Genetic Algorithm for Optimization of Water Distribution Systems

Environmental Modelling & Software 14 (1999) 437–446

Genetic algorithm for optimization of water distribution systems

Indrani Gupta*, A. Gupta, P. KhannaNational Environmental Engineering Research Institute, Nagpur-440020, India

Received 20 January 1998; accepted 15 July 1998

Abstract

A methodology based on genetic algorithm has been developed for lower cost design of new, and augmentation of existing waterdistribution networks. The results have been compared with those of non-linear programming technique through application toseveral case studies. The genetic algorithm results in a lower cost solution. Parameters governing the convergence of the solutionsin non-linear and genetic algorithms are also discussed. 1999 Elsevier Science Ltd. All rights reserved.

Keywords:Genetic algorithm; Optimization; Water distribution system; Non-linear programming

Software availabilityProgram title: GENEDevelopers: Indrani Gupta, A. Gupta

and P. KhannaContact address: National Environmental

Engineering ResearchInstitute, Nagpur-440020,India

Hardware: HP 9000/730 PA-RISCWorkstation under HP-UX8.07 multi-user operatingsystem

Source language: FORTRAN

1. Introduction

Water distribution system (WDS) design belongs to agroup of inherently intractable problems commonlyreferred to as NP-hard (Templeman, 1982; Parker andRardin, 1988). Essentially NP-hard means that a rigorousalgorithm to find an optimum design using discretediameters is not a practical possibility. Severalresearchers have reported algorithms for minimising thecost through the application of mathematical techniques,such as linear, non-linear or dynamic programming. It

* Corresponding author.

1364-8152/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S1364-8152 (98)00089-9

is well known that when diameters are assumed as thedecision variables (DV), the constraints are implicitfunctions of the DV, the feasible region is non-convex,and the objective function is multimodal. Hence, con-ventional optimization methods result in a local optimumwhich is dependent on the starting point in the searchprocess.

The application of stochastic optimization techniquessuch as genetic algorithm (GA) and simulated annealingto WDS optimization is of recent origin. Simpson et al.(1994) have presented a methodology for finding the bestcost alternative for pipe networks using a three operatorGA comprising reproduction, crossover and mutation.An inherent problem in that the model is the large com-putational time in comparison to the non-linear program-ming techniques. Loganathan et al. (1995) proposed anouter flow search inner optimization procedure to ident-ify lower cost design solutions. In that approach eachpipe network is subjected to an outer search scheme thatselects alternative flow configurations in an attempt tofind an optimal flow division among pipes. For eachselected set of pipe flows a linear program is used to findthe associated optimal pipe diameters and energy heads.

A new GA based methodology for optimaldesign/augmentation of pipe networks is described inthis paper. The methodology was compared with a non-linear programming (NLP) technique based on interiorpenalty function (IPF) with the Davidon-Fletcher-Powell(DFP) method. The NLP technique was first evaluatedby application to a case study which has been previously

438 I. Gupta et al. /Environmental Modelling & Software 14 (1999) 437–446

attempted by several researchers (Loganathan et al.,1995, 1990; Fujiwara et al., 1987; Quindry et al., 1979;Alperovits and Shamir, 1977). The optimal cost obtainedfrom the NLP technique was 0.57% higher than the sol-ution achieved by Loganathan et al. (1995). The sol-utions achieved by other researchers are 1.9–18.3%higher than the solution obtained by Loganathan et al.(1995).

Further, a comparison between the results of the GAand NLP techniques for augmentation of several mediumsize networks showed that the GA in general provideda lower cost solution, than that obtained from the NLPtechnique. The hydraulic simulator ANALIS (Bassin etal., 1992) which is based on graph theory, was used inboth the NLP and GA solutions to calculate pressureheads, flows and velocities in the design of branched,looped and combined systems.

2. Deterministic optimization techniques

A number of investigators have dealt with the problemof optimization of WDS by applying mathematical pro-gramming techniques.

