1. Introduction
The water distribution network (WDN) is one of the essential public infrastructure systems. It requires frequent inspection, occasional maintenance, and swift repairs throughout its service life to maintain its performance levels [
1]. Therefore, it is essential to maintain effective maintenance and repair plans and to refresh the system’s physical assets, particularly in regions where widespread failure is possible. A complicated WDN is susceptible to failure and may result in the collapse of the pipeline, the failure of neighboring utilities such as underground electrical lines, the disruption of traffic and local economic activity, and even fatalities. Environmental variables, the age of the assets, and component deterioration may cause system failures. These faults can occur at any time, anywhere, and have a variety of threatening consequences for other infrastructure systems, such as transportation networks and building foundations [
2].
The sensitivity of WDNs to failure is the driving force behind the deployment of asset management, which aims to achieve an optimal price and service level for customers. The most expensive component of the WDN is the pipeline utility. Pipe failure is affected by the fundamental properties of the utility, such as pipe material, diameter, length, age, and previous failure history. Establishing a correlation between the failure rate and these variables is essential for assessing network status and preventing catastrophic failure. In the WDN, decision makers must implement infrastructure asset management to ensure that infrastructure performance achieves service level, risk management, and cost management in the context of the asset life-cycle at the most cost-effective level. If asset management is a crucial component of any pipeline utility, then accurate information on the service life of the pipes is a crucial aspect of asset management. Understanding the current condition and remaining lifetime of a network gives numerous benefits, including optimizing hydraulic operation and maintenance techniques, reducing network expenses and costs, and identifying the optimal level of service [
3].
Throughout its lifetime, the WDN may be subjected to unanticipated loading or lining conditions, which might cause pipe damage. The expected incidence of pipe failure for older water systems employing pipe materials such as asbestos–cement, cast iron, and ductile iron can be approximated by analyzing the considerable data associated with long previous failure records. On the other hand, for relatively new pipe materials, such as polyvinyl chloride (PVC), it is hard to make accurate predictions based on limited historical data [
4]. Failure frequency tends to be biased when parameter estimation is not carried out properly, leading to selection bias for pipe feasibility. As the sample size increases, the uncertainty becomes smaller, while the bias remains constant [
5]. This highlights the significance of applying an appropriate method to estimate the parameter values when evaluating brief failure data series. This article aims to determine pipe failure prediction to optimize pipe renewal time in the WDN. This research methodology investigates the most appropriate parameters for predicting pipe failure for optimization. In particular, the non-homogeneous Poisson process (NHPP) with Bayesian inference is used to predict pipe failure rates. In the following explanation, Bayesian inference is compared with frequentist inference. The Markov chain Monte Carlo (MCMC) approach is presented for Bayesian inference, while maximum likelihood (ML) is used for frequentist inference. We investigate how failure prediction can provide a cost-effective pipe renewal strategy and determine the most economical renewal lifetime. The model calculates the most cost-effective life-cycle costs based on the present value of renewal costs, repair costs, and predicted failure.
2. Literature Review
Pipe failure prediction is required for the development of preventative techniques in infrastructure management. Shamir and Howard’s [
6] article on establishing the optimal pipe renewal time is a crucial source for determining the appropriate renewal interval. It comprised a deterministic model for optimizing the economic effectiveness of repair versus renovation times. Since then, deterministic, statistical, and machine learning methods have been applied in the research of pipe performance and resulting lifetime. Lauer [
7] presented an excellent general strategy for network maintenance with insight into failure prediction. Rostum [
8] offered a thorough statistical study of the pipe failure process based on the NHPP model. Kleiner et al. [
9] proposed economies of scale into a model of pipe renewal, while Fuchs-Hanusch et al. [
10] modified the pipe life-cycle equation by integrating leak detection costs that increase with pipe age. Scholten et al. [
11] developed a failure model using the exponential–Weibull distribution. It was also used in multi-criteria decision analysis to rank various long-term rehabilitation alternatives. Amaitik et al. [
12] supported using neural networks for failure prediction and pipe renovation, whereas Kabir [
13] utilized a Bayesian framework to accomplish the same goal. Motiee et al. [
14] examined four alternative regression models, Di Nardo et al. [
15] applied fractal theory to evaluate the robustness of pipe failures, and Kutylowska [
16] predicted pipe failure rates with support vector machines. Specifically, research on the Weibull proportional hazard model (WPHM) can be found in the works of Le Gat and Eisenbeis [
17].
