Abstract
The performance of a water distribution system of providing a required flow rate at all the nodes with required pressure heads throughout its design life is affected by uncertainties associated with different parameters such as future water demands, pipe roughness coefficient values, required pressure heads at nodes, etc. The objective of this paper is to present a comprehensive review on the nature of uncertainties (random or fuzzy), various models and methods used for their quantification, and different ways of handling them in the design of water distribution networks. While probabilistic based approaches are used for handling uncertainty of random type, the possibilistic based approach considers uncertainty of fuzzy nature. Some key issues and serious limitations of the existing approaches for modeling uncertain parameters related to water distribution networks are identified. The uncertainty in water demands is due to both their random nature and lack of information about their values. Therefore, a combination of both types of approaches, called the fuzzy random approach, is found to be more effective. The fuzzy random approach can provide optimal design solutions that are not only costeffective but also has higher reliability to cope with severe future uncertainties.
NOTATIONS AND SYMBOLS
 ACO
ant colony algorithm
 CE
cross entropy
 DP
dynamic programming
 FORM
firstorder reliability method
 FOSM
first order second moment
 FRV
fuzzy random variable
 GA
genetic algorithm
 GRG2
generalized reduced gradient method
 LHS
latin hypercube sampling
 LP
linear programming
 MCS
Monte Carlo Simulation
 MFs
membership functions
 NLP
nonlinear programming
probability distribution function
 PSO
particle swarm algorithm
 SAA
simulated annealing algorithm
 SFLA
shuffled frog leaping algorithm
INTRODUCTION
Water distribution systems (WDS) are considered as an important public infrastructure that is designed to supply acceptable quality of water in adequate quantities with the required residual heads at all consumer tapping points. The networks are complex as they involve interaction between a large number of components such as pipes, pumps, valves, etc (Mays 2000). The design of the systems has to consider a large number of parameters, some of them are known more precisely than others; for example, length of pipes, as its value remains constant over a given period (Bhave & Gupta 2006). Conversely, reservoir water levels, nodal demands, and pipe roughness values are considered as uncertain parameters as their values change with time. The uncertainties in these parameters are characterized due to their random nature and/or lack of knowledge about the same. Figure 1 depicts various types of uncertainties.
The aleatoric uncertainty that is caused by natural variability requires statistically determined parameters for the representation of uncertainty, which are represented by probability distribution function (PDFs). A huge reliable data set is required to define the PDF of an uncertain variable, which is treated as a random variable. The PDFs can be of various forms such as normal, uniform, Gaussian, etc., (Kang et al. 2009; MacLeod & Filion 2009). The approach thus holds good for such events/processes whose occurrence is uncertain; that is, whether the event/process can occur or not: thus it is a binary characterization (either 0 or 1) approach.
Conversely, an epistemic uncertainty, which is caused due to lack of knowledge, does not require crisp statistical measures of input parameter distribution. The possibility based approach; that is, the fuzzy approach, considers an uncertain parameter as the fuzzy parameter and considers the possibility of occurrence of an event between 0 and 1. The fuzzy sets are different from traditional sets as their boundaries are not precisely defined. Hence, the membership function obtained for a parameter is not exact as it considers a certain degree of inaccuracy or subjectivity. The membership function (μ_{A}) may be of triangular or trapezoidal type (depending upon the likely variation of a parameter) and is not of binary type, and hence different from PDF. The fuzzy approach proves to be more appropriate to handle uncertainty associated with lack of information.
The fuzzy random approach is considered as the simultaneous representation of randomness and fuzziness associated with an uncertainty (Fu & Kapelan 2011; Shibu & Reddy 2014). Such uncertain parameters are collectively known as fuzzy random variables (FRVs). FRV is an extension to the random variable, whose outcomes are considered as fuzzy numbers rather than crisp real values (Kwakernaak 1978; Puri & Ralescu 1986; Moller & Beer 2004; Shapiro 2009). This allows a combination of both the probabilistic and fuzzy approaches to represent uncertain parameters.
Among all the uncertain parameters involved in the design of WDS, the future water demand is considered to be the most uncertain parameter. The forecasting of water demand is affected by several factors such as population projection, seasonal variations in demand, peak factor, etc. The population projection is itself very uncertain due to various methods used for the forecast, migration of people from one place to another, birth rate, death rate, etc. These factors make the accurate projection of future population the most difficult challenge. This lack of information (fuzziness) introduces inherent uncertainty in predicting the values of nodal demand. Similarly, the nodal demand at a particular place keeps fluctuating from time to time and from one place to another at a given time randomly. Also, there are several uncertain events such as leakages in pipes, thefts, fire demand, etc. that further introduce uncertainty of random nature.
This article aims to summarize and critically review the approaches adopted to quantify uncertainty in future water demand and ways to handle the same in WDS design. By synthesizing the updated research findings, this study highlights new directions for considering the uncertainty in optimal design of the WDS.
