In this paper, WDNs were optimized with pipe diameters as decision variables. Cost was considered as the objective function that must be minimized (Eq. 1), and the reliability criteria were modeled in the form of a two-objective function (Eq. 2). 1Minf1=∑i=1NciDi×li2Maxf2=Reliabilityfactor, where f1 is network cost, f2 is network reliability, ci is cost for unit length of pipe with diameter, Di length li, and N is pipe numbers in the network.

The constraints to the optimization problem were as follows:

Explicit system constraints such as the conservation of mass of flow, conservation of energy, and conservation of mass of constituent, which all were controlled by water network simulator software, EPANET (Mala-Jetmarova et al., 2015; Rossman, 2000).

A range of reliability criteria has been introduced to different degrees of complexity. Usually, these criteria give some suggestion of the ability of a WDN to handle changing conditions and are straightforward to analyze, so they are practical for optimization studies that compare the performance of a network design. This section presents the definition of the key criteria and their derivatives and their advantages and disadvantages.

In this paper, GANetXL was employed in the following two steps: it was used in the first step for a single-objective optimization based on GA and in the second step for two-objective optimization based on NSGA-II. GA and NSGA-II parameters such as population size, the number of generations, selection method, crossover and mutation operators, crossover and mutation probability, and the type of algorithm were tested, and reasonably well-performing parameters were selected for final optimization runs (see the Supplement).

The two-loop network was originally presented by Alperovits and Shamir (1977). The network consisted of seven nodes and eight pipes with two loops and was fed by gravity from a reservoir with a 210 m fixed head. The minimum pressure head requirement of the other nodes was 30 m above the nodal elevations.

In the first step, as a result of single-objective optimization of the two-loop network using GA technique in GANetXL, the minimum cost obtained was USD 419 000, with 35 000 number of function evaluations (NFEs), which was the same as the minimum costs obtained by GA (Savic and Walters, 1997), simulated annealing (SA; Cunha and Sousa, 1999), shuffled frog leaping algorithm (SFLA; Eusuff and Lansey, 2003), harmony search (HS; Geem, 2009), and scatter search (SS; Lin et al., 2007), with 250 000, 25 000, 11 323, 5000, and 3215 NFEs, respectively.

As a result, minimum cost was USD 419 000 for a one-objective optimization of this network, using GANetXL after 1000 generations that were equal to the minimum costs obtained by GA, SA, SFLA, HS, and SS.

In the second step, Fig. 1a–d show the obtained Pareto front for a two-objective optimization of two-loop network using NSGA-II in GANetXL, considering Ir, In, MRI, and Im as the second objective function, respectively. All of the solutions in this Pareto front were feasible (and all the network constraints are satisfied). As it is observed, the cost changes in the range of USD [0.424 × 106–4.400 × 106] and Ir, In, MRI, and Im criteria change in the ranges [0.338–0.903], [0.287–0.903], [0.040–0.107], and [0.122–12.856], respectively. The range of numbers presented was based on the minimum and maximum values of the solutions presented in the graphs. The lowest and the highest point on the Pareto front in each of the charts were the criteria for deriving the range of reliability indices and the cost range. In the cost range of USD [0.424 × 106–1 × 106], the cost-In Pareto front showed more and varied solutions in comparison to other graphs. However, with the increase in cost, non-dominated solutions decreased, and the current continuity in Pareto front disappears while cost-Ir (Fig. 1a) and cost-MRI (Fig. 1c) Pareto fronts had a better performance. In the cost-Im graph (Fig. 1d), the variety of obtained solutions in the lower and upper bound of Pareto front was lower than other graphs.

Figure 2 shows the surplus pressure of the minimum pressure head requirement in the nodes of two-loop network for solutions with maximum reliability criteria and minimum cost. As observed, the surplus pressure of the nodes in the solutions with minimum cost was lower than the solutions of maximum reliability criteria (Ir, In, MRI, and Im). Also, with the design based on single-objective function (minimum cost), the surplus pressure was closer to the minimum allowed pressure in nodes 3, 6, and 7, showing that these nodes were the critical nodes of the network. As a result, if the two-loop network was designed only based on the minimum cost in critical periods such as pipe failures, then there would be problems or issues at these nodes.

The reliability evaluation should be analyzed under all feasible extreme conditions. Failure of multiple pipes and the failure of the reservoir connection line during a firefighting event and/or power or pumping station failures should be evaluated simultaneously. Although an infinite number of failure scenarios were likely, the probability of simultaneous failures in multiple pipes was too low (Tabesh et al., 2001). Pipe failure independency can be assumed (Su et al., 1987), and any likely dependency will be negative. For example, if a pipe failure occurs in the network, the pressure will decrease, and consequently, the probability of another pipe failure will decrease as well. However, in case the system is a large-scale WDN, the influence of pressure might not be significant. Other pipe failure reasons, such as damages or traffic loading, may lead to pipe failures that are completely independent events (Shafiqul Islam et al., 2013).

In this paper, to evaluate reliability of the candidate solutions of maximum Ir, In, MRI, and Im criteria, the nodal pressures were investigated under pipe failure conditions.

