Uncertainty analysis of water supply networks using the fuzzyset theory and NSGA-II

Ali Haghighi n, Arezoo Zahdei AslCivil Engineering Department, Faculty of Engineering, Shahid Chamran University of Ahvaz, Ahvaz, Iran

a r t i c l e i n f o

Article history:Received 25 August 2013Received in revised form5 February 2014Accepted 12 February 2014Available online 12 March 2014

xxxKeywords:Water supply networksFuzzy set theoryNSGA-IIUncertainty

a b s t r a c t

This work introduces an approach for taking into account the uncertainty of pipe friction coefficients andnodal demands in the hydraulic analysis of water supply networks. For this purpose, uncertainties arerepresented by fuzzy numbers and incorporated into the network's governing equations. Inputuncertainties are spread out on the network and influence its hydraulic responses, including pipevelocities and nodal pressures. To estimate the responses' uncertainty, input fuzzy numbers arediscretized in some levels of membership function. Then, a multiobjective optimization problem isdeveloped for each level to find the extreme values of the node pressures and pipe velocities. The raisedproblem is solved using the method of Non Dominated Sorting Genetic Algorithm (NSGA-II) coupled tothe network hydraulic simulation model. The proposed approach is applied to an example and a real pipenetwork. It is found that small uncertainties in input variables can significantly influence the network'sresponses as well as its performance reliability. It is also concluded that NSGA-II has a great role insolving the problem systematically, and improves the computational efficiency of the whole process ofnetwork fuzzy analysis.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

For designing a new water distribution network or for condi-tion assessment; calibration and extension of an existing one, it isrequired to simulate the network hydraulics. In this regard, nodalpressures and pipe velocities are the responses of interest, whichmay be calculated against several design scenarios or calibrationvariables. In fact, any decision on the network is taken based onthe aforementioned responses. Hence, the reliability of the resultsobtained by the hydraulic simulation models is quite crucial toevery decision making on the network development and rehabi-litation. The results' reliability highly depends on accuracy of themathematical model in one side, and the precision of inputvariables in another side. The latter is always a major concernwith mathematical simulations and can significantly affect thecredibility of the results. Some physical parameters used in thehydraulic modeling of water networks cannot be precisely mea-sured and need to be estimated by engineering judgments. Thisissue results in uncertainty in input variables and consequently, inthe network's responses. Major uncertainties in the analysis ofpipe networks are resulted from the estimation of pipe frictionfactors and nodal demands, which are not only imprecise in naturebut also significantly change over time. For instance, the network

future demands are roughly estimated according to the availableexperiences and based on statistic data of the population growthand water consumption patterns. Furthermore, pipe friction coef-ficients as a function of pipe roughness and fluid specifications arecontinuously under the influences of increasing corrosion anddeposition and changes in quantity and quality of water in thesystem. Depending on the flow characteristics and the network'sgraph configuration, input uncertainties are spread out over thesystem and impress the network's responses and performance.For considering these issues, it is required to develop a hydraulicsimulation model that can handle such uncertainties in the flowcalculations.

Apart from how the uncertainty resources are identified andquantified in a system, there is a challenging issue to apply themas imprecise values to the governing equations. In recent years,this importance has been the subject of interest of many investi-gators in various fields of engineering. For an engineering system,it is aimed at developing a mathematical model to find out that inxxxwhat extent the input uncertainties are spread over the outputresponses. For a long time, the theory of statistics and probabilitywas the predominant approach for handling uncertainties insimulations. In this context, the Monte Carlo method has beenone of the most popular approaches for representing and analyz-ing uncertainties in engineering systems as well as in pipe net-works (Bargiela and Hainsworth, 1998; Bao and Mays (1990); Duanet al., 2010; Aghmiuni et al., 2013). Generally, in order to develop

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/engappai

Engineering Applications of Artificial Intelligence

http://dx.doi.org/10.1016/j.engappai.2014.02.0100952-1976 & 2014 Elsevier Ltd. All rights reserved.

n Corresponding author.

Engineering Applications of Artificial Intelligence 32 (2014) 270–282

a reliable probability distribution for every parameter havinginherent uncertainty, extensive measurements and empirical dataare required. Using the Monte Carlo simulation, the system'sresponses are over and over calculated, each time using a set ofrandom values from the probability functions. It finally producesdistributions of possible outcome values. Depending upon thenumber of uncertainties and the range of variations specified forthem, a Monte Carlo simulation could involve more than tens ofthousands of recalculations to be completed. In many engineeringproblems faced with significant uncertainties, e.g., pipe networksanalysis and design, the Monte Carlo approach can be computa-tionally very expensive and burdensome.

An alternative approach for the uncertainty analysis is theapplication of fuzzy set theory and fuzzy logic. Fuzzy set theorywas introduced by Zadeh (1965). Principles of the theory in bothterms of mathematics and applications were then developed byhim and his colleagues as well as by Mamdani and Assilian (1975),Sugeno (1985), Zimmerman (1985), Buckley (1987), Pedrycz(1989), Kandel (1992), Bit et al. (1992), and Bardossy andDuckstein (1995). The fuzzy set theory was initially intended tobe an extension of dual logic and/or classical sets theory(Zimmermann, 2010) however, during the last decades, the con-cept of fuzziness has been highly developed in the direction of apowerful ‘fuzzy’ mathematics. Many scientific and engineeringapplications can be named which have successfully exploited thefuzzy set theory as a strict mathematical framework. By fuzzymathematics, vague conceptual phenomena can be more preciselystudied. In this view, the fuzzy set theory can be taken intoaccount as a serious rival to the probability theory for theuncertainty analysis in engineering problems. This capability wasfirst pointed out by Zadeh (1973) when he said: “as the complexityof a system increases, our ability to make precise and yetsignificant statements about its behavior diminishes until athreshold is reached beyond which precision and significance (orrelevance) become almost mutually exclusive characteristics.”At present, the fuzzy set theory is a popular tool for analyzingfuzzy systems, which are equivocal in nature and contain ambi-guity in variables and process. Furthermore, this theory has beenwidely used in different areas of water resources engineering, forexample in hydrology (Pesti et al., 1996; Pongracza et al., 1999;Ozelkan and Duckstein, 2001; Bardossy et al., 2002; Srinivas et al.,2008; Srivastava et al., 2010), water quality (Sasikumar andMujumdar, 1998; Nasiri et al., 2007; Ghosh and Mujumdar,2010), ground water (Guan and Aral, 2004; Dixon, 2005;Kurtulus and Razack, 2010), urban flood management (Changet al., 2008; Fu et al., 2011), reservoirs operation (Saad et al.,1996; Cheng and Chau, 2001; Karaboga et al., 2004), riverengineering (Mujumdar and Subbarao, 2004; Ozger, 2009; Kisi,2010) and pipe hydraulics (Revelli and Ridolfi, 2002; Branisavljevicand Ivetic, 2006; Gupta and Bhave, 2005; Haghighi and Keramat,2012; Spiliotis and Tsakiris, 2012). Focusing on the theme of thisstudy, number of selected investigations that applied the fuzzy settheory to hydraulic simulation of piping systems are brieflyreviewed through the next paragraphs.

