Alternative Decision Making in Water Distribution Network with NSGA-II

TECHNICAL NOTES

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Alternative Decision Making in Water Distribution Networkwith NSGA-II

Md. Atiquzzaman1; Shie-Yui Liong, M.ASCE2; and Xinying Yu3

Abstract: Cost of effective design or rehabilitation of the pipe networks depends largely on the available budget and satisfying pressuresexpected at the demand nodes. With the available market pipe sizes, one or more solutions of the same cost, the least cost, andsimultaneously meeting the pressure constraints can be obtained from EPANET, a water distribution network simulation model, coupledwith an optimization algorithm. Often, however, the least cost may be prohibitive from the overall budget consideration especially indeveloping countries and a less optimal solution is therefore the only option. The study shows a scheme in which solution selection ismade, among solutions, which �1� is within allowable budget; and �2� yields acceptable total pressure deficit distributed more equally atseveral nodes instead of loaded on one or a few nodes only. A multiobjective optimization algorithm �NSGA-II� is coupled with waterdistribution network simulation software �EPANET� to provide the much needed Pareto front of the cost and nodal pressure deficit. Atwo-looped simple network is used to demonstrate the application of the scheme.

DOI: 10.1061/�ASCE�0733-9496�2006�132:2�122�

CE Database subject headings: Algorithms; Optimization; Water distribution systems; Costs.

Introduction

A water distribution network �WDN� is designed or rehabilitatedto supply a sufficient amount of water with adequate nodalpressure to the user. Optimal pipe diameters are often chosenfrom a set of available market sizes to minimize the total networkcost satisfying the pressure constraints. Hence cost is the primeconsideration. The least network cost may, however, exceed theavailable budget in developing countries. Lower cost solutionsimply at the same time, however, that pressure deficit occurs atone or more nodes. For sound decision making optimal and othersolution options should be made available. Solution options must,however, be accompanied with much needed useful informationsuch as the consequence �pressure deficit, for example� forchoosing a lower ranked �hence lower network cost� solutionoption.

In the past decade or so, many single objective optimizationalgorithms have been introduced to solve the WDN problems.The widely used algorithms are, for examples, linear program-

1Research Scholar, Dept. of Civil Engineering, National Univ. ofSingapore, Blk E1A, #07-03, 1 Engg Dr. 2, Singapore 117576.

2Associate Professor, Dept. of Civil Engineering, National Univ. ofSingapore, Blk E1A, #07-03, 1 Engg Dr. 2, Singapore 117576. E-mail:[email protected]

3Research Scholar, Dept. of Civil Engineering, National Univ. ofSingapore, Blk E1A, #07-03, 1 Engg Dr. 2, Singapore 117576.

Note. Discussion open until August 1, 2006. Separate discussionsmust be submitted for individual papers. To extend the closing date byone month, a written request must be filed with the ASCE ManagingEditor. The manuscript for this technical note was submitted for reviewand possible publication on September 7, 2004; approved on July 15,2005. This technical note is part of the Journal of Water ResourcesPlanning and Management, Vol. 132, No. 2, March 1, 2006. ©ASCE,
ISSN 0733-9496/2006/2-122–126/$25.00.
122 / JOURNAL OF WATER RESOURCES PLANNING AND MANAGEMENT

J. Water Resour. Plann. Mana

ming �LP� �Alperovits and Shamir 1977�, enumeration techniques�Gessler 1985�, nonlinear programming �NLP� �Chiplunkar et al.1986�, genetic algorithm �GA� �Savic and Walters 1997�,simulated annealing �SA� �Cunha and Sousa 1999�, shuffled frogleaping algorithm �SFLA� �Eusuff and Lansey 2003�, ant colonyoptimization algorithms �ACOAs� �Maier et al. 2003�, shuffledcomplex evolution �Liong and Atiquzzaman 2004�, etc. Thesealgorithms must, however, transform the multiobjective formula-tion into a single objective by using penalty factors. It is widelyknown that depending on the problem structure the choice ofits appropriate value affects significantly, especially for highdimensional problems, the resulting optimal solution �Wu andSimpson 2002; Wu and Walski 2004�. Moreover, this singleobjective optimization algorithm does not offer the required trade-off curve �Pareto front�, between cost and pressure deficits, tohelp the water manager in better decision making. Many of thetechniques �e.g., LP, NLP� have not been widely accepted forpractical applications �Walski 1985�. The reasons for limitedacceptance are discussed in Walski �1995� and Wu et al. �2001�.

