## Abstract

The design of the water distribution network (WDN) is very difficult mainly due to the nonlinear relation between head and flow. The distribution network should also be cost-effective. In any simulation–optimization approach, the computational time requirement is very high for very complex problems. The optimization module plays a crucial role in reducing the computational time. This study aims to apply a simulation–optimization approach to designing a WDN. EPANET is selected as the simulation model and Real-Coded Genetic Algorithm (RCGA) is selected as the optimization module. To link the simulation module with the optimization module, a program is written in MATLAB. The developed simulation–optimization approach was applied to two benchmark network problems to check the suitability of the method. The parameters of the RCGA were optimized for the two networks. The computational efficiency of the developed simulation–optimization model is checked based on the number of function evaluations. For both networks, the number of function evaluations to get the optimum network design was less than the number of function evaluations required for other methods mentioned in the literature.

## HIGHLIGHTS

Simulation–optimization approach is adopted for the design of WDNs.

Developed an interface in MATLAB to link simulation and optimization.

EPANET is used as the simulation module and RCGA as the optimization module.

The parameters of Real Coded Genetic Algorithm is optimized.

The developed simulation optimization model is applied to two benchmark problems.

## INTRODUCTION

Proper design of water distribution networks (WDN) plays a very important role in distributing adequate quality and quantity of water to consumers (Kanakoudis 2004). Two types of simulation approaches are used to analyze WDNs, namely steady-state and extended-period simulation. In steady-state simulation, the flows and other attributes do not change over time; whereas, in the extended-period simulation, the variation of flows and other attributes are simulated over various time periods (Cesario 1995). The simulation of a WDN is a difficult process. The main task in the analysis of a WDN involves solving simultaneous nonlinear equations (Kessler & Shamir 1989; Eiger *et al.* 1994; Dandy *et al.* 1996) These equations include a continuity equation for every node, an energy equation, and the equation relating pipe flow and the head-losses. There are many useful and efficient computer programs available for WDN simulation. One of the most popular computer programs is EPANET (Rossman 1993). To determine the optimum parameters in a WDN, the traditional approach is to use the trial and error method by applying a simulation model. The trial and error approach is a cumbersome task because, for the determination of optimum parameters, it is necessary to adjust the variables based on the results from the simulation model until some pre-defined specifications are satisfied. The complexity of this process increases as the number of decision variables increases (Wu & Simpson 2001) One solution to these problems is to use simulation–optimization approach. The combined simulation–optimization application in a WDN is generally based on a surrogate method or direct linking of the simulation and optimization model. In the surrogate method (Broad *et al.* 2005, 2010), the response of the complex simulation model is captured by simple models like artificial neural networks (ANNs) and then these simple models will be linked with the optimization model. In direct linking of simulation–optimization approach for any application (Mohan *et al.* 2007; Batista do Egito *et al.* 2023), simulation models are directly linked with an optimization model.

Kessler & Shamir (1989) described that the nonlinear relationship between flow and head-loss and the existence of discrete variables, such as pipe market sizes, make the optimum network design exceedingly complex. The objective function is also nonlinear and non-convex. Many of the traditional optimization methods namely Linear Programming, Nonlinear Programming, and Dynamic Programming (Alperovits & Shamir 1977; Chiplunkar *et al.* 1986; Bhave 1988; Kessler & Shamir 1989; Lansey & Mays 1989; Eiger *et al.* 1994) cannot be used to determine the global optimum results of WDN problems. Nowadays, the global optimization techniques namely Genetic Algorithm (GA) (Goldberg 1989), Simulated Annealing (Cunha & Sousa (1999), Particle Swarm Optimization (PSO), and Ant Colony Optimization Algorithms are widely used for optimizing WDN. Recently many researchers have adopted other metaheuristic algorithms namely Grass Hopper Optimization Algorithm, Sparrow Search Algorithm, and Artificial Bee Colony Algorithm to a few optimization problems (Chen *et al.* 2023; Wang *et al.* 2023; Yao *et al.* 2023). The most commonly used metaheuristic approach is genetic algorithms for a wide variety of problems (Goldberg 1989; Simpson *et al.* 1994; Savic & Walters 1997; Gupta *et al.* 1999; Wu & Simpson 2002; Pramada *et al.* 2018a, 2018b; Sangroula *et al.* 2022; Surono *et al.* 2022). Savic & Walters (1997) developed a GA program called GANET for the least-cost design of WDNs. In their study, the combined simulation–optimization problem of the least-cost design of WDNs was formulated and it was shown that GA was suitable for solving WDN problems. Vairavamoorthy & Ali 2000 proposed a methodology for the optimal design of WDNs using GA and used Pipe Index Vector for controlling the GA search.

