# Water Pipe Replacement Scheduling Based on Life Cycle Cost Assessment and Optimization Algorithm

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Optimal Water Pipe Replacement Based on LCC Assessment

_{N}), which represents the cost at which the summation of the initial and running costs is the minimum, and is defined as

_{i}is the length of pipe i.

#### 2.2. Smoothing the Investment Time Series

#### 2.3. Pipe Replacement Scheduling Using a Multiobjective Optimization Algorithm

#### 2.3.1. Overview of NSGA-II

- NSGA-II randomly generates an initial population consisting of a number of chromosomes. Each chromosome is the value of the design parameter (i.e., the replacement time of an individual pipe).
- Each individual in the population generated in the previous step is ranked based on an evaluation of the objective function. Then, the individuals are sorted based on their rank by using the crowding distance.
- Parents are randomly selected from a mating pool to create the new generation. The mating pool consists of solutions with a higher crowding distance and rank.
- To produce offspring, parents from the previous step undergo crossover and mutation procedures. In the crossover procedure, two parents breed to produce an offspring that inherits its genes from both parents. In the mutation procedure, some values of the genes in each offspring are changed, thus providing the offspring with the opportunity to have at least one different gene value than their parents.

#### 2.3.2. Construction of Optimization Model

_{N}(i.e., LCC of the network) and LLCC

_{N}. By applying this objective function, the optimized LCC

_{N}is kept as close as possible to LLCC

_{N}, implying that the imposed cost is minimized. Equation (8) minimizes the standard deviation (SD) of the annual investments to smooth the investment time series. Equation (9) minimizes the average pipe age in the WDN to more reliably maintain the network [34].

## 3. Results

#### 3.1. Case Study and Assumptions

- All pipes need to be replaced at least once in the simulation time horizon in consideration of the diameter and the first installation year of the pipes.
- Pipes that already passed their replacement time (${t}^{*}$) will be replaced on priority in the first year of the scheduling plan.

#### 3.2. Pareto Front Obtained by Multiobjective Optimization

^{®}Core i9-10920X CPU @ 3.5 GHz with 128 GB memory. Three decision-making scenarios were considered during the optimization procedure as described below.

_{N}and, consequently, a less smooth investment time series that is likely to violate the budget constraint.

#### 3.3. Comparison of Solutions for WDN Investment

#### 3.3.1. Future Investment Based on LLCC_{N}

#### 3.3.2. Future Investment Based on Scenario 1

_{N}and therefore has the lowest imposed cost. The third scheduling plan (Solution 3, shown in Figure 10c) yields the youngest network with the lowest pipe average age, which can be considered as a more reliable plan than other solutions. The use of the knee-point (Solution 4) provides a balanced scheduling plan that considers all three objective functions simultaneously, as shown in Figure 10d.

_{N}schedule to provide the smallest fluctuation in annual expenditure. The running and initial costs and their summation after changing the replacement time represent the change in expenditure due to the selection of a different budget and replacement boundary. When the replacement time is postponed by 5 years from the expected ${t}^{*}$ in the first scenario, a 5-year running cost is added to the new plan, therefore 5 years are added to its life cycle. On the other hand, when a pipe is replaced 5 years sooner than the expected ${t}^{*}$, 5 years are omitted from its life cycle, which reduces the running cost and pipe age, but increases the replacement cost. The TAI, defined as the total cost divided by the time horizon (i.e., 108 years), provides an estimation of the annual WDN maintenance cost and enables a more realistic budget to be assigned for WDN rehabilitation. Note that the LCC does not depend on the time scale and represents the continuous life cycle of an asset. However, the total cost depends on the time horizon to be investigated; upon changing the time horizon, the total cost changes because the cycle is cut in a particular year. In this case, only the cost up to that specific year is counted. Note that the time horizon of the case study WDN was determined based on the above-mentioned assumption in which all pipes need to be replaced at least once.

_{N}, and the MODE is 0. This solution was obtained to prioritize keeping the replacement time the same as ${t}^{*}$(MODE = 0) while dispersing the peaks as necessary. As shown in Figure 10b, this time series follows the same pattern as the time series before smoothing (Figure 9), but in a smoother manner, and imposed the minimum cost to the network compared to the other solutions, with a maximum annual investment of 2.22 M$, and the LCC marginally increased by 0.08% compared to the before-smoothing-plan. For Solution 3, shown in Figure 10c, the priority was to keep the network younger; as a consequence, the majority of pipes are replaced 5 years sooner than ${t}^{*}$, which results in a more reliable plan than provided by the other three solutions. The average age in this solution decreased by 14.7% compared to the before-smoothing-plan. In Solution 4, which used the knee-point to fairly consider all three objectives, the replacement time can be observed to be dispersed evenly around the provided boundary, as shown in Figure 10d.