Revelli and Ridolfi (2002) published an outstanding paperin which for the first time the fuzzy set theory was used forconsideration of uncertainty in analysis of water distributionnetworks. They coupled a hydraulic simulation model with agradient-based mathematical optimization method. There, thepipe friction factors and nodal demands were introduced by fuzzynumbers. Through that work, to find the extreme values of eachresponse, the optimization is applied twice to the hydraulicsimulation model in each level of uncertainty. The procedure ismany times performed until all node pressures and pipe dis-charges in the network are analyzed in all levels of uncertainty. Itwas concluded there that the fuzzy responses cannot be explicitly

obtained without using optimization. In fact, for every extremenodal pressure and pipe discharge, there is a critical combinationof input uncertainties, which need to be found with the aid ofoptimization. In spite of the use of a fast classical optimizationtechnique in that investigation, the whole procedure takes a lot oftime for computations as well as it may become trapped in localsolutions, particularly in large and complex networks.

Branisavljevic and Ivetic (2006) guided a valuable study toinvestigate the influences of uncertainties on different responsesof pipe networks. They found that when the pipe friction factorsare considered as the input uncertainties and simulated by fuzzyvariables, the network's responses can be classified in two groups.First, there are node pressures that depend on the input variablesin a monotonic way and second, there are pipe velocities thatdepend on the input variables in a non-monotonic way. The nodepressures can be analyzed by the hydraulic simulation model in adeterministic iterative way. This is while, for the pipe velocities,the hydraulic model needs to be coupled to an optimization solver.As a consequence, in a general formwhere both types of responsesare of interest, and system may include other uncertainties like inwater demands, the problem becomes more complicated, and theoptimization is inevitably required.

Gupta and Bhave (2005) paid a special attention to the com-plexity of the raised optimization in the previous studies andproposed an approach to simplify the problem. They consideredthe fuzziness of both friction factors and water demands. It wasassumed that the maximum impact of the input uncertainties onthe responses occurs when the input parameters are at theirextreme values. On this basis, a deterministic iterative approachwithout optimization was proposed to obtain the extreme valuesof responses. Although the optimization was omitted in thatapproach, but it still needs heavy combinatorial computations tofind the extreme responses by changing the input variables fromone extreme to another extreme point for all input variables in alllevels of uncertainty.

Spiliotis and Tsakiris (2012) introduced a fuzzy approach foraddressing the uncertainty of water demands by the Newton–Raphson method. It was shown that if pipe friction factors areconsidered to be constant, the principle of monotony of thecontinuity equation in the network junctions can be exploited. Ifso, the optimization procedure for handling fuzzy variables wouldnot be required. Thereby, the investigators suggested an explicitalgorithm for the fuzzy analysis of pipe networks under fuzzywater demands. However, the method needed to manipulate thegoverning equations and to develop a particular analysis solver forthe network's hydraulics. This can be burdensome for large andcomplex networks. In addition, the uncertainty of friction factors isstill a major concern with that method.

Haghighi and Keramat (2012) introduced a model for studyinguncertainty in the transient analysis of pipe networks. It wasconsidered that the water demands; pipe friction factors and wavespeeds include uncertainty. These parameters were represented byfuzzy numbers and introduced to the transient analysis solver. Ineach level of uncertainty, two objective functions were defined atevery network junction to minimize the lowest and maximize thehighest pressures in the excited transient flow. The simulatedannealing method was used there for optimization of the objectivefunctions. The solver was many times applied to the transientsimulation model to evaluate the fuzzy responses of the networkagainst the input uncertainties. Burden of computations in bothparts of transient analysis and optimization was problematic inthat work and made the whole process computationally expen-sive. It was concluded that in a generalized fuzzy analysis, differentinput variables such as friction factors, node demands and reser-voir levels may include different uncertainties and can be simu-lated by different shapes of fuzzy numbers. This issue, especially

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282 271

in complex and large networks, causes changes in the responsesin a non-monotonic fashion with the input variables. In fact, forevery response, there are two different combinations of inputuncertainties that result in its extreme high and low values.Searching for these critical combinations and their correspondingcritical responses needs the use of optimization techniques.

This research considers that the iterative and nonsystematicuse of single objective optimization for fuzzy analysis of pipenetworks highly increases the complexity and computational loadof the problem solution. To handle this issue, uncertainties arerepresented by fuzzy numbers with a given membership functionto apply to the governing equations. Fuzzy variables are discre-tized in some levels, and the network's extreme responses areevaluated in each level separately. To analyze the hydraulicresponses' uncertainty, a multiobjective optimization problem isdeveloped. The raised problem consists of a large number ofobjective functions to evaluate the minimum and maximumvalues of all node pressures and pipe velocities. For solving theproblem, the method of Non Dominated Sorting Genetic Algorithm(NSGA-II) with some modifications is coupled to a hydraulicsimulation model. It is applied against an example pipe networkand a real case study, and the obtained results are finallydiscussed. The methodology and main components of the modelare in details described in the following.

2. Fuzzy stet theory and the uncertainty representation

For designing a pipe network it is required to compute thenodal pressures and pipe velocities against design scenarios tocheck whether they satisfy the design criteria or not. In commonapproaches, a fixed unique value is assigned to every designvariable upon some information on system, technical manualsand previous experiences. The network is analyzed against theunique input variables and returns the unique output responses.This is while, with no doubt, every design variable may includeuncertainty. Major uncertainties in the hydraulic analysis of waterdistribution networks are with the estimation of pipe frictioncoefficients and nodal demands. Depending on the networkconfiguration and the governing flow specifications, input uncer-tainties are stochastically distributed over the system and result inuncertainty in the responses too. For a reliable design, estimationof the responses' uncertainty is of the central importance. Thisissue is followed up here by using the fuzzy set theory.

In the fuzzy set theory, the uncertain parameters are repre-sented by fuzzy numbers. A fuzzy number N is, in fact, a set of realnumbers so that N� R. For a normalized fuzzy number N, there isa continuous membership function μNðxÞA ½0; 1� that describes towhich grade a variable xAN belongs to N. When μNðxÞ ¼ 0 it meansthat x is “not included” and when μNðxÞ ¼ 1 it means that x is a“fully included” crisp number in the fuzzy set. In caseof 0o μNðxÞo1, x is a number that is partially belonging to Nwith the membership of grade μNðxÞ. Fig. 1. illustrates a triangularnormalized fuzzy number defined by three special values for xconsisting of xa, xc and xb such that xaoxcoxb. As seen in Fig. 1,the membership function μ at xa and xb is zero while at xc is one.xc is the most likely value of N and the interval ½xa; xb�, where themembership function at its extreme points is zero is called thesupport of the fuzzy number N. One of the useful approaches forhandling fuzzy numbers in complex engineering systems is theα�cut decomposition method. An operation so-called α�cut,where αAμN , is applied to the fuzzy set to facilitate the interpreta-tion of the fuzziness in engineering systems, which are originallydeveloped on the basis of fixed crisp variables. α¼ 0 and 1 respec-tively correspond to the support interval and the most likely valueof N. In general, every α�cut on N creates a sub-fuzzy number Nα

whose support is interval ½xa;α; xb;α� and the most likely value isstill xc . In this study, the fuzzy number is used to representuncertainty in a variable like x which is estimated to be about xc.However, due to the uncertainty at level α, it can also take a valuebetween xa;α and xb;α . Lower α�cuts introduce more uncertaintiesfrom the input variables to the system under consideration. In thefuzzy set theory, this issue is known as the fuzzy numberconvexity, which mathematically implies that for two arbitraryα1 and α2 if α1oα2 then Nα2 �Nα1 and in other words, xa;α1 oxa;α2and xb;α1 4xb;α2 .

