This study aims at the development of an optimization model based on artificial immune systems (AIS) to minimize cost designs of water distribution networks (WDNs). Clonal selection algorithm (Clonalg), a class of AIS, was used as an optimization technique in the model, and its mutation operation was modified to increase the diversity (search capability). EPANET, a widely known WDN simulator, was used in conjunction with the proposed model. The model was applied to four WDNs of Two-loop, Hanoi, Go Yang, New York City, and the results obtained were compared with other heuristic and mathematical optimization models in the related literature, such as harmony search, genetic algorithm, immune algorithm, shuffled complex evolution, differential evolution, and non-linear programming-Lagrangian algorithm. Furthermore, the modified Clonalg was compared with the classic Clonalg in order to demonstrate the impact of the modification on the diversity. The proposed model appeared to be promising in terms of cost designs of WDNs.

INTRODUCTION

At the present time, design engineers should consider the cost and functionality while designing any project. In this regard, cost optimization techniques are mostly used in the designing of water distribution networks (WDNs). Many conventional optimization techniques have been developed to solve minimal-cost problems, such as linear programming gradient (Alperovits & Shamir 1977; Fujiwara et al. 1987; Kessler & Shamir 1989; Bhave & Sonak 1992; Eiger et al. 1994), dynamic programming (Schaake & Lai 1969), and non-linear programming (NLP) (Fujiwara & Khang 1990; Varma et al. 1997). However, obtaining a least-cost WDN with discrete-value pipe diameters is extremely difficult using conventional optimization techniques (Yates et al. 1984). On the other hand, heuristic algorithms have recently been utilized to solve WDN optimization problems, such as genetic algorithm (GA), simulated annealing, Tabu search, ant colony optimization, scatter search, shuffled complex evolution (SCE), and differential evolution (DE) (Simpson et al. 1994; Cunha & Sousa 1999; Maier et al. 2003; Cunha & Ribeiro 2004; Liong & Atiquzzaman 2004; Zecchin et al. 2006; Lin et al. 2007; Dong et al. 2012). Geem (2006), Savic & Walters (1997), and Kim et al. (1994) proposed cost optimization models using a harmony search (HS) algorithm, GA, and a projected Lagrangian algorithm (NLP), respectively, for the following four famous WDN problems of Two-loop, Hanoi, Go Yang, and New York City in the related literature. Another heuristic algorithm is artificial immune systems (AIS) simulating the natural immune system (NIS) (Harmer et al. 2002; De Castro & Timmis 2003; Koster et al. 2003; Bezerra et al. 2004). AIS have been successfully applied to various test functions and optimization problems (Tazawa et al. 1996; Hajela et al. 1997; Chu et al. 2008).

This study develops a cost optimization model based on AIS in order to contribute to the related literature. The model aims to determine optimal diameters of pipes in WDNs under hydraulic conditions such as water demand, and minimum pressure requirement. Clonal selection algorithm (Clonalg), a class of AIS, was used as an optimization technique in the model in order to minimize the cost designs of WDNs. The main difference to the classic Clonalg, is the modification proposed to its mutation operation. While running the algorithm, new genes were generated for each antibody during the mutation operation instead of generating new individuals (antibody) in each iteration. This modification provides to increase a diversity. The model was applied to the four benchmark WDNs, and results obtained were compared with the other algorithms in the related literature, such as HS, GA, NLP, immune algorithm (IA), SCE, and DE. Furthermore, the modified Clonalg was compared with the classic Clonalg in order to demonstrate the impact of the modification on the diversity.

CLONALG

AIS are based on the principles of the NIS. There are two selection theories of ‘Clonal Selection Theory’ and ‘Negative Selection Theory’, involving these principles in the immune system. Clonalg, a class of AIS, is based on Clonal Selection Theory (De Castro & Von Zuben 2000, 2002). This theory uses the following three major principles of Charles Darwin's law of evolution: diversity, variation, and natural selection (De Castro & Von Zuben 2002). Figure 1 illustrates the Clonal Selection Theory.

Figure 1

Clonal Selection Theory (De Castro & Von Zuben 2000).

Figure 1

Clonal Selection Theory (De Castro & Von Zuben 2000).

Clonalg can be described for optimization problems as follows (De Castro & Von Zuben 2002):

  • (1) An antibody set (Ab) is randomly constituted and there is an objective function g(.) to be optimized (maximized or minimized). An antibody's antigenic affinity corresponds to the value of this objective function for a given antibody, so that each antibody (Abi) represents an element of the input set (Ab).

  • (2) For each Abi in Ab, the affinity value (fi) is determined.

  • (3) The n amount of antibodies with the highest affinity is selected.

  • (4) The n amount of antibodies selected is cloned (reproduced) independently and proportionally to their antigenic affinities. The clones constitute a repertoire C. The higher antigenic affinity means the higher number of clones generated for each of the n antibodies.

  • (5) The repertoire C is exposed to an affinity maturation process (mutation) inversely proportional to the antigenic affinity. The matured clones constitute a population C*. The higher antigenic affinity means a smaller mutation rate.

  • (6) For each matured clone in C*, the affinity value (fi*) is determined.

  • (7) From the population C*, the n amount of the matured clones with the highest affinity is reselected and added to Ab.

  • (8) Finally, the d amount of the antibodies with the lowest affinity is replaced with new individuals (Abd).

Figures 2 and 3 illustrate the flow charts of the classic Clonalg and the modified Clonalg for optimization problems, respectively.

Figure 2

Flow chart of Clonalg for optimization (De Castro & Von Zuben 2002).

Figure 2

Flow chart of Clonalg for optimization (De Castro & Von Zuben 2002).

Figure 3

Flow chart of the modified Clonalg for optimization.

Figure 3

Flow chart of the modified Clonalg for optimization.