UNCERTAINTY IN WDS PARAMETERS
WDS involves a large number of uncertainties arising due to the nature of information available about the values of parameters, economy, and random failure of components. Handling these uncertainties in the design of WDS has gained tremendous attention in last decades (Lansey et al. 1989; Bao & Mays 1990; Babayan et al. 2005; Basupi & Kapelan 2015a; Hwang & Lansey 2017). It provoked the decisionmakers and designers to make a switch from the leastcost design of WDS to performancebased and multiobjective design to understand how a system performs under uncertainty. Among the numerous sources of uncertainty (Bhave 2003): (1) future water demand and pressure head requirement at nodes and (2) pipe roughness values due to pipe aging are the most uncertain parameters (Hudson 1966). The uncertainty in both these parameters affects the hydraulic performance of the system and is thus referred to as hydraulic uncertainty.
Uncertainty in water demands
Water demand is the most uncertain parameter in the design of WDS, as it is influenced by numerous factors such as population growth, climate change, socioeconomic changes, etc (CPHEEO, G. 1999). The complex interaction of these factors makes the prediction of water demand a difficult task (Wang et al. 2017). Hence, accurate prediction of water demand either in short term (hourly, daily), or longterm (monthly or yearly) time horizons has become a challenge.
The uncertainty in population forecast (Wilson 2012) is the most influential factor in predicting the demand values. The difference in actual and projected values is mainly attributed to the methodology used for prediction. Traditionally, the population projection is done by using Numerical methods (Arithmetic Increase, Geometric Increase, Incremental Increase, etc.) and Graphical methods (Graphical Comparison, Logistic Curve, and Graphical Extension, etc.) (Punmia 1977). To analyse the forecast error, the population of one city, Kanpur, in India, is forecasted using census data for 30 years, which is usually considered as design period for WDS. Census data from 1951 to 2011 was available. Using the data from 1951 to 1981, population projection is carried out for the years 1991, 2001 and 2011 by different methods as shown in Figure 2.
As seen clearly, there is a difference between the actual population and the forecasted one, showing uncertainty in the population projection. Different methods give different forecasted values; some methods underestimate the population, some give fairly close results, while other methods give an overestimated value, the maximum error being 23%.
Accurate and reliable water demand forecasting is necessary to minimize uncertainty in the design of WDS. Water demand predictions are commonly based on three approaches: (1) End use forecasting (which requires a tremendous amount of data and assumptions), (2) econometric forecasting (based on statistically estimating historical relationships and assuming the same relationship in future) and (3) time series forecasting (forecast water consumption directly, without forecasting other parameters) (Khatri & Vairavamoorthy 2009; Wang et al. 2017; Xie et al. 2017). The traditional approaches only consider the linear variation in climate change, population growth, and socioeconomic factors, which requires the long term continuous historical data, while in modelbased (statistical) approaches, nonlinear variations of these parameters are considered as well (Khatri & Vairavamoorthy 2009; Wang et al. 2017). Hence, the statistical approach gives fairly accurate predictions and has gained a lot of attention from researchers in the present scenario. However, before applying these prediction models, accurate demand pattern recognition is very important. Buchberger & Wells (1996) developed Poisson Rectangular Pulse (PRP) model to obtain the realistic demand pattern and asses the water quality in networks. Also, Buchberger et al. (2003) used the PRP model for simulating water demands of residential consumers and studied the changes in flow rates, flow reversals, and flow types. The Artificial neural network (ANN), time series, regression, fuzzy or hybridbased approach model (Gato et al. 2005), etc., have been used to model the water demand forecast. The predictive performance of such models is evaluated through indicators such as Root mean square error (RMSE), Mean Absolute Percentage Error (MAPE), etc.
All the models explained or compared have their individuality and each of them provides a varied result, which may lead to the uncertainty in the system as well. Hence, its actual quantification is an essential yet difficult task to develop a more reliable system. An overview of the same is provided in Table 1.
Author .  Review duration .  Description . 

Donkor et al. (2012)  2000–2010 

Anele et al. (2017)  3 decades 

Ghalehkhondabi et al. (2017)  2005–2015 

Author .  Review duration .  Description . 

Donkor et al. (2012)  2000–2010 

Anele et al. (2017)  3 decades 

Ghalehkhondabi et al. (2017)  2005–2015 

Quantification of uncertainty in water demand
The inaccuracy or uncertainty in input parameters is carried forward through the model and affects the system state variables, which are the performance deciding parameters of any system. Thus, the quantification of uncertainty or error in the forecasted values is the most critical issue in reliabilitybased designs of systems (Khatri & Vairavamoorthy 2009). Not much work has been done in the area of quantification of demand uncertainty previously. Till date, in the literature published on the uncertaintybased design of WDSs, the uncertainty in water demand has been assumed with coefficients of variation (COV i.e. levels of uncertainty) without any sound explanation. This may lead to the largest approximation of the system ultimately resulting in overdesign or underdesign of the system.
The approach is independent of quantifying the uncertainty in the system. The uncertainty value for head and flow was identified at various stages.