The Hanoi network in Vietnam, first presented by Fujiwara and Khang (1990), was a new design, as all-new pipes were selected. The network consists of 32 nodes and 34 pipes organized in three loops. The system was gravity fed by a single reservoir. The network details are given in Fujiwara and Khang (1990). The minimum required pressure head for all nodes was 30 m, and the elevation for all nodes was zero. There were six available pipe diameters to be selected for each new pipe, and the pipe cost per meter for the six available pipe diameters had been listed in previous studies (Savić et al., 2011; Atiquzzaman and Liong, 2004; Zecchin et al., 2006; Pant and Snasel, 2021).

In the first step, as a result of single-objective optimization, the GA method in GANetXL obtained a minimum cost of USD 6.097 × 106 with 100 000 NFEs for this network, while in previous research the methods of GA (Savic and Walters, 1997), ant colony optimization (ACO; Zecchin et al., 2006), and shuffled complex evolution (SCE; Atiquzzaman and Liong, 2004) reported costs of USD 6.195 million, USD 6.134 million, and USD 6.22 million, with 1 × 106, 25 402, and 85 571 NFEs, respectively.

In the second step, Fig. 3a–d show non-dominated solutions of the Hanoi network which, when calculated by NSGA-II considering minimum cost versus maximum reliability criteria and all of the solutions in the Pareto front, was feasible. As observed in Fig. 4, the minimum values of Ir, In, MRI, and Im are 0.228, 0.256, 0.555, and 0.090, and the maximum values were 0.353, 0.353, 0.825, and 19.916, respectively. Cost values change in a range of USD [6.251 × 106–10.791 × 106] for cost-Ir and in USD [6.584 × 106–10.969 × 106] for the cost-In space, showing that the increase in cost-In to cost-Ir was due to the Cj factor in the formula (Eq. 8), which causes uniformity diameters in the design phase. In this example, the monotony and variety of represented solutions were observed in all Pareto fronts; the reason could be found in the increase in the network size and possible solutions for the network design.

Figure 5 shows the surplus pressure in comparison with the minimum allowed pressure in the nodes of the Hanoi network for solutions of the maximum reliability criteria and minimum cost. In the cost-based optimization, surplus pressure in node nos. 13, 30, and 31 was less than 1 m, which shows that these nodes were the most critical ones of this network. Ir, In, and MRI criteria have similar performance for all the nodes, but the Im criterion determines more surplus pressure for most of the nodes compared to other criteria in this network and unlike the two-loop network.

The real-life WDN was located in Iran, and it had 37 pipes, 24 nodes, and a reservoir with a 962 m fixed head (Fig. 7). The design purpose of this network was municipal water supply to the city and improving the existing condition of the WDN (Moghaddam et al., 2020). For this purpose, a series of pipes which had diameters more than 100 mm were used for future conditions. For designing this network, polyethylene pipes (PE 80) with a Hazen–Williams coefficient of 130 were used. The nodes and pipes characteristics are presented in Moghaddam et al. (2020). In the design of the network, node pressure and velocity constraints were between 14–60 m and 0.2–2 m s-1, respectively (Department of Technical Affairs, Ministry of Energy, 2013). There were more constraints in this example than in the other ones.

In the first step, as a result of single-objective optimization using GA in GANetXL, the minimum cost was estimated at 7.54 × 108 Rials with 100 000 NFEs, which shows a cost decrease of 46.14 % in comparison to the solution of the consultant company with 14×108 Rials (Rasekh et al., 2010).

In the second step, the results of Fig. 5a–d show that the Ir, MRI, and Im criteria had a better performance than the In criterion for this network in terms of non-dominated solutions. All three of these criteria had similar solutions of maximum and minimum cost in the Pareto front. All of the solutions in the Pareto front (Fig. 5) which were obtained by NSGA-II are feasible and satisfied the velocity and pressure constraints.

The results shown in Fig. 6 demonstrate that in the cost-based optimization, the surplus pressure in the node nos. 13 and 23 was less than 1 m, which explains that these nodes were the most critical ones in the network. Ir and MRI criteria had similar and more successful performance compared to Im in terms of the surplus pressure for all the nodes in the network. In had a smaller capability than other criteria to create surplus pressure in the network.

The results of the investigations in Fig. S4 (see the Supplement) show that only the failure in pipe no. 18 can influence the pressure nodes. Consequently, this pipe was one of the most sensitive pipes in this network. However, a reliability performance in the failure conditions was similar to the no-failure conditions in Fig. 7. Finally, for this network that includes a low diameter in existing pipes, In did not have a suitable performance because making the uniformity in the pipes connected to a node leads to a decrease in the diameter of new pipes. Thus, the capability of the surplus pressure decreased due to the increase in head loss in the pipes.

Figure 7 shows the velocity variations in the pipes for the solutions with minimum cost and maximum Ir, MRI, and Im criteria obtained using GA and NSGA-II in GANetXL. As observed, when the cost was the basis for the design and optimization of the real-life network, the velocity variation was high in the pipes. This can lead to some issues in the network. But in the presented solutions with maximum reliability criteria (Ir, MRI, and Im), velocity variations were not only low but almost uniform.