For uncertainty analysis of pipe networks, it is considered thatpipe friction coefficients ~C as well as nodal demands ~q includeuncertainty and are treated in simulations by the triangular fuzzynumbers. The capital sign ″� ″ is to emphasize that the variablehas uncertainty and is represented by the fuzzy number. The mostlikely value ~Cc and ~qc is assigned to every variable on the basis ofavailable information and possible estimations. Then, upon theknowledge one has about the accuracy or inaccuracy of the dataand relying on engineering judgments the support of each vari-able, i.e., ½ ~Ca; ~Cb� and ½ ~qa ; ~qb � is introduced. As a consequence, atriangular fuzzy number for every input fuzzy variable is formed.This makes it possible to translate the qualitative terms like“about, close to, probably and approximately” with the variables,in mathematical expressions to take them into account in numer-ical analyses. When fuzzy variables ~C and ~q are introduced to anetwork, the applied analysis model needs to evaluate the fuzzyresponses including, pipe velocities ~V and nodal pressures ~P . Thistask implies that the model must be able to find out that in whatextent the input uncertainties are spread over the system andreflected in the responses. This issue cannot be straightforwardlyfulfilled by the common approaches and available simulationmodels like EPANET, which works on the basis of unique inputvariables and gives unique output responses. For analysis of pipenetworks having fuzzy variables, the hydraulic simulation modelneeds to be equipped with other mathematical techniques asdiscussed in the following.

3. Fuzzy analysis model

Two conservative rules of flow continuity in nodes and energyin loops govern the steady-state flow distribution in pipe net-works. Flow continuity in every node and energy conservation inevery loop is respectively met by the following equations.

∑Q inþ∑Qout ¼ q for every node ð1Þ

∑hf þ∑EP ¼ 0 for every loop ð2Þ

where Q in and Qout ¼ respectively, the flow discharges toward andout of the node, q¼ the node demand, EP ¼ the energy added tothe system by pumps and hf ¼ head losses computed for all pipesin the loop. hf can be estimated by Darcy–Weisbach (DW),Manning and Hazen–Williams (HW) equations however, the latter

Fig. 1. A triangular normalized fuzzy number.

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282272

is more popular in practice. The HW is an empirical equationwhich, can be written in a general form as follows.

hf ¼ωLQα

CαDβ ¼ ðZu�ZdÞþPu�Pd

γ

� �ð3Þ

in which, C ¼ the HW coefficient, ω¼ numerical constantdepending on the problem's units. ω is practically ranging from10.5 to 10.9 in the metric system. α and β are the HW exponentswhich are respectively 1.85 and 4.87, D¼ pipe diameter, L¼ pipelength, Z ¼ elevation from datum, P ¼ pressure, γ ¼ specific gravityand subscripts u and d respectively denote the upstream anddownstream ends of the pipe.

The above equations are developed for all nodes and loops innetwork and result in a system of nonlinear equations. This systemmay be arranged with respect to either nodal heads or pipedischarges as the problem unknowns. In any case, the equationsmust be solved using a numerical method such as the lineartheory or gradient method (Todini and Pilati, 1987) which isutilized in EPANET. The governing equations and the solver acceptonly the unique values for the input parameters like the HWcoefficients ðCÞ and nodal demandsðqÞ. When these variablesinclude uncertainty and are represented by fuzzy numbers ~C and~q, the system's unknowns are obtained as fuzzy pressures ~P andfuzzy discharges ~Q and velocities ~V too (Fig. 2). In this condition,instead of unique values for input variables, an interval of possiblevalues is introduced to the network. Thereby, for each response,there will be infinite combinations of the input fuzzy variables.The aim of fuzzy analysis is not to find all these combinations for aresponse but is to obtain the interval that the responses of thesecombinations belong to. For this purpose, the following algorithmproposed by Revelli and Ridolfi (2002) is adopted here.

The input fuzzy numbers, ~C and ~q, are discretized with alimited number of α�cuts including α¼ 0 and α¼ 1 and a fewnumbers between them.

1 First, the cut α¼ 1 is taken into account at which, all inputvariables take their most likely values, ~Cc and ~qc . EPANET is runand the most likely values of network's responses, ~Pc and ~V c ,are obtained.

2 Another α�cut is considered, according to that the rangeof variations of input variables, ½ ~Ca;α; ~Cb;α� and ½ ~qa;α ; ~qb;α�, isdetermined from the input fuzzy numbers (Fig. 1).

3 In every α�cut, the fuzzy analysis of the pipe network istreated as an optimization problem. For every response, twosingle-objective optimization problems are introduced suchthat the response is the objective function that is once max-imized and once minimized while, the input fuzzy numbers are

the decision variables. The α�cut under consideration deter-mines the limits of the decision space. This mathematicalprogramming can be written as follows,Maximize|Minimize

~Pi ¼ φð ~C1; 2; 3;…; m; ~q1;2;3;…;n Þ; i¼ 1; 2; 3;…; n ð4Þ

~V j ¼ φð ~C1; 2; 3;…; m; ~q1;2;3;…;n Þ; i¼ 1; 2; 3;…; m ð5Þsubject to,

~Ca;αr ~Cr ~Cb;α and ~qa;αr ~qr ~qb;α ð6Þwhere n and m¼ respectively, the number of nodes and pipesin network and φ is, in fact, the hydraulic simulation model,EPANET, which acts as a function of input uncertainties andreturns the network's responses. After that the above program-ming problem was optimized for the whole network, theobtained maximum and minimum objective functions resultin the range of variations of the network's responses corre-sponding to the α�cut at hand, i.e., ½ ~Pa;α; ~Pb;α� and ½ ~V a;α ; ~V b;α�for all nodes and pipes. Fig. 2 schematically illustrates theconnection between the optimization and EPANET in thedescribed scheme.

4 Next α�cut is taken into account and the procedure iscontinued until the support of fuzzy numbers ðα¼ 0Þ is calcu-lated too, and the shape of the membership function for allfuzzy responses is completed.

When the above algorithm is viewed as a close package, it canbe mentioned that the fuzzy analysis of pipe networks is a multi-criteria decision making problem with many objective functions.However, the challenges between the objectives do not matterhere. One approach to solve this problem is to optimize eachobjective function separately by a single objective solver. If so, forevery α�cut and for every node pressure or pipe velocity, it isrequired to carry out the optimization twice, to find the upper andlower extremities of the response (Fig. 2). In this context, Revelliand Ridolfi (2002) used the mathematical method of QuadraticProgramming (QP) and Haghighi and Keramat (2012) used themetaheuristic of Simulated Annealing (SA). By using this approach,for fuzzy analysis of the whole network the optimization must be2� ðmþnÞ times applied to the problem for every α�cut.This issue is the main concern with the procedure that makesthe fuzzy analysis computationally inefficient and burdensome toimplement.

