 364 | 26,196 | 79 | 150 | 64,474,069 | |

C | 3585 | 276 | 25,327 | 84 | 149 | 65,612,889 | |

D | 3315 | 363 | 25,640 | 81 | 150 | 64,937,535 | |

E | 3705 | 276 | 24,954 | 85 | 152 | 65,461,382 | |

F | 3390 | 364 | 26,014 | 69 | 163 | 70,376,530 |

Criterion | C_01 | C_02 | C_03 | C_04 | C_05 | C_06 |

Weight | 298 | 241 | 187 | 219 | 219 | 187 |

Min/Max | Min | Min | Min | Min | Min | Min |

**Table 6.**Final results based on equations for the six criteria to evaluate the performance of a solution.

Scenario | Best Ranking | Criteria | |||||
---|---|---|---|---|---|---|---|

C_01 | C_02 | C_03 | C_04 | C_05 | C_06 | ||

(min) | (min) | (min%) | (min) | (-) | (L) | ||

1 | C (Diameter and Velocity) | 2970 | 6900 | 41,098 | 211.01 | 760 | 77,370,498 |

2 | F (Impact of network) | 3960 | 4140 | 18,608 | 37.45 | 120 | 59,438,741 |

3 | A (Diameter) | 3960 | 5460 | 26,224 | 36.5 | 73 | 82,986,155 |

4 | A (Diameter) | 2565 | 5460 | 32,555 | 79.71 | 166 | 74,506,054 |

5 | A (Diameter) | 3300 | 5460 | 25,279 | 79.82 | 150 | 64,664,998 |

Criterion | C_01 | C_02 | C_03 | C_04 | C_05 | C_06 |
---|---|---|---|---|---|---|

Weight granted | 22% | 18% | 14% | 16% | 16% | 14% |

Stability interval (WSI) | 13.86–23.93% | 0–100% | 10.53–100% | 12.99–28.17% | 7.39–17.23% | 13.04–22.45% |

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**MDPI and ACS Style**

Bałut, A.; Brodziak, R.; Bylka, J.; Zakrzewski, P.
Ranking Approach to Scheduling Repairs of a Water Distribution System for the Post-Disaster Response and Restoration Service. *Water* **2019**, *11*, 1591.
https://doi.org/10.3390/w11081591

**AMA Style**

Bałut A, Brodziak R, Bylka J, Zakrzewski P.
Ranking Approach to Scheduling Repairs of a Water Distribution System for the Post-Disaster Response and Restoration Service. *Water*. 2019; 11(8):1591.
https://doi.org/10.3390/w11081591

**Chicago/Turabian Style**

Bałut, Alicja, Rafał Brodziak, Jędrzej Bylka, and Przemysław Zakrzewski.
2019. "Ranking Approach to Scheduling Repairs of a Water Distribution System for the Post-Disaster Response and Restoration Service" *Water* 11, no. 8: 1591.
https://doi.org/10.3390/w11081591