Giraldo-Gonzales and Martinez [
6] exposed several statistical models, such as Poisson regression, linear regression, and EPR, to predict the number of pipe group failures. Due to their explicit polynomial expressions, which offer a decent correlation between covariates and dependent variables, these models are recommended. Linear regression is an extension of regression analysis that includes covariates as explanatory variables in the prediction equation. In the linear regression model, whether the value of the covariate increases or decreases, the value of the dependent variable changes at a constant rate. Consequently, the fundamental disadvantage of statistical models is their dependence on the availability of comprehensive data. Probabilistic statistical models need the development of a time-dependent failure rate function, in which the time until the next failure varies based on the conditions of the previous failure. Based on previous research, this can be achieved by performing the time-dependent Poisson model and the Markov process in conjunction with the Bayesian principle. Atique and Attoh-Okine [
18] constructed a pipe failure model using Bayesian inference and the Copula parameter to describe bivariate distribution by sampling the distribution, including the dependent variable. Lin and Yuan [
19] created an NHPP model and presented a two-scale process with two-time variables applying Markov chain Monte Carlo (MCMC).
The importance of asset management in urban water utilities continues due to their technological, economic, and environmental ramifications being substantial and numerous. Specifically, providing accurate estimates for the service life of pipes is a crucial component of the asset management problem. In Mailhot et al.’s [
20] model, another optimization-based rehabilitation planning technique with a cost-conscious focus is considered as an indicator of the pipe’s structural integrity to calculate the best renewal criterion. Hong et al. [
21] proposed an analytical approach for the optimal pipe renewal based on the annual cost as a proportion of the overall cost by lowering the total expected cost over a set service life. A mathematical model developed by Luong and Nagarur [
22] can assist in determining whether to repair or replace the pipe and how to deploy maintenance expenditures most effectively. The optimization formulation’s objective function is the total system availability over the long term. Grigg [
23] offered a risk-based approach to pipe renewal to avoid utilizing a suboptimal budget. Lansey et al. [
24], Kim and Mays [
25], and Shin et al. [
8] established more cost-minimization-focused models. Other optimization models integrate system cost and reliability as competing objectives. Dandy and Engelhardt [
26] developed a trade-off curve for reliability and cost for the effective pipe renewal option.
Notably, several researchers have implemented life-cycle cost (LCC) analysis in WDN. This powerful notion highlights the analytical tools that assist decision makers in making the most cost-effective choices among the options provided to them at various life-cycle stages and, therefore, with varying costs. Shamir and Howard [
27] constructed an exponential relationship between a pipe failure rate and its age to calculate the pipe renewal interval that minimizes total repair and renewal costs. Lee et al. [
28] classified every network object to give an inventory-based technique for the LCC analysis of a WDN. This methodology was created to assist decision makers in determining when and which WDN components require repair.
Marzouk and Osama [
29] developed a method to assist decision makers in WDN management with their short-term and long-term plans. Four objective functions were considered: the risk index, the infrastructure condition, the level of asset service, and the LCC. The failure probability was simulated using a fuzzy Monte Carlo simulation. The research discovered that economic variables influenced asset failure results significantly, whereas pipe size influenced the overall failure consequence index. Jayaram and Srinivasan [
28] suggested an innovative multi-objective approach for lowering LCC and improving network performance. Roshani and Filion [
30] created the OptiNET model to reduce renewal time and pipe diameter while reducing LCC. Capital and operating expenses were evaluated as the goal function for determining the optimal renewal age. According to Frangopol and Soliman [
31], LCC analysis could significantly reduce long-term expenditures while improving the resilience and sustainability of the infrastructure. Based on the LCC assessment, Ghobadi et al. [
32] present a new pipe renewal scheduling strategy to smooth the investment time series for a large-scale WDN.
