Development of a new self-adaptive F-NSGA-III algorithm with fuzzy structure for designing urban water distribution networks

The urban water distribution networks are essential infrastructures in industrial and urban operations, fulfilling crucial functions and incurring significant expenses (Mu et al. 2021). An essential requirement for a contemporary urban water supply network is to meet the qualitative and quantitative requirements of water in accordance with the current standards (Brentan et al. 2021). Sufficient and healthy water supply required by the urban population is one of the important aspects in urban water distribution networks (Surendra et al. 2021).

An urban water supply network must be able to meet the expectations and water demands of consumers in terms of quality (physical, chemical, and bacteriological properties of water (Nouiri 2017) and quantity (flow and water pressure) based on existing standards (Ahmadianfar et al. 2017; Dai 2021).

The common method of designing water distribution networks is that designers choose different options and simulate them. The designers select many alternatives and proceed to simulate them. Given the intricate interconnections among several elements, it is challenging to ascertain the modifications that result in the enhancement of the network architecture, especially in networks of limited size. Furthermore, employing this approach does not provide certainty regarding the optimality of the obtained answer.

While artificial intelligence-based approaches are extensively used in the water resources sector (Nourmohammadi Dehbalaei et al. 2023; Soltani & Azari 2024), their use in the optimal design of water distribution networks is limited.

Consequently, numerous endeavors have been undertaken to address the optimization issue of water distribution networks, resulting in the extensive adoption of contemporary optimization techniques. Given the substantial financial investment involved in establishing the water supply network, it is crucial to prioritize the design of networks that are both cost-effective and highly dependable (Berardi et al. 2022; Zarei et al. 2022; Sirsant et al. 2023). The EPANET simulator model and the NSGA-II optimization algorithm were employed by Atiquzzaman et al. (2006) to optimize water distribution networks. Their findings revealed that the effective design cost is influenced by the budget amount and the anticipated supply pressure in the nodes.

Zheng et al. (2015) investigated the enhancement of efficiency in evolutionary multi-objective algorithms by partitioning a water distribution network into several sub-networks using decomposition techniques. The application of this approach in two water distribution networks shown that the suggested method exceeds the performance of traditional comprehensive search methods (network optimization without deconstruction).

An investigation conducted by Mora-Melia et al. (2017) examined the impact of population size on the effectiveness of evolutionary algorithms in the design of water networks. Their discovery revealed that the extraction of the optimal configuration, which is determined by the quality of the solutions and the convergence speed of the algorithm, is contingent upon the size of the population.

A novel meta-exploratory method for the optimal design of water distribution networks was investigated by Moosavian & Kasaee Roodsari (2014) in the context of football league competition. The application of this algorithm in three benchmark pipe networks demonstrated its convergence to the global optimum with enhanced dependability and accelerated speed in comparison to alternative meta-exploration approaches.

In their study, Khedr et al. (2015) examined the process of recalibration in the water distribution system by employing a three-objective optimization technique with the PA-DDS algorithm. The results demonstrated the robust performance of the algorithm during the calibration of the system.

To enhance the design of water distribution systems, Bi et al. (2015) introduced a novel exploratory technique known as the pre-screened exploratory sampling method (PHSM). The results demonstrated that the PHSM has superior computational efficiency and the capacity to identify nearly ideal solutions. Furthermore, its efficiency proves to be positively correlated with the enlargement of the network size.

In their study, Marques et al. (2015) employed a multi-objective simulated annealing technique integrated with the hydraulic simulator to improve the water distribution system. Their approach involved altering the hydraulic restrictions. This approach is capable of addressing environmental impacts and future uncertainties by taking into account competing objectives.

The genetic algorithm (GA) was employed by Do et al. (2017) to investigate the assessment of demand in water distribution systems. Notably, this approach does not necessitate the same number of known inputs as the number of variables. Furthermore, the nodes within the model can exhibit distinct demand patterns.

The present work utilized the NSGA-III method, which was adapted from the NSGA-II algorithm by Deb et al. (2002), Jalili et al. (2023), shabanlou (2018) and Jalilian et al. (2022), with a more recent structure (Deb & Jain 2014).