Description of Ab 
formula
1
where NAb is the total number of the antibodies in Ab, xij is the gene of Abi, corresponding to a variable of the objective function, and nd is the number of genes of Abi. In this study, xij corresponds to a diameter of the pipe.
The number of clones generated for all the n selected antibodies can be estimated as follows (De Castro & Von Zuben 2002): 
formula
2
where Nc is the total number of the clones in C, β is a multiplying coefficient, ‘’ is the rounding operator for an integer.
The mutation rate can be computed as follows (De Castro & Von Zuben 2002): 
formula
3
where αi is the mutation rate for the clones exposed to the maturation process, ρ is a decay coefficient, and fi is the affinity value normalized over the interval [0,1].
In this study, new genes are generated for each clone with a certain probability depending on a given problem (probability rate) in step 5, instead of step 8. This modification provides to increase the diversity. Furthermore, instead of using Equation (2) as proposed by De Castro & Von Zuben (2002), the following equation was used to calculate Nc (De Castro & Von Zuben 2002): 
formula
4
This modification results in that both all antibodies in Ab are cloned and the same number of the clones is generated for each antibody in Ab. Thus, there is not any selection of n amount of antibodies with the highest affinity (step 3).

MODEL FORMULATION

The cost design of WDNs consists of diameters and lengths of pipes constituting the network. Thus, the objective function depends on the pipe's diameter and length. The objective function used in the model is of the following form (Geem 2006; Chu et al. 2008):

 
formula
5
where f(Di, Li) is the cost of pipe i with its diameter Di and length Li, and N is the number of pipes in the network. While determining the diameters, the hydraulic conditions of water demands and minimum pressures required at all the junction nodes of the WDN should be satisfied. In this regard, the cost function is to be minimized by considering the following constraints (Geem 2006; Chu et al. 2008).
For each node, the continuity equation should be satisfied 
formula
6
where Qin and Qout are the inflow and outflow rate of the node, respectively, and Qe is the external inflow rate or demand at the node.
For each loop in the network, the energy conservation equation is utilized 
formula
7
where ΔHi is the head loss in pipe i, and Ep is the energy added to the water in the network by a pump.
For each node, the minimum pressure required is expressed as follows: 
formula
8
where Hj is the pressure head at node j, Hjmin is the minimum required pressure head at node j, and M is the number of nodes in the network.
The model uses EPANET for the hydraulics analysis. EPANET has three head loss equations – Hazen–Williams, Darcy–Weisbach and Chezy–Manning – to compute the friction head loss in the pipes. The Hazen–Williams equation was used in this study (Rossman 2000) 
formula
9
where is the head loss (length; ft or m), C is Hazen–Williams roughness coefficient, Q is the flow rate (volume/time; cfs), D is the pipe diameter (ft), and L is the pipe length (ft).
Savic & Walters (1997) introduced a numerical conversion constant depending on the units used. With this constant, the Hazen–Williams equation can be re-arranged as follows: 
formula
10
where ω is a conversion constant, α is a coefficient equal to 1.852 (Rossman 2000), and β is a coefficient equal to 4.871 (Rossman 2000). While considering the minimum pressure head requirements, the higher constant ω values require a larger diameter to deliver the same amount of water than the lower values since the greater head loss means higher cost for the WDN designs (Geem 2006). The constant ω value in EPANET v.2.0 is 10.667 (Eusuff 2004; Van Dijk et. al 2008). This value ω can be derived as follows (Rossman 2000; Reehuis 2010): 
formula
where convQ is a conversion factor from defined unit for Q to cfs, and convD is a conversion factor from defined unit for D to ft. When Q and D are expressed in m3/s and m, respectively, and ω becomes: 
formula
where: 
formula
 
formula
The penalty function is described in addition to the objective function in case of violating the constraints. This function prevents searching from taking place in the infeasible solution space where pipes with small diameters that cannot satisfy the minimum pressure requirement at each node are located. The penalty function is of the following form (Geem 2006): 
formula
where fp is the penalty function, max is the maximum function giving the larger value, sgn is the sign function extracting the sign of a real number, and a and b are the penalty coefficients. The penalty function is added to the total design cost Ct: 
formula

APPLICATIONS

The model was applied to four benchmark WDNs. The commercial diameters and their corresponding costs used in the applications are given in Table 1.

Table 1

Candidate diameters and corresponding costs for the networks

Network Candidate Diameter Corresponding Cost 
Two-loop 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 in inches 2, 5, 8, 11, 16, 23, 32, 50, 60, 90, 130, 170, 300, 550 in USD/meter 
Hanoi 12, 16, 20, 24, 30, 40 in inches 45.726, 70.4, 98.378, 129.333, 180.748, 278.28 in USD/meter 
New York 0, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204 in inches 93.5, 134, 176, 221, 267, 316, 365, 417, 469, 522, 577, 632, 689, 746, 804 in USD/foot 
GoYang 80, 100, 125, 150, 200, 250, 300, 350 in millimeters 37,890, 38,933, 40,563, 42,554, 47,624, 54,125, 62,109, 71,524 in won/meter 
Network Candidate Diameter Corresponding Cost 
Two-loop 1, 2, 3, 4, 6, 8, 10, 12, 14, 16, 18, 20, 22, 24 in inches 2, 5, 8, 11, 16, 23, 32, 50, 60, 90, 130, 170, 300, 550 in USD/meter 
Hanoi 12, 16, 20, 24, 30, 40 in inches 45.726, 70.4, 98.378, 129.333, 180.748, 278.28 in USD/meter 
New York 0, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204 in inches 93.5, 134, 176, 221, 267, 316, 365, 417, 469, 522, 577, 632, 689, 746, 804 in USD/foot 
GoYang 80, 100, 125, 150, 200, 250, 300, 350 in millimeters 37,890, 38,933, 40,563, 42,554, 47,624, 54,125, 62,109, 71,524 in won/meter 

Two-loop water distribution network

This network consists of seven nodes and eight pipes with two loops, and is fed by the gravity from a reservoir with a 210 m fixed head (Alperovits & Shamir 1977). Node and pipe data and the layout of Two-loop network are shown in Table 2 and Figure 4, respectively.

Table 2

Node and pipe data of Two-loop network

Node Elevation (m) Demand (m3/h) Min. Pressure (m) Pipe Length (m) C 
210 Reservoir Reservoir 1000 130 
150 100 30 1000 130 
160 100 30 1000 130 
155 120 30 1000 130 
150 270 30 1000 130 
165 330 30 1000 130 
160 200 30 1000 130 
    1000 130 
Node Elevation (m) Demand (m3/h) Min. Pressure (m) Pipe Length (m) C 
210 Reservoir Reservoir 1000 130 
150 100 30 1000 130 
160 100 30 1000 130 
155 120 30 1000 130 
150 270 30 1000 130 
165 330 30 1000 130 
160 200 30 1000 130 
    1000 130 
Figure 4

Layout of Two-loop network.