In spite of all the aforementioned complexities for the fuzzyanalysis of pipe networks, the problem has some particularfeatures that help solve it more efficiently if they are appropriately

EPANET

Fuzzy HW coefficients

Fuzzy nodal demands

Optimization

Acombinationof and

Network’s configuration and other fixed design parameters

Crisp velocities

Crisp pressures

Finally

One extremity of

cut operator

Fig. 2. A cycle of optimization-hydraulic simulation for the fuzzy analysis of pipe networks.

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282 273

utilized. First, all objective functions have the same decisionvariables and decision space. Second, the purpose of the optimiza-tion is only finding the extreme values of pressures and velocitiesin every α�cut and the trade-off between them is not important.Third, all objectives are evaluated by a common function named asφ in the above mathematical programming. Function φ is, in fact,the EPANET hydraulic simulation model which returns all nodepressures and pipes velocities once it is run against a combinationof input uncertainties (Fig. 2). In other words, every singlesimulation run of EPANET contains all objective values in a certainset of decision variables. Utilizing this characteristic in the opti-mization allows to evaluate a candidate solution simultaneouslyfor all objective functions that, leads to exploit the hydraulic solvermuch more efficiently. In approaches that the problem is treatedas an iterative single objective optimization, most computationalefforts and results of the hydraulic solver are wasted since at eachtime only one objective is optimized, and the others are ignored(Fig. 2). Consideration of the above three features implies that thecurrent fuzzy analysis problem can be more efficiently solved if itis treated as a multiobjective optimization. For this purpose,the optimization box in Fig. 2. is equipped with a multiobjectiveoptimization solver which is fed back with all objective functionsonce EPANET is run. In this study, to optimize functions of (3) and(4) a version of NSGA-II method is utilized.

4. NSGA-II for fuzzy analysis

The Non-dominated Sorting Genetic Algorithm abbreviated asNSGA (Srinivas and Deb, 1995) was one of the first evolutionaryalgorithms that utilized the principle of Pareto optimality insolving multiobjective problems. Deb et al. (2002) raised somecriticisms to the NSGA and developed a powerful approach knownas NSGA-II. In comparison with the previous version, the NSGA-IIhas a less computational complexity, considers elitism, system-atically preserves the diversity of Pareto-optimal solutions andadaptively handles the problem constraints. These features havemade the NSGA-II very successful and popular in a wide range ofengineering problems.

As earlier discussed, for the fuzzy analysis of a pipe network itis required to solve a special multiobjective problem. Herein, amultiobjective genetic algorithm on the basis of the NSGA-II isutilized. Before that, the fundamentals of the method are brieflydescribed and the main steps of the standard NSGA-II is quicklyreviewed.

The main concept behind every non-dominated sorting multi-objective optimization is to identify and promote the solutionsthat dominate other solutions. The basic definition for the notionof dominancy in multiobjective optimization is that; the solution xdominates the solution y if both following conditions are simulta-neously met:

1. None of the objectives in x is worse than in y.2. At least one objective in x is better than in y.

In solving real-world problems other aspects may be alsoadded to the above definition as Deb et al. (2002) did whendeveloping the NSGA-II. They proposed a new operator that notonly fulfils the above criterion but also, in lower levels, preservesthe diversity of solutions in the Pareto fronts as well as thefeasibility of the search. Through the next paragraph, the mainskeleton of the NSGA-II is briefly described. Afterward, thedifferences between the multiobjective optimization raised inthe fuzzy analysis of pipe networks with the common multi-objective problems are addressed and accordingly, the appliedNSGA-II in this study is introduced.

Standard NSGA-II: The optimization is begun by an initialpopulation that is randomly generated and then evaluated by allobjective functions. The population is sorted based on the non-domination principle. Each chromosome is then assigned a ranknumber equal to its non-domination level. For each level, a Paretofront is introduced in which the corresponding chromosomes arestored. Solutions belonging to the best non-dominated fronts aredirectly transferred to the mating pool to create the next genera-tion. In conventional multiobjective problems, it is also importantto maintain a good spread of the solutions in the optimum fronts.To this end, NSGA-II employs the crowded-comparison scheme forpreserving diversity among the population members. For everychromosome in a Pareto front, a crowding distance is measured asthe distance of the biggest cuboid contacting the two neighboringsolutions as shown in Fig. 3. For each objective function, theboundary solutions having the minimum and maximum functionvalues (points A and B in Fig. 3) are assigned a huge distance value,e.g., infinity. These solutions are, in fact, the extremities of thenon-dominating front and must be emphasized more than inter-mediate solutions. As a consequence, every chromosome in thepopulation will have two attributes, including the non-dominationrank (or Pareto front level) and crowding distance. Thereby, ahybrid domination criterion namely the crowded-comparisonoperator is applied to the selection process so that it is guided atthe various stages of the algorithm toward a uniformly spread-outPareto-optimal front (Deb et al., 2002). Simply speaking, thecrowded-comparison operator is so applied to the population thatbetween two solutions with different non-domination rank, thesolution with lower rank is selected. However, if both solutions havea same rank, the one with higher crowded distance is preferred.For the selection process, the binary tournament method is appliedto the parent population P. Then; the selected parents are matedto generate the new offspring. After mutating new members, the

A

B

The crowding-distance

Fig. 3. The crowding-distance in the standard NSGA-II.

A

B

The closeness-distance ,

Fig. 4. The closeness–distance in the proposed algorithm.

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282274

offspring population Q is formed and merged into the parentpopulation resulting in a combined population R¼ P[Q [ . Thepopulation R is sorted based on the non-domination, and the Paretofronts with different levels are formed. Then, the best half of thecombined population R is transferred to the next generation. Since allparents and offspring chromosomes are included in R, elitism isautomatically provided. The procedure is continued until the desiredconvergence is achieved.

Deb et al. (2002) also proposed an adaptive approach forconstraint handing in NSGA-II for solving constrained multiobjec-tive optimization problems. Similar to the crowded-comparisonoperator, their constraint handing method also uses the binary

0PGenerate

Tournament Selection

Stopping Criteria?

Population Sorting

Non-dominated Sorting

Closeness- Distance Sorting

No

Start

EPANET

Crossover and Mutation

Children Population Qi

Combined Population Pi-1UQi

New Population Pi

Stop

Yes

i =i+1

i =0

EPANET

Population Sorting

Non-dominated Sorting

Closeness- Distance Sorting

Fig. 5. The flowchart of NSGA-II applied to fuzzy analysis of pipe networks.

q = 150 l/s

q = 300 l/s

q = 260 l/s

Z = 0Z = 0

Z = 0

Z = 100

42

3

1

[4]

[1]

[5]

[3] [2]

D=350

mm

L=900

m

D=500 mm

L=1200 m

D=350 mm L=1000 m

D=500 mm

L=1500 m

D=500 mm

L=1100 m

Fig. 6. Example pipe network (Revelli and Ridolfi, 2002).

Table 1Maximum uncertainty for the Hazen–Williams coefficients.

