During the last three decades, the water resources engineering field has received a tremendous increase in the development and use of meta-heuristic algorithms like evolutionary algorithms (EA) and swarm intelligence (SI) algorithms for solving various kinds of optimization problems. The efficient design and operation of water resource systems is a challenging task and requires solutions through optimization. Further, real-life water resource management problems may involve several complexities like nonconvex, nonlinear and discontinuous functions, discrete variables, a large number of equality and inequality constraints, and often associated with multi-modal solutions. The objective function is not known analytically, and the conventional methods may face difficulties in finding optimal solutions. The issues lead to the development of various types of heuristic and meta-heuristic algorithms, which proved to be flexible and potential tools for solving several complex water resources problems. This paper provides a review of state-of-the-art methods and their use in planning and management of hydrological and water resources systems. It includes a brief overview of EAs (genetic algorithms, differential evolution, evolutionary strategies, etc.) and SI algorithms (particle swarm optimization, ant colony optimization, etc.), and applications in the areas of water distribution networks, water supply, and wastewater systems, reservoir operation and irrigation systems, watershed management, parameter estimation of hydrological models, urban drainage and sewer networks, and groundwater systems monitoring network design and groundwater remediation. This paper also provides insights, challenges, and need for algorithmic improvements and opportunities for future applications in the water resources field, in the face of rising problem complexities and uncertainties.

  • This paper reviews working principles and applications of evolutionary algorithms.

  • This paper reviews working principles and applications of swarm intelligence algorithms.

  • Discusses the issues and merits of various meta-heuristic algorithms.

  • Provides insights and future research directions.

Graphical Abstract

Graphical Abstract
Graphical Abstract

Over the last few decades, there has been rising concern about global warming and associated changes in rainfall, streamflows, and water availability in the river basins. Many regions are often facing a shortage of water, as they receive rainfall in a particular season only, but the water demands have to be satisfied for the entire year. There is an ever-increasing demand for water to meet the diverse needs of society majorly for domestic, industrial, and agricultural purposes. Also, the efficient use of limited water for different users imposes substantial difficulties with conflicting goals (Reddy & Kumar 2006). Therefore, planning, construction, development, and operational activities of water resources projects warrant for solutions using systematic procedures. They can help planners to develop improved designs and operational systems, decide innovative management policies, improve and calibrate simulation models, and resolve conflicts between conflicting stakeholders (Maier et al. 2014). The complexity of systems models in water resources engineering has increased tremendously, with several socio-environmental–ecological issues and requires better alternative methods. In general, systems aim to reduce the total system cost or failure risk, maximize net benefits by providing an efficient design or operation policy. One of the important engineering tools that can be used in such events is the optimization tool, which helps to find a set of values of the decision variables subject to the various constraints that will produce the desired optimum response for the chosen objective function. As computers have become more powerful, the size and complexity of problems that can be simulated and solved by optimization techniques have correspondingly expanded.

Today there are a variety of optimization techniques existing to tackle different issues in practical problems of water resources. Some techniques (like exact methods) may provide optimal solutions for smaller problems, and others like meta-heuristic techniques may provide near-optimal solutions while solving large-scale water resources problems. A taxonomy or classification of optimization methods is given in Figure 1. But no single optimization method or algorithm is unanimously declared as the winner that can be applied efficiently to all types of problems. The method chosen for any particular case will depend primarily on (Reddy & Kumar 2012): (1) complexity of the problem, and the character of the objective function whether it is known explicitly, (2) the number and nature of the constraints, equality and inequality constraints, (3) the number of continuous and discrete variables, etc.