Several researchers employed linear programming tooptimise a WDS. Principal approaches include those ofAlperovits and Shamir (1977), Quindry et al. (1981) andKessler and Shamir (1989). The technique given by Alp-erovits and Shamir (1977) requires that a set of variables(pipe flows) be set to particular values before the linearprogramme can be formulated. Information availablefrom the solution of linear programming problem can beused to calculate a gradient which is then used to changepipe flows. Quindry et al. (1981) have decomposed thelooped network problem to branched systems. The limi-tation of such simplified solution has been critically dis-cussed by Templeman (1982). Kessler and Shamir(1989) also use linear programming gradient procedure.

Several non-linear programming packages have beendeveloped for network design problems. These packagesinclude GRG2 (Lasdon and Waren, 1983), MINOS(Murtagh and Saunders, 1987), GINO (Liebman et al.,1986), and GAMS (Brooke et al., 1988) which are allbased on the generalised reduced gradient method. Chi-plunkar et al. (1986) presented an algorithm based oninterior penalty function (IPF) with the Davidon-Fletcher-Powell (DFP) method. Lansey and Mays (1989)used GRG2 to find the optimum design and to simulatepumps, tanks and multiple loading cases. Lansey et al.(1989) considered uncertainty in nodal demands, Hazen-Williams coefficients and minimum nodal heads, anddeveloped a methodology for optimal design withrecourse to chance constrained optimization. Duan et al.(1990) extended the work of Lansey and Mays (1989)further and developed a model that (i) identifies the num-bers and locations of pumps and tanks by implicit enu-

meration, (ii) uses GRG2 to find optimum pipe sizes forthe pump and tank layout specified in (i), and (iii) usesa separate model to compute various measures of sys-tem reliability.

Gupta et al. (1993) developed the software packageWATDIS based on IPF and DFP methods. In thatapproach the problem was formulated as a cost minimiz-ation problem wherein the objective function F(x) com-prised the cost of power and annualised cost of pipes,pumps, and reservoirs satisfying the hydraulic loop lawswith constraints on minimum diameter and residualhead. The non-linear non-convex problem was convertedto an unconstrained problem by appending the con-straints to the objective function through penalty andweighting factors using the IPF method. An independentweighting factor was assigned to each constraint in orderto ensure the normalisation required by the significantlydifferent contributions of diameter, reservoir height, andresidual head constraints to the unconstrained objectivefunction.

Recently, Loganathan et al. (1995) presented a designheuristic for global cost minima design. That methodwas used to solve a standard eight pipe problem, eachpipe being 1000 m long with a Hazen Williams coef-ficient of 130. The pipe sizes and associated costs usedin the study are presented in Table 1. By assuming aminimum diameter of 1 inch and minimum flow con-straint of 1 m3/hour the method identified a design forthe network costing US $405 301.

The same problem with the same minimum diameterand flow constraints was solved by the authorsemploying WATDIS. Since the single cost equation wasof exponential form and did not show a good fit (thecoefficient of determination5 0.932), a piecewise linearfunction was used to represent the cost. The optimal costobtained employing WATDIS is $407 625 which is theactual cost of the network finally calculated from costof pipes per unit length. This is 0.57% higher than theone reported by Loganathan et al. (1995). The details ofthe pipe cost and solution are presented in Fig. 1, andTables 1–4. The cost of the same network as determinedin a number of other methods in previous studies is

Table 1Pipe sizes and associated costs

Diameter Unit cost Diameter Unit cost(in.) (US$/m) (in.) (US$/m)

1 2 12 502 5 14 603 8 16 904 11 18 1306 16 20 1708 23 22 300

10 32 24 550

Note: 1 in.5 25.4 mm

439I. Gupta et al. /Environmental Modelling & Software 14 (1999) 437–446

Fig. 1. Optimal looped network.

Table 2Pipe details for optimal solution

Pipe Node Length Diameter Flow Headloss StatusNo (m) (in) (lps) (m)

