Due to the economic crisis and water scarcity happening in recent years, many researchers have focused on the water distribution network optimization problem. On this specified subject, all recent research has applied stochastic meta-heuristic algorithms to solve these sets of problems. In this study, the application of a novel deterministic physically inspired heuristic algorithm for minimizing the cost of pipe-sizing in the water distribution system (WDS) is investigated. In fact, the algorithm used in this research is the modified central force optimization algorithm to solve the water distribution network problem called CFOnet. The approach is applied to optimize the design of the Kadu and Khorramshahr networks. For this purpose, CFOnet method is programmed in MATLAB and interfaced with the hydraulic simulation model, EPANET. The obtained solutions in this study are compared with those stochastic methods in the WDS optimization literature. This comparison shows that CFOnet is more efficient in obtaining lower cost than other optimization methods for solving the two mentioned WDSs, while it enjoys the merits of a deterministic optimization method.

## INTRODUCTION

A significant portion of the water supply system cost is relevant to the water distribution network (Swamee & Sharma 2008). Thus, finding the optimal cost for implementation of these networks is economically helpful in water supply system design. In the past few decades, researchers have focused on utilizing meta-heuristic algorithms, such as genetic algorithm (GA), simulated annealing (SA) and particle swarm optimization (PSO), to evaluate water network designs, and have shown that these algorithms are successful in finding the most economic cost of water networks. But, due to their stochastic nature, there is no guarantee that the global optimum is found. Also, the obtained solution in each run is not always the same; therefore, several runs are necessary to ensure that the solutions are identified as good quality. Moreover, the required computation time to reach near-optimal configuration is a great limiting problem in using these algorithms in real water distribution systems (WDSs). These limitations have led researchers to combine these algorithms with the deterministic math-based approaches. Cisty (2010) and Haghighi *et al.* (2011) proposed GA-LP (combination of GA and linear programming) and GA-ILP (combination of GA and integer-linear programming) models, respectively, to solve the WDS problem. They have shown that their proposed models are better from the computational efficiency point of view compared with completely stochastic algorithms.

Recently, a new approach to the optimization has been introduced, called central force optimization (CFO). This method is a new meta-heuristic algorithm for multidimensional optimization which performs based on the rules of gravity (Formato 2007). Since this physical law, in which masses move towards the gravitational field, is deterministic, CFO's equations are inherently deterministic. Therefore, in CFO, multiple runs for the sake of finding its performance are not required, and every CFO run with the same setup will result the same (Formato 2007). This algorithm has been successfully applied to a variety of problems, among them: antenna optimization (Formato 2010a); drinking water distribution networks (Haghighi & Ramos 2012); and improving the global search ability of Standard CFO (Ding *et al.* 2012). This research has shown high efficiency of this optimization method, and also in some cases, the researchers have enhanced CFO performance through creating some modifications in the method.

Since the problem of WDS has never been solved by any completely deterministic meta-heuristic method, utilization of the CFO has been investigated in this research. Since this method is still under development, in the present research, a modified CFO algorithm is introduced to optimize the design cost of water distribution networks (CFOnet). For this purpose, CFOnet is linked to the hydraulic simulation model, EPANET 2.0 (Rossman 2000), in MATLAB. The obtained optimum design costs for two WDSs are compared with the solutions published using other approaches.

## METHODS

where is head loss in the pipe *i* and is the total number of loops in the system.

^{3}/s), is the Hazen–Williams roughness,

*D*is the diameter of the pipe (m), and

_{i}*L*is the length of the pipe (m).

_{i} is a numerical value which depends on the units used in the equation and acts as a dimensionless conversion factor. Herein the *ω* in EPANET 2.0 is equal to 10.667 (Eusuff & Lansey 2003).

### Optimization model formulation

*i*with the diameter and the length; and is the total number of pipes in the network.

*i*; and are allowable minimum and maximum velocities in the pipes, respectively; is the pressure in node; and are allowable minimum and maximum pressure points in nodes, respectively; and is the total number of nodes in the system.