#### 3.3.3. Future Investment Based on Scenario 2

_{N}(MODE = 0) and consequently a less smooth curve showing the highest maximum annual investment of 2.18 M$ can be observed in Figure 11b. It is observed that the overall investment of Scenario 2 marginally increases compared to Scenario 1, because the replacement time boundary was relaxed by five additional years. The youngest scheduling plan was provided by Solution 3, which showed a 23.53% reduction compared to the before-smoothing-plan and is shown in Figure 11c.

#### 3.3.4. Future Investment Based on Scenario 3

#### 3.3.5. Comparison between Representative Solutions

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Annual investment time series without smoothing (the dashed line indicates an annual budget limit).

**Figure 5.**Case study network layout according to (

**a**) different pipe diameters and (

**b**) different pipe ages.

**Figure 8.**Two-dimensional projections of nondominated Pareto solutions for three scenarios: (

**a**) comparison between average age and imposed LCC

_{N}, (

**b**) comparison between average age and SD of annual investment, and (

**c**) comparison between imposed LCC

_{N}and SD of annual investment

_{.}

**Figure 10.**Future investment time series for Scenario 1: (

**a**) Solution 1—min. SD of annual investment, (

**b**) Solution 2—min. imposed LCC, (

**c**) Solution 3—min. system age, and (

**d**) Solution 4—knee-point.

**Figure 11.**Future investment time series for Scenario 2: (

**a**) Solution 1—min. SD of annual investment, (

**b**) Solution 2—min. imposed LCC, (

**c**) Solution 3—min. system age, and (

**d**) Solution 4—knee-point.

**Figure 12.**Future investment time series for Scenario 3: (

**a**) Solution 1—min. SD of annual investment, (

**b**) Solution 2—min. imposed LCC, (

**c**) Solution 3—min. system age, and (

**d**) Solution 4—knee-point.

**Table 1.**Initial cost data of ductile iron pipe [35].

Diameter (mm) | Pipe Cost ($/m) | ||
---|---|---|---|

Material | Construction | Total | |

80 | 15 | 65 | 80 |

100 | 28 | 66 | 94 |

150 | 41 | 76 | 117 |

200 | 59 | 86 | 145 |

250 | 81 | 96 | 177 |

300 | 103 | 105 | 208 |

350 | 125 | 114 | 239 |

400 | 149 | 127 | 276 |

450 | 156 | 136 | 292 |

500 | 182 | 148 | 330 |

**Table 2.**LLCC (least life cycle cost) and corresponding replacement age (${t}^{*}$) for different diameters.

Diameter (mm) | ${\mathit{t}}^{*}\text{}\left(\mathbf{Year}\right)$ | CI ($/km/Year) | CR ($/km/Year) | LLCC ($/km/Year) |
---|---|---|---|---|

80 | 35 | 2286 | 1725 | 4010 |

100 | 37 | 2541 | 1878 | 4418 |

150 | 42 | 2786 | 2080 | 4865 |

200 | 49 | 2959 | 2223 | 5182 |

250 | 57 | 3105 | 2275 | 5380 |

300 | 67 | 3104 | 2304 | 5408 |

350 | 78 | 3064 | 2264 | 5327 |

400 | 91 | 3033 | 2203 | 5236 |

450 | 104 | 2808 | 2065 | 4873 |

500 | 122 | 2705 | 1991 | 4696 |

Standard Deviation (M$) | LLCC_{N}(M$/year) | Average Age (Year) | $\mathbf{Max}\text{}{\mathit{I}}_{\mathit{y}\mathit{e}\mathit{a}\mathit{r}}^{\mathit{t}}$ $\left(\mathbf{M}\text{\$}\right)$ | Running Cost (M$) | Initial Cost (M$) | Total Cost (M$) | TAI (M$/Year) |
---|---|---|---|---|---|---|---|

0.72 | 1.504 | 27.2 | 3.36 | 70.28 | 90.65 | 160.93 | 1.490 |

**Table 4.**Details of four selected points for Scenario 1 (replacement time boundary = $\pm $5 years, annual budget limit = 2.5 M$).