Determining the best design for a WDN has always been very problematic for water resources managers. In any simulation–optimization approach, the computational time requirement for very complex problems is one of the biggest issues. High-performance computing facilities and parallel computing can significantly reduce the computational time for complex problems. Jetmarova *et al.* (2017) stated that even with efficient optimization algorithms or with high-performance computing or parallel computing, WDN designs are still not computationally efficient. In these cases, choosing an optimization module with less function evaluation will help to reduce the computational time. Function evaluation is the average number of objective function/fitness function evaluations until the stopping criterion has been reached within each run. Recently, population-based search algorithms have been extensively used in the field of WDN design. Even though Real-Coded Genetic Algorithm (RCGA) is very effective in handling various engineering problems, there are only limited studies related to the selection of parameters of RCGA to get the optimum design of the WDN. The main objective of the present study is the selection of parameters of RCGA in a simulation–optimization framework for getting the global optimal results of WDN with a smaller number of function evaluations.

## MATERIALS AND METHODS

In this study, EPANET is used as the simulation module and RCGA as the optimization module.

### Simulation model – EPANET

A WDN consists of various components namely pipes, nodes, pumps, valves, storage tanks or reservoirs, etc. EPANET is used as the simulation model in this study. EPANET is a computer program that performs an extended-period simulation of hydraulic and water quality behavior within pressurized pipe networks (Rossman 1993). It has functionalities for tracking the flow in each pipe, the nodal pressure, the elevation of water in each tank, and the transport of chemicals through the network during the simulation period.

### Optimization model

The cost comprises only of the pipe cost which varies with respect to the diameters of the networks in this study.

The constraints comprise the continuity equation at the nodes, the energy balance equation along a loop, and maintaining the minimum pressure head requirements at the node.

The above optimization model is solved using RCGA. GA is a search technique based on the principles of natural selection. Holland developed the GA original theory in 1975. GAs, which fall under the domain of evolutionary algorithms, simulate the process of natural selection. In the domains of engineering and computer science, they are effective global optimization approaches that address challenging issues in engineering. In order to find more accurate approximations of a solution, a population of potential solutions is incorporated into the search space. At first, a large number of the population are generated at random. The objective function of these populations is then assessed. If the termination conditions are not satisfied after one generation, the process of choosing individuals based on their level of fitness in the feasible domain produces a new set of approximations. This technique produces improved children (offspring). There are a few parameters required to run a genetic algorithm, namely population size, the mutation probability, and the crossover probability. The usual way to get these parameters is to do a lot of experimentation to find a set of values that solves a particular problem. A broad rule of thumb, to start with, is to use a mutation probability of 0.05, a crossover rate of 0.6, and a population size of about 50. (Goldberg *et al.* 2005) RCGA was first implemented by Wright in 1991 (Wright 1991). RCGAs do not use any coding (eg. binary coding) of the problem variables; instead, they work directly with the variables. RCGA is simple and straightforward compared to binary-coded GA.

### Interfacing program to link EPANET with optimization module

## RESULTS AND DISCUSSION

The developed methodology for the optimal design of a WDN is applied to two benchmark problems such as a Two-Loop WDN reported by Alperovits & Shamir (1977) and GoYang WDN reported by Kim *et al.* (1994), and its performance is evaluated by comparing the results with past literature.

### Benchmark problem 1 – Two-Loop WDN

Two-Loop WDN was first reported by Alperovits and Shamir as the least-cost design of WDN. Thereafter, many researchers used this network to test their approach. In this study, network configuration, with relevant network data, was taken from Alperovits & Shamir (1977).