*v*(

*P*) are penalty functions for violation of velocity and pressure constraints, respectively, and is the violation factor. Herein

_{j}*C*is computed with calibrated and penalty functions are considered as below:

_{t}### CFO algorithm

*j*is the iteration number, which is valued in the range from one to (the maximum number of iterations). Index

*p*is the probe number, which is valued in the range from one to (the total numbers of probes). Each of the other probes also has associated with it the fitness . Likewise, , and

*G*are the CFO constants, which usually are equal to 2 (Formato 2007). Also, is the Unit Step function, which determines whether probe

*k*applied the attraction force on probe

*p*or not.

*U*in maximizing the objective function is defined by the below equation, according to which probe

*k*could attract probe

*p*if it has a greater mass than that of probe : And is the distance between the position of probes

*p*and, which is given by: where and are the positions of probe

*k*and

*p*in the dimension

_{,}respectively. The acceleration causes probe

*p*to move from position at step to position at step

*j*according to the below equation: Figure 1 demonstrates the gravitational metaphor for a three-dimensional space with four probes in a maximizing problem in which the probes' fitness is shown by the black filled circles; the size of the circles is proportional to the mass of probes (their fitness value). Movement of each probe is restricted to the bounded feasible region. However, when some of the probes fly out, a retrieving mechanism such as the below scheme is used to act on these errant probes and reposition them in the feasible region:

where is the repositioning factor and More detailed information on CFO can be seen in (Formato 2010a, 2010b).

### CFOnet algorithm and hydraulic model implementation

To evaluate the CFO method for solving the water network problem, two WDSs are selected. Kadu and Khorramshahr WDSs, which include a range of different decision space topologies and make a suitable evaluation of CFO method performance for solving the WDS problem, are assessed. The CFO method is implemented to optimize the design cost of these WDSs.

Given the poor performance of CFO to optimize these networks, it seems reasonable to apply a few modifications to this method. The modified CFO algorithm is called CFOnet, and the differences from the original CFO are explained herein.

_{:}In addition, after iteration, and are reduced to a quarter of their initial value. Therefore, according to the normalized acceleration values, probe positions have changed by Equation (17) and then new diameters are obtained. Due to the discreteness of the WDS problem, it is decided to eliminate the parameter in Equations (18) and (19), and the errant probes are repositioned by the following simple definitions: The input file for EPANET is updated using new diameters. This process is continued until the number of iterations reaches , which in this research is set to 10,000. As for prevention of probe trapping phenomena in the local optimum solutions, a mutation operator is defined in CFOnet. In this case, during running the algorithm, when the best cost remains constant after a specified number of iterations, the deterministic mutation operator acts on the probes. For this purpose, firstly, the existing commercial diameters matrix replicates to the number of pipes quantity in the considered network, which makes a larger matrix called the sized diameters matrix. The mutated probes are moved to the new positions, which are produced by applying Swap, Insertion and Reversion operators on the sized diameters matrix. The mutated rate is considered as 15% herein. The CFOnet operation flowchart is shown in Figure 3. Also, the main differences between CFOnet and conventional CFO are demonstrated in Table 1 for more clarification.

Operator . | . | . | . |
---|---|---|---|

method . | Repairing the errant probes . | Normalizing the accelerations . | Mutation of the probes . |

CFO | Returning the errant probes considering (Equations (18) and (19)) | Unavailable | Unavailable |

CFOnet | Returning the errant probes without using (Equations (22) and (23)) | Normalizing the accelerations regarding the existing commercial diameters | Replacing the probes with new probes which are defined by user deterministically |

Operator . | . | . | . |
---|---|---|---|

method . | Repairing the errant probes . | Normalizing the accelerations . | Mutation of the probes . |

CFO | Returning the errant probes considering (Equations (18) and (19)) | Unavailable | Unavailable |

CFOnet | Returning the errant probes without using (Equations (22) and (23)) | Normalizing the accelerations regarding the existing commercial diameters | Replacing the probes with new probes which are defined by user deterministically |

### Kadu network

*et al.*(2008), which comprises 26 nodes, 34 pipes and two reservoirs. The heads of reservoirs 1 and 2 are 100 and 95 m, respectively. The additional information on this network can be found in Kadu

*et al.*(2008). The HW coefficient is considered to be 130 for all pipes. Table 2 represents 14 commercially available diameters, which makes 14

^{34}possibilities to solve this network. The layout of the network is shown in Figure 4.