From To

1 1 2 1000.00 18 311.111 6.756 N2 2 3 210.72 12 102.215 1.305 N

789.28 10 102.215 11.881 NS3 2 4 932.17 16 181.118 4.104 N

67.83 14 181.118 0.572 NS4 4 5 57.55 2 0.277 0.039 N

942.45 1 0.277 18.486 NS5 4 6 836.22 16 147.508 2.517 N

163.78 14 147.508 0.945 NS6 6 7 989.13 10 55.841 4.860 N

10.87 8 55.841 0.158 NS7 3 5 899.81 10 74.438 7.529 N

100.19 8 74.438 2.485 NS8 7 5 535.81 2 0.286 0.381 N

464.19 1 0.286 9.663 NS

Legend: N, New pipe; NS, New pipe in series with the previous pipe

$412 931 (Loganathan et al., 1990), $415 271 (Fujiwaraet al., 1987), $441 522 (Quindry et al., 1979), and$479 525 (Alperovits and Shamir, 1977). These costs are1.9%, 2.5%, 8.9% and 18.3% higher respectively thanthe cost of $405 301 achieved by Loganathan et al.(1995). Accordingly, the performance of WATDIS was

Table 3Nodal details

Node no. Ground Residual Hydraulic Demandlevel head grade (lps)(m) (m)

1 R 210.00 00.00 210.00 2 311.11112 D 150.00 53.24 203.24 27.77783 D 160.00 30.06 190.06 27.77784 D 155.00 43.57 198.57 33.33335 D 150.00 30.04 180.04 75.00006 D 165.00 30.11 195.11 91.66677 D 160.00 30.09 190.09 55.5556

Legend: R, Reservoir location; D, Demand node

judged to be fairly good in comparison to that of otheralgorithms reported in the literature. The WATDIS wasthus considered an adequate basis for evaluation of theGA described in this paper.

3. Overview of genetic algorithms

GAs are nature based stochastic computational tech-niques. The major advantages of these algorithms aretheir broad applicability, flexibility and their ability tofind optimal or near optimal solutions with relativelymodest computational requirements. GAs, pioneered by


Table 4Summary of new pipe lines

Pipe Dia Total length Cost(in.) (m) (dollars)

1 1406.6 2813.32 593.4 2966.88 111.1 2554.4

10 2678.2 85703.012 210.7 10536.014 231.6 13896.616 1768.4 159155.118 1000.0 130000.0

Total cost: US$407 625

Holland (1975), have proven useful in a variety of searchand optimization problems in engineering, science andcommerce (Goldberg, 1989). The algorithms are basedon the principle of the survival of the fittest which triesto retain genetic information from generation to gener-ation. GAs work with a rich database of population andsimultaneously climb many peaks in parallel during thesearch so that the probability of being trapped in a localminimum is reduced significantly.

To implement a GA, a population of initial solutionsmust first be generated randomly or heuristically. GA isan iterative process where each iteration has two steps,the evaluation step and the generation step. In the evalu-ation step, fitness of the individual which is a measureof the quality of a candidate is determined. The gener-ation step includes a selection operator and a modifi-cation operator. Two individuals (parents) are chosenfrom the population using a scheme which favours thefitter individuals. The two selected parents, are recom-bined to form two children, typically using the mech-anism of crossover. The crossover operator exchangesa sub-string of the codes of the parents at a randomlydetermined point or points. Mutation is applied to eachchild individually after crossover with a small prob-ability typically between 0.1–0.2. A mutation operatormay then randomly change some of the values of thegenes constituting an individual.

4. GA based pipe network optimization

GAs typically require problem system states to be rep-resented as strings called chromosomes. For example, ifeight different pipe sizes are available then a binary sub-string of three bits is used to represent the options. Thisprocess requires that the binary coding be converted todiscrete pipe diameters when evaluating the cost of thenetwork. However in the GA based methodologydescribed in this paper it was considered unnecessaryto represent the solution as a chromosome to avoid theconversion of binary coding to discrete pipe sizes.

The technique developed in this research involves thefollowing steps for least cost design/augmentation ofWDS.

1. Read the network data, cost data, required minimumresidual head, probability of mutation, populationsize of solutions (range 50–350), maximum no. ofgenerations (MG, range 10–30), penalty factor(range 0.9–1.0 million), tolerance (range 5–10 m),average head-loss per unit length (HL), maximumno. of iterations for diameter adjustment (MI), mini-mum desirable velocity in a pipe (MV).

2. Generate the population of initial solutions usingrandom number generator. The network is stratifiedinto upper, middle and lower diameter sets. Thisstratification of the network is based on the judge-ment of the design engineer. For example, the pipeswhich are located at the nodes most distant from thesource are grouped into lower dimensional sizes.The lower diameter set may consist of 50, 80, 100,125 and 150 mm which helps in pruning the searchspace and facilitating faster convergence to the opti-mum.