Solution Number | Standard Deviation (M$) | Imposed LCC (k$/Year) | Average Age (Year) | MODE (Year) | $\mathbf{Max}\text{}{\mathit{I}}_{\mathit{y}\mathit{e}\mathit{a}\mathit{r}}^{\mathit{t}}$ $\left(\mathbf{M}\text{\$}\right)$ | Running Cost (M$) | Initial Cost (M$) | Total Cost (M$) | TAI (M$/Year) |
---|---|---|---|---|---|---|---|---|---|

1 | 0.298 | 3.8 | 23.5 | −5 | 1.99 | 69.70 | 90.66 | 160.36 | 1.485 |

2 | 0.381 | 1.3 | 24.2 | 0 | 2.22 | 70.53 | 89.48 | 160.00 | 1.481 |

3 | 0.315 | 3.5 | 23.2 | −5 | 2.08 | 69.72 | 90.61 | 160.33 | 1.485 |

4 | 0.332 | 2.2 | 23.6 | 0 | 2.15 | 70.15 | 90.22 | 160.37 | 1.485 |

**Table 5.**Details of four selected points for Scenario 2 (replacement time boundary = $\pm $10 years, annual budget limit = 2.2 M$).

Solution Number | Standard Deviation (M$) | Imposed LCC (k$/Year) | Average Age (Year) | MODE (Year) | $\mathbf{Max}\text{}{\mathit{I}}_{\mathit{y}\mathit{e}\mathit{a}\mathit{r}}^{\mathit{t}}$ $\left(\mathbf{M}\text{\$}\right)$ | Running Cost (M$) | Initial Cost (M$) | Total Cost (M$) | TAI (M$/Year) |
---|---|---|---|---|---|---|---|---|---|

1 | 0.243 | 15 | 22.4 | −10 | 1.80 | 68.27 | 92.79 | 161.06 | 1.491 |

2 | 0.342 | 4 | 22.2 | 0 | 2.18 | 69.83 | 90.56 | 160.39 | 1.485 |

3 | 0.268 | 10 | 20.8 | −10 | 2.08 | 68.03 | 92.82 | 160.86 | 1.489 |

4 | 0.278 | 8 | 21.2 | −3 | 1.97 | 68.48 | 92.07 | 160.56 | 1.487 |

**Table 6.**Details of four selected points for Scenario 3 (replacement time boundary =$\pm $16 years, annual budget limit = 2.0 M$).

Solution Number | Standard Deviation (M$) | Imposed LCC (k$/Year) | Average Age (Year) | MODE (Year) | $\mathbf{Max}\text{}{\mathit{I}}_{\mathit{y}\mathit{e}\mathit{a}\mathit{r}}^{\mathit{t}}$ $\left(\mathbf{M}\text{\$}\right)$ | Running Cost (M$) | Initial Cost (M$) | Total Cost (M$) | TAI (M$/year) |
---|---|---|---|---|---|---|---|---|---|

1 | 0.193 | 51 | 22.2 | −10 | 1.79 | 66.47 | 98.61 | 165.08 | 1.523 |

2 | 0.290 | 16 | 21.1 | 0 | 1.97 | 68.56 | 93.00 | 161.57 | 1.496 |

3 | 0.240 | 36 | 19.9 | −12 | 1.90 | 65.82 | 97.85 | 163.67 | 1.515 |

4 | 0.226 | 28 | 20.7 | −5 | 1.91 | 67.43 | 94.87 | 162.31 | 1.503 |

Scenarios | Budget Constraint (M$) | Replacement Time Span [LB,UB] (Year) | Minimum Standard Deviation (M$) | Minimum Imposed LCC (k$/Year) | Minimum Average Age (Year) |
---|---|---|---|---|---|

1 | 2.5 | −5, 5 | 0.298 | 1.3 | 23.2 |

2 | 2.2 | −10, 10 | 0.243 | 4.0 | 20.8 |

3 | 2.0 | −16, 16 | 0.193 | 16.0 | 19.9 |

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**MDPI and ACS Style**

Ghobadi, F.; Jeong, G.; Kang, D.
Water Pipe Replacement Scheduling Based on Life Cycle Cost Assessment and Optimization Algorithm. *Water* **2021**, *13*, 605.
https://doi.org/10.3390/w13050605

**AMA Style**

Ghobadi F, Jeong G, Kang D.
Water Pipe Replacement Scheduling Based on Life Cycle Cost Assessment and Optimization Algorithm. *Water*. 2021; 13(5):605.
https://doi.org/10.3390/w13050605

**Chicago/Turabian Style**

Ghobadi, Fatemeh, Gimoon Jeong, and Doosun Kang.
2021. "Water Pipe Replacement Scheduling Based on Life Cycle Cost Assessment and Optimization Algorithm" *Water* 13, no. 5: 605.
https://doi.org/10.3390/w13050605