Reliability assessment of WDS is concerned with measuring its performance in terms of both quantity and quality. The uncertainty in quantity of water supplied may be due to flow and/or pressure at which the water is supplied, while the quality is assessed based on the concentration of waterborne substances that adversely affects the health. Buchberger & Wu (1995) developed a PRP queuing model to illustrate the temporal and spatial variability in the flow regime through dead end trunk lines, in determining the uncertainty in water quality parameters. Also, Lee & Buchberger (2001) suggested a model for uncertainty in water quality at dead ends of a network. The model can also analyze the type of flow i.e. stagnant, laminar, turbulent and minimum and maximum flow. Further, after the terrorists' attack on the twin towers, a lot of research has been carried out on early detection of intentional contamination events so as to reduce the impact of contaminated water (Rathi et al. 2015). Apart from parameter uncertainty, there exists Model Uncertainty (Moller & Beer 2004) i.e. the uncertainty in the mapping. This is introduced by uncertain input parameters that act exclusively within the model and thus are called uncertain model parameters (Mavromatidis et al. 2018).
Once the uncertainty in water demand (and other input parameters) is quantified, the next task is to characterize them and to know how uncertainty in such parameters affects the output parameters. The detailed review of these uncertainty handling approaches in the analysis and design of WDS is presented in the following section.
CHARACTERISATION OF UNCERTAINTY IN ANALYSIS AND DESIGN OF WDS
Probabilistic approaches
In the probabilistic approach, an uncertain parameter is treated as a random parameter having an assumed PDF. The type and nature of distribution for an uncertain parameter is fixed based on the available past statistical data. Once it is done, the analysis of the system aims at deriving the PDF of an output parameter (such as pipe flows and nodal heads) using different techniques. The samplingbased methods like Monte Carlo Simulation (MCS) and Latin Hypercube Sampling (LHS) are commonly used. MCS is an enumeration technique which requires generating a large number of samples randomly and evaluating each of them for the output parameters. LHS is a stratified sampling method which selects a random sample of each random variable in a more stratified manner, thus requiring less number of samples for accurate predictions. Analytical based methods like FirstOrderSecondMoment (FOSM) or First OrderReliabilityMethod (FORM), works on the principle of estimation of variance around mean value of the parameter. Further, bounds based approach is used where the lower and upper bounds on the probability curve need to be fixed based on experience and judgment of the designer.
Probabilistic analysis and reliability evaluation of WDSs
Probabilistic analysis involves understanding how the uncertainties in input parameters (nodal demands and pipe roughness coefficients) are propagated in output parameters (nodal heads and pipe discharges) affects the performance of the system evaluated using reliability parameters.
Bargiela & Hainsworth (1989) addressed the uncertainty in calculated flow and pressure values due to uncertainty in input parameters as these values are used by the operator to control the flow and pressure in the network. The quantification of uncertainty was termed confidence limit analysis and was compared using various methods such as MCS, optimization method and sensitivity matrix technique.
Pasha & Lansey (2005) examined the impact of alternative sources of uncertainty such as the decay coefficient, pipe roughness, pipe diameter and nodal demand on the system, and water quality. It was observed that the magnitude of the uncertain concentration was dependent upon the location in the system and location relative to sources. Later, Kretzman & Zyl (2006) developed a methodology using a software package called Monte Carlo SimulationII (MocasimII) for the stochastic analysis of WDS. The software allows the quantification of the relationship between the reliability of the supply system and the capacity of its storage tank using MCS.
Sumer & Lansey (2009) developed a methodology to estimate the impact of uncertainty in pipe roughness values on decisions that were developed using the model for a system expansion design by employing a steady state hydraulic model. The model parameter uncertainty was evaluated using FOSM, which was then propagated to model prediction uncertainties through a second FOSM for a defined set of demand conditions.
Hwang et al. (2017) compared the results obtained from FOSM with MCS based on WDS peak demand conditions, topology and pipe diameter. WDS calibration and abnormality detection was considered as the first case and network design as the second case.
The normal analysis was further extended to reliability assessment of WDS. Bao & Mays (1990) developed the methodology for quantification of hydraulic reliability of WDS using MCS under uncertainty in future water demand, pressure head requirement and pipe roughness coefficients (C). The pressure heads (H_{d}) were determined through the KYPIPE hydraulic network simulator. It was observed that the uncertainties in water demand create much impact on nodal or system reliability. Xu & Goulter (1998) suggested a twostage methodology for the assessment of the reliability of WDS by considering the uncertainty in nodal demand, pipe capacity, and reservoir level along with mechanical uncertainty of system components. Xu & Goulter (1999b) identified critical nodes to impose the reliability constraint on those nodes in a cost minimization problem. This was carried out by mean value firstorder second moment (MVFOSM) method along with uncertain design. The model was solved by integrating the FORM and GRG2 optimization program. The methodology provided an improved solution in the design of reliable systems.
SeifollahiAghmiuni et al. (2013a) and SeifollahiAghmiuni et al. (2013b) proposed a probabilistic model based on the MCS method to considered uncertain parameters in reliability assessment. While SeifollahiAghmiuni et al. (2013a) considered uncertainty in only the HW coefficient, SeifollahiAghmiuni et al. (2013b) considered uncertainties in both nodal demands and HW coefficients. They observed that ignoring variation in uncertain parameters during design affects the performance of the network significantly in long term operation.