Pipe No. ~Ca;α ¼ 0~C crisp

~Cb;α ¼ 0

1 63.19 76 82.872 73.26 80.5 87.743 79.87 88 95.664 67.1 75 80.365 72.58 78 86.94

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282 275

tournament selection in such a way that two solutions are pickedfrom the population, and the better one is selected. In the presenceof constraints, a new definition for domination is defined andreplaced for the previous definition. In constrained problems,solution x dominates y only if,

1. x is feasible and y is not.2. x and y are both infeasible, but x has a smaller overall

constraint violation (with respect to all objective functionsand constraints).

3. x and y are both feasible, and x dominates y by the previouslyintroduced criteria.

This was the main structure of the NSGA-II which has been sofar applied to many complex multiobjective optimization pro-blems successfully. However, the current problem in some sensesis different from the common multiobjective problems. The maindifferences were already pointed out. In this problem, the numberof objective functions is very large but the trade-off between themdoes not matter. The main goal of the current multiobjectiveoptimization is only to find the optimum boundary solutions, andthe diversity preservation among the solutions is not important.On the contrary, it is desired here to guide the solutions in Paretofronts toward the boundary solutions, i.e., the maximum andminimum values of each objective function. This issue requiresan absolutely different definition for the crowded-comparisonoperator in the selection process. With respect to this issue and

the fact that the problem is unconstrained, a new version of NSGA-II for the fuzzy analysis of pipe networks is introduced as follows.

1 An initial random population of size N is generated.2 The flow simulation model is called to analyze the network

against the population. All M objective function values, pres-sure and flow responses, are evaluated for every individual by asingle one run of EPANET.

3 The population is ranked based on the non-domination, andthe Pareto fronts from level 1 to l are formed.

4 Instead of the crowding distance in the standard NSGA-II, anew metric is defined here to measure the distance betweenevery solution and the extreme points in the Pareto front. Thismetric named as the closeness-distance is computed for everyindividual i (i¼ 1 to N) with respect to every the objectivefunction j (j¼ 1 to M) by the following equation,

di;j ¼f ij� fmin

j

fmaxj � fmin

j

ð7Þ

where fminj are fmax

j are respectively the minimum and max-imum values of the objective function j in the population, andf ij is the value of the objective function j in the solution i. Thecloseness-distance schematically shown in Fig. 4. di;j is anormalized dimensionless value which differs from zero inthe boundary solution A to 1 in the most distant solution,i.e., solution B in the Pareto front (Fig. 4). After that forall individuals the closeness-distances with respect to all

0

0.5

1

μ

P2 (m)

0

0.5

1

μ

P3 (m)

0

0.5

1

60 65 70 75 80 55 60 65 70 75 55 60 65 70 75

μ

P4 (m)

NSGA-II QP

Fig. 7. Membership function of the fuzzy pressures in example 1.

0

0.5

1

μ

Q1 (l/s)

0

0.5

1

μ

Q2 (l/s)

NSGA-II QP

0

0.5

1

0.4 0.425 0.45 0.475 0.5 0.175 0.205 0.235 0.265 0 0.05 0.1 0.15

μ

Q3 (l/s)

0

0.5

1

μ

Q4 (l/s)

0

0.5

1

0.15 0.2 0.25 0.02 0.06 0.1

μ

Q5 (l/s)

Fig. 8. Membership function of the fuzzy discharges in example 1.

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282276

objective functions were computed, the minimum of di;1 to M isassigned to i as its selected closeness-distance. Thereby, theboundary solutions, points A and B in Fig. 4, as well as thesolutions close to them are emphasized through the selectionprocess.

5 The selection of parents is done according to the non-domination and closeness-distance criteria. Using the tourna-ment method, two individuals x and y are randomly picked upfrom the population. With respect to the non-domination ranknumber and the selected closeness-distance, between x and ythe one wins the tournament that (a) belongs to a better levelof Pareto front and (b) has a smaller closeness-distance if theyboth belong to a same front.

6 An appropriate crossover technique is applied to the selectedparents to create N offspring. In this work, the blend crossovermethod (BLX-α) proposed by Eshelman and Shaffer (1993) isadopted.

7 A few genes of the new chromosomes are randomly mutated.8 The children and parents' populations are combined together.

The resulted population is sorted based on the Pareto non-domination criterion.

9 Old individuals are replaced by better offspring, and the newgeneration is formed.

10 If the stopping criteria are met the optimization is terminatedotherwise; it goes to step 2.

At the end of optimization, the boundary solutions in the firstPareto-optimal front represent the extreme fuzzy responses of thenetwork according to the α-cut introduced to the input variables.The described procedure has been also presented by a flowchart inFig. 5.

1.09

3.692.55

2

8 7 6

13 121415

171819203231

27 26 25 24 23 22 21

35 34 33

40 39 38

4544

45

11 [2][9][13]

[6][8]

[4][5][7][10][11]

[16][18][20]

[15][17][19][27]

[23][26]

[22][24][25][36][40][44]

[30][32][35][39]

[43]

[42]

[29][31][33][34][37][38][41]

[48][51]

[46][47][49][50]

[54][57][59]

[53][55][56][58]

[63][65]

[62][64]

2.291.95

5.213.15

2.414.325.834.09

4.34

7.01

67.13

2.676.394.593.9511.49.97

17.46

1.524.273.67

4.695.02

3.1

4.74

2811.26

3.564.122.73.91

374.05

42

43.89

4.9 4.034.933.014.04

2.51

5.54469.93

11.6930

10

9

16

3

[12] [3]

[14]

[21]

[28]

[45]

[61]

[52]

29

36

41

43[60]

[10] Pipe name

Node name8 7

4.09Nodal demand (l/s)

Reservoir

Legend

Fig. 9. Pipe network of the case study.

Table 2Pipe lengths and inner diameters.

Pipe No. Diameter (mm) Length (m) Pipe No. Diameter (mm) Length (m)

1 572.8 25 34 181.8 204.12 322.6 210.1 35 409 1413 254.4 100 36 454.4 2004 227.2 158.8 37 322.6 1105 363.6 50 38 322.6 906 454.4 100 39 113.6 1417 254.4 192.9 40 227.2 2008 363.6 115.5 41 181.8 2009 409 250.8 42 181.8 91

10 181.8 107.1 43 145.4 5011 100 100 44 145.4 20012 145.4 100 45 227.2 15013 363.6 149.2 46 100 202.114 145.4 200 47 322.6 10015 181.8 156.1 48 322.6 15016 454.4 100 49 113.6 9017 322.6 126.9 50 145.4 20018 322.6 231.2 51 145.4 15019 322.6 223.1 52 254.4 15020 227.2 200 53 145.4 200.121 227.2 100 54 227.2 15022 100 223.7 55 322.6 11023 322.6 115.6 56 100 9024 409 281.8 57 113.6 15025 322.6 100 58 181.8 20026 409 100 59 363.6 10027 454.4 100 60 113.6 5028 181.8 141 61 227.2 175.129 181.8 142 62 145.4 308.430 100 163 63 181.8 156.231 100 362.9 64 254.4 290.632 113.6 141 65 322.6 138.433 113.6 95.91

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282 277

5. Examples

In this part, the proposed algorithm is applied against two pipenetworks. The first example is a small network taken from theliterature, and the second one is a real case study.