3. Counter 15 1.4. For all solutions of the population carry out the fol-

lowing:(i) Counter 25 1.(ii) For designing a new network go to step (iii).

In the case of augmentation of an existing net-work combine the existing diameter set withnew parallel lines to obtain equivalent pipediameters. Assuming the coefficient of rough-ness of the equivalent pipe to be the same asthat of the new pipe, the diameter of the com-bined pipe is given as D5 [CR2/CR1 * Dold

a

1 Dnewa ] b where,a is 4.8099/1.8099;b is

1.8099/4.8099; CR2, CR1 are the coefficients ofroughness for old and new pipes respectively;Dold, Dnew are the diameters of old and newpipes respectively derived from modifiedHazen-Williams formula (Jain et al., 1978).

(iii) Invoke the hydraulic analysis subroutineANALIS (Bassin et al., 1992) to computeflow, velocity and residual head.

(iv) Increase the pipe diameter to the next uppercommercial diameter size if the head-loss headper unit length > HL. Reduce the size to thenext lower commercial diameter size if velo-city , MV and such that the absolute valueof (total no of increments-total number ofreductions), 4.

(v) Repeat steps (ii) and (iii). If the solution is(a) In-feasible but was feasible earlier restore

the original solution and go to step 5(b) Feasible then store the solution.

(vi) Increment Counter 2.


(vii) If Counter 2 is less than MI, go to step (ii)otherwise go to step 5.

5. Store the solutions in the set of new populations forwhich the residual heads at the nodes are greaterthan the desired residual head tolerance. This stephelps reduce the number of hydraulic analysesrequired.

6. Evaluate cost of the solutions in the new populationand store the feasible solution having the mini-mum cost.

7. Increment Counter 1.8. If Counter 1 is more than MG go to step 159. If a solution does not satisfy the minimum residual

head constraint, evaluate a penalty cost as the pro-duct of the penalty factor and head violated at thecritical node. In the present study the penalty factorhas been taken between 0.9–1.0 million per meterof head.

10. Compute the total cost as the sum of network costand penalty cost.

11. Compute the fitness for each solution as f51/total cost.

12. Perform crossover of solutions of the new popu-lation taken two at a time based on their fitnessvalues as described earlier to produce two offspring.

13. Mutate each offspring based on the mutation rate.14. The offspring constitute the new population. Go to

step 4.15. Write the stored solution set for each generation and

write network details for the best solution.

The above algorithm is an improvement of the method-ology presented by Simpson et al. (1994) as evident inthe following:

1. The set of solutions is stored in discrete pipe sizesand not in binary alphabet as is usual in a GA. Thedecoding required to calculate the fitness for each setof solutions is therefore avoided. The simple ideas ofcrossover and mutation are applied to the discretepipe diameters directly.

2. The network is stratified into a number of groups ofpipe sizes for generation of the initial population ofsolutions. This process helps in reducing the numberof redundant solutions.

3. The solutions which are within tolerable limits areplaced in the set of new solutions there by reducingthe total number of hydraulic analyses.

4. The set of solutions are modified by velocity andaverage head loss adjustments which helps in bring-ing rationality to the solutions.

The maximum size of the distribution system that canbe designed using the software is 200 pipes, 175 nodes,and 2 reservoirs. The software can handle a maximumpopulation of 200 solutions and 30 generations. Incorpo-rating these limitations the size of the executable pro-gram is 122.8 KB. The software can be easily modified

to design larger systems by re-dimensioning the vari-ables in the program.

5. Comparison of GA and NLP-IPF basedtechniques

GA and NLP based techniques are powerful toolswhich have been effectively applied to water distributionsystem optimization problems. The effectiveness of thetechniques with respect to convergence relies on theadaptation of inherent features and properties of the dis-tribution system in the problem formulation. Both thetechniques require few parameter adjustments throughtrial and error to obtain the best solution. The fitnessfunction is most crucial aspect of any GA. Otherimportant parameters include the size of the populationof solutions, the strategy for the stratification of solutionspace in more than one set of diameters, tolerance level,mutation rate, and penalty factor. In the case of the NLPwith IPF, the parameters which control convergence rateinclude the penalty parameter in the unconstrainedobjective function and its subsequent values, and the steplength in the finite difference scheme. In this research,following advantages and disadvantages of GAs andNLPs were observed:

5.1. Advantages of the GA over the NLP technique

GA deals with a population of solutions which arespread over the solution space. It simultaneously climbsmany peaks in parallel during the search so that the prob-ability of trapping into a local minimum is reduced con-siderably. In case of NLP technique, the solution ishighly dependent on the initial solution and it convergesalways to a local minimum based on the initial solution.