Figure 4

Layout of Two-loop network.

Hanoi water distribution network

This network consists of 32 nodes, 34 pipes and three loops, and is fed by the gravity from a reservoir with a 100 m fixed head (Fujiwara & Khang 1990). Node and pipe data and the layout of Hanoi network are shown in Table 3 and Figure 5, respectively.

Table 3

Node and pipe data of Hanoi network

Node Elevation (m) Demand (m3/h) Min. Pressure (m) Pipe Length (m) C 
100 Reservoir Reservoir 100 130 
890 30 1350 130 
850 30 900 130 
130 30 1150 130 
725 30 1450 130 
1005 30 450 130 
1350 30 850 130 
550 30 850 130 
525 30 800 130 
10 525 30 10 950 130 
11 500 30 11 1200 130 
12 560 30 12 3500 130 
13 940 30 13 800 130 
14 615 30 14 500 130 
15 280 30 15 550 130 
16 310 30 16 2730 130 
17 865 30 17 1750 130 
18 1345 30 18 800 130 
19 60 30 19 400 130 
20 1275 30 20 2200 130 
21 930 30 21 1500 130 
22 485 30 22 500 130 
23 1045 30 23 2650 130 
24 820 30 24 1230 130 
25 170 30 25 1300 130 
26 900 30 26 850 130 
27 370 30 27 300 130 
28 290 30 28 750 130 
29 360 30 29 1500 130 
30 360 30 30 2000 130 
31 105 30 31 1600 130 
32 805 30 32 150 130 
    33 860 130 
    34 950 130 
Node Elevation (m) Demand (m3/h) Min. Pressure (m) Pipe Length (m) C 
100 Reservoir Reservoir 100 130 
890 30 1350 130 
850 30 900 130 
130 30 1150 130 
725 30 1450 130 
1005 30 450 130 
1350 30 850 130 
550 30 850 130 
525 30 800 130 
10 525 30 10 950 130 
11 500 30 11 1200 130 
12 560 30 12 3500 130 
13 940 30 13 800 130 
14 615 30 14 500 130 
15 280 30 15 550 130 
16 310 30 16 2730 130 
17 865 30 17 1750 130 
18 1345 30 18 800 130 
19 60 30 19 400 130 
20 1275 30 20 2200 130 
21 930 30 21 1500 130 
22 485 30 22 500 130 
23 1045 30 23 2650 130 
24 820 30 24 1230 130 
25 170 30 25 1300 130 
26 900 30 26 850 130 
27 370 30 27 300 130 
28 290 30 28 750 130 
29 360 30 29 1500 130 
30 360 30 30 2000 130 
31 105 30 31 1600 130 
32 805 30 32 150 130 
    33 860 130 
    34 950 130 
Figure 5

Layout of Hanoi network.

Figure 5

Layout of Hanoi network.

New York City water distribution network

This network consists of 20 nodes, 21 pipes and one loop, and is fed by the gravity from a reservoir with a 300 ft fixed head. The objective of the problem is to add new pipes parallel to existing pipes in the network since the existing pipes cannot satisfy the pressure head requirements at nodes 16, 17, 18, 19, and 20 (Schaake & Lai 1969). Node and pipe data and the layout of the New York City network are shown in Table 4 and Figure 6, respectively.

Table 4

Node and pipe data of New York City network

Node Demand (cfs) Min. Head (ft) Pipe Length (ft) C 
Reservoir Reservoir 11,600 100 
92.4 255.0 19,800 100 
92.4 255.0 7300 100 
88.2 255.0 8300 100 
88.2 255.0 8600 100 
88.2 255.0 19,100 100 
88.2 255.0 9600 100 
88.2 255.0 12,500 100 
170.0 255.0 9600 100 
10 1.0 255.0 10 11,200 100 
11 170.0 255.0 11 14,500 100 
12 117.1 255.0 12 12,200 100 
13 117.1 255.0 13 24,100 100 
14 92.4 255.0 14 21,100 100 
15 92.4 255.0 15 15,500 100 
16 170.0 260.0 16 26,400 100 
17 57.5 272.8 17 31,200 100 
18 117.1 255.0 18 24,000 100 
19 117.1 255.0 19 14,400 100 
20 170.0 255.0 20 38,400 100 
   21 26,400 100 
Node Demand (cfs) Min. Head (ft) Pipe Length (ft) C 
Reservoir Reservoir 11,600 100 
92.4 255.0 19,800 100 
92.4 255.0 7300 100 
88.2 255.0 8300 100 
88.2 255.0 8600 100 
88.2 255.0 19,100 100 
88.2 255.0 9600 100 
88.2 255.0 12,500 100 
170.0 255.0 9600 100 
10 1.0 255.0 10 11,200 100 
11 170.0 255.0 11 14,500 100 
12 117.1 255.0 12 12,200 100 
13 117.1 255.0 13 24,100 100 
14 92.4 255.0 14 21,100 100 
15 92.4 255.0 15 15,500 100 
16 170.0 260.0 16 26,400 100 
17 57.5 272.8 17 31,200 100 
18 117.1 255.0 18 24,000 100 
19 117.1 255.0 19 14,400 100 
20 170.0 255.0 20 38,400 100 
   21 26,400 100 
Figure 6

Layout of New York network.

Figure 6

Layout of New York network.

Go Yang water distribution network

This network consists of 22 nodes, 30 pipes and nine loops, and is fed by a pump from a reservoir with a 71 m fixed head (Kim et al. 1994). The pressure head produced by the pump is 15.61 m so that the total head of the water supplied from the reservoir becomes 86.61 m. Node and pipe data and the layout of Go Yang network are shown in Table 5 and Figure 7, respectively.