A multiobjective evolutionary algorithm �MOEA�, which hasbeen used since the 1980s �Farmani et al. 2004�, on the otherhand, can truly deal with several objectives and numerous linearand nonlinear constraints. As the constraints are handledseparately, no penalty coefficient is needed in the solution process�Walters et al. 1999�. Hence a MOEA yields a set of optimalsolutions �Pareto front� that can enhance the decision-makingprocess in the pipe network problems �Nicolini 2004�. Recently,Prasad and Park �2004� introduced multiobjective geneticalgorithms �NSGA� to the design of a water distribution networkconsidering system reliability. Two objectives, mainly minimiza-tion of cost and maximization of a measure of reliability, wereconsidered in their model. The solution process was demonstratedthrough two example problems from the literature. Nicolini
�2004� presented three elitist multiobjective nondominated sorting
© ASCE / MARCH/APRIL 2006

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GAs �NSGAs�, namely, enhanced nondominated sorting geneticalgorithm �ENGA�, nondominated sorting genetic algorithm�NSGA-II�, and controlled NSGA-II in pipeline optimization toachieve a true Pareto front and compared the results obtained bythese algorithms. He concluded that the NSGA-II performedbetter than ENGA based on the solution of a simple designproblem. Khu and Keedwell �2004� also introduced NSGA-II indetermining an optimal rehabilitation alternative. They combinedNSGA-II with an expert heuristic search method for furtherreduction of network cost by allowing pressure violations at somenodes. It should be noted that Khu and Keedwell �2004� andNicolini �2004� did not consider solutions having the samenetwork cost with, however, a similar or the same total nodalpressure deficit. For the same network cost and pressure deficit,often there are a number of possible solutions with differentcombinations of pipe diameters. Should the network cost resultingfrom the optimal solution exceed the budget allocated, there is a

Table 1. Cost of Various Pipe Sizes

Diameter�mm �in.��

Cost�units/length�

25.4 �1� 2

50.8 �2� 5

76.2 �3� 8

101.6 �4� 11

152.4 �6� 16

203.2 �8� 23

254.0 �10� 32

304.8 �12� 50

355.6 �14� 60

406.4 �16� 90

457.2 �18� 130

508.0 �20� 170

558.8 �22� 300

609.6 �24� 550

Fig. 1. Two-looped network

JOURNAL OF WATER RESOURCES P


need to find the next best alternative solution which is within thebudget and yet with some tolerable minimal pressure deficit atsome demand nodes.

This paper presents a multiobjective GA, the nondominatedsorting genetic algorithm �NSGA-II; Deb et al. 2000�, coupledwith EPANET network solver �Rossman 1993� to yield theoptimal solution and other less optimal solutions accompaniedwith information such as total pressure deficits, the nodes atwhich pressure deficits occur, and their individual nodal pressuredeficit. A simple two-looped network for single loading condition�peak demands�, with predesigned pumps, tanks, and valves,is considered to demonstrate the application of NSGA-II inobtaining such solutions. It should be noted that fire flows, pipebreaking, and changing pipe location are not taken into account inthe design process.

Nondominated Sorting Genetic Algorithm „NSGA-II…

NSGA-II �Deb et al. 2000� is a multiobjective optimizationalgorithm and provides a trade-off between the various objectivesconsidered. The trade-off information is very useful in making asound decision for alternative options.

The method generates an initial set P with N solutionsrandomly. It creates an offspring population �Qt� from the parentpopulation �Pt� of size N. These parent and offspring populationsare combined together where a fast nondominated sorting isperformed to classify the entire population �Pt+Qt�. This resultsin different nondominated fronts F1, F2, etc. The new parentpopulation �Pt+1� is then produced where the solutions from thefirst front F1 are accounted for and the process is continued untilthe size becomes N. Hence the solutions in the finally approvedfront are rearranged based on the crowded comparison criterionthat keeps diversity by locating a point in a less crowded regionof population. The new offspring population is regenerated andthe procedure is repeated in the subsequent generation. Furtherdetails can be found in Deb et al. �2000�.