Reliabilitybased design of WDS
Lansey et al. (1989) produced a pioneering work in the optimal design of WDSs with uncertainties. A chance constraints optimization problem was framed considering various uncertainties, which were modeled using a nonlinear programming model and solved using a generalized reduced gradient method (GRG2). Several design solutions were obtained for different reliability levels. Xu & Goulter (1999b) used FORM (advanced secondmoment method) to estimate the capacity reliability of the network defined as the probability that the nodal demand was met at or over the prescribed minimum pressure under uncertain nodal demands and pipe roughness coefficients. An iterative method was used in which an initial design was successively modified using generalized reduced gradient 2 (GRG2) technique to achieve a desired level of reliability. Further, the final solution consisted of noncommercial sizes and the method required excessive computing time to quantify the effect of uncertainty.
While Kapelan et al. (2004) considered stochastic leastcost design problem, subjected to a prespecified level of design reliability. The uncertain parameter was assigned by PDF and modeled using LHS method. The model was solved using GA and robust GA, (RRGA) and results were compared. Babayan et al. (2005) replaced MCS by integrationbased uncertainty quantification technique the stochastic model was replaced by deterministic one by changing the probabilistic pressure requirement criterion by adding a margin of safety at critical nodes for satisfying the pressure head requirements. The deterministic model was then solved using a modified GA. Several designs were obtained by changing the margin of safety and their robustness was compared for selecting a solution with desired robustness. Babayan et al. (2007) further suggested two new approaches to fulfill the minimum cost objective, subjected to a target level of system robustness. The first approach was an integration approach in which stochastic problem formulation was replaced with deterministic one and after that optimization problem was solved with the standard GA. The second was the sampling approach (LHS) in which stochastic problem was solved by GA directly to evaluate the fitness of each solution. Perelman et al. (2013) suggested a nonprobabilistic robust counterpart (RC) approach. The deterministic formulation was equivalent to a stochastic approach. The RC was then coupled with the CrossEntropy (CE) optimization technique in seeking a robust solution.
There exists research in multiobjective design as well. Xu & Goulter (1998) used intermediate results of FORM for identifying the most critical node. The use of FORM increased the computational time. To overcome this, Tolson et al. (2004) introduced a performance function to use with FORM so that the critical node location and reliability at the critical node could be determined without accessing the intermediate results of the FORM. Kapelan et al. (2005) formulated a design problem in which the uncertain variables were modeled using PDFs and corresponding PDFs of nodal heads were calculated using LHS. The model was solved using RNSGAII (a modified form of NSGAII). The methodology provided the whole Pareto optimal front in a single optimization model run, which proves to be efficient.
The twostage methodology developed by Giustolisi et al. (2009) consists of finding a deterministic design using GA approach, in the first stage and obtaining a robust solution using the deterministic solution as an initial solution in the second stage. The optimized multiobjective genetic algorithm (OPTIMOGA) was featured. It was observed that a strict design constraint concerning service levels can render the WDS more robust concerning unexpected variations in demand and even pipe roughness. MacLeod & Filion (2009) developed a model that considers water demand projected at the end of the 20 year planning period as an uncertain parameter, which was modeled as a random variable with an error PDF. The design alternative was selected in order to meet design demand at a minimum pressure at some cost. Jung et al. (2011) suggested a multiobjective genetic algorithm (MOGA) to solve the problem. The nodal pressure variability was calculated using critical node and all nodes in the network and the results were compared among the two based on Disturbance Index (DI) values. It was found that the system robustness is not sufficiently minimized when only the critical nodes were considered.
Basupi & Kapelan (2015b) combined the sampling techniques (MCs or LHS), decision tree analysis and GA to carry out the optimization. Both deterministic and flexible designs were considered for the design. It was observed that the flexible intervention strategies under uncertainty outperform corresponding deterministic strategy, but the flexible strategy can cause a drawback if the future water demand meets the designed level.
Table 2 represents a brief overview of the studies on uncertainty based design of the WDS using a probabilistic approach.
Uncertain parameter .  % uncertainty (in terms of SD, means and COV) .  Distribution function .  Sampling techniques .  Optimization techniques .  References .  Remarks . 

For probabilistic analysis of WDS  
q, H, C_{HW}  Normal  FORM  GRG2  Xu & Goulter (1999a)  GRG2 require a decision variable to be modeled as a continuous variable, which is unrealistic, hence it was easily trapped in local minimum in complex and discrete search spaces.  
q, C_{HW}  COV0.1,0.2  Normal  MCS  SFLA  SeifollahiAghmiuni et al. (2013a, 2013b)  Introduced dynamic design methods helps the designer simultaneously for design and rehabilitation, which also decreases the cost. 
q, C_{HW,} peak demand factor (A)  COVq = 0.15, C_{HW} = 0.05, peak demand = 0.1  Normal  FOSM & MCS  –  Hwang et al. (2017)  Accuracy of FOSM in uncertainty analysis is determined by comparing with MCS. Observed that FOSM works well for looped than that for a branch system. 