Diameter (inches) . | Cost (Rs/m) . | Diameter (inches) . | Cost (Rs/m) . | Diameter (inches) . | Cost (Rs/m) . |
---|---|---|---|---|---|

150 | 1,115 | 400 | 4,255 | 750 | 11,874 |

200 | 1,600 | 450 | 5,172 | 800 | 13,261 |

250 | 2,154 | 500 | 6,092 | 900 | 16,151 |

300 | 2,780 | 600 | 8,189 | 1,000 | 19,395 |

350 | 3,475 | 700 | 10,670 |

Diameter (inches) . | Cost (Rs/m) . | Diameter (inches) . | Cost (Rs/m) . | Diameter (inches) . | Cost (Rs/m) . |
---|---|---|---|---|---|

150 | 1,115 | 400 | 4,255 | 750 | 11,874 |

200 | 1,600 | 450 | 5,172 | 800 | 13,261 |

250 | 2,154 | 500 | 6,092 | 900 | 16,151 |

300 | 2,780 | 600 | 8,189 | 1,000 | 19,395 |

350 | 3,475 | 700 | 10,670 |

### Khorramshahr network

^{58}possibilities. The amount is considered to be 120 for all the pipes. Also, the minimum hydraulic head requirement is set to 30 m for all of the nodes. Additional information on this network can be found in Samani & Zanganeh (2010). With regard to the results of that research, the minimum and maximum allowable flow velocities in pipes are considered to be 0.1 m/s and 2.04 m/s, respectively. The layout of the network is shown in Figure 5.

Diameter . | Cost (units/m) . | Diameter . | Cost (units/m) . | Diameter . | Cost (units/m) . |
---|---|---|---|---|---|

25 | 205 | 124 | 450 | 222 | 590 |

75 | 411 | 142 | 470 | 248 | 610 |

111 | 430 | 160 | 490 | 279 | 700 |

Diameter . | Cost (units/m) . | Diameter . | Cost (units/m) . | Diameter . | Cost (units/m) . |
---|---|---|---|---|---|

25 | 205 | 124 | 450 | 222 | 590 |

75 | 411 | 142 | 470 | 248 | 610 |

111 | 430 | 160 | 490 | 279 | 700 |

## RESULTS AND DISCUSSION

^{6}(Rs) to 291 × 10

^{6}(Rs) after running the first 11 iterations of CFOnet. In this network the calibrated for CFOnet method is 1,000 iterations. The comparison of the CFOnet results with those obtained using other optimization techniques in the literature with the same is shown in Table 4.

Pipe . | Diameter (mm) . | |||
---|---|---|---|---|

Kadu et al. (2008)
. | Haghighi et al. (2011)
. | Mohammadi-Aghdam et al. (2015)
. | CFOnet . | |