5.1. Example 1

To evaluate the proposed fuzzy approach, a small pipe networkoriginally introduced by Revelli and Ridolfi (2002) is taken intoaccount. The network as shown in Fig. 6, is consisting of two loops,five pipes and four nodes. The pipe diameters and lengths as wellas nodal demands and elevations are given in Fig. 6. In thisexample, it is supposed that only pipe friction coefficients(Hazen–Williams) include uncertainty and need to be treated byfuzzy numbers. The maximum uncertainty of Hazen–Williamscoefficients, i.e., the support of input fuzzy numbers in α¼ 0 ispresented in Table 1. In the present analysis, fuzzy responses ofnodal pressures at nodes 2, 3 and 4 and pipe discharges in all pipesare computed in three levels of α¼ 0; 0:5 and 1. Revelli and Ridolfi(2002) utilized a single-objective optimization model to solve thisproblem. They applied the Quadratic Programming (QP) method asa classical optimization technique, to maximize and minimize eachhydraulic response in each α-cut. On the other hand, in that work,for finding the maximum and minimum values of three nodalpressures and five pipe discharges in two α-cuts (0 and 0.5),overall, the QP solver was carried out 32 times independently. TheQP results are shown in Figs. 7 and 8 by dash lines.

Herein, the example is also solved using the NSGA-II method.For this purpose, the network is analyzed in two steps, includingall hydraulic responses in cuts α¼ 0 and 0:5. As a result, in eachα-cut, the NSGA-II must optimize 16 objective functions

simultaneously. As earlier discussed, a new feature namely thecloseness-distance was added to the NSGA-II to improve its efficiencyin solving fuzzy problems. This feature would lead the Paretosolutions toward the end points (A and B in Fig. 3) which are theresponses of interest in this fuzzy analysis. To evaluate the perfor-mance of this feature in solving the problem at hand, both standardand new NSGA-II schemes were applied to the network analysis. Forboth algorithms, the population size was considered to be 200 andthe mutation ratio was set 0.02. Both schemes ultimately resulted inthe same responses. However, the new version was faster than thestandard version. The closeness–distance operator in the new versionsignificantly speeds up the algorithm convergence in finding theextreme points on the optimal Pareto front presented in Figs. 7 and 8.Several primary runs were done for this example, and it wasconcluded that, averagely, the new version of NSGA-II converges tothe results shown in Figs. 7 and 8 in less than 100 generations while,the standard NSGA-II gives the same results after about 250 genera-tions. This turns out that the new feature works well in analyzingfuzzy systems in which the diversity of solutions is not important.

Furthermore, as seen in Figs. 7 and 8, the recently obtainedresults by NSGA-II are very close to the previously obtained by theQP. Not only that, but in some cases like Q4 and Q5, NSGA-II hasgiven slightly better results. This confirms that the proposed fuzzy

10

12

14

16

18

20

22

24

26

2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23

P (m

)

Node Name

Minimum Value of Fuzzy Pressure Crisp Value of Fuzzy Pressure

Maximum value of Fuzzy Pressure

10

12

14

16

18

20

22

24

26

24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46

P (m

)

Node Name

Fig. 10. Maximum uncertainty in the nodal pressures.

0

0.5

1

1.5

2

2.5

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22

V (m

/s)

V (m

/s)

V (m

/s)

Pipe Name

Minimum Value of Fuzzy Velocity Crisp Value of Fuzzy VelocityMaximum value of Fuzzy Velocity

0

0.5

1

1.5

2

2.5

23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45

Pipe Name

0

0.5

1

1.5

2

2.5

46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65

Pipe Name

Fig. 11. Maximum uncertainty in the pipe velocities.

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282278

approach is capable of uncertainty analysis in the network with anefficient and reliable algorithm.

5.2. Example 2 (case study)

A real pipe network is now analyzed by the described fuzzyapproach. The network has been constructed in a 120 ha area tosupply the water demands of an industrial park near the PersianGulf in southwest of Iran. The network has 46 nodes and 65 pipesand a reservoir with 27 m piezometric head that feeds thedemands by gravity. The network's layout configuration, includingthe pipe and node names as well as the most likely demands aredepicted in Fig. 9. The network has been constructed in a quite flatarea which is about 2 m higher than the mean sea level. The pipesare made of polyethylene (PE80) and the most likely HW coeffi-cient for all of them was considered to be 130 when designing thenetwork. The pipe lengths and inner diameters are also presentedin Table 2. As the design criteria for sizing the pipe diameters andreservoir elevation, the minimum required nodal pressure andpipe velocity were respectively considered to be 18 m and 0.3 m/s.The network was designed according to the future estimateddemands, shown in Fig. 9, and the aforementioned frictioncoefficient. As discussed in the manuscript, both water demandand friction coefficient are significantly under the influence ofuncertainties during the lifetime of the system. The operators arecurious to know how the system reacts to the input uncertainties,and its performance reliability is affected. The fuzzy analysis

method is used to respond this question. For this purpose, thepipe friction coefficients and the nodal demands are representedby triangular fuzzy numbers (Fig. 1) while, their most likely valuesare considered to be those used in the original design. The HWcoefficients and the nodal demands are respectively assumed tohave 710% and 715% uncertainty with respect to their mostlikely quantities. On this basis, the support of every fuzzy numberis obtained. For example, for a pipe HW coefficient with the mostlikely value Cc ¼ 130, an isosceles triangular fuzzy number iscreated whose the support interval is ½117, 143]. Now, we aregoing to find out that what will happen to these uncertaintieswhen they are introduced to the network, how they are spread outover the network and in what extent; they influence the network'shydraulic performance. In this case study, except the reservoir,there are 45 nodes and 65 pipes. If all nodal pressures and pipevelocities are the responses of interest, there will be 110 objectivefunctions in each α�cut. Since every objective function needs tobe once maximized and once minimized, the number of requiredoptimizations becomes 220 for every α�cut. For the current pipenetwork, the fuzzy numbers are discretized in three α�cutsincluding α¼ 0, 0.5 and 1. Except for α¼ 1 that is correspondingto the most likely values, for fuzzy analysis of the whole networkin α¼ 0 and 0.5, the problem is totally consisting of 440 singleobjective functions, which indeed introduce a very large nonlinearoptimization problem. To make the problem more manageableduring the optimization, it is divided into eight independentsub-problems in every α�cut. Before that, to calculate the mostlikely values of the pipe velocities ~V c and nodal pressures ~Pc

0

0.5

1

μ

P37 (m)

0

0.5

1

μ

P41 (m)

0

0.5

1

μ

P42 (m)

0

0.5

1

μ

P43 (m)

0

0.5

1

μ

P45 (m)

0

0.5

1

13 15 17 19 21 13 15 17 19 21

13 15 17 19 21 13 15 17 19 21

13 15 17 19 21 13 15 17 19 21

μ

P46 (m)

Fig. 12. Membership function of the fuzzy pressures.

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282 279

corresponding to α¼ 1, the hydraulic solver, EPANET, is run againstthe most likely values, ~Cc and ~qc . Afterward, in each of theremaining α�cuts (α¼ 0.5 and 0) the extreme values of½ ~Pa;α; ~Pb;α� and ½ ~V a;α ; ~V b;α� for the whole network are calculatedthrough solving the following eight multiobjective optimizationsub-problems. Each sub-problem has a different set of objectivefunctions but the same decision variables and search space.