The commonly used methods in water resources include linear programming (LP), dynamic programming (DP), and nonlinear programming (NLP) methods. The LP method can guarantee global-optimal solutions for linear problems and has wider applications in water resources, like for irrigation planning, reservoir operation, conjunctive use planning, crop water allocation, seawater intrusion, command area management, etc. (Yeh 1985). But many practical water resources applications may involve nonlinear functions in optimization modeling for solving the problems. So the popular LP method cannot work in the case of models with nonlinear functions. The DP method (Bellman 1957) is popularly used for solving sequential decision making or multi-state decision-making problems in water resources. It can handle any kind of functional relationships in the model and can provide optimal solutions based on chosen interval values for the state and decision variables. The main applications in water resources are water allocation, reservoir operation, capacity expansion of water infrastructural facilities, water conveyance/shortest route-finding problems, etc. (Yakowitz 1982). Being a complete enumeration technique, the DP faces computational difficulties while solving large-size problems due to an increase in the number of state variables and the corresponding discrete states, since in the DP method, a linear increase in the number of state variables causes an exponential increase in computational time requirement. So, when DP is applied to larger-size problems, it has the main hurdle of the ‘curse of dimensionality’. The gradient-based NLP methods can solve problems with smooth nonlinear objectives and constraints. However, in large and highly nonlinear models, these algorithms may fail to find feasible solutions or converge to local optimum depending upon the degree of nonlinearity and an initial guess (Reddy & Kumar 2012). Hence, these traditional optimization techniques do not ensure global optimum and also have other limitations like requirements of objective functions to be continuous functions and easily differentiable, continuous variables, etc. Lack of ability to obtain a global optimum in the case of traditional nonlinear-optimization techniques and intensity of computational requirements in the case of dynamic programming motivated the search for new approaches, which would conglomerate efficiency and ability to find the global optimum.

In the recent past, nontraditional search and optimization methods based on natural and biological evolution, also called bio-inspired techniques, such as EA and swarm intelligence (SI) algorithms have been receiving increased attention in view of their potential as global optimization techniques for solving complex problems in water resources engineering (Reddy & Kumar 2012). Since, the first applications of genetic algorithms in the water resources area (McKinney & Lin 1994; Ritzel et al. 1994) and their acceptance as optimizers have increased tremendously for several practical applications in the water resources planning and management (Maier et al. 2014). However, there is a lack of synthesis between common algorithm challenges, common problem behaviors, and needed improvements for different key applications in the field. This paper will help the researchers to comprehend the algorithms and their applications in the planning and management of water resource systems. In the following first, the basic principles of EA and then some of the major types of EA are discussed.

The EAs are rapidly expanding in the area of artificial intelligence research. During the last two decades, there has been a growing interest in algorithms, which are based on the principle of natural evolution (i.e., the survival of the fittest) (Fogel et al. 1966). The EAs are the population-based random search techniques guided with some heuristics (also called as meta-heuristic techniques). The EAs consist of a population of individuals, each representing a search point in the space of feasible solutions and is exposed to a collective learning process which proceeds from generation to generation (Brownlee 2011). The population is randomly initialized and then subjected to the process of selection, recombination, and mutation through several generations, such that the newly created generations evolve towards more favorable regions of the search space. The progress in the search is achieved by evaluating the fitness of all individuals in the population, selecting the individuals with a better fitness value, and combining them to create new individuals with an increased likelihood of improved fitness. After some generations, the program converges, and the best individual represents the optimum (or near-optimum) solution. There exist several EAs, but the basic structure of any evolutionary algorithm is very much the same (Reddy & Kumar 2012). A sample structure is shown in Figure 2.

The key steps involved in EA include:

  • 1.

    Seeding the population using random generation

  • 2.

    Evaluate the fitness of each individual in the population

  • 3.

    Repeat the evolution steps until stopping criterion satisfied:

    • (a)

      Select the individuals for reproduction

    • (b)

      Perform genetic operations to generate the offspring

    • (c)

      Evaluate the individual fitness of the offspring

    • (d)

      Replace the least fit individuals with new best fit individuals

  • 4.

    Report the best solution of the fittest individual.

The two most important issues in the evolution process are population diversity and selective pressure. These factors are strongly related to each other, i.e., an increase in the selective pressure decreases the diversity of the population and vice versa. In other words, strong selective pressure ‘supports’ the premature convergence of the search and a weak selective pressure can make the search ineffective. Different evolutionary techniques use different scaling methods and different selection schemes (e.g., probabilistic or proportional selection, ranking, tournament) to strike a balance between these two issues. Furthermore, these algorithms can be easily combined with local search and other exact methods. In addition, it is often straightforward to incorporate domain knowledge in the evolutionary operators and in the seeding of the population. Moreover, EAs can handle problems with any combination of the challenges that may be encountered in real-world applications, such as local optima and multiple objectives. The popular EAs are such as genetic algorithm (GA), evolutionary strategies (ES), evolutionary programming (EP), differential evolution (DE), etc. Some studies also proposed hybrid systems by combining features of more than one EA, and they exhibited significant results in several water resources applications. In the following, a brief overview of popular EAs is given with an emphasis on showing how the various types of algorithms differ, and the stages involved in defining each one.