1 | 1,000 | 1,000 | 900 | 900 |

2 | 900 | 900 | 900 | 900 |

3 | 400 | 400 | 500 | 350 |

4 | 350 | 350 | 250 | 300 |

5 | 150 | 150 | 150 | 150 |

6 | 250 | 250 | 200 | 300 |

7 | 800 | 800 | 900 | 800 |

8 | 150 | 150 | 150 | 150 |

9 | 400 | 400 | 600 | 600 |

10 | 500 | 500 | 700 | 600 |

11 | 1,000 | 1,000 | 900 | 900 |

12 | 700 | 700 | 700 | 700 |

13 | 800 | 800 | 500 | 500 |

14 | 400 | 400 | 450 | 500 |

15 | 150 | 150 | 150 | 150 |

16 | 500 | 500 | 450 | 500 |

17 | 350 | 350 | 300 | 350 |

18 | 350 | 350 | 450 | 400 |

19 | 150 | 150 | 500 | 500 |

20 | 200 | 150 | 150 | 150 |

21 | 700 | 700 | 600 | 600 |

22 | 150 | 150 | 150 | 150 |

23 | 400 | 450 | 150 | 150 |

24 | 400 | 400 | 400 | 450 |

25 | 700 | 700 | 500 | 500 |

26 | 250 | 250 | 150 | 200 |

27 | 250 | 250 | 350 | 350 |

28 | 200 | 200 | 350 | 250 |

29 | 300 | 300 | 150 | 250 |

30 | 300 | 300 | 300 | 250 |

31 | 200 | 200 | 200 | 150 |

32 | 150 | 150 | 150 | 150 |

33 | 250 | 200 | 200 | 150 |

34 | 150 | 150 | 150 | 150 |

Method used | GA | GA-ILP | PSO | CFOnet |

Cost (Rs) | 131,678,935 | 131,312,815 | 130,666,043 | 126,535,915 |

Eval. | 36,000 | 4,440 | 22,000 | 259,476 |

Pipe . | Diameter (mm) . | |||
---|---|---|---|---|

Kadu et al. (2008)
. | Haghighi et al. (2011)
. | Mohammadi-Aghdam et al. (2015)
. | CFOnet . | |

1 | 1,000 | 1,000 | 900 | 900 |

2 | 900 | 900 | 900 | 900 |

3 | 400 | 400 | 500 | 350 |

4 | 350 | 350 | 250 | 300 |

5 | 150 | 150 | 150 | 150 |

6 | 250 | 250 | 200 | 300 |

7 | 800 | 800 | 900 | 800 |

8 | 150 | 150 | 150 | 150 |

9 | 400 | 400 | 600 | 600 |

10 | 500 | 500 | 700 | 600 |

11 | 1,000 | 1,000 | 900 | 900 |

12 | 700 | 700 | 700 | 700 |

13 | 800 | 800 | 500 | 500 |

14 | 400 | 400 | 450 | 500 |

15 | 150 | 150 | 150 | 150 |

16 | 500 | 500 | 450 | 500 |

17 | 350 | 350 | 300 | 350 |

18 | 350 | 350 | 450 | 400 |

19 | 150 | 150 | 500 | 500 |

20 | 200 | 150 | 150 | 150 |

21 | 700 | 700 | 600 | 600 |

22 | 150 | 150 | 150 | 150 |

23 | 400 | 450 | 150 | 150 |

24 | 400 | 400 | 400 | 450 |

25 | 700 | 700 | 500 | 500 |

26 | 250 | 250 | 150 | 200 |

27 | 250 | 250 | 350 | 350 |

28 | 200 | 200 | 350 | 250 |

29 | 300 | 300 | 150 | 250 |

30 | 300 | 300 | 300 | 250 |

31 | 200 | 200 | 200 | 150 |

32 | 150 | 150 | 150 | 150 |

33 | 250 | 200 | 200 | 150 |

34 | 150 | 150 | 150 | 150 |

Method used | GA | GA-ILP | PSO | CFOnet |

Cost (Rs) | 131,678,935 | 131,312,815 | 130,666,043 | 126,535,915 |

Eval. | 36,000 | 4,440 | 22,000 | 259,476 |

The CFO method finds the optimum cost of 282,474,720 (Rs) after 25,452 function evaluations. This solution has a fairly big difference from the best results in the literature, so it is decided to eliminate it from Table 4. With regard to the CFO result, it shows a relatively poor performance to solve this problem. This is due to CFO's high convergence speed, so that all probes quickly move to the central gravitational field and lose their accelerations. Therefore, CFO unavoidably is unable to decrease the cost anymore.