For probabilistic based design of WDS  
q, H, C_{HW}  S.D.q = 0, 0.10, 0.25 H = 0, 5, 10 and C_{HW} = 0, 5, 10  Normal  MCs  GRG2  Lansey et al. (1989)  Generalized reduced gradient procedure used to solve nonlinear optimization problem, which resulted in a nondiscrete pipe diameter. Hence it was suggested to use different optimization approach. 
q  S.D.10%  Gaussian  LHS  GA  Babayan et al. (2004)  – 
q, C_{HW}  S.D.C_{HW} = 10%  Normal  FORM and MCS  –  Tolson et al. (2004)  Computationally inefficient as pareto optimal curve is obtained by solving a series of single objective optimization problem. 
q  S.D.10%  Gaussian  LHS  RNSGAII  Kapelan et al. (2005)  Pareto optimal solutions obtained with reduced computational effort. 
q  S.D.10%  Gaussian  MCS  NSGAII  Babayan et al. (2005)  In order to overcome the complexity observed by FORM, a new model is prepared, which is suitable for the complex network. 
q, C_{HW}  S.D.C_{HW} = 20%  Beta  LHS  GA  Giustolisi et al. (2009)  Require less computational time, as single objective deterministic problem served as the initial population of a multiobjective GA. 
q, C_{HW}  –  Beta  LHS  MOGA  Jung et al. (2011)  Instigate the use of reliability and robustness index to solve optimization problem and FOSM for uncertainty quantification. 
q  S.D. – q = 5, 10, 20%  Gaussian  MCS & LHS  NSGAII  Basupi & Kapelan (2015a, 2015b)  Flexible design approaches along with uncertainty is introduced and is found to be better than the deterministic approach. 
Uncertain parameter .  % uncertainty (in terms of SD, means and COV) .  Distribution function .  Sampling techniques .  Optimization techniques .  References .  Remarks . 

For probabilistic analysis of WDS  
q, H, C_{HW}  Normal  FORM  GRG2  Xu & Goulter (1999a)  GRG2 require a decision variable to be modeled as a continuous variable, which is unrealistic, hence it was easily trapped in local minimum in complex and discrete search spaces.  
q, C_{HW}  COV0.1,0.2  Normal  MCS  SFLA  SeifollahiAghmiuni et al. (2013a, 2013b)  Introduced dynamic design methods helps the designer simultaneously for design and rehabilitation, which also decreases the cost. 
q, C_{HW,} peak demand factor (A)  COVq = 0.15, C_{HW} = 0.05, peak demand = 0.1  Normal  FOSM & MCS  –  Hwang et al. (2017)  Accuracy of FOSM in uncertainty analysis is determined by comparing with MCS. Observed that FOSM works well for looped than that for a branch system. 
For probabilistic based design of WDS  
q, H, C_{HW}  S.D.q = 0, 0.10, 0.25 H = 0, 5, 10 and C_{HW} = 0, 5, 10  Normal  MCs  GRG2  Lansey et al. (1989)  Generalized reduced gradient procedure used to solve nonlinear optimization problem, which resulted in a nondiscrete pipe diameter. Hence it was suggested to use different optimization approach. 
q  S.D.10%  Gaussian  LHS  GA  Babayan et al. (2004)  – 
q, C_{HW}  S.D.C_{HW} = 10%  Normal  FORM and MCS  –  Tolson et al. (2004)  Computationally inefficient as pareto optimal curve is obtained by solving a series of single objective optimization problem. 
q  S.D.10%  Gaussian  LHS  RNSGAII  Kapelan et al. (2005)  Pareto optimal solutions obtained with reduced computational effort. 
q  S.D.10%  Gaussian  MCS  NSGAII  Babayan et al. (2005)  In order to overcome the complexity observed by FORM, a new model is prepared, which is suitable for the complex network. 
q, C_{HW}  S.D.C_{HW} = 20%  Beta  LHS  GA  Giustolisi et al. (2009)  Require less computational time, as single objective deterministic problem served as the initial population of a multiobjective GA. 
q, C_{HW}  –  Beta  LHS  MOGA  Jung et al. (2011)  Instigate the use of reliability and robustness index to solve optimization problem and FOSM for uncertainty quantification. 
q  S.D. – q = 5, 10, 20%  Gaussian  MCS & LHS  NSGAII  Basupi & Kapelan (2015a, 2015b)  Flexible design approaches along with uncertainty is introduced and is found to be better than the deterministic approach. 
q, water demand; C_{HW}, pipe roughness; H, pressure heads.
Fuzzy approaches
In the fuzzy approach (Zadeh 1965), an uncertain parameter is considered as an independent fuzzy parameter which is assigned a membership function; that is, triangular, trapezoidal, etc. The membership function for a fuzzy parameter is selected based on the available information, and designer's knowledge and judgment. The fuzzy analysis involves deriving membership functions of output (dependent) parameters, such as pipe flows, pipe velocities, and nodal pressures. (Revelli & Ridolfi 2002; Branisavljevic & Ivetic 2006; Gupta & Bhave 2007; Shibu & Reddy 2011; Spiliotis & Tsakiris 2012; Gupta et al. 2014, etc). Conversely, fuzzybased design involves minimum cost design considering fuzzy parameters (Farmani et al. 2005a, 2005b; Shibu & Reddy 2012; Dongre & Gupta 2017).