(1) Minimization of P1 to P23 to find ~Pa;α values.(2) Minimization of P24 to P45 to find ~Pa;α values.(3) Maximization of P1 to P23 to find ~Pb;α values.(4) Maximization of P24 to P45 to find ~Pb;α values.(5) Minimization of V1 to V33 to find ~V a;α values.(6) Minimization of V34 to V65 to find ~V a;α values.(7) Maximization of V1 to V33 to find ~V b;α values.(8) Maximization of V34 to V65 to find ~V b;α values.

The NSGA-II is separately applied to either of the above sub-problems. In all optimization runs the population size is set to be250, and the mutation ratio is considered to be 0.025. Followingthe algorithm shown in Fig. 5, the fuzzy analysis in every α�cut isstarted by a randomly generated population and is terminatedby checking the convergence criteria. Averagely in all runs, the bestPareto-optimal front was obtained after a bout 600–700 generations.

After optimization of all sub-problems, the uncertainty distri-bution over the network is obtained. One of the most importantresults from this analysis is the support of the fuzzy responses.

The support values represent the maximum uncertainty in theresponses corresponding to the maximum uncertainty in the inputvariables. Figs. 10 and 11 respectively present the extreme valueswithin the support intervals for the fuzzy pressures and velocitiesalong with their most likely values already used to design thenetwork. These figures clearly show that how the input uncertain-ties are spread out on the network and influence the results. As itis rationally expected, the nodes more distant from the reservoirare more affected by uncertainty, while the closer nodes are moreresistant to the input uncertainties. The network nodes averagelyinclude �10:73% (ranging from �0:16% to �24:43%) andþ6:13% (ranging from þ0:08% to þ14:60%) uncertainty withrespect to the original design values. Furthermore, to better seehow the input triangular fuzzy numbers are distributed to thesystem and appear in the responses; some nodal pressures,including large uncertainty have been selected to show inFig. 12. Some nodal pressures have about �24% uncertaintyresulting in that, the reliability in these nodes with respect tothe minimum required pressure may fail. For example, in nodes 37,41, 42, 43, 45 and 46, the most likely pressure (used in the originaldesign) is respectively 19.11, 18.71, 18.45, 18.42, 19.12 and 18.5 m.However, due to the introduced uncertainty their pressures maybecome 14.84, 14.25, 13.96, 13.92, 14.95 and 14.03 m, all of whichare lower than the required minimum value (18 m). Also, byinspecting the entire network it is found that about 33.3% ofnodes may experience lower than 18 m pressure values.

A similar argument can be provided on the pipe velocities.As seen in Fig. 11, the input uncertainties have made more

0

0.5

1

μ

V11 (m/s)

0

0.5

1

μ

V30 (m/s)

0

0.5

1

μ

V39 (m/s)

0

0.5

1

μ

V43 (m/s)

0

0.5

1

μ

V46 (m/s)

0

0.5

1

0 0.3 0.6 0.9 1.2 0 0.3 0.6 0.9 1.2

0 0.3 0.6 0.9 1.2 0 0.3 0.6 0.9 1.2

0 0.3 0.6 0.9 1.2 0 0.3 0.6 0.9 1.2

μ

V50 (m/s)

Fig. 13. Membership function of the fuzzy velocities.

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282280

significant changes to the flow distribution and consequently, tothe pipe velocities rather than the nodal pressures. The networkpipes averagely include �39:71% (ranging from �12:7%to �100%) and þ47:88% (ranging from þ14:29% to þ127:27%)uncertainty with respect to the original design values. In Fig. 13,the fuzzy numbers of some selected pipes having the mostuncertainty in the whole network are presented. As seen, thevelocity in all of these pipes may become zero due to the inputuncertainty while, the recommended minimum velocity is 0.3 m/s.The most likely values in Figs. 11 and 13 clearly show that theconstraint of minimum velocity has been strictly met in the originaldesign so that all pipes in the network (in the absence of uncertainty)has velocity greater than 0.3 m/s. However, because of uncertainty, itis found that about 31% of pipes may experience less than 0.3 m/svelocity.

6. Conclusion

Estimation of water demands and pipe friction coefficientsintroduces significant uncertainties to analysis and design of waternetworks. These uncertainties are spread out over the system andmake the network's responses uncertain too. This study intro-duced a mathematical model for taking into account the inputuncertainties in the network hydraulic simulation. For this pur-pose, an analysis package utilizing the fuzzy set theory, NSGA-IImethod and the hydraulic simulation model of EPANET wasdeveloped. EPANET accepts only unique (crisp) values for nodaldemands and pipe friction coefficients and gives only uniquevalues for nodal pressures and pipe velocities. When the inputuncertainties are represented by the fuzzy numbers and intro-duced to the network's governing equations by using the α�cutmethod, a multi-criteria decision making problem is formed,which needs optimization techniques to be solved. The raisedproblem has many objective functions, which do not challengewith each other and therefore, the trade-off between them doesnot matter. The problem is neither single nor multi-objectiveoptimization such that, in the previous investigations, it wastreated as a multi-single-objective optimization. The conventionalapproach to solve the problem is to apply a single-objective solver,every time to only one of the objective functions. In this study, amultiobjective optimization method on the basis of the NSGA-IIwas utilized to solve the problem. In the proposed scheme, thenotion of dominance in viewpoints of density and diversity ofPareto solutions was modified. A new metric named as thecloseness–distance was substituted for the crowded-comparisonoperator in the standard NSGA-II. This new metric guides thePareto solutions toward the extreme points on the Pareto frontwhich are, in fact, the global optimum values of the objectivefunctions. The model was then applied against an example and arelatively large pipe network. It was found that small uncertaintiesin the network can result in large uncertainties in the hydraulicresponses and significantly influence the system's performancereliability. The flow distribution in the network and consequently,the pipe velocities were more dramatically affected by the inputuncertainties rather than the nodal pressures. Also, with respect tothe minimum required pressure and pipe velocity, it was resultedthat about 33:3% of nodal pressures and 31% of pipe velocitiesmay fail during the network operation due to the introduceduncertainties. Solving the case study showed that the NSGA-II isvery well-suited to the fuzzy analysis of the pipe networks and canbe successfully utilized for solving large problems. In fact, theapplication of the NSGA-II makes the problem solution moresystematic and computationally more efficient so that, many ofthe fuzzy hydraulic responses can be simultaneously analyzedin only one single simulation run. This would be a promising

outcome for development and application of this approach toother engineering systems.

References

Aghmiuni, S.S., Haddad, O.B., Omid, M.H., Mariño, M.A., 2013. Effects of piperoughness uncertainty on water distribution network performance during itsoperational period. J. Water Resour. Manage. 27, 1581–1599.

Bao, Y., Mays, L.W., 1990. Model for water distribution system reliability. J. Hydraul.Eng. 116 (9), 1119–1137.

Bardossy, A., Duckstein, L., 1995. Fuzzy rule-based modeling with application togeophysical, Economic Biological and Engineering System. CRC, London

Bardossy, A., Stehlic, J., Caspary, H.J., 2002. Automated objective classification ofdaily crirculation patterns for precipitation and temperature downscalingbased on optimized fuzzy rules. Climate Res. 23, 11–22.