Table 5

Node and pipe data of Go Yang network

Node Elevation (m) Demand (m3/day) Min. Pressure (m) Pipe Length (m) C 
71.0 Reservoir Reservoir 165 100 
56.4 153.0 15 124 100 
53.8 70.5 15 118 100 
54.9 58.5 15 81 100 
56.0 75.0 15 134 100 
57.0 67.5 15 135 100 
53.9 63.0 15 202 100 
54.5 48.0 15 135 100 
57.9 42.0 15 170 100 
10 62.1 30.0 15 10 113 100 
11 62.8 42.0 15 11 335 100 
12 58.6 37.5 15 12 115 100 
13 59.3 37.5 15 13 345 100 
14 59.8 63.0 15 14 114 100 
15 59.2 445.5 15 15 103 100 
16 53.6 108.0 15 16 261 100 
17 54.8 79.5 15 17 72 100 
18 55.1 55.5 15 18 373 100 
19 54.2 118.5 15 19 98 100 
20 54.5 124.5 15 20 110 100 
21 62.9 31.5 15 21 98 100 
22 61.8 799.5 15 22 246 100 
    23 174 100 
    24 102 100 
    25 92 100 
    26 100 100 
    27 130 100 
    28 90 100 
    29 185 100 
    30 90 100 
Node Elevation (m) Demand (m3/day) Min. Pressure (m) Pipe Length (m) C 
71.0 Reservoir Reservoir 165 100 
56.4 153.0 15 124 100 
53.8 70.5 15 118 100 
54.9 58.5 15 81 100 
56.0 75.0 15 134 100 
57.0 67.5 15 135 100 
53.9 63.0 15 202 100 
54.5 48.0 15 135 100 
57.9 42.0 15 170 100 
10 62.1 30.0 15 10 113 100 
11 62.8 42.0 15 11 335 100 
12 58.6 37.5 15 12 115 100 
13 59.3 37.5 15 13 345 100 
14 59.8 63.0 15 14 114 100 
15 59.2 445.5 15 15 103 100 
16 53.6 108.0 15 16 261 100 
17 54.8 79.5 15 17 72 100 
18 55.1 55.5 15 18 373 100 
19 54.2 118.5 15 19 98 100 
20 54.5 124.5 15 20 110 100 
21 62.9 31.5 15 21 98 100 
22 61.8 799.5 15 22 246 100 
    23 174 100 
    24 102 100 
    25 92 100 
    26 100 100 
    27 130 100 
    28 90 100 
    29 185 100 
    30 90 100 
Figure 7

Layout of Go Yang network.

Figure 7

Layout of Go Yang network.

RESULTS

The following two conditions were used to stop running the model: (1) when the absolute error between the maximum and minimum values of the objective function is less than 0.1; and (2) when the maximum iteration number is reached.

The model was run 20 times for each WDN. Random seed (random number generation) was applied while constituting the initial set in each run. In the applications of the classic Clonalg, n and d were assigned as 90% of NAb and 10% of NAb, respectively, since De Castro & Von Zuben (2002) suggested that values of d range from 5 to 20% of NAb (population).

In the study, a PC with Intel Core Duo 2.16 Ghz and Matlab R2009a 7.8 programming language were used for the analyses. Optimal costs obtained by the modified Clonalg are 419,000 USD (using ω = 10.4973, ω = 10.5088, ω = 10.667), 6,016,520 and 6,081,087 USD (using ω = 10.5088, ω = 10.667, respectively), 36,660,000 and 38,637,600 USD (using ω = 10.5088, ω = 10.667, respectively), 176,958,824, 176,994,561 and 177,009,557 won (using ω = 10.5088, ω = 10.5879, ω = 10.667, respectively) for Two-loop, Hanoi, New York City and Go Yang, respectively. On the other hand, optimal costs for these networks in the related literature are 419,000 USD (Savic & Walters 1997; Keedwell & Khu 2005; Geem 2006), 6,056,000 USD (Vairavamoorthy & Ali 2000; Wu & Walski 2005; Geem 2006), 36,660,000 USD (Geem 2006), and 177,010,000 won (Dong et al. 2012), respectively. In addition, optimal costs obtained by the classic Clonalg for these networks are 419,000 USD (using ω = 10.667), 6,085,284 USD (using ω = 10.667), 49,240,000 USD (using ω = 10.667), and 177,009,557 won (using ω = 10.667), respectively. Parameters of the modified Clonalg and the classic Clonalg, comparisons of the results, pressures at the nodes, and performances of the modified Clonalg and the classic Clonalg for all the networks are given in Tables 678910111213141516171819202122232425.

Table 6

Parameters of the modified Clonalg and the classic Clonalg for Two-loop network

Algorithm ω NAb β ρ Probability Rate Max. Iteration 
Modified 10.4973 150 0.02 0.25 500 
10.5088 150 0.02 0.25 500 
10.667 150 0.1 0.25 500 
Classic 10.667 150 0.1 – 500 
Algorithm ω NAb β ρ Probability Rate Max. Iteration 
Modified 10.4973 150 0.02 0.25 500 
10.5088 150 0.02 0.25 500 
10.667 150 0.1 0.25 500 
Classic 10.667 150 0.1 – 500 
Table 7

Comparison of the results for Two-loop network

  Modified Clonalg (This study) HS Geem (2006)  GA Keedwell & Khu (2005)  GA Savic & Walters (1997)  
Pipe ω = 10.4973 ω = 10.5088 ω = 10.667 ω = 10.4973 ω = 10.5879 ω = 10.667 ω = 10.5088 
18 18 18 18 18 18 18 
10 10 10 10 10 10 10 
16 16 16 16 16 16 16 
16 16 16 16 16 16 16 
10 10 10 10 10 10 10 
10 10 10 10 10 10 10 
Cost($) 419,000 419,000 419,000 419,000 419,000 419,000 419,000 
  Modified Clonalg (This study) HS Geem (2006)  GA Keedwell & Khu (2005)  GA Savic & Walters (1997)  
Pipe ω = 10.4973 ω = 10.5088 ω = 10.667 ω = 10.4973 ω = 10.5879 ω = 10.667 ω = 10.5088 
18 18 18 18 18 18 18 
10 10 10 10 10 10 10 
16 16 16 16 16 16 16 
16 16 16 16 16 16 16 
10 10 10 10 10 10 10 
10 10 10 10 10 10 10 
Cost($) 419,000 419,000 419,000 419,000 419,000 419,000 419,000 
Table 8

Comparison of the modified Clonalg and the classic Clonalg for Two-loop network

  Modified Clonalg Classic Clonalg 
Pipe ω = 10.667 ω = 10.667 
18 18 
10 10 
16 16 
16 16 
10 10 
10 10 
Cost (won) 419,000 419,000 
  Modified Clonalg Classic Clonalg 
Pipe ω = 10.667 ω = 10.667 
18 18 
10 10 
16 16 
16 16 
10 10 
10 10 
Cost (won) 419,000 419,000 
Table 9