Fuzzy analysis of WDSs
Two types of approaches have been used to obtain the membership function of dependent parameters with known membership functions of independent fuzzy parameters: (1) optimizationbased approaches; and (2) analysis based approaches. Revelli & Ridolfi (2002) suggested an optimizationbased method in which 2X (X is an element for which the membership function is to be derived) optimization problems are formulated, one for minimum and another for the maximum value of the dependent parameter at each α* cut subjected to the hydraulic constraints. The model was solved by the quadratic programming technique. The constrained optimization model was converted to an unconstrained one by Lagrange function of the objective function and was approximated at each iteration using its Hessian. Branisavljevic & Ivetic (2006) considered GA as an optimization methodology for 2X optimization problems. Haghighi & Asl (2014) used modified NSGAII as an optimization tool. Optimizationbased approaches are more useful when dependent parameters change nonmonotonically with independent parameters. However, the computational work is greater and increases with the required accuracy in results. Sabzkouhi used many optimizations (MO)PSO that provided minimum and maximum values at all nodes simultaneously for any αcut in an attempt to reduce computational efforts.
Gupta & Bhave (2007) suggested a simple method based on repeated analysis. They observed that the dependent parameters (both nodal pressures and pipe flows) show monotonic variation with the change in the values of fuzzy independent parameters. Also, the maximum impact of uncertainty of independent parameters occurs when they are at their extreme maximum and/or extreme minimum values. Taking such monotonic change into account, an impact table is prepared. Selection of appropriate values of fuzzy parameters from this table results in extreme (either minimum or maximum) values of dependent parameters. This approach completely avoids optimization and requires simple analysis to derive the resulting membership functions of dependent parameters.
Shibu & Reddy (2011) suggested a mathematical model, based on the fuzzycross entropy optimization method considering the monotonous change in dependent parameters with independent parameters for obtaining membership functions for different α values. Moosavian & Lence (2018) proposed an approximate method to obtain the membership function, which outperforms the impact table approach on account of the computation time and number of simulations as compared to the impact table approach of Gupta & Bhave (2007). However, the obtained membership functions are approximate. Further, the method is found to be very well suitable for membership functions of pressure heads but lacks in providing the membership function of pipe flow/velocity.
Spiliotis & Tsakiris (2012) suggested an approach by considering the monotony of mass conservation equations at nodes. The mass continuity equations at all nodes were thus represented as a function of nodal heads and were solved using the NewtonRaphson method.
Gupta et al. (2014) developed an effective means of analysis which takes into account the pressuredeficient condition using Node flow analysis (NFA). Initially, Fuzzy demand dependent analysis (FDDA) was carried out using the approach suggested by Gupta & Bhave (2007). Later, the nodal outflows were considered pressuredependent and FNFA was carried out. It was observed that FNFA, which analyzed supply shortfall along with pressure deficiency, is better in finding vulnerable zones as compared to FDDA.
While maintaining the flow velocity at a medium or low level, Tsakiris & Spiliotis (2017) studied the effect of the uncertain parameter on both branched and looped WDS. For analysis of looped WDS, two approaches were adopted. First is the extension of the Revelli & Ridolfi 2002 method. The other referred to a global objective function by taking into account all the branches simultaneously. The head losses at each branch were calculated. This resulted in a condensed subset of fuzziness, which was found to be convenient for operational use.
Reliabilitybased design of WDS
Xu & Goulter (1999a) suggested the heuristic technique; that is, the fuzzy linear programming approach, to directly consider the fuzzy nodal demands and fuzzy nodal head requirement. They obtained a model that can take a wide range of demand values. The approach was later improved by Bhave & Gupta (2004) by including loop head loss constraints in the form of path head loss constraints. This is to maintain the hydraulic consistency in the network for critical demand. Initially a branched configuration was obtained; later, the links were assigned weights (Dubois & Prade 1986) to obtain the flow distribution as inversely proportional to path lengths. Upon comparing the results, the methodology by Bhave & Gupta (2004) was found to be cheaper and less time consuming than Xu & Goulter (1999a).
Farmani et al. (2005a, 2005b) opted for NSGAII to solve the problem. The fuzzy rules were used to describe the hydraulic reliability of the system for solutions under demand uncertainty. The system robustness was also optimized simultaneously with the total cost.
Shibu & Reddy (2012) framed an optimization model in two stages. Initially, a deterministic model was formulated to find the minimum and maximum values of the objective function to fix the boundaries of the membership function. In the second stage, a fuzzy optimization model was solved using CE to model fuzzy water demand. Gupta et al. (2013) suggested a GA based methodology for the optimal design of network under fuzzy nodal demands with levelone redundancy (failure of any one pipe without disrupting service). The network simulation was carried out under all pipe working conditions (APWC) and onepipe failure condition (1PFC) for each loading condition.
Lence et al. (2017) framed a multiobjective optimization model to minimize the network cost and maximize the number of reliability surrogates, which was solved by soccer league competition (SLC) method (Moosavian & Roodsari 2014a, 2014b).
Dongre & Gupta (2017) converted a fuzzy constrained optimization model into a deterministic model by considering the relationship between fuzzy demands and fuzzy nodal heads. The methodology can meet pressure requirements even for the worstcase scenario and can obtain high reliability for the network.
Table 3 presents a summary of fuzzy optimizationbased approaches for the design of WDS.
Uncertain input parameters .  % Uncertainty .  Uncertainty characterization .  Optimization techniques .  References .  Remarks . 