The GA uses discrete pipe diameters for generationof each solution set while in NLP technique, the diam-eters are generated as real numbers requiring furtherrounding to commercial sizes. The process often con-verts the solution away from optimum particularly forlarge size networks even after rounding using pro-fessional judgement.

The GA uses a more rational fitness function to selectthe members of the next generation while the NLP relieson derivatives of the unconstrained objective function.

5.2. Advantages of the NLP over the GA technique

NLP converges much faster particularly for mediumand large size networks as compared to GA.

It is possible to incorporate additional techniques inthe NLP for example splitting of link in two sectionswith next lower and upper commercially available diam-eter sizes such that their combined hydraulic character-


istics are the same as that of the non-commercial diam-eter.

6. Case study

In order to establish the efficacy of GA based algor-ithm in comparison with NLP technique several net-works were optimized employing both the techniques.

Fig. 2 delineates network 1. This network consists of38 pipes (30 existing and 8 new) and 23 nodes including21 demand nodes. Water is supplied through a reservoir

Fig. 2. Network diagram for case study 1.

of 20 m height at node no. 8. Constraints on minimumnodal pressure and pipe diameter are 12 and 0.08 mrespectively. The coefficients of roughness for old andnew pipes are 0.7 and 0.9 respectively in the modifiedHazen-Williams equation (Jain et al., 1978). The terrainis flat with little variation in relative elevation. The initialstarting solution for the NLP algorithm is the best alter-native out of five attempted solutions. Application of thisoptimization algorithm resulted in a solution in termsof continuous diameters which has been rounded off tocommercial diameters. The population size of the sol-utions used for the GA based algorithm is 200. In the


present work subroutine ran2 from Numerical Recipes(1992) which is based on the method of L’Eeuyer (1988)has been used for generation of random numbers. Thepipe details of the best solutions obtained through GAand NLP techniques are presented in Tables 5 and 6.Diameters of pipes to be placed in parallel to existingpipes and new pipe lines are given in column 5 whilethe flow and headloss in each pipe is given in columns6 and 7. Nodal details and comparison of the pipe cost

Table 5Pipe details of optimal solution for the case study network-1 employing GA

Pipe no. Node Length (m) Diameter (mm) Flow (lps) Headloss (m) Status

From To

1 2 1 400 80 3.4084 4.954 O2 3 2 300 125 3.5449 0.466 O3 3 4 450 125 4.5319 1.091 O4 4 5 400 80 3.2815 4.625 O5 1 6 200 80 0.0433 0.001 O6 7 2 350 100 3.9342 1.921 O7 8 3 350 150 13.4472 2.528 O8 9 4 350 100 2.6559 0.944 O9 10 5 350 80 1.7098 1.244 O

10 7 6 400 100 7.3922 6.876 O11 9 10 400 125 10.3529 4.325 O12 11 6 220 80 3.5223 2.892 O13 7 12 280 150 15.3784 2.578 O14 14 9 280 150 5.3645 0.383 O15 10 15 280 125 0.9377 0.039 O16 14 15 400 100 6.0240 4.748 O17 11 16 450 80 1.3952 1.107 O18 12 16 450 80 2.1951 2.513 O19 15 19 380 80 1.4814 1.042 O20 8 7 300 150 9.1183 1.072 O