Researchers have shown interest in the NSGA-III algorithm because to its enhanced implementation of the initial population generation and parent chromosome selection procedure, as compared to NSGA-II (Swathi et al. 2019).

A study conducted by Chaudhari et al. (2022) demonstrated that the NSGA-III algorithm offers a broader spectrum of optimal operating conditions compared to NSGA-II.

Jafari et al. (2022) introduced a novel goal function targeting the minimization of pollution impact based on its significance. They employed a robust methodology to address a five-objective problem by utilizing the NSGA-III algorithm.

An analysis of prior research indicates that the majority of studies have focused on using multi-objective algorithms using a two-dimensional response space to optimize urban water distribution systems. The present study aims to design a novel self-adaptive multi-objective algorithm including a fuzzy structure. The performance of this algorithm will be evaluated in comparison to the existing work of other scholars. Thorough investigations have not been carried out to compare these algorithms in terms of the quality and diversity of optimal solutions in the final iteration, as well as their rate of convergence. The innovation of this research is the implementation of the new F-NSGA-III algorithm with a more recent structure and its comparison with the NSGA-II algorithm in optimizing urban water distribution networks. The F-NSGA-III algorithm employs a novel framework for evaluating optimal solutions. Through modifications in the generation of the initial population and the selection process of parent chromosomes, it is anticipated that the F-NSGA-III algorithm will exhibit superior convergence speed compared to the F-NSGA-II algorithm in the context of intricate water distribution networks. The current study has meticulously examined and explored this matter. Another novel aspect of this study is the incorporation of fuzzy definitions for the objective functions and research constraints within the F-NSGA-III algorithms. This is particularly significant because the objective functions differ, with one being the pressure in meters and the other being the cost in dollars. Yet another novel aspect of this study is the alteration of the structure of the objective functions. The analyses of the examined publications reveal that the cost function equation remains constant and unchanged in each repetition. It is solely dependent on the length and diameter of the chosen pipes. The cost goal function in this study is modeled as a variable by incorporating the penalty resulting from surpassing the permissible pressure limits in the nodes. This novel function employs random numbers to adjust the penalty amount based on the degree of pressure limit violations. By incorporating randomization or staircase mode for the penalty amount, the convergence speed in the problem will be explored and compared to the normal mode. Consequently, this study has provided useful recommendations for the adaptation of the proposed algorithm to other networks by employing various approaches, comparing and adjusting the structure of the objective functions.

Deb & Jain (2014) created the NSGA-III algorithm, which modifies a few selection methods, as an alternative to the NSGA-II algorithm. NSGA-III and NSGA-II have a similar basic framework, albeit with notable modifications to the selection method. The distinctions between the two are as follows: (1) Optimization issues with two objective functions are handled by the NSGA-II algorithm; however, the NSGA-III algorithm can handle problems with three to fifteen objective functions. (2) The NSGA-III algorithm allows the decision space to be expanded and displays the decision space with three objective functions – F1, F2, and F3 — in three dimensions as opposed to the NSGA-II algorithm's two-dimensional decision space with only two objective functions – F1 and F2 — in front of each other. (3) The crowding distance approach is used in the NSGA-II algorithm to rank the best response, whereas the Niche method is used in the NSGA-III algorithm. (4) The starting population is created using selection operators in the NSGA-II algorithm, however in the NSGA-III method, no selection operators are utilized and the offspring are formed randomly. (5) The initial population in the NSGA-III algorithm need only be a multiple of 4, and it is not required to be greater than the decision variable. This is regarded as an advantage since it shortens the problem's run time. In contrast, the NSGA-II algorithm has a selection structure and requires the initial population to be at least twice the decision variable. Below is a quick explanation of NSGA-III's primary methodology. The first step in NSGA-III is to define a collection of reference points. Next, N is the population size, since an initial population of N members is formed at random. Until the termination requirement is satisfied, the following stages are repeated. Although it has been demonstrated that the NSGA-III algorithm performs effectively in three-objective optimization problems, its main application is in the solution of many optimization problems with more than three objectives. While NSGA-II stresses somewhat varied solution techniques per year and does not use any specific and predetermined guidelines, the NSGA-III algorithm uses predefined guidelines to pick diverse solutions within the population. Thus, for solving two-objective optimization problems, NSGA-III may outperform other comparable algorithms or even the NSGA-II method if the first factor is considered and additional selection pressure is applied. In NSGA-III, the parameters, namely maximum number of iterations, number of population, percent of crossover and percent of mutation are adjustable parameters. These parameters have a significant effect on the performance of the algorithm and by trial and error, the most appropriate value for these parameters was determined.