For fuzzy analysis of WDS  
C_{HW}  C_{HW} = ±10 of most likely value  Triangular  CE  Shibu & Reddy (2011)  Assumes monotonous change along with roughness coefficient and observed on smaller network. 
q, C_{HW}  q = ±15% C_{HW} = ±10% of most likely value  Triangular  NSGAII  Haghighi & Asl (2014)  Provides problem solution more systematically and computationally efficient, hydraulic responses simultaneously analyzed in single simulation run. 
q, C_{HW}  q & C_{HW} = ±15% of most likely value  Triangular  MOPSO  Sabzkouhi & Haghighi (2016)  Computational efforts found to be less than that of previous modified GA approach but still large. 
q, C_{HW,} D  –  Triangular  –  Tsakiris & Spiliotis (2017)  Both the branch and looped network is solved. 
q, C_{HW}  q = 10%, 20% C_{HW} = ±10  Triangular  –  Moosavian & Lence (2018)  Requires less computational time and suitable for larger network. Provides approximate membership functions only for nodal heads. 
For fuzzy based design of WDS  
q  Triangular  Linear programming  Xu & Goulter (1999)  Developed model is based on robust program which considers larger variation in demand but limited to smaller network.  
q  –  Trapezoidal  Linear programming  Bhave & Gupta (2004)  Loop head loss constraints are considered, hence the optimization problem increases the number of loops in the network fourfold. 
q  –  Triangular  GA  Gupta et al. (2013)  Assumes monotonous change in nodal outflows with respect to change in nodal demand. 
q  q = +10%  –  SLC  Lence et al. (2017)  – 
q, C_{HW}  q = 10% C_{HW} = ±10 of most likely value  Triangular  GA  Dongre & Gupta (2017)  Computational burden observed to be less. Applied on two benchmarks and demonstrated its effectiveness in improving design. 
Uncertain input parameters .  % Uncertainty .  Uncertainty characterization .  Optimization techniques .  References .  Remarks . 

For fuzzy analysis of WDS  
C_{HW}  C_{HW} = ±10 of most likely value  Triangular  CE  Shibu & Reddy (2011)  Assumes monotonous change along with roughness coefficient and observed on smaller network. 
q, C_{HW}  q = ±15% C_{HW} = ±10% of most likely value  Triangular  NSGAII  Haghighi & Asl (2014)  Provides problem solution more systematically and computationally efficient, hydraulic responses simultaneously analyzed in single simulation run. 
q, C_{HW}  q & C_{HW} = ±15% of most likely value  Triangular  MOPSO  Sabzkouhi & Haghighi (2016)  Computational efforts found to be less than that of previous modified GA approach but still large. 
q, C_{HW,} D  –  Triangular  –  Tsakiris & Spiliotis (2017)  Both the branch and looped network is solved. 
q, C_{HW}  q = 10%, 20% C_{HW} = ±10  Triangular  –  Moosavian & Lence (2018)  Requires less computational time and suitable for larger network. Provides approximate membership functions only for nodal heads. 
For fuzzy based design of WDS  
q  Triangular  Linear programming  Xu & Goulter (1999)  Developed model is based on robust program which considers larger variation in demand but limited to smaller network.  
q  –  Trapezoidal  Linear programming  Bhave & Gupta (2004)  Loop head loss constraints are considered, hence the optimization problem increases the number of loops in the network fourfold. 
q  –  Triangular  GA  Gupta et al. (2013)  Assumes monotonous change in nodal outflows with respect to change in nodal demand. 
q  q = +10%  –  SLC  Lence et al. (2017)  – 
q, C_{HW}  q = 10% C_{HW} = ±10 of most likely value  Triangular  GA  Dongre & Gupta (2017)  Computational burden observed to be less. Applied on two benchmarks and demonstrated its effectiveness in improving design. 
q, water demand; C_{HW}, pipe roughness; H, pressure heads; D, pipe diameter.
Fuzzy random approach
The probabilistic and fuzzy approaches discussed above consider uncertainties of either an epistemic or aleatoric nature and treats the uncertain variables as either a random variable or fuzzy variable respectively. Kwakernaak (1978) observed that both approaches are fundamentally different from each other as the prior one is based upon huge statistical information while the latter is due to insufficient data. However, as a fact, there are uncertainties that are partly random and partly fuzzy; that is, which are due to the random nature of events and also due to lack of knowledge about the same. Hence, a lot of efforts have been made (Puri & Ralescu 1986; Liu & Liu 2003) for an independent but simultaneous representation of both these uncertainties and this is referred to as the fuzzy random approach and such variables are termed as fuzzy random variables (FRV). It works on the principle that the input uncertain parameter are considered to be a fuzzy sets, which are handled through a probabilistic approach; that is, either by MCS or LHS, and the output is in the form of a fuzzy value. In the area of WDSs, the work done by Fu & Kapelan (2011) and Shibu & Reddy (2014) is found to be very important and reliable.