Performances of the modified Clonalg and the classic Clonalg for Two-loop network

Algorithm ω Min. Cost (USD) Max. Cost (USD) Success rate in 20 runs (%) Average Run Time (min) Average Iteration Number 
Modified 10.4973 419,000 419,000 100 5.5 448 
10.5088 419,000 419,000 100 486 
10.667 419,000 420,000 95 116 
Classic 10.667 419,000 459,000 15 0.7 124 
Algorithm ω Min. Cost (USD) Max. Cost (USD) Success rate in 20 runs (%) Average Run Time (min) Average Iteration Number 
Modified 10.4973 419,000 419,000 100 5.5 448 
10.5088 419,000 419,000 100 486 
10.667 419,000 420,000 95 116 
Classic 10.667 419,000 459,000 15 0.7 124 
Table 10

Pressures at the nodes of Two-loop network

  Pressures (m) 
Node ω = 10.4973 ω = 10.5088 ω = 10.667 
Reservoir Reservoir Reservoir 
53.35 53.34 53.24 
30.77 30.75 30.46 
43.63 43.62 43.44 
34.22 34.19 33.80 
30.67 30.66 30.44 
30.86 30.84 30.55 
  Pressures (m) 
Node ω = 10.4973 ω = 10.5088 ω = 10.667 
Reservoir Reservoir Reservoir 
53.35 53.34 53.24 
30.77 30.75 30.46 
43.63 43.62 43.44 
34.22 34.19 33.80 
30.67 30.66 30.44 
30.86 30.84 30.55 
Table 11

Parameters of the modified Clonalg and the classic Clonalg for Hanoi network

Algorithm ω NAb β ρ Probability Rate Max. Iteration 
Modified 10.5088 30 0.05 1.7 0.01 20,000 
10.667 40 1.7 0.01 500 
Classic 10.667 40 1.7 – 1000 
Algorithm ω NAb β ρ Probability Rate Max. Iteration 
Modified 10.5088 30 0.05 1.7 0.01 20,000 
10.667 40 1.7 0.01 500 
Classic 10.667 40 1.7 – 1000 
Table 12

Comparison of the results for Hanoi network

  Modified Clonalg (This study) Van Dijk et al. (2008)  HS Geem (2006)  Wu & Walski (2005)  Liong & Atiquzzaman (2004)  Vairavamoorthy & Ali (2000)  GA Savic & Walters (1997)  
Pipe ω = 10.5088 ω = 10.667 ω = 10.667 ω = 10.5088 ω = 10.667 ω = 10.667 ω = 10.5088 ω = 10.5088 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 30 40 40 
30 40 40 40 40 30 40 40 
10 30 30 30 30 30 30 30 
11 24 24 24 24 24 30 24 24 
12 24 24 24 24 24 24 24 24 
13 16 20 24 20 20 16 20 20 
14 12 16 12 16 16 12 16 16 
15 12 12 12 12 12 12 12 12 
16 12 12 12 12 12 24 12 12 
17 20 16 16 16 16 30 16 16 
18 20 24 24 20 20 30 20 20 
19 24 20 24 20 20 30 20 20 
20 40 40 40 40 40 40 40 40 
21 20 20 20 20 20 20 20 20 
22 12 12 12 12 12 12 12 12 
23 40 40 40 40 40 30 40 40 
24 30 30 30 30 30 30 30 30 
25 30 30 30 30 30 24 30 30 
26 20 20 20 20 20 12 20 20 
27 16 12 12 12 12 20 12 12 
28 12 12 12 12 12 24 12 12 
29 16 16 16 16 16 16 16 16 
30 12 12 12 12 12 16 12 16 
31 12 12 12 12 12 12 12 12 
32 20 16 20 16 16 16 16 12 
33 16 16 16 16 16 20 16 16 
34 24 24 24 24 24 24 24 20 
Cost($) 6,016,520 6,081,087 6,110,000 6,056,000 6,056,000 6,220,000 6,056,000 6,073,000 
  Modified Clonalg (This study) Van Dijk et al. (2008)  HS Geem (2006)  Wu & Walski (2005)  Liong & Atiquzzaman (2004)  Vairavamoorthy & Ali (2000)  GA Savic & Walters (1997)  
Pipe ω = 10.5088 ω = 10.667 ω = 10.667 ω = 10.5088 ω = 10.667 ω = 10.667 ω = 10.5088 ω = 10.5088 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 40 40 40 
40 40 40 40 40 30 40 40 
30 40 40 40 40 30 40 40 
10 30 30 30 30 30 30 30 
11 24 24 24 24 24 30 24 24 
12 24 24 24 24 24 24 24 24 
13 16 20 24 20 20 16 20 20 
14 12 16 12 16 16 12 16 16 
15 12 12 12 12 12 12 12 12 
16 12 12 12 12 12 24 12 12 
17 20 16 16 16 16 30 16 16 
18 20 24 24 20 20 30 20 20 
19 24 20 24 20 20 30 20 20 
20 40 40 40 40 40 40 40 40 
21 20 20 20 20 20 20 20 20 
22 12 12 12 12 12 12 12 12 
23 40 40 40 40 40 30 40 40 
24 30 30 30 30 30 30 30 30 
25 30 30 30 30 30 24 30 30 
26 20 20 20 20 20 12 20 20 
27 16 12 12 12 12 20 12 12 
28 12 12 12 12 12 24 12 12 
29 16 16 16 16 16 16 16 16 
30 12 12 12 12 12 16 12 16 
31 12 12 12 12 12 12 12 12 
32 20 16 20 16 16 16 16 12 
33 16 16 16 16 16 20 16 16 
34 24 24 24 24 24 24 24 20 
Cost($) 6,016,520 6,081,087 6,110,000 6,056,000 6,056,000 6,220,000 6,056,000 6,073,000 
Table 13