The optimization problem in this research is defined in the form of a multi-objective function so that the first objective is to minimize the cost of the system design versus the second objective, i.e. minimize the total pressure lack in the entire system.

The constraints of the optimization of water distribution networks incude two categories. The first category, which are in the form of equality, are the continuity equation and the energy conservation equation, which are implemented automatically in the EPANET software, and there is no need to define them as constraints in the evaluator function. The second category are inequalities including shape, diameter and velocity in pipes.

From Expression (11), it is understood that larger values reflect higher objectives satisfaction.

The outcomes obtained from the optimization of the TLN and Lansey networks with the aim of extracting optimal diameters using the F-NSGA-II and F-NSGA-III algorithms are analyzed in two ways. In the first method, the results of implementing these algorithms are compared by defining the cost function based on the relationship between diameter and cost and with the number of repetitions of 5,000 and 10,000. In the second method, the results obtained from their implementation, taking into account the penalty for exceeding the allowable pressure limits in the cost function, are carried out until the best solution is reached and compared with the results of the previous method. It should be noted that in the second method, the penalty coefficient in the equation of the second objective function is chosen randomly in each iteration of the algorithm.

Based on Table 3, the chosen solutions resulting from the implementation of the F-NSGA-II algorithm in the first method with 5,000, 10,000, and 20,000 iterations have zero pressure deficiency, but the cost of implementing the network in this method is high and the algorithm has not yet been able to get the best cost. In the second method, the F-NSGA-II and F-NSGA-III algorithms have reached zero pressure deficiency and the desired cost of $419,000 after 35 and 20 iterations, respectively. In this method, the pressure in the nodes and the speed in the pipes are determined within the permissible limits in the entire network. Based on this table, the F-NSGA-III algorithm is able to achieve the optimal solution in a lower number of iterations and a shorter duration and reduces about 34% of the time to solve the TLN problem. The selected solutions resulting from the implementation of the F-NSGA-II algorithm in both the first and second methods have zero pressure deficiency. But in the first method, in the optimal solution proposed by F-NSGA-II, even after 10,000 iterations, the velocity in pipe number 8 is lower than the minimum allowed velocity.

In general, the results show that the execution cost in the second method is lower than the first method. The time to reach convergence in the second method is much less than the first method. Thus, in order to optimize the TLN, taking into account the cost of surpassing the allowable pressure limits in the cost function of the F-NSGA-II and F-NSGA-III algorithms yields better results than running the algorithm without taking into account the cost of exceeding the allowable pressure limit.

Several researchers have optimized this network using various algorithms, including GA, Scattered Search Algorithm (SS), Shuffled Frog Leaping Algorithm (SFLA), Particle Swarm Optimization-Differential Evolution Algorithm (PSO-DE), Honey-Bees Mating Optimization Algorithm (HBMO), and Multi-Objective Particle Swarm Optimization Algorithm (MOPSO). This research achieves a minimum cost of 419,000 dollars, and it also considers a minimum pressure of 30 m, which is also achieved in the current research. However, this research goes beyond the cost objective function by taking into account the cost of exceeding the permissible range of pressure and the lack of pressure. As a result, more optimal solutions are obtained, and the speed of convergence significantly increases. The best solution is obtained with the fewest number of iterations, demonstrating the superiority of this method when compared to other studies. The comparison between the findings of this study and those of other researchers is displayed in Table 4.