200 25.1820 1.072 NP21 8 9 390 150 13.0697 2.675 O22 8 13 280 200 11.6166 0.389 O

350 66.0862 0.389 NP23 12 11 390 100 3.1193 1.407 O

150 11.7808 1.407 NP24 13 12 310 125 10.1977 3.262 O25 13 14 380 125 6.7662 1.903 O

200 30.3353 1.903 NP26 13 17 370 125 4.8921 1.030 O

200 21.9332 1.030 NP27 14 18 370 100 4.5619 2.655 O

125 10.6128 2.655 NP28 17 16 500 80 2.9420 4.745 O29 17 18 480 80 2.5547 3.528 O

100 5.9432 3.528 NP30 18 19 400 80 2.6466 3.134 O

100 6.1570 3.134 NP31 17 14 520 80 1.4524 0.873 N32 16 20 280 100 6.5322 2.442 N33 17 21 290 100 10.1888 5.654 N34 18 22 290 100 6.3678 2.415 N35 19 23 290 100 3.7745 0.937 N36 21 20 510 80 2.0036 1.532 N37 21 22 370 100 1.7211 0.289 N38 22 23 410 80 2.3598 1.656 N

O, Existing pipe; N, New pipe; NP, New and parallel pipe

for the optimal solutions are presented in Tables 7 and8. The CPU times for network 1 were 6 minutes 40seconds and 2 minutes and 10 seconds respectively forGA and NLP based techniques. The least cost solutionobtained with the GA and NLP techniques are IndianRs. 2,301,330 (1US $5 43.5 Indian Rs.) and2 485 690 respectively.

Network 2 represents an alternate problem to network1 with same network structure having significantly dif-


Table 6Pipe details of the optimal solution for the case study network-1 employing non-linear programming

Pipe no. Node Length (m) Diameter (mm) Flow (lps) Headloss (m) Status

From To

1 2 1 400 80 3.13 4.253 O2 3 2 300 125 6.04 1.223 O3 3 4 450 125 2.55 0.386 O4 4 5 400 80 3.42 4.996 O5 6 1 200 80 0.23 0.019 O6 7 2 350 100 1.16 0.212 O7 8 3 350 150 13.96 2.706 O8 9 4 350 100 2.09 0.612 O

100 2.69 0.612 NP9 10 5 350 80 1.57 1.062 O

10 7 6 400 100 5.81 4.445 O80 4.13 4.445 NP

11 9 10 400 125 10.64 4.546 O12 11 6 220 80 1.25 0.446 O13 7 12 280 150 3.81 0.206 O14 9 14 280 150 7.81 0.756 O15 10 15 280 125 1.37 0.078 O16 14 15 400 100 5.38 3.868 O17 16 11 450 80 2.00 2.132 O18 12 16 450 80 1.75 1.662 O19 15 19 380 80 1.27 0.787 O20 8 7 300 150 18.12 3.717 O

80 4.38 3.717 NP21 8 9 390 150 12.53 2.480 O

150 16.12 2.480 NP22 8 13 280 200 22.07 1.242 O

250 51.34 1.242 NP23 12 11 390 100 5.40 3.794 O

80 3.83 3.794 NP24 13 12 310 125 9.15 2.681 O

100 6.50 2.681 NP25 13 14 380 125 6.94 1.994 O

150 14.49 1.994 NP26 13 17 370 125 5.97 1.477 O

200 26.77 1.477 NP27 14 18 370 100 3.89 1.995 O

125 9.06 1.995 NP28 17 16 500 80 2.23 2.866 O

100 5.18 2.866 NP29 17 18 480 80 2.12 2.511 O

125 8.91 2.511 NP30 18 19 400 80 2.42 2.661 O

100 5.62 2.661 NP31 17 14 520 80 1.09 0.516 N32 16 20 280 100 7.15 2.875 N33 17 21 290 100 9.47 4.954 N34 18 22 290 100 7.44 3.201 N35 19 23 290 80 2.80 1.598 N36 21 20 510 80 1.39 0.787 N37 21 22 370 80 1.62 0.758 N38 22 23 410 100 3.33 1.058 N

O, Existing pipe; N, New pipe; NP, New and parallel pipe

ferent demand pattern. The GA and NLP solutionsresulted in Indian Rs. 2.99 million and 3.006 millionrespectively. Networks 3 and 4 were optimized similarlywhile networks 5 and 6 are two other different networks

optimized employing GA and NLP techniques. Table 9summarizes the details of these networks and optimalcosts. Out of these six sets of solutions except for thecase of network 3, all optimal costs obtained through


Table 7Nodal details of optimal solution for case study network-1 employing GA and NLP techniques