Bargiela, A., Hainsworth, G.D., 1998. Pressure and flow uncertainty in water system.J. Water Resour. Plan. Manage. 115 (2), 212–229.

Bit, A.K., Biswal, M.P., Alam, S.S., 1992. Fuzzy programming approach to multi-criteria decision making transportation problem. Fuzzy Sets Syst. 50, 135–141.

Branisavljevic, N., Ivetic, M., 2006. Fuzzy approach in the uncertainty analysis ofwater distribution network of Becej. Civil Eng. Environ. Syst. 23 (3), 221–236.

Buckley, J.J., 1987. Fuzzy programming and the multicriteria decision problem. In:Kacprzyk, J. (Ed.), Optimization Models Using Fuzzy Sets and Possibility Theory.D. Reidel Publishing Co., Boston, pp. 226–244

Chang, F.j., Chang, K.Y., Chang, L.C., 2008. Counter propagation fuzzy neuralnetwork for city flood control system. J. Hydrol. 358, 24–34.

Cheng, C.T., Chau, K.W., 2001. Fuzzy iteration methodology for reservoir floodcontrol operation. J. Am. Water Resour. Assoc. 37, 1381–1385.

Deb, K., Pratap, A., Agarwal., S., Meyarivan, T., 2002. A fast and elitist multi-objectivegenetic algorithm: NSGA-II. IEEE Trans. Evolut. Comput. 6 (2), 182–197.

Dixon, B., 2005. Applicability of Neuro-fuzzy techniques in predicting ground watervulnerability: sensitivity analysis. J. Hydrol. 309, 17–38.

Duan, H.F., Tung, Y.K., Ghidaoui, M.S., 2010. Probabilistic analysis of transient designfor water supply systems. J. Water Resour. Plan. Manage. 136 (6), 678–687.

Eshelman, L.J., Shaffer, D.J., 1993. Real-coded genetic algorithms and intervalschemata. In: Whitley, D.L. (Ed.), Foundations of Genetic Algorithms 2. MorganKaufman, San Mateo, CA, pp. 187–202

Fu, G., Butler, D., Khu, S.T., Sun, S.Z., 2011. Imprecise probabilistic evaluation ofsewer flooding in urban drainage systems using random set theory. J. WaterResour. Res. 47, 1–13.

Ghosh, S., Mujumdar, P.P., 2010. Fuzzy waste load allocation model: a multiobjective approach. J. Hydroinf. 12, 96–109.

Guan, J.B., Aral, M.M., 2004. Optimal design of groundwater remediation systemsusing Fuzzy set theory. Water Resour. Res. 40, 1–20.

Gupta, B., Bhave, P.R., 2005. Fuzzy parameters in pipe network analysis. Civil Eng.Environ. Syst. 24 (1), 33–54.

Haghighi, A., Keramat, A.R., 2012. A fuzzy approach for considering uncertainty intransient analysis of pipe networks. J. Hydroinf. 14 (4), 1024–1035.

Kandel, A., 1992. Fuzzy Expert System. CRC, Boca Ratan, FLKaraboga, D., Bagis, A., Haktanir, T., 2004. Fuzzy logic based operation of spillway

gates of reservoirs during floods. J. Hydrol. Eng. 9, 544–549.Kisi, O., 2010. River suspended sediment concentration modeling using a neural

differential evolution approach. J. Hydrol. 389, 227–235.Kurtulus, B., Razack, M., 2010. Modeling daily discharge of a large karstic aquifer

using soft computing methods: Artificial neural network and neuro-fuzzy.J. Hydrol. 381, 101–111.

Mamdani, E.H., Assilian, S., 1975. An experiment in linguistic synthesis with a fuzzylogic controller. Int. J. Man-Mach. Stud. 7, 1–13.

Mujumdar, P.P., Subbarao, V.V.R., 2004. Fuzzy waste load allocation model for riversystems: simulation-optimization approach. J. Comput. Civil Eng. 18, 120–131.

Nasiri, F., Maqsood, I., Huang, G., Fuller, N., 2007. Water quality index: a fuzzy riverpollution decision support expert system. J. Water Resour. Plan. Manage. 133,95–105.

Ozelkan, E.C., Duckstein, L., 2001. Fuzzy conceptual rainfallrunoff models. J. Hydrol.253, 41–68.

Ozger, M., 2009. Comparison of fuzzy inference systems for streamflow prediction.Hydrol. Sci. J., 54261–54273

Pesti, G., Shrestha, B.P., Duckstein, L., Bogardi, I., 1996. A fuzzy ruled-basedapplication to drought assessment. Water Resour. Res. 32, 1741–1747.

Pedrycz, W., 1989. Fuzzy Control and Fuzzy Systems. Wiley, New YorkPongracza, R., Bogardib, I., Ducksteinc, L., 1999. Application of fuzzy rule-based

modeling technique to regional drought. J. Hydrol. 224, 100–114.Revelli, R., Ridolfi, L., 2002. Fuzzy approach for analysis of pipe network. J. Hydraul.

Eng. 128, 93–101.Saad, M., Bigras, P., Turgeon, A., Duguette, R., 1996. Fuzzy learning decomposition

for scheduling of hydroelectric power systems. Water Resour. Res. 32, 179–186.Sasikumar, K., Mujumdar, P.P., 1998. Fuzzy optimization model for water quality

management of a river system. J. Water Resour. Plann. Manage. 124, 79–88.Spiliotis, M., Tsakiris, G., 2012. Water distribution network analysis under fuzzy

demand. Civil Eng. Environ. Syst. 29 (2), 107–122.Srinivas, N., Deb, K., 1995. Multiobjective optimization using nondominated sorting

in genetic algorithm. Evol. Comput. 2, 221–248.

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282 281

Srinivas, V.V., Tripathia, S., Raoa, A.R., Govindarajua, R.S., 2008. Regional floodfrequency analysis by combining self-organizing feature map and fuzzyclustering. J. Hydrol. 348, 148–166.

Srivastava, G., Panda, S.N., Mondal, P., Liu, J., 2010. Forecasting of rainfall using ocean-atmospheric indices with a fuzzy neural technique. J. Hydrol. 395, 190–198.

Sugeno, M., 1985. Industrial Applications of Fuzzy Control. Elsevier, New YorkTodini, E., Pilati, S., 1987. A Gradient Method for the Analysis of Pipe Networks.

International Conference on Computer Application for Water Supply andDistribution. Lecicester Polytechnic, UK

Zadeh, L.A., 1965. Fuzzy Sets. Inf. Control 8, 338–353.Zadeh, L.A., 1973. Outline of a new approach to the analysis of complex systems and

decision processes. IEEE Trans. Syst. Man Cybernet. 3, 28–44.Zimmerman, H.J., 1985. Fuzzy Set Theory and Its Application. Dordrecht, BostonZimmermann, H.J., 2010. Fuzzy set theory. WIREs Comput. Stat. 2, 317–332.

A. Haghighi, A.Z. Asl / Engineering Applications of Artificial Intelligence 32 (2014) 270–282282