Optimization Formulation

Designs of a water distribution network involve multiple objec-tive functions subject to several constraints. The optimizationalgorithm adopted searches the optimal solution in the solution

Fig. 2. Pareto front of optimal solutions for two-looped network

space. The commonly used optimization formulation contains

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some objectives such as minimizing both the network cost�excluding energy cost� and the total pressure deficit subject to thefollowing constraints: �1� the decision variables are to be selectedfrom a set of commercially available diameters; and �2� the nodalpressure at each node should be greater than or equal to apredefined minimum value. The scheme proposed in this studyslightly modifies the generally used optimization formulation.The formulation is expressed as follows:

Minimize: NC = �k=1

NP

ck�Dk� � Lk �1�

and

PD = �j=1

NJ

�Hmin,j − Hj� �2�

Subject to

Dk � �D1,D2, . . . ,Dn� �3a�

Hj � Hmin,j�j = 1,2, . . . ,TN� �3b�

PDL � PD � PDU �3c�

Table 2. Top 10 Solutions with Their Costs, Total Pressure Deficits, and

Ranknumber

Cost�$�

Totalpressuredeficit�m� 1 2

1 419,000 0 457.2 254

2 415,000 0.67 457.2 254

3 412,000 2.86 457.2 254

4 403,000 3.29 457.2 254

5 398,000 5.46 457.2 203.2

6 394,000 5.54 457.2 254

7 385,000 6.15 457.2 254

8 376,000 9.13 457.2 203.2

9 374,000 11.33 457.2 203.2

10 371,000 11.46 457.2 203.2

Table 3. Samples of Solution Cost and Distribution of Nodal HeadDeficits: Second Run

Cost�$�

Total pressuredeficit�m�

Nodes with pressure deficit�m�

2 3 4 5 6 7

404,000 5.87 0 0 0 0 3.00 2.87

404,000 5.89 0 0 0 0 0.89 5.00

416,000 0.68 0 0.68 0 0 0 0

416,000 1.01 0 1.01 0 0 0 0

416,000 3.03 0 0 0 0 1.57 1.46

416,000 3.30 0 0 0 0 3.15 0.15



NCL � NC � NCU �3d�

where NC=network cost; ck�Dk�=cost per unit length of the kthpipe of diameter Dk ;Lk=length of the kth pipe; NP=total numberof pipes in the system; PD=total head deficit; Hmin,j =minimumhead requirement at node j; Hj =nodal head found after simula-tion at node j; �D1 ,D2 , . . . ,Dn�=set of commercially availablesizes; TN=total number of junctions; NJ=total number ofjunctions in the system where pressure violation occurs;PDL=lower limit of total head deficit; PDU=upper limit of totalhead deficit; NCL=lower limit of total network cost; andNCU=upper limit of total network cost.

It should be noted, as explained in the following applicationsection, the scheme runs the coupled NSGA-II and EPANET twotimes. The first run is with the above formulation, however, with-out constraints Eqs. �3c� and �3d�; this is a general commonlyused formulation. The second run considers all constraints�3a�–�3d�; constraints �3c� and �3d� imply the limits of totalpressure deficit and network cost with which designers feelcomfortable.

Application

The network considered, Fig. 1, is a two-looped simple network,first presented by Alperovits and Shamir �1977�, consisting ofeight pipes �each 1,000 m long with Hazen-Williams C value of130�, seven nodes, and a single reservoir. The minimum pressurerequirements are 30 m for each node. Table 1 contains thecommercially available pipe sizes and corresponding cost per unitlength. There are 14 commercial diameters to be selected. If onlydiscrete pipe sizes are considered, a total of 148�1.48�109�possible solutions exist �Savic and Walters 1997; Wu et al. 2001�.The solution space is first explored using NSGA-II without theconstraints given in Eqs. �3c� and �3d�. The resulting Pareto frontis shown in Fig. 2. Table 2 shows examples of the best10 solutions with cost ranging from $419,000 to $371,000. It isobvious that the optimal solution, highest ranking �$419,000�,meets the pressure requirements. As the ranking decreases, thepressure deficit gets larger. This paper shows a scheme as to howto select the next best alternative in the event the cost �$419,000�from the optimal solution is beyond the available budget. The

izes: First Run

Diameter of the pipes�mm�

3 4 5 6 7 8

6.4 101.6 406.4 254 254 25.4

6.4 152.4 406.4 254 203.2 25.4

6.4 152.4 355.6 304.8 254 25.4

6.4 152.4 355.6 304.8 203.2 25.4

7.2 203.2 355.6 254 152.4 25.4

6.4 152.4 355.6 254 254 25.4

6.4 152.4 355.6 254 203.2 25.4

6.4 203.2 355.6 254 152.4 25.4

6.4 254 355.6 254 50.8 25.4

6.4 254 355.6 254 25.4 25.4

Pipe S

40

40

40

40

45

40

40

40

40

40

scheme is as follows.