As shown in Figure 6, the convergence behavior of deterministic proposed CFOnet is improved as compared with the CFO, so that the best cost decreases to 126,535,915 (Rs) after 259,476 function evaluations. This obtained solution is better than the widely obtained answers as shown in Table 4. By simulating Kadu network with the obtained optimum diameters using EPANET, it can be observed that all the pressure values meet the requirements in all nodes of the network shown in Table 5. It should be noted that since the computers utilized in the mentioned researches were not identical in terms of processing speed, the run times are not considered herein.

Node . | Pressure . | Node . | Pressure . | Node . | Pressure . | Node . | Pressure . |
---|---|---|---|---|---|---|---|

1 | Reservoir 1 | 8 | 89.17 | 15 | 88.29 | 22 | 80.46 |

2 | Reservoir 2 | 9 | 91.13 | 16 | 82.06 | 23 | 82.71 |

3 | 98.29 | 10 | 88.19 | 17 | 90.24 | 24 | 83.24 |

4 | 95.06 | 11 | 86.36 | 18 | 86.33 | 25 | 80.05 |

5 | 87.83 | 12 | 85.12 | 19 | 85.36 | 26 | 80.00 |

6 | 86.18 | 13 | 84.65 | 20 | 82.95 | ||

7 | 87.78 | 14 | 94.15 | 21 | 83.18 |

Node . | Pressure . | Node . | Pressure . | Node . | Pressure . | Node . | Pressure . |
---|---|---|---|---|---|---|---|

1 | Reservoir 1 | 8 | 89.17 | 15 | 88.29 | 22 | 80.46 |

2 | Reservoir 2 | 9 | 91.13 | 16 | 82.06 | 23 | 82.71 |

3 | 98.29 | 10 | 88.19 | 17 | 90.24 | 24 | 83.24 |

4 | 95.06 | 11 | 86.36 | 18 | 86.33 | 25 | 80.05 |

5 | 87.83 | 12 | 85.12 | 19 | 85.36 | 26 | 80.00 |

6 | 86.18 | 13 | 84.65 | 20 | 82.95 | ||

7 | 87.78 | 14 | 94.15 | 21 | 83.18 |

^{6}(thousand Iran rials) from 109 × 10

^{6}(thousand Iran rials). In this network for CFOnet is calibrated as 300 iterations. The lowest cost found by CFO is 22,974,703 (thousand Iran rials) for this network. This solution is obtained after 25,410 function evaluations, and CFO is unable to improve it anymore. So with regard to the CFO result, it is decided to eliminate the CFO curve in Figure 7. By applying the CFOnet model, after 41,454 evaluations, the best obtained cost became 1,474,807 (thousand Iran rials). Samani & Zanganeh (2010) obtained the best cost as 1,535,371 (thousand Iran rials) after 14 iterations for this network. It can be concluded that CFOnet produces a significant improvement in the best cost as compared with the one obtained by LP technique as shown in Table 6. Also, the obtained optimal diameters for this network are listed in Table 7. All the hydraulic characteristics are satisfactory in the optimum design of Khorramshahr network, as reported in Tables 8 and 9.

Authors . | Samani & Zanganeh (2010) . | Present work . |
---|---|---|

Method used | LP | CFOnet |

Cost (thousand Iran rials) | 1,535,371 | 1,474,807 |

Authors . | Samani & Zanganeh (2010) . | Present work . |
---|---|---|

Method used | LP | CFOnet |

Cost (thousand Iran rials) | 1,535,371 | 1,474,807 |

Pipe . | Samani & Zanganeh (2010) . | Present work . | Pipe . | Samani & Zanganeh (2010) . | CFOnet . |
---|---|---|---|---|---|