Genetic algorithms

GA is the most popular algorithm of EAs. GA was inspired by population genetics (like heredity and gene frequencies), and evolution at the population level besides the Mendelian understanding of the structure (like chromosomes, genes, alleles) and process/mechanisms (like recombination and mutation) (Fogel et al. 1966; Goldberg 1989). The basic GA was introduced by Holland (1975) based on the binary encoding of the solution parameters, utilizes multi-point crossover and bit-flip mutation for the evolution of the solutions. Later, several variants of GA (binary/real coding) with a variety of genetic operators were developed and used in various water resources applications. The working of GA involves random initialization of population (i.e., members are randomly generated to cover the entire search space uniformly) to start the process. Then, evaluation of the objective functions, selection of parents, and applying genetic operations, recombination operator for creation of offspring, and mutation operation for perturbing the individuals to produce a new population are conducted. The steps are repeated until a termination condition is satisfied. A sample pseudo-code of GA, describing the key steps in the algorithm, is depicted in Figure 3(a).

Recently, real-coded GA has been receiving more recognition and applications. There exist several variants of crossover operators (arithmetic, simulated binary crossover, Blend crossover, etc.) and mutation operators (e.g., Gaussian, polynomial, random mutation, etc.). The basic selection operator for GAs was proportional (or roulette-wheel) selection, but because of its known drawbacks of premature convergence to locally optimal solutions, tournament selection and ranking selection are commonly used nowadays. For numerical optimization, a real-coded GA with Gaussian mutation, arithmetic crossover, and tournament selection is a common choice. Moreover, an operation called elitism is remarkably important for the performance of a GA. The usage of elitism is to leave a certain proportion of the best individuals in every generation untouched by the variation operators. This is to some extent, similar to evolution strategies, where a population of parents generates a new offspring by a mutation in each iteration. The population of the next generation is created by selection from the elite parents and newly created offspring. Nicklow et al. (2010) presented an overview of the GA method and its applications in water resources management. By utilizing the strengths of EAs, quick convergence, and yielding efficient solutions for single-objective optimization, researchers also developed multi-objective algorithms by integrating Pareto optimality principles into single-objective genetic algorithms, such as Nondominated Sorting Genetic Algorithms (NSGA-II; Deb et al. 2002), Multi-objective Evolutionary Algorithm (MOEA; Reddy & Kumar 2006), etc. Reed et al. (2013) discussed the principles of different MOEAs methods and their applications in water resources.

Different variants of GA were developed over the years, like Micro GA (Krishnakumar 1990), Cellular GA (Manderick & Speissens 1989), NSGA (Srinivas & Deb 1994), Contextual GA (Rocha 1995), Grouping GA (Falkenauer 1996), Quantum-inspired GA (Narayanan & Moore 1996), Linkage learning GA (Harik 1997), Island GA (Whitley et al. 1998), NSGA-II (Deb et al. 2002), Interactive GA (Takagi 2001), Jumping gene GA (Man et al. 2004), Dynamic rule-based GA (He & Hui 2006), Hierarchical cellular GA (Janson et al. 2006), NSGAIII (Deb & Jain 2014), Tribe competition-based GA (Ma & Xia 2017), Fluid GA (Jafari-Marandi & Smith 2017), Block-based GA (Tseng et al. 2018), etc. Historical development of GA variants is also depicted in Figure 4. Although many of these variants use the same basic principles of natural selection and survival of the fittest, they engage different strategies and improved mechanisms in pursuit of better guidance of the search and aiding the enhanced convergence of the method. For example, for multi-objective optimization, the initial version of NSGA (proposed in 1994) was improved over the years and later proposed NSGA-II in 2002 and NSGAIII in 2014 by incorporating additional mechanisms (like nondominated sorting, crowding distance measures, etc.) to handle different issues and complexities of multi-objective optimization problems.

Differential evolution

Storn & Price (1996) proposed DE as a variant of EAs to achieve the goals of robustness in optimization and quick convergence to an optimal solution for numerical optimization. The DE contrasts from other EAs in the evolution process. Here mutation is the main operator, and the crossover is the secondary operator for the generation of new solutions (Reddy & Kumar 2012). After random initialization of the population, the objective functions are evaluated, and the following steps are repeated until a termination condition is satisfied. At each generation, two operators, namely mutation and crossover, are applied to each individual, to produce a new population. In DE, the mutation is the main operator, and each individual is updated using a weighted difference of a selected parent solution and crossover acts as background operator where the crossover is performed on each of the decision variables with a small probability. The offspring replaces the parent only if it improves the fitness value; otherwise, the parent is copied in the new population. The pseudo-code of the DE algorithm is given in Figure 3(b).