Node . | Elevation in m . | Demand in m^{3}/h
. |
---|---|---|

1 | 210 | −1,120 |

2 | 150 | 100 |

3 | 160 | 100 |

4 | 155 | 120 |

5 | 150 | 270 |

6 | 165 | 330 |

7 | 160 | 200 |

Node . | Elevation in m . | Demand in m^{3}/h
. |
---|---|---|

1 | 210 | −1,120 |

2 | 150 | 100 |

3 | 160 | 100 |

4 | 155 | 120 |

5 | 150 | 270 |

6 | 165 | 330 |

7 | 160 | 200 |

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

1 | 2 |

2 | 5 |

3 | 8 |

4 | 11 |

6 | 16 |

8 | 23 |

10 | 32 |

12 | 50 |

14 | 60 |

16 | 90 |

18 | 130 |

20 | 170 |

22 | 300 |

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

1 | 2 |

2 | 5 |

3 | 8 |

4 | 11 |

6 | 16 |

8 | 23 |

10 | 32 |

12 | 50 |

14 | 60 |

16 | 90 |

18 | 130 |

20 | 170 |

22 | 300 |

1 inch = 2.54 cm.

In the paper by Alperovits & Shamir (1977), costs are given in arbitrary units. When we apply to a specific case study, specific monetary units may be given depending on the country. To generalize the study, for both benchmark problems adopted in this study costs are given in arbitrary units (Table 2).

Population size . | Number of generations . | Total cost (unit) . | Optimal cost at ith iteration
. |
---|---|---|---|

20 | 100 | 572,000 | 17 |

30 | 100 | 475,000 | 14 |

40 | 100 | 443,000 | 19 |

60 | 100 | 419,000 | 23 |

Population size . | Number of generations . | Total cost (unit) . | Optimal cost at ith iteration
. |
---|---|---|---|

20 | 100 | 572,000 | 17 |

30 | 100 | 475,000 | 14 |

40 | 100 | 443,000 | 19 |

60 | 100 | 419,000 | 23 |

Crossover (pc) . | Total cost (unit) . | Optimal cost at ith iteration
. |
---|---|---|

0.6 | 419,000 | 25 |

0.7 | 419,000 | 23 |

0.8 | 428,000 | 19 |

0.9 | 446,000 | 28 |

Crossover (pc) . | Total cost (unit) . | Optimal cost at ith iteration
. |
---|---|---|

0.6 | 419,000 | 25 |

0.7 | 419,000 | 23 |

0.8 | 428,000 | 19 |

0.9 | 446,000 | 28 |

Mutation (pm) . | Total cost (unit) . | Optimal cost at ith iteration
. |
---|---|---|

0.01 | 419,000 | 23 |

0.02 | 424,000 | 13 |

0.03 | 428,000 | 19 |

0.04 | 450,000 | 34 |

Mutation (pm) . | Total cost (unit) . | Optimal cost at ith iteration
. |
---|---|---|

0.01 | 419,000 | 23 |

0.02 | 424,000 | 13 |

0.03 | 428,000 | 19 |

0.04 | 450,000 | 34 |

After doing the sensitivity analysis the final input parameters of RCGA are: population size = 60; number of generations = 100; crossover probability = 0.7; mutation probability = 0.01.

*et al.*(2017) used the FEM-PSO simulation–optimization technique and the number of function evaluations was 7,400. RCGA methodology used in this study found that the optimal cost obtained was exactly matching with the results reported in the past. RCGA-based methodology used in the present study required only 1,380 function evaluations to obtain the optimal solution. Thus, the RCGA-based methodology is performing well. Table 7 shows the Two-Loop Network Node results such as pressure heads for optimal diameters in the present study.

Pipe no. . | Savic & Walters (1997) Diameter (inches) . | Cunha & Sousa (1999) Diameter (inches) . | Suribabu & Neelakantan (2006) Diameter (inches) . | Rao et al. (2017) Diameter (inches)
. | RCGA present study Diameter (inches) . |
---|---|---|---|---|---|

1 | 18 | 18 | 18 | 18 | 18 |

2 | 10 | 10 | 10 | 10 | 10 |

3 | 16 | 16 | 16 | 16 | 16 |

4 | 4 | 4 | 4 | 4 | 4 |

5 | 16 | 16 | 16 | 16 | 16 |

6 | 10 | 10 | 10 | 10 | 10 |

7 | 10 | 10 | 10 | 10 | 10 |

8 | 1 | 1 | 1 | 1 | 1 |

Optimal cost (unit) | 419,000 | 419,000 | 419,000 | 419,000 | 419,000 |

Number of function evaluation | 6,750 | – | 5,138 | 7,400 | 1,380 |

Pipe no. . | Savic & Walters (1997) Diameter (inches) . | Cunha & Sousa (1999) Diameter (inches) . | Suribabu & Neelakantan (2006) Diameter (inches) . | Rao et al. (2017) Diameter (inches)
. | RCGA present study Diameter (inches) . |
---|---|---|---|---|---|