Comparison of the modified Clonalg and the classic Clonalg for Hanoi network

  Modified Clonalg Classic Clonalg 
Pipe ω = 10.667 ω = 10.667 
40 40 
40 40 
40 40 
40 40 
40 40 
40 40 
40 40 
40 40 
40 40 
10 30 30 
11 24 24 
12 24 24 
13 20 20 
14 16 16 
15 12 12 
16 12 12 
17 16 16 
18 24 24 
19 20 20 
20 40 40 
21 20 20 
22 12 12 
23 40 40 
24 30 30 
25 30 30 
26 20 20 
27 12 12 
28 12 12 
29 16 16 
30 12 12 
31 12 12 
32 16 20 
33 16 16 
34 24 24 
Cost ($) 6,081,087 6,085,284 
  Modified Clonalg Classic Clonalg 
Pipe ω = 10.667 ω = 10.667 
40 40 
40 40 
40 40 
40 40 
40 40 
40 40 
40 40 
40 40 
40 40 
10 30 30 
11 24 24 
12 24 24 
13 20 20 
14 16 16 
15 12 12 
16 12 12 
17 16 16 
18 24 24 
19 20 20 
20 40 40 
21 20 20 
22 12 12 
23 40 40 
24 30 30 
25 30 30 
26 20 20 
27 12 12 
28 12 12 
29 16 16 
30 12 12 
31 12 12 
32 16 20 
33 16 16 
34 24 24 
Cost ($) 6,081,087 6,085,284 
Table 14

Performances of the modified Clonalg and the classic Clonalg for Hanoi network

Algorithm ω Min. Cost (USD) Max. Cost (USD) Success rate in 20 runs (%) Average Run Time (min) Average Iteration Number 
Modified 10.5088 6,016,520 6,025,984 95 141 20,000 
10.667 6,081,087 6,089,927 90 271.6 500 
Classic 10.667 6,085,284 6,431,702 37.8 1000 
Algorithm ω Min. Cost (USD) Max. Cost (USD) Success rate in 20 runs (%) Average Run Time (min) Average Iteration Number 
Modified 10.5088 6,016,520 6,025,984 95 141 20,000 
10.667 6,081,087 6,089,927 90 271.6 500 
Classic 10.667 6,085,284 6,431,702 37.8 1000 
Table 15

Pressures at the nodes of Hanoi network

  Pressures (m) 
Node ω = 10.5088 ω = 10.667 
Reservoir Reservoir 
97.18 97.14 
62.24 61.67 
57.84 56.92 
52.40 51.02 
46.70 44.81 
45.38 43.35 
43.84 41.61 
42.63 40.23 
10 39.07 39.20 
11 37.53 37.64 
12 34.16 34.21 
13 30.01 30.01 
14 32.37 35.52 
15 30.08 33.72 
16 30.04 31.30 
17 38.56 33.41 
18 45.54 49.93 
19 59.28 55.09 
20 50.99 50.61 
21 41.78 41.26 
22 36.70 36.10 
23 44.70 44.52 
24 38.81 38.93 
25 34.94 35.34 
26 30.56 31.70 
27 30.06 30.76 
28 39.10 38.94 
29 30.13 30.13 
30 30.35 30.42 
31 30.45 30.70 
32 32.84 33.18 
  Pressures (m) 
Node ω = 10.5088 ω = 10.667 
Reservoir Reservoir 
97.18 97.14 
62.24 61.67 
57.84 56.92 
52.40 51.02 
46.70 44.81 
45.38 43.35 
43.84 41.61 
42.63 40.23 
10 39.07 39.20 
11 37.53 37.64 
12 34.16 34.21 
13 30.01 30.01 
14 32.37 35.52 
15 30.08 33.72 
16 30.04 31.30 
17 38.56 33.41 
18 45.54 49.93 
19 59.28 55.09 
20 50.99 50.61 
21 41.78 41.26 
22 36.70 36.10 
23 44.70 44.52 
24 38.81 38.93 
25 34.94 35.34 
26 30.56 31.70 
27 30.06 30.76 
28 39.10 38.94 
29 30.13 30.13 
30 30.35 30.42 
31 30.45 30.70 
32 32.84 33.18 
Table 16

Parameters of the modified Clonalg and the classic Clonalg for New York City network

Algorithm ω NAb β ρ Probability Rate Max. Iteration 
Modified 10.5088 100 0.02 2.4 0.1 1000 
10.667 100 0.1 2.4 0.1 500 
Classic 10.667 300 – 500 
Algorithm ω NAb β ρ Probability Rate Max. Iteration 
Modified 10.5088 100 0.02 2.4 0.1 1000 
10.667 100 0.1 2.4 0.1 500 
Classic 10.667 300 – 500 
Table 17

Comparison of the results for New York City network

  Modified Clonalg (This study) IA Chu et al. (2008)  Van Dijk et al. (2008)  HS Geem (2006Montesinos et al. (1999)  GA Savic & Walters (1997)  
Pipe ω = 10.5088 ω = 10.667 ω = 10.5088 ω = 10.667 ω = 10.5088 ω = 10.667 ω = 10.5088 
96 144 108 144 96 108 
10 
11 
12 
13 
14 
15 120 
16 96 96 96 96 96 84 96 
17 96 96 96 96 96 96 96 
18 84 84 84 84 84 84 84 
19 72 72 72 72 72 72 72 
20 
21 72 72 72 72 72 72 72 
Cost ($) 36,660,000 38,637,600 37,130,000 38,637,600 36,660,000 38,800,000 37,130,000 
  Modified Clonalg (This study) IA Chu et al. (2008)  Van Dijk et al. (2008)  HS Geem (2006Montesinos et al. (1999)  GA Savic & Walters (1997)  
Pipe ω = 10.5088 ω = 10.667 ω = 10.5088 ω = 10.667 ω = 10.5088 ω = 10.667 ω = 10.5088 
96 144 108 144 96 108 
10 
11 
12 
13 
14 
15 120 
16 96 96 96 96 96 84 96 
17 96 96 96 96 96 96 96 
18 84 84 84 84 84 84 84 
19 72 72 72 72 72 72 72 
20 
21 72 72 72 72 72 72 72 
Cost ($) 36,660,000 38,637,600 37,130,000 38,637,600 36,660,000 38,800,000 37,130,000 
Table 18

Comparison of the modified Clonalg and the classic Clonalg for New York City network

  Modified Clonalg Classic Clonalg 
Pipe ω = 10.667 ω = 10.667 
36 
48 
36 
36 
144 84 
48 
10 36 
11 36 
12 36 
13 
14 36 
15 36 
16 96 96 
17 96 96 
18 84 84 
19 72 72 
20 
21 72 72 
Cost ($) 38,637,600 49,240,000 
  Modified Clonalg Classic Clonalg 
Pipe ω = 10.667 ω = 10.667 
36 
48 
36 
36 
144 84 
48 
10 36 
11 36 
12 36 
13 
14 36 
15 36 
16 96 96 
17 96 96 
18 84 84 
19 72 72 
20 
21 72 72 
Cost ($) 38,637,600 49,240,000 
Table 19