There are several variants of DE, depending on the number of weighted differences between solution vectors considered for perturbation, and the type of crossover operator (binary or exponential) used (Storn & Price 1997). For example, in DE/rand-to-best/1/bin variant of DE: perturbation is made with the vector difference of best vector of the previous generation (best) and current solution vector, plus single vector differences of two randomly chosen vectors (rand) among the population. The DE variant uses the binomial (bin) variant of crossover operator, where the crossover is performed on each of the decision variables whenever a randomly picked number between 0 and 1 is within the crossover constant (CR) value. More details of DE can be found elsewhere (Price et al. 2005; Das et al. 2016). To generate Pareto-optimal solutions for multi-objective problems, researchers also developed different variants of multi-objective differential evolution algorithms (MODE; Reddy & Kumar 2007c).

Different variants of DE were developed over the years, like DE (Storn & Price 1997), Self-adaptive DE (SaDE; Qin & Suganthan 2005), jDE (Brest et al. 2006), Chaotic DE (Wang & Zhang 2007), adaptive DE with optional external archive (JADE; Zhang et al. 2009), ensemble of mutation strategies DE (EPSDE; Mallipeddi et al. 2011), Composite DE (CoDE; Wang et al. 2011b), Multi-population DE (Yu & Zhang 2011), Adaptive Cauchy DE (ACDE; Choi et al. 2013), improved JADE (Yang et al. 2014), Extended adaptive Cauchy DE (Choi & Ahn 2014), jDErpo (Brest et al. 2014), Restart DE algorithm with Local search mutation (RDEL; Ali 2014), Colonial competitive DE (Ghasemi et al. 2016), Memory-based DE (Parouha & Das 2016), Stochastic Quasi-Gradient (SQG)-DE (Sala et al. 2017), Unified DE (UDE; Trivedi et al. 2017), Opposition-based Compound Sinusoidal DE (OCSinDE; Draa et al. 2019), etc. The evolution of various DE variants is also depicted in Figure 4. The different variants of DE use different improved strategies and self-adaptive schemes for enhancing the convergence and consistency in solutions for single (or multiple) objective optimization problems. More details can be found in the referred papers.

Evolutionary strategies

ES model evolution as a process of the adaptive behavior of the individual or in other words ES focus mutational transformations that maintain the behavioral linkage between each parent and its offspring, respectively, at the level of the individual (Rechenberg 1973; Fogel 1994). ES uses real variables and aims at numerical optimization. Because of that, the individuals incorporated could be a set of strategic parameters. ES rely mainly on the mutation operator (Gaussian noise with zero means). ES evolves by making a series of discrete adjustments (i.e., mutations) to an experimental structure. After each adjustment, the new structure, i.e., the offspring, is evaluated and compared to the previous structure, i.e., the parent. The better of the two is then chosen and used in the next cycle. As selection in this evolutionary cycle is made from one parent and one offspring, the algorithm is known as a ‘(1 + 1)’ ES.

These two-membered ES modify (i.e., mutate) an n-dimensional real-valued vector of object variables by adding a normally distributed random variable with expectation zero and standard deviation to each of the object variables xi. The standard deviation is the same for all components of x, i.e., , where x′ is the offspring of x and Ni(0; 1) is the realization of a normally distributed random variable with expectation 0 and standard deviation 1. Since the introduction of ES, two additional strategies have been developed: () and (). Both of these ES work on populations rather than single individuals and are referred to as multi-membered ES. A () ES creates off-springs from parents and selects the best individuals from the combined set of parents plus off-springs to make the next population. A () ES, on the other hand, creates off-springs and selects the best individuals from the off-springs alone (for ).

Different variants of ES were developed over the years, like Derandomized Self-adaptation ES (Ostermeier et al. 1994a), CSA-ES (Ostermeier et al. 1994b), CMA-ES (Hansen & Ostermeier 2001), Weighted multi-recombination ES (Arnold 2006), Meta-ES (Jung et al. 2007), Natural ES (Wierstra et al. 2008), Exponential natural ES (Glasmachers et al. 2010), Limited memory CMA-ES (Loshchilov 2014), Fitness inheritance CMA-ES (Liaw & Ting 2016), RS-CMSA ES (Ahrari et al. 2