Reliabilitybased design of WDS
Fu & Kapelan (2011) have a pioneering work in the design and rehabilitation of WDS using FRV. Uncertain future water demand was characterized by applying the fuzzy random approach and a nodal head was represented as the fuzzy sets. The measures were defined as adopting an algorithm developed through MCS and was effectively merged with NSGAII. Shibu & Reddy (2014) adopted the CE Optimization approach for solving the multiobjective model. Since the objectives are conflicting with each other, the authors considered a single objective function of minimization of cost subjected to the constraint of reliability. The fuzzy number, which serves as prior knowledge in analyzing the parameters of PDF, was obtained by the LHS and hydraulic simulation was carried out through EPANET (Shang et al. 2008). The authors assumed that nodal demand is normally distributed, and the fuzzy mean was considered as the mean of PDF with standard deviation as 10% of the original demand. A fuzzy mean was shown by a triangular membership function having original demand as its kernel and ±5% deviation at the support. After implementing this hybrid approach, Fu & Kapelan (2011) observed that if the highreliability solution acquires under a low level of uncertainty, it will result in extremely low reliability. This forms a conclusion that neglecting uncertainty leads to underdesign of the system. Shibu & Reddy (2014) introduced % of variation in mean demand in four varied ways in terms of mean and coefficient of variation (Cv). The results obtained satisfied the demand at each node and fulfill the physical constraints such as conservation of mass and energy. The authors compared their results with those obtained by Fu & Kapelan (2011) and found a cheaper solution with the same reliability.
GENERAL DISCUSSION
In this section, some key issues that are unfolding from this review have been highlighted. It is evident that neglecting uncertainty in input parameters may significantly affect the overall performance of the system, usually depicted through reliability parameters. The quantification of uncertainty is the most crucial part to proceed with the uncertainty based design of systems. In most of the studies, the percentage of uncertainty in water demand has been assumed, without any explanation for the same. The type of uncertainty in the characterization of water demand is majorly based on assumptions. These assumptions are made due to the lack of actual data available and also due to lack of knowledge about such parameters. Anyhow, to achieve more clarity regarding uncertainty, adequate work should be carried out in the direction of uncertainty quantification in order to support those assumptions. This is especially necessary as incorporation of uncertainty increases the cost of the network. However, for some parameters uncertainty quantifications may be difficult, such as for the roughness coefficient.
The various types of representations for handling the uncertain parameters have been observed in the literature review as random, fuzzy, and fuzzy random. Generating the data for variation in pipe roughness coefficients is difficult and can be more appropriately represented as fuzzy variable due to impreciseness. Water demand, which has both randomness at a given time and impreciseness in prediction, can be considered as a fuzzyrandom variable. It is necessary that uncertainty in both the parameters is considered in reliabilitybased design of WDNs.
The reliabilitybased optimal design requires evaluation of the performance of the network under various conditions. Hence, it requires more computational efforts. The probabilistic approaches use MCS or LHC to generate various scenarios and when combined with evolutionary techniques like GA, make the algorithm difficult to use for design of large networks. Further, probabilistic approaches necessitate statistically determined parameters for the representation of uncertainty in variables, which requires an extensive amount of data to obtain the statistics, hence making the process more computationally demanding. The fuzzy approaches are better, but they have the same drawback when MCH and LHS are used with GA in reliabilitybased design. Further, some of the approaches are observed to be capable of considering only nodal demands. There is a need for a simple approach that can include all types of uncertain parameters and can be used for design of large networks with less computational efforts.
SUMMARY AND CONCLUSIONS
Amongst the several types of uncertain parameters, water demand and the pipe roughness coefficient are observed to affect the network performance significantly and recommended for consideration in reliabilitybased design of WDNs. The proper quantification of the uncertainty in these parameters is very important as it has a direct impact on cost of the network. In literature, various models and methods like ANN, ARIMA, and Fuzzy, have been used for water demand forecasting. However, most of these models predict future water demands only for short term periods, also the reliability or accuracy of the models are in question. The design and expansion of any network requires water demand prediction for a long term and developing a proper model for the same is required.
These uncertainties are modeled using probabilistic, fuzzy or fuzzy – probabilistic approaches. The probabilitybased approach demands a huge amount of data for defining the PDF of a parameter and MCS, FOSM, FORM or LHS for deriving the PDF of an output parameter. The approach proves to be quite exhaustive; however, it may be applied when sufficient data are available. The fuzzy approach proves to be ideal for insufficient data. Many researchers have treated water demand as either a random variable or a fuzzy variable. However, the uncertainty in water demand is due to both its random nature and imprecise knowledge about the same. Hence, it is more appropriate to treat such parameters as fuzzyrandom variables. Even though several approaches are available for fuzzy demand, no approach is available for considering both fuzzy demand and the fuzzy pipe roughness coefficient in the design of WDNs. Fuzzy analysis using impact table (Gupta & Bhave 2007) is fast in developing membership function and the approach of Fu & Kapelan (2011) considers water demand as FRV. This can be clubbed with the impact table method to include both uncertainties in nodal demands and pipe roughness coefficients. Also, the methodology of Dongre & Gupta (2017) that uses impact table method can be extended to include both nodal demands and pipe roughness coefficients as uncertain parameters.
Overall, this review aims to encourage the modelers and designers with the selection of suitable approaches for the characterization of the uncertain parameters involved in their models. An additional ambition of this review is to contribute towards a change of the status quo of uncertainty characterization approaches from probabilistic and fuzzy to combined fuzzyprobabilistic approach that enables more reliable design of the system.