Node no. Gr. level (m) Peak demand (l/s) Residual head (m) Residual head (m)employing GA employing NLP

1 D 101.45 3.365 12.60 12.372 D 101.00 4.071 18.01 17.073 D 102.00 5.370 17.47 17.294 D 101.50 3.906 16.88 17.415 D 101.75 4.991 12.01 12.166 D 100.50 10.958 13.55 13.347 D 101.40 7.595 19.53 16.888 R 102.00 2 138.520 20.00 20.009 D 100.90 5.425 18.43 18.62

10 D 101.25 7.705 13.75 13.7211 D 102.00 9.983 14.94 12.2812 D 100.80 8.481 17.55 17.2813 D 101.40 3.578 20.21 19.3614 D 101.70 11.991 18.01 17.0615 D 101.50 5.480 13.46 13.4016 D 101.80 0.000 14.04 14.6117 D 101.30 3.744 19.28 17.9818 D 100.95 8.501 16.10 15.8219 D 100.00 6.510 13.92 14.1120 D 101.00 8.536 12.39 12.5421 D 100.00 6.464 14.93 14.3322 D 100.50 5.729 14.14 13.0723 D 100.50 6.134 12.48 12.01

R: Reservoir location; D, Demand node

Table 8Comparison of cost of new pipe lines in GA and NLP based techniques for case study network-1

Pipe dia. (mm) Unit cost Length of pipe in m using Cost of pipe in Rs.(Indian Rs./m)

GA NLP GA NLP

80 228 1440 2780 328320 633840100 278 2400 2830 667200 786740125 336 370 850 124320 285600

Total costs (Indian Rs.): 2 301 330; 2,485,690Note: 1US$5 43.5 Indian Rs.

Table 9Network details and comparison of optimal design costs employing GA and NLP techniques for various case studies

Network no. No. of pipes No. of Total pipe Total Coeff. of friction Optimal* cost (million)node length (m) demand

(lps)

Total New Old New GA NLP

1 38 30 23 13,820 138.5 0.70 0.90 2.301 2.4862 38 30 23 13,820 155.9 0.65 0.85 2.991 3.0063 52 30 31 18,385 218.8 0.70 0.90 5.374 5.3324 52 30 31 18,385 200.5 0.65 0.85 5.140 5.1545 28 23 18 10,210 107.7 0.65 0.85 1.622 1.7426 13 8 11 24,136 145.0 0.65 0.85 28.244 28.364

*Indian RupeesNote: 1US$5 43.5 Indian Rs.


GA technique were found to be lower than thoseobtained through NLP technique.

7. Conclusions

The paper presents the applicability of genetic algor-ithm in the design of water distribution systems. Thealgorithm has been compared with a NLP technique withIPF method which was found to be fairly efficient incomparison to the techniques presented thus far in theliterature. The solution set obtained from GA and NLPtechniques for several medium size networks showedthat GA provides a better solution in general, in compari-son with that obtained with the NLP technique. The dif-ferences in costs, however, is marginal which showsboth the techniques are efficient.

The convergence of GA was considerably improvedby providing initial information on network stratifi-cation. Generation of initial feasible solution, as requiredin NLP technique, is avoided in GA. Considerableefforts are needed to generate separate initial solutionsto try acceptable number of trials in NLP methodologyin order to ensure optimality particularly in case of largernetworks. Trials in GA are made by providing differentseeds which does not require elaborate computations ingenerating a feasible solution. NLP technique alsorequires rounding of pipe diameters to the available com-mercial sizes while GA selects discrete diameters. Foridentical problem dimensions of pipes (200), nodes (175)and reservoirs (2), the size of the executable programsfor GA and NLP techniques are 122.8 and 126.9 KBrespectively.

Experience, however, indicates that both the tech-niques warrant several trials to obtain the best solution.The GA requires the trials in order to try out differentseeds as well as initial guess on network stratificationwhile the NLP requires different initial solutions fromwhich to begin the search for the optimum. Generationof information on network stratification is significantlyeasier in comparison with generation of initial solutions.As a result, GA provides a convenient technique in per-forming more trials in comparison with NLP techniquein order to obtain cost effective design of water distri-bution systems.

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Genetic Algorithm for Optimization of Water Distribution Systems