5. Conclusions
This article aims to determine pipe failure prediction to optimize WDN renewal. This research methodology investigates the most appropriate parameters for predicting pipe failure for optimization. The non-homogeneous Poisson process (NHPP) with Bayesian inference is used to predict pipe failure numbers. In the following explanation, Bayesian inference is compared with frequentist inference. The Markov chain Monte Carlo (MCMC) approach is presented for Bayesian inference, while maximum likelihood (ML) is used for frequentist inference. We investigate how failure prediction can provide a cost-effective pipe renewal strategy and determine the most economical renewal lifetime. The model calculates the most cost-effective life-cycle costs based on the NPV value of renewal costs, repair costs, and predicted failure.
The counting process is applied to analyze the number of failures in an interval of failure time. Based on the homogeneity examination, it can be determined that the failure intensity is inhomogeneous. From the two equations generated from the MCMC and ML estimation, it can be seen that although the parameters have different values, they tend to have the same relationship with the pipe failures. Pipe diameter and pipe failure have a negative relationship, whereas pipe age has a positive relationship. This result is consistent with earlier pipe failure analysis findings by several researchers. The failure prediction model tends to have a middle rating of the peak number of failures indicated by the observation data. On the other hand, we discover that the number of failures between the MCMC and ML estimation is not significantly different, despite the fact that MCMC has a higher total number of failures, which is closer to the observed data. The total number of predicted and observed failures are comparable, and it is exciting to observe the model’s performance in the event of failure. There are some apparent errors among the observed values and those predicted by MCMC and ML. Related to this, the MCMC predictions are better than the ML predictions because it MCMC has a lower mean squared error (MSE) value.
This article also proposes visual assessment techniques for evaluating the model concerning the resulting errors. In this evaluation, cumulative and annual charts are practical visual assessment techniques that can be used to compare model outcomes to actual failures visually. From pair plot validation, both models produce predictive values that are not much different from the observed values, which can be seen from the distribution of the predicted values around the Y=X line. The table demonstrates that the model is relatively appropriate at predicting failure in each quartile. There is a tendency for the predicted failures to be consistent with the observed failures. The pair plot shows that most of the failures occurred in the first quartile, and fewer occurred in the second. It is critical to highlight that the failure rate examines the group of pipes rather than the history of pipe failure for each pipe. The group of pipes in the first quartile has the highest priority for renewal. From the previous research, prediction models developed with unbalanced data sets will have a greater tendency to anticipate the number of errors, which can reduce the prediction accuracy. The solution to this problem is to extend the length of the data series by accumulating errors over a longer period of time. Based on the analysis, the recommended model is appropriate for predicting failure with serial data from ten years. Pipe failure predictions should be more accurate in situations with more than ten years of data.
The LCC analysis of the MCMC estimation has shown some points; although the initial cost (CI) is different for each diameter, the trend decreases and flattens as the pipe age approaches 50 years. At the age of the pipe above 30 years, the running cost (CR) takes over the total cost, depending on the pipe failure rate from the pipe failure analysis results. The higher the pipe failure rate, the more the point of intersection of the CI and CR curves will shift to the left. This condition is similar to the optimal value of LCC, where the higher the failure rate, the more the optimal point will shift to a shorter age. Based on the lowest LCC curve value for each diameter, the optimal pipe renewal time is obtained at the age of 35 years. The LCC analysis of the ML estimation shows that the lowest LCC value is in the pipe age range of 25 to 35 years. At diameters of 63 mm and 90 mm, the lowest LCC is at the age of 35 years. When the pipe diameter is 110 mm, the lowest LCC is at the age of 30 years, and it increases at the pipe diameter of 150 mm, where the lowest LCC is in the age range of 25 years. As the diameter increases, the age of the pipe decreases. The LCC analysis shows a different trend to the MCMC parameter estimation results. However, the ML estimation results for pipe diameters of 63 mm and 90 mm show the optimum pipe lifetime are relatively identical to the MCMC estimation results. If we look at the value of the LCC in the same pipe age period, the ML estimation results show a higher value, and this difference will be more significant as the time variable increases, even though the analysis in both models is conducted with the same value of cost components. The difference, in this case, is the pipe failure rate from the estimation results. The MCMC estimation results generally produce a lower LCC and a slightly longer optimum pipe life than the ML estimation. This will affect the decision making regarding pipe renewal in WDNs.