1 | 18 | 18 | 18 | 18 | 18 |

2 | 10 | 10 | 10 | 10 | 10 |

3 | 16 | 16 | 16 | 16 | 16 |

4 | 4 | 4 | 4 | 4 | 4 |

5 | 16 | 16 | 16 | 16 | 16 |

6 | 10 | 10 | 10 | 10 | 10 |

7 | 10 | 10 | 10 | 10 | 10 |

8 | 1 | 1 | 1 | 1 | 1 |

Optimal cost (unit) | 419,000 | 419,000 | 419,000 | 419,000 | 419,000 |

Number of function evaluation | 6,750 | – | 5,138 | 7,400 | 1,380 |

1 inch = 2.54 cm

Node ID . | Head (m) . | Pressure (m) . |
---|---|---|

Junction 2 | 203.24 | 53.24 |

Junction 3 | 190.61 | 30.61 |

Junction 4 | 200.52 | 45.52 |

Junction 5 | 184.07 | 34.07 |

Junction 6 | 197.52 | 32.52 |

Junction 7 | 192.62 | 32.62 |

Reservoir 1 | 210 | 0.00 |

Node ID . | Head (m) . | Pressure (m) . |
---|---|---|

Junction 2 | 203.24 | 53.24 |

Junction 3 | 190.61 | 30.61 |

Junction 4 | 200.52 | 45.52 |

Junction 5 | 184.07 | 34.07 |

Junction 6 | 197.52 | 32.52 |

Junction 7 | 192.62 | 32.62 |

Reservoir 1 | 210 | 0.00 |

### Benchmark problem 2 – GoYang WDN

*et al.*(1994). The GoYang WDN includes 22 demand nodes, thirty pipes, and one constant pump of 4.52 kW linking to one reservoir. The reservoir has a constant head of 71 m. The Hazen–Williams head-loss equation is adopted with the roughness coefficient value for each pipe as 100. The minimum pressure head above the ground elevation of each node is maintained at 15 m. The schematic of the network is shown in Figure 8. The nodal elevation and nodal demand data for the GoYang network are given in Table 8. The data of the commercially available set of diameters and its corresponding unit cost are given in Table 9.

Node . | Elevation in m . | Demand in m^{3}/day
. | Node . | Elevation in m . | Demand in m^{3}/day
. |
---|---|---|---|---|---|

1 | 71 | −2,550 | 12 | 58.6 | 37.5 |

2 | 56.4 | 15 | 13 | 59.3 | 37.5 |

3 | 53.8 | 70.5 | 14 | 59.8 | 63 |

4 | 54.9 | 58.5 | 15 | 59.2 | 445.5 |

5 | 56 | 75 | 16 | 53.6 | 108 |

6 | 57 | 67.5 | 17 | 54.8 | 79.5 |

7 | 53.9 | 63 | 18 | 55.1 | 55.5 |

8 | 54.5 | 48 | 19 | 54.2 | 118.5 |

9 | 57.9 | 42 | 20 | 54.5 | 124.5 |

10 | 62.1 | 30 | 21 | 62.9 | 31.5 |

11 | 62.8 | 42 | 22 | 61.8 | 799.5 |

Node . | Elevation in m . | Demand in m^{3}/day
. | Node . | Elevation in m . | Demand in m^{3}/day
. |
---|---|---|---|---|---|