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The coupled EPANET and NSGA-II modules are now run onceagain with the two constraints equations �3c� and �3d�. Thetolerated pressure deficit and budget will automatically set theupper limits of the total pressure limit and the network cost. Thelower limit of the total pressure deficit can be set as 0 while thatfor the network cost is recommended to be set not too low so thatall solutions obtained are focused in the desired search space.

To demonstrate the main idea of this study, the limits set forthe total pressure deficit are 0 and 18 m while that for the networkcost is $400,000 and $419,000. Table 3 shows that:1. There is more than one solution with the same network cost

�e.g., $416,000� and yet different total pressure deficits.Should the network cost be within the available budget, it isquite obvious that the solution with the least pressure deficit�and within tolerated deficit� is most desirable; and

2. For the same network cost �e.g., $404,000� several solutionswith about the same total pressure deficit exist; somesolutions with the total pressure deficit are shared by severalnodes “equally” �nodes 6 and 7 with individual pressuredeficit of 3.00 and 2.87 m, respectively� while othersolutions with total pressure deficit loaded mainly on onenode �node 7 with pressure deficit of 5.00 m�. The pipe sizesof these solutions are shown in Table 4. With informationsuch as that presented in Table 3, the decision maker can thenmake a more sound judgment for various budget scenariosand design considerations.

Conclusion

A scheme has been suggested in this study to assist decisionmakers in selecting the best alternative water distribution networkdesign solution which is within: �1� available budget; and �2�tolerated pressure deficit. A multiobjective optimizationalgorithm, NSGA-II, is coupled with water distribution networksimulation software, EPANET. The coupled modules, withappropriately formulated objectives and constraints, yield usefulalternative solutions which are particularly useful when theassociated network cost of the optimal solution is beyond theavailable budget. The alternative solutions provide not onlyassociated costs but also the individual nodal pressure deficitswhich allow the designers to relook at the originally set pressureconstraints whether the magnitude of the pressure violation maybe tolerated.

The application of the model is limited to a single loadingcondition �peak demands� of the network. However, multipleloading conditions including fire flow scenario with pumps,

Table 4. Samples of Solution Cost, Total Head Deficits and Pipe Sizes:

Cost�$�

Totalpressuredeficit�m� 1 2 3

404,000 5.87 457.2 203.2 406.4

404,000 5.89 457.2 355.6 355.6

416,000 0.68 457.2 254 406.4

416,000 1.01 457.2 254 406.4

416,000 3.03 457.2 203.2 457.2

416,000 3.30 457.2 304.8 355.6

valves, etc. can also be considered in the network problem. In

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addition, similar solutions can be found if the network is designedto provide the excess pressure among the nodes and to ensurereliability.

Notation

The following symbols are used in this technical note:ck�Dk� � cost per unit length of the kth pipe of

diameter Dk;�D1 ,D2 , . . . ,Dn�

� set of commercially available sizes;Fi � nondominated front of rank i;Hj � nodal head found after simulation at node j;

Hmin,j � minimum head requirement at node j;Lk � length of the kth pipe;

NC � network cost;NCL � lower limit of total network cost;NCU � upper limit of total network cost;

NJ � total number of junctions where pressureviolation occurs;

NP � total number of pipes in the system;PD � total head deficit;

PDL � lower limit of total head deficit;PDU � upper limit of total head deficit;

Pt � parent population;Qt � offspring population; and

TN � total number of junctions.

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Run

Diameter of the pipes�mm�

4 5 6 7 8

254.0 406.4 254.0 50.8 25.4

50.8 355.6 152.4 304.8 203.2

76.2 406.4 254 254 25.4

25.4 406.4 254 254 76.2

203.2 355.6 254 152.4 25.4

25.4 406.4 304.8 254 25.4

Second

multi-objective optimization of anytown pipe rehabilitation, tank

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Alternative Decision Making in Water Distribution Network with NSGA-II