1 | 222 | 222 | 30 | 111 | 25 |

2 | 160 | 222 | 31 | 25 | 111 |

3 | 142 | 124 | 32 | 25 | 25 |

4 | 124 | 160 | 33 | 25 | 75 |

5 | 111 | 111 | 34 | 75 | 75 |

6 | 25 | 25 | 35 | 142 | 111 |

7 | 142 | 111 | 36 | 75 | 160 |

8 | 75 | 160 | 37 | 75 | 160 |

9 | 75 | 124 | 38 | 75 | 25 |

10 | 75 | 25 | 39 | 25 | 111 |

11 | 111 | 75 | 40 | 75 | 25 |

12 | 75 | 25 | 41 | 75 | 75 |

13 | 75 | 25 | 42 | 111 | 75 |

14 | 25 | 111 | 43 | 111 | 160 |

15 | 25 | 160 | 44 | 25 | 25 |

16 | 75 | 25 | 45 | 25 | 25 |

17 | 25 | 25 | 46 | 75 | 25 |

18 | 142 | 75 | 47 | 75 | 25 |

19 | 25 | 75 | 48 | 111 | 111 |

20 | 75 | 160 | 49 | 25 | 25 |

21 | 75 | 25 | 50 | 25 | 160 |

22 | 75 | 75 | 51 | 75 | 111 |

23 | 111 | 160 | 52 | 75 | 111 |

24 | 75 | 25 | 53 | 111 | 124 |

25 | 75 | 25 | 54 | 25 | 25 |

26 | 25 | 75 | 55 | 75 | 75 |

27 | 25 | 25 | 56 | 75 | 75 |

28 | 124 | 25 | 57 | 25 | 25 |

29 | 75 | 160 | 58 | 75 | 75 |

Pipe . | Samani & Zanganeh (2010) . | Present work . | Pipe . | Samani & Zanganeh (2010) . | CFOnet . |
---|---|---|---|---|---|

1 | 222 | 222 | 30 | 111 | 25 |

2 | 160 | 222 | 31 | 25 | 111 |

3 | 142 | 124 | 32 | 25 | 25 |

4 | 124 | 160 | 33 | 25 | 75 |

5 | 111 | 111 | 34 | 75 | 75 |

6 | 25 | 25 | 35 | 142 | 111 |

7 | 142 | 111 | 36 | 75 | 160 |

8 | 75 | 160 | 37 | 75 | 160 |

9 | 75 | 124 | 38 | 75 | 25 |

10 | 75 | 25 | 39 | 25 | 111 |

11 | 111 | 75 | 40 | 75 | 25 |

12 | 75 | 25 | 41 | 75 | 75 |

13 | 75 | 25 | 42 | 111 | 75 |

14 | 25 | 111 | 43 | 111 | 160 |

15 | 25 | 160 | 44 | 25 | 25 |

16 | 75 | 25 | 45 | 25 | 25 |

17 | 25 | 25 | 46 | 75 | 25 |

18 | 142 | 75 | 47 | 75 | 25 |

19 | 25 | 75 | 48 | 111 | 111 |

20 | 75 | 160 | 49 | 25 | 25 |

21 | 75 | 25 | 50 | 25 | 160 |

22 | 75 | 75 | 51 | 75 | 111 |

23 | 111 | 160 | 52 | 75 | 111 |

24 | 75 | 25 | 53 | 111 | 124 |

25 | 75 | 25 | 54 | 25 | 25 |

26 | 25 | 75 | 55 | 75 | 75 |

27 | 25 | 25 | 56 | 75 | 75 |

28 | 124 | 25 | 57 | 25 | 25 |

29 | 75 | 160 | 58 | 75 | 75 |

Node . | Pressure . | Node . | Pressure . | Node . | Pressure . |
---|---|---|---|---|---|