Performances of the modified Clonalg and the classic Clonalg for New York City network

Algorithm ω Min. Cost (USD) Max. Cost (USD) Success rate in 20 runs (%) Average Run Time (min) Average Iteration Number 
Modified 10.5088 36,660,000 37,764,900 85 14.7 780 
10.667 38,637,600 39,671,200 35 20.7 203 
Classic 10.667 49,240,000 59,370,500 83.1 500 
Algorithm ω Min. Cost (USD) Max. Cost (USD) Success rate in 20 runs (%) Average Run Time (min) Average Iteration Number 
Modified 10.5088 36,660,000 37,764,900 85 14.7 780 
10.667 38,637,600 39,671,200 35 20.7 203 
Classic 10.667 49,240,000 59,370,500 83.1 500 
Table 20

Pressures at the nodes of New York network

  Pressures (ft) 
Node ω = 10.5088 ω = 10.667 
Reservoir Reservoir 
294.39 294.21 
286.59 286.15 
284.31 283.79 
282.30 281.70 
280.75 280.07 
278.31 277.51 
276.45 276.67 
273.76 273.78 
10 273.73 273.74 
11 273.86 273.87 
12 275.15 275.14 
13 278.12 278.10 
14 285.59 285.56 
15 293.34 293.33 
16 260.27 260.08 
17 272.87 272.87 
18 261.40 261.18 
19 255.37 255.05 
20 260.92 260.73 
  Pressures (ft) 
Node ω = 10.5088 ω = 10.667 
Reservoir Reservoir 
294.39 294.21 
286.59 286.15 
284.31 283.79 
282.30 281.70 
280.75 280.07 
278.31 277.51 
276.45 276.67 
273.76 273.78 
10 273.73 273.74 
11 273.86 273.87 
12 275.15 275.14 
13 278.12 278.10 
14 285.59 285.56 
15 293.34 293.33 
16 260.27 260.08 
17 272.87 272.87 
18 261.40 261.18 
19 255.37 255.05 
20 260.92 260.73 
Table 21

Parameters of the modified Clonalg and the classic Clonalg for Go Yang network

Algorithm ω NAb β ρ Probability Rate Max. Iteration 
Modified 10.5088 50 0.01 100 
10.5879 50 0.01 100 
10.667 30 0.01 100 
Classic 10.667 30 – 100 
Algorithm ω NAb β ρ Probability Rate Max. Iteration 
Modified 10.5088 50 0.01 100 
10.5879 50 0.01 100 
10.667 30 0.01 100 
Classic 10.667 30 – 100 
Table 22

Comparison of the results for Go Yang network

  Modified Clonalg (This study) DE Dong et al. (2012)  GA Dong et al. (2012)  HS Geem (2006)    
Pipe ω = 10.5088 ω = 10.5879 ω = 10.667 ω = 10.667 ω = 10.667 ω = 10.5879 NLP Kim et al. (1994)  
200 150 150 N/A N/A 150 200 
125 150 150 N/A N/A 150 200 
100 125 125 N/A N/A 125 125 
100 150 125 N/A N/A 150 125 
80 100 100 N/A N/A 100 100 
100 100 80 N/A N/A 100 100 
80 80 80 N/A N/A 80 80 
80 80 80 N/A N/A 100 80 
80 80 80 N/A N/A 80 80 
10 80 80 80 N/A N/A 80 80 
11 80 80 80 N/A N/A 80 80 
12 80 80 80 N/A N/A 80 80 
13 80 80 80 N/A N/A 80 80 
14 80 80 80 N/A N/A 80 80 
15 80 80 80 N/A N/A 80 80 
16 80 80 80 N/A N/A 80 80 
17 80 80 80 N/A N/A 80 80 
18 80 80 80 N/A N/A 80 100 
19 80 80 80 N/A N/A 80 125 
20 80 80 80 N/A N/A 80 80 
21 80 80 80 N/A N/A 80 80 
22 80 80 80 N/A N/A 80 80 
23 80 80 100 N/A N/A 80 80 
24 80 80 80 N/A N/A 80 80 
25 80 80 80 N/A N/A 80 80 
26 80 80 80 N/A N/A 80 80 
27 80 80 100 N/A N/A 80 80 
28 80 80 80 N/A N/A 80 80 
29 80 80 80 N/A N/A 80 100 
30 80 80 80 N/A N/A 80 80 
Cost(won) 176,958,824 176,994,561 177,009,557 177,010,000 177,061,000 177,135,800 179,142,700 
  Modified Clonalg (This study) DE Dong et al. (2012)  GA Dong et al. (2012)  HS Geem (2006)    
Pipe ω = 10.5088 ω = 10.5879 ω = 10.667 ω = 10.667 ω = 10.667 ω = 10.5879 NLP Kim et al. (1994)  
200 150 150 N/A N/A 150 200 
125 150 150 N/A N/A 150 200 
100 125 125 N/A N/A 125 125 
100 150 125 N/A N/A 150 125 
80 100 100 N/A N/A 100 100 
100 100 80 N/A N/A 100 100 
80 80 80 N/A N/A 80 80 
80 80 80 N/A N/A 100 80 
80 80 80 N/A N/A 80 80 
10 80 80 80 N/A N/A 80 80 
11 80 80 80 N/A N/A 80 80 
12 80 80 80 N/A N/A 80 80 
13 80 80 80 N/A N/A 80 80 
14 80 80 80 N/A N/A 80 80 
15 80 80 80 N/A N/A 80 80 
16 80 80 80 N/A N/A 80 80 
17 80 80 80 N/A N/A 80 80 
18 80 80 80 N/A N/A 80 100 
19 80 80 80 N/A N/A 80 125 
20 80 80 80 N/A N/A 80 80 
21 80 80 80 N/A N/A 80 80 
22 80 80 80 N/A N/A 80 80 
23 80 80 100 N/A N/A 80 80 
24 80 80 80 N/A N/A 80 80 
25 80 80 80 N/A N/A 80 80 
26 80 80 80 N/A N/A 80 80 
27 80 80 100 N/A N/A 80 80 
28 80 80 80 N/A N/A 80 80 
29 80 80 80 N/A N/A 80 100 
30 80 80 80 N/A N/A 80 80 
Cost(won) 176,958,824 176,994,561 177,009,557 177,010,000 177,061,000 177,135,800 179,142,700 
Table 23