According to Table 5, based on the selected answer of the implementation of the F-NSGA-II algorithm in the first method, there is zero pressure deficiency in both 5,000 and 10,000 repetitions. In 5,000 repetitions, the speed at node number 5 is lower than the minimum allowed. At iteration 10,000, the speed at node number 2 is greater than the maximum allowed, which is neither desirable. In the second technique, the pressure deficiency in the network is zero and the speed in all of the network pipes is within the allowed range following the application of the F-NSGA-II and F-NSGA-III algorithms, after 130 and 100 repetitions, respectively. The second technique takes a lot less time to attain convergence than the first method, and it costs less to execute the optimal solution found in the last iteration of the second method. Consequently, when running the F-NSGA-II and F-NSGA-III algorithms to optimize the Lansey network, taking into account the cost of surpassing the permitted pressure limits in the cost function yields more appropriate optimal solutions than when the algorithm is run without taking this into account. Table 5 demonstrates that the F-NSGA-III method reduces the issue solving time by approximately 25% in the Lansey network by reaching the optimal solution faster and with fewer iterations than the F-NSGA-II approach.

With the diameter of the pipes taken into consideration, the Lansey sample network developed by Lansey et al. (2001) had a cost equal to $3,119,022. After optimization by the F-NSGA-II and F-NSGA-III algorithms, a cost equal to $1,137,942 was obtained, with no shortage of pressure in the network and a high convergence speed. This was achieved using a GA to select the network. Many studies utilizing various methods have also been conducted to construct the Lansey network optimally. The comparison between the findings of this study and those of other researchers is displayed in Table 6. The analysis yielded a minimal cost ranging from $3,119,022 to $1,137,942. All studies have found that there is no pressure in the Lansey network when the minimum pressure is 30 m, and this finding is also confirmed by the current investigation. According to what was said in the current study, the optimal solution is obtained with a significantly lower number of iterations and a faster rate of convergence, demonstrating the superiority of the approach used in this study when compared to other experiments. This is because, in addition to the cost objective function, the absence of pressure is also taken into consideration, as is the cost of exceeding the allowable pressure range.

In this study, urban water distribution networks were optimized through the application of the F-NSGA-III and F-NSGA-II multi-objective algorithms. Two sample networks, TLN and Lansey, were implemented for optimal design using these algorithms twice: once, without taking into account the cost of exceeding the minimum and maximum standard pressure threshold with two repetitions of 5,000 and 10,000 (the first method), and once more taking into consideration the cost of exceeding the minimum and maximum standard pressure values in the cost function (the second method). The best solution generated and the time it took for both sample networks to converge were used to assess their performance. Overall, the findings demonstrated that these algorithms had a high degree of ability to identify optimal solutions. By determining the appropriate pipe diameter, they were able to optimize the network in terms of both cost and pressure. In the event of their execution, optimal solutions were reached in both algorithms at a low number of iterations, which considerably shortened the time to attain convergence, taking into consideration the cost of exceeding the minimum and maximum standard pressure thresholds in the cost function. Also, this increased the number of optimal solutions in the last iteration. If the program is implemented considering the cost of exceeding the allowed pressure range (second method) in the TLN and Lansey networks, in terms of producing the best solution, both algorithms showed good performance and the solution with zero pressure deficiency and the lowest cost obtained in other researches were achieved, and in terms of the time to reach convergence, both algorithms reached the optimal solution and a high convergence speed with a much lower number of iterations than other studies. The results showed that the F-NSGA-III algorithm is able to achieve the optimal solution in a lower number of iterations and a shorter duration than the F-NSGA-II algorithm and in the TLN and Lansey networks it reduces the solution time in about 34 and 25%, respectively. The results showed that both F-NSGA-II and F-NSGA-III algorithms have been successful in finding solutions with zero pressure deficiency in the network and the lowest cost by considering the cost of exceeding the allowed pressure limits in the cost function. But the performance of the F-NSGA-III algorithm is better in terms of problem solving time. This is especially important in large and complex networks, because in such networks, the use of the F-NSGA-III algorithm developed in this research will save significant time and money. The fuzzy structure considered in this algorithm has made the convergence time in the F-NSGA-II and F-NSGA-III algorithms significantly shorter than the NSGA-II algorithm that was previously used in the research conducted by Zarei et al. (2022). The method used in this research can be implemented to design or update real urban networks, which has limitations. Due to the complexity of these networks and the increase in the number of pipes, the execution time of the algorithm may be long. It is suggested to use multi-objective optimization algorithms with parallel structure developed for using multi-core computer systems in these networks to reduce the problem solving time.

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The authors declare there is no conflict.