1 | 71 | −2,550 | 12 | 58.6 | 37.5 |

2 | 56.4 | 15 | 13 | 59.3 | 37.5 |

3 | 53.8 | 70.5 | 14 | 59.8 | 63 |

4 | 54.9 | 58.5 | 15 | 59.2 | 445.5 |

5 | 56 | 75 | 16 | 53.6 | 108 |

6 | 57 | 67.5 | 17 | 54.8 | 79.5 |

7 | 53.9 | 63 | 18 | 55.1 | 55.5 |

8 | 54.5 | 48 | 19 | 54.2 | 118.5 |

9 | 57.9 | 42 | 20 | 54.5 | 124.5 |

10 | 62.1 | 30 | 21 | 62.9 | 31.5 |

11 | 62.8 | 42 | 22 | 61.8 | 799.5 |

Sr.No . | Diameter (mm) . | Cost (unit/m) . |
---|---|---|

1 | 80 | 37,890 |

2 | 100 | 38,933 |

3 | 125 | 40,563 |

4 | 150 | 42,554 |

5 | 200 | 47,624 |

6 | 250 | 54,125 |

7 | 300 | 62,109 |

8 | 350 | 71,524 |

Sr.No . | Diameter (mm) . | Cost (unit/m) . |
---|---|---|

1 | 80 | 37,890 |

2 | 100 | 38,933 |

3 | 125 | 40,563 |

4 | 150 | 42,554 |

5 | 200 | 47,624 |

6 | 250 | 54,125 |

7 | 300 | 62,109 |

8 | 350 | 71,524 |

Similar to the Two-Loop network, sensitivity analysis was carried out for the GoYang WDN and the input parameters chosen are: population size = 150; number of generations = 200; crossover probability = 0.8; mutation probability = 0.01. tournament selection, blend crossover and random mutation operators are used in this study.

*et al.*(1994) used nonlinear programming and obtained an optimum cost of 179,142,700 units, Geem (2006) used a harmony search algorithm and obtained an optimal cost of 177,135,800 units, whereas the present study using RCGA optimal cost is obtained as 176,098,456 unit. Geem (2006) used harmony search to obtain the optimum network and 10,000 function evaluations were required; whereas in the present study using RCGA, only 8,850 function evaluations were required to obtain the optimal solution. In the present study, the optimal cost and the number of function evaluation is less compared to other methods. Table 11 shows the node pressure for optimal diameters for the GoYang network for the present study.

Pipe no. . | Pipe length (m) . | Original diameter (mm) . | Kim et al. (1994) diameter (mm)
. | Geem (2006) diameter (mm) . | Present study (RCGA) diameter (mm) . |
---|---|---|---|---|---|

1 | 165 | 200 | 200 | 150 | 80 |

2 | 124 | 200 | 200 | 150 | 125 |

3 | 118 | 150 | 125 | 125 | 125 |

4 | 81 | 150 | 125 | 150 | 125 |

5 | 134 | 150 | 125 | 100 | 80 |

6 | 135 | 100 | 100 | 100 | 100 |

7 | 202 | 80 | 80 | 80 | 80 |

8 | 135 | 100 | 80 | 100 | 80 |

9 | 170 | 80 | 80 | 80 | 80 |

10 | 113 | 80 | 80 | 80 | 80 |

11 | 335 | 80 | 80 | 80 | 80 |

12 | 115 | 80 | 80 | 80 | 80 |

13 | 345 | 80 | 80 | 80 | 80 |

14 | 114 | 80 | 80 | 80 | 80 |

15 | 103 | 100 | 80 | 80 | 80 |

16 | 261 | 80 | 80 | 80 | 80 |

17 | 72 | 80 | 80 | 80 | 80 |

18 | 373 | 80 | 100 | 80 | 80 |

19 | 98 | 80 | 125 | 80 | 80 |

20 | 110 | 80 | 80 | 80 | 80 |

21 | 98 | 80 | 80 | 80 | 80 |

22 | 246 | 80 | 80 | 80 | 80 |

23 | 174 | 80 | 80 | 80 | 100 |

24 | 102 | 80 | 80 | 80 | 80 |

25 | 92 | 80 | 80 | 80 | 80 |

26 | 100 | 80 | 80 | 80 | 100 |

27 | 130 | 80 | 80 | 80 | 100 |

28 | 90 | 80 | 80 | 80 | 80 |

29 | 185 | 80 | 100 | 80 | 80 |

30 | 90 | 80 | 80 | 80 | 80 |

Total cost (unit) | – | 179,428,600 | 179,142,700 | 177,135,800 | 176,098,456 |

Number of function evaluation | – | – | – | 10,000 | 8,850 |

Pipe no. . | Pipe length (m) . | Original diameter (mm) . | Kim et al. (1994) diameter (mm)
. | Geem (2006) diameter (mm) . | Present study (RCGA) diameter (mm) . |
---|---|---|---|---|---|