Comparison of the modified Clonalg and the classic Clonalg for Go Yang network

  Modified Clonalg Classic Clonalg 
Pipe ω = 10.667 ω = 10.667 
150 150 
150 150 
125 125 
125 125 
100 100 
80 80 
80 80 
80 80 
80 80 
10 80 80 
11 80 80 
12 80 80 
13 80 80 
14 80 80 
15 80 80 
16 80 80 
17 80 80 
18 80 80 
19 80 80 
20 80 80 
21 80 80 
22 80 80 
23 100 100 
24 80 80 
25 80 80 
26 80 80 
27 100 100 
28 80 80 
29 80 80 
30 80 80 
Cost (won) 177,009,557 177,009,557 
  Modified Clonalg Classic Clonalg 
Pipe ω = 10.667 ω = 10.667 
150 150 
150 150 
125 125 
125 125 
100 100 
80 80 
80 80 
80 80 
80 80 
10 80 80 
11 80 80 
12 80 80 
13 80 80 
14 80 80 
15 80 80 
16 80 80 
17 80 80 
18 80 80 
19 80 80 
20 80 80 
21 80 80 
22 80 80 
23 100 100 
24 80 80 
25 80 80 
26 80 80 
27 100 100 
28 80 80 
29 80 80 
30 80 80 
Cost (won) 177,009,557 177,009,557 
Table 24

Performances of the modified Clonalg and the classic Clonalg for Go Yang network

Algorithm ω Min. Cost (won) Max. Cost (won) Success rate in 20 runs (%) Average Run Time (min) Average Iteration Number 
Modified 10.5088 176,958,824 176,994,561 90 3.9 17 
10.5879 176,994,561 177,010,359 25 4.0 17 
10.667 177,009,557 177,010,359 15 7.2 14 
Classsic 10.667 177,009,557 177,104,733 15 0.9 17 
Algorithm ω Min. Cost (won) Max. Cost (won) Success rate in 20 runs (%) Average Run Time (min) Average Iteration Number 
Modified 10.5088 176,958,824 176,994,561 90 3.9 17 
10.5879 176,994,561 177,010,359 25 4.0 17 
10.667 177,009,557 177,010,359 15 7.2 14 
Classsic 10.667 177,009,557 177,104,733 15 0.9 17 
Table 25

Pressures at the nodes of Go Yang network

Node Pressures (m) 
 ω = 10.5088 ω = 10.5879 ω = 10.667 
Reservoir Reservoir Reservoir 
28.93 24.99 24.95 
28.86 26.30 26.25 
25.39 24.07 24.02 
23.07 22.75 22.37 
20.40 20.66 20.57 
27.42 25.20 25.14 
25.76 24.35 24.29 
20.14 19.98 19.90 
10 15.23 15.42 15.07 
11 15.02 15.05 15.04 
12 17.93 18.16 17.79 
13 17.16 17.38 17.15 
14 15.09 15.30 15.00 
15 15.25 15.46 15.14 
16 28.32 25.69 25.63 
17 26.59 23.85 23.79 
18 26.23 23.56 23.50 
19 27.21 24.35 24.28 
20 26.61 23.42 23.36 
21 19.71 16.09 16.03 
22 19.30 15.88 15.81 
Node Pressures (m) 
 ω = 10.5088 ω = 10.5879 ω = 10.667 
Reservoir Reservoir Reservoir 
28.93 24.99 24.95 
28.86 26.30 26.25 
25.39 24.07 24.02 
23.07 22.75 22.37 
20.40 20.66 20.57 
27.42 25.20 25.14 
25.76 24.35 24.29 
20.14 19.98 19.90 
10 15.23 15.42 15.07 
11 15.02 15.05 15.04 
12 17.93 18.16 17.79 
13 17.16 17.38 17.15 
14 15.09 15.30 15.00 
15 15.25 15.46 15.14 
16 28.32 25.69 25.63 
17 26.59 23.85 23.79 
18 26.23 23.56 23.50 
19 27.21 24.35 24.28 
20 26.61 23.42 23.36 
21 19.71 16.09 16.03 
22 19.30 15.88 15.81 

CONCLUSIONS

The value of ω has a significant impact on the cost designs of the WDNs. In order to compare the modified Clonalg and the other algorithms (HS, GA, NLP, IA, SCE, and DE) in the related literature, the same values of ω (10.4973, 10.5088 and 10.5879) were applied for the analyses in addition to EPANET v.2.0's ω value of 10.667. Results showed that the modified Clonalg could find lower costs than the other algorithms with the same values of ω, and appeared to be significantly successful and feasible for the cost designs of the WDNs.

In the application of the Two-loop network, the modified Clonalg obtained a higher success rate and a lower maximum cost (worst cost) than the classic Clonalg, although both algorithms could find the same minimum cost (419,000 USD). In the application of the Hanoi network, the modified Clonalg could find both a lower minimum cost with a higher success rate and a lower maximum cost than the classic Clonalg. In the application of the New York City network, the modified Clonalg could find both a lower minimum cost with a higher success rate and a lower maximum cost than the classic Clonalg. In the application of the Go Yang Network, the modified Clonalg and the classic Clonalg could find the same minimum cost (177,009,557 won) with the same success rate. But the modified Clonalg obtained a lower maximum cost than the classic Clonalg. All of these demonstrate that the modification in the mutation process increases the diversity (search capability) and improves the performance of the algorithm.

The modified Clonalg uses Equation (4) instead of Equation (2) for the cloning (the classic Clonalg uses Equation (2)). Equation (4) increases the running time in comparison with Equation (2) since more clones are generated by it. Therefore, the modified Clonalg took a longer time than the classic Clonalg in the analyses (the application of the New York City network is excepted since higher values of NAb and β for the classic Clonalg were assigned than for the modified Clonalg).

In future studies, performance of this model needs to be explored under various restrictions such as a velocity, maximum pressure in the node, variation in water demands depending on time, in addition to the minimum pressure requirements for the WDNs.

ACKNOWLEDGEMENT

The author thanks Fatih Evrendilek for his textual assistance.

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