1 | Reservoir | 14 | 33.58 | 27 | 39.70 |

1j | 50.34 | 15 | 49.23 | 28 | 35.10 |

2 | 50.27 | 16 | 33.30 | 29 | 32.45 |

3 | 47.54 | 17 | 40.66 | 30 | 31.23 |

4 | 46.55 | 18 | 42.29 | 31 | 39.19 |

5 | 46.45 | 19 | 42.83 | 32 | 30.52 |

6 | 45.93 | 20 | 32.19 | 33 | 35.78 |

7 | 36.62 | 21 | 41.04 | 34 | 37.43 |

8 | 50.14 | 22 | 45.00 | 35 | 36.73 |

9 | 33.32 | 23 | 42.15 | 36 | 34.78 |

10 | 46.46 | 24 | 41.61 | 37 | 36.32 |

11 | 46.35 | 25 | 41.27 | 38 | 35.18 |

12 | 46.26 | 26 | 40.42 | 39 | 33.79 |

13 | 42.90 | 26j | 39.54 |

Node . | Pressure . | Node . | Pressure . | Node . | Pressure . |
---|---|---|---|---|---|

1 | Reservoir | 14 | 33.58 | 27 | 39.70 |

1j | 50.34 | 15 | 49.23 | 28 | 35.10 |

2 | 50.27 | 16 | 33.30 | 29 | 32.45 |

3 | 47.54 | 17 | 40.66 | 30 | 31.23 |

4 | 46.55 | 18 | 42.29 | 31 | 39.19 |

5 | 46.45 | 19 | 42.83 | 32 | 30.52 |

6 | 45.93 | 20 | 32.19 | 33 | 35.78 |

7 | 36.62 | 21 | 41.04 | 34 | 37.43 |

8 | 50.14 | 22 | 45.00 | 35 | 36.73 |

9 | 33.32 | 23 | 42.15 | 36 | 34.78 |

10 | 46.46 | 24 | 41.61 | 37 | 36.32 |

11 | 46.35 | 25 | 41.27 | 38 | 35.18 |

12 | 46.26 | 26 | 40.42 | 39 | 33.79 |

13 | 42.90 | 26j | 39.54 |

Pipe . | Velocity . | Pipe . | Velocity . | Pipe . | Velocity . |
---|---|---|---|---|---|

1 | 1.65 | 21 | 1.32 | 41 | 0.97 |

2 | 1.50 | 22 | 2.01 | 42 | 0.54 |

3 | 1.67 | 23 | 0.36 | 43 | 1.00 |

4 | 0.58 | 24 | 1.01 | 44 | 1.04 |

5 | 1.09 | 25 | 0.72 | 45 | 1.27 |

6 | 2.04 | 26 | 0.87 | 46 | 0.84 |

7 | 0.40 | 27 | 2.04 | 47 | 0.55 |

8 | 1.79 | 28 | 1.57 | 48 | 1.83 |

9 | 0.62 | 29 | 0.57 | 49 | 1.65 |

10 | 0.21 | 30 | 0.56 | 50 | 0.1 |

11 | 1.94 | 31 | 1.13 | 51 | 0.31 |

12 | 1.16 | 32 | 0.73 | 52 | 0.51 |

13 | 1.90 | 33 | 0.86 | 53 | 0.51 |

14 | 0.47 | 34 | 0.68 | 54 | 0.37 |

15 | 0.52 | 35 | 1.00 | 55 | 0.31 |

16 | 1.16 | 36 | 0.81 | 56 | 0.86 |

17 | 2.04 | 37 | 0.30 | 57 | 0.42 |

18 | 0.51 | 38 | 0.18 | 58 | 0.41 |

19 | 0.11 | 39 | 1.29 | ||

20 | 1.42 | 40 | 0.69 |

Pipe . | Velocity . | Pipe . | Velocity . | Pipe . | Velocity . |
---|---|---|---|---|---|