1 | 165 | 200 | 200 | 150 | 80 |

2 | 124 | 200 | 200 | 150 | 125 |

3 | 118 | 150 | 125 | 125 | 125 |

4 | 81 | 150 | 125 | 150 | 125 |

5 | 134 | 150 | 125 | 100 | 80 |

6 | 135 | 100 | 100 | 100 | 100 |

7 | 202 | 80 | 80 | 80 | 80 |

8 | 135 | 100 | 80 | 100 | 80 |

9 | 170 | 80 | 80 | 80 | 80 |

10 | 113 | 80 | 80 | 80 | 80 |

11 | 335 | 80 | 80 | 80 | 80 |

12 | 115 | 80 | 80 | 80 | 80 |

13 | 345 | 80 | 80 | 80 | 80 |

14 | 114 | 80 | 80 | 80 | 80 |

15 | 103 | 100 | 80 | 80 | 80 |

16 | 261 | 80 | 80 | 80 | 80 |

17 | 72 | 80 | 80 | 80 | 80 |

18 | 373 | 80 | 100 | 80 | 80 |

19 | 98 | 80 | 125 | 80 | 80 |

20 | 110 | 80 | 80 | 80 | 80 |

21 | 98 | 80 | 80 | 80 | 80 |

22 | 246 | 80 | 80 | 80 | 80 |

23 | 174 | 80 | 80 | 80 | 100 |

24 | 102 | 80 | 80 | 80 | 80 |

25 | 92 | 80 | 80 | 80 | 80 |

26 | 100 | 80 | 80 | 80 | 100 |

27 | 130 | 80 | 80 | 80 | 100 |

28 | 90 | 80 | 80 | 80 | 80 |

29 | 185 | 80 | 100 | 80 | 80 |

30 | 90 | 80 | 80 | 80 | 80 |

Total cost (unit) | – | 179,428,600 | 179,142,700 | 177,135,800 | 176,098,456 |

Number of function evaluation | – | – | – | 10,000 | 8,850 |

Node . | Present study (RCGA) pressure (m) . | Node . | Present study (RCGA) pressure (m) . |
---|---|---|---|

1 | 26.62 | 12 | 17.38 |

2 | 26.33 | 13 | 1,522 |

3 | 24.32 | 14 | 15.35 |

4 | 23.17 | 15 | 26.07 |

5 | 20.68 | 16 | 24.69 |

6 | 26.04 | 17 | 24.4 |

7 | 24.72 | 18 | 25.28 |

8 | 19.9 | 19 | 24.54 |

9 | 15.34 | 20 | 17.52 |

10 | 15.62 | 21 | 17.18 |

11 | 18.1 | 22 | 0 |

Node . | Present study (RCGA) pressure (m) . | Node . | Present study (RCGA) pressure (m) . |
---|---|---|---|

1 | 26.62 | 12 | 17.38 |

2 | 26.33 | 13 | 1,522 |

3 | 24.32 | 14 | 15.35 |

4 | 23.17 | 15 | 26.07 |

5 | 20.68 | 16 | 24.69 |

6 | 26.04 | 17 | 24.4 |

7 | 24.72 | 18 | 25.28 |

8 | 19.9 | 19 | 24.54 |

9 | 15.34 | 20 | 17.52 |

10 | 15.62 | 21 | 17.18 |

11 | 18.1 | 22 | 0 |

## CONCLUSIONS

Research related to the least-cost design of WDNs is presented in this study. RCGA and EPANET were considered as the optimization module and simulation module, respectively. The developed model is applied to two benchmark WDN problems, i.e Two-Loop and GoYang networks. It was found that the RCGA approach is a good tool after parameter optimization for the least-cost design of WDN as the results obtained from the application of the RCGA methodology for the two benchmark networks proved that RCGA can give optimal solutions with less function evaluation. The number of function evaluations will increase with increasing complexity of the system. For very complex systems with large numbers of pipes, it is suggested to use RCGA as the optimization module. RCGA are useful when the search space is very large and there are a large number of parameters involved. In reallife, WDNs are large and the search space of decision parameters (pipe diameter) is therefore also large. The advantage of RCGA is that uncertainty in demand and optimal positioning of the tank/reservoir can be easily incorporated into the model. For future studies, a real case study can be used to check the computational efficiency and also to include multiple objectives. Also, the metaheuristic algorithms Grass Hopper Optimization Algorithm, Sparrow Search Algorithm, and Artificial Bee Colony Algorithm can be applied to the benchmark problems and their computational efficiency checked in the simulation–optimization framework.

## AUTHOR CONTRIBUTIONS

V.H.S.K. did formal assessment and was involved in conceptualization, data collection, framework of methodology, model development, and software analysis. S.K.P. did formal assessment, conceptualized, performed methodology, collected resources, and was involved in supervision, initial draft writing, review, and editing.

## DATA AVAILABILITY STATEMENT

All relevant data are included in the paper or its Supplementary Information.

## CONFLICT OF INTEREST

The authors declare there is no conflict.

## REFERENCES

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