1 | 1.65 | 21 | 1.32 | 41 | 0.97 |

2 | 1.50 | 22 | 2.01 | 42 | 0.54 |

3 | 1.67 | 23 | 0.36 | 43 | 1.00 |

4 | 0.58 | 24 | 1.01 | 44 | 1.04 |

5 | 1.09 | 25 | 0.72 | 45 | 1.27 |

6 | 2.04 | 26 | 0.87 | 46 | 0.84 |

7 | 0.40 | 27 | 2.04 | 47 | 0.55 |

8 | 1.79 | 28 | 1.57 | 48 | 1.83 |

9 | 0.62 | 29 | 0.57 | 49 | 1.65 |

10 | 0.21 | 30 | 0.56 | 50 | 0.1 |

11 | 1.94 | 31 | 1.13 | 51 | 0.31 |

12 | 1.16 | 32 | 0.73 | 52 | 0.51 |

13 | 1.90 | 33 | 0.86 | 53 | 0.51 |

14 | 0.47 | 34 | 0.68 | 54 | 0.37 |

15 | 0.52 | 35 | 1.00 | 55 | 0.31 |

16 | 1.16 | 36 | 0.81 | 56 | 0.86 |

17 | 2.04 | 37 | 0.30 | 57 | 0.42 |

18 | 0.51 | 38 | 0.18 | 58 | 0.41 |

19 | 0.11 | 39 | 1.29 | ||

20 | 1.42 | 40 | 0.69 |

Generally, the obtained results using the proposed CFOnet method as compared with the conventional CFO have a better quality in the two investigated networks. CFOnet shows the solution about 55.2% better than CFO in Kadu network. Moreover, this improvement in Khorramshahr network was about 93.6%. In both networks CFOnet shows a considerable performance rather than the CFO. The main cause of this improvement is due to applying the deterministic mutation operator in CFOnet, which replaces the trapped probes in the local optimum point with new previously defined probes.

It can be observed from this research that CFOnet obtains the better quality solutions in two specified WDSs as compared with the best solutions found before. With regard to Table 4, the optimum cost obtained by CFOnet for Kadu network compared with those obtained by GA, GA-ILP and PSO methods shows an approximate improvement of 3.9%, 3.64% and 3.16%, respectively. Also, the optimum cost obtained by CFOnet for Khorramshahr network as compared with the LP method result (Table 6) shows an approximate improvement of 3.94%.

It should be noted that the number of objective function evaluations in the previous research cannot be compared directly with this study, due to the fact that their calculations are stochastic based and the results are obtained after several runs. By contrast, every CFOnet run with the same parameters returns entirely the same solutions (Formato 2007) and this method is performed only once. In addition, in CFOnet there is no need to calibrate several parameters (only is set), so it saves time significantly.

## CONCLUSIONS

The present research assessed the optimization of WDSs. Application of some of the stochastic algorithms to solve this problem was shown in the literature previously. In this research, as a remedy for the deficiencies of utilizing stochastic optimization algorithms, a proposed deterministic approach (CFOnet) was used to find the optimal design cost of WDSs. The major differences between CFOnet and the conventional CFO method were in injecting new probes within the optimization process via a deterministic mutation operator and also in normalizing the acceleration values of probes in a defined range. Since probes in the proposed CFOnet method shared their own global and individual information, this method could be accounted as an intelligent swarm algorithm. Also, the mutation operator enhances increasing the global exploration ability and convergence velocity of this approach. Evaluating CFOnet to optimize the design of Kadu and Khorramshahr networks showed that this model gives the desired results in this kind of problem. For these two networks, the least lowest cost found by CFOnet was better than the best previous investigation. So the results of this comparison indicated that the CFOnet method was more efficient in solving the WDS optimization problem than other methods, while enjoying the merits of a deterministic optimization method. This study is a first effort to optimize WDSs utilizing a deterministic evolutionary algorithm; hence, only the pipe-sizing optimization of two looped networks with a single demand loading was presented. Deterministic optimization of the complex looped networks with multi-objective functions is highly recommended for the subject of future research. Also, the application of the CFOnet to select the optimum rehabilitation alternatives of WDSs during the operational period can be considered for future studies. The CFOnet code could be available upon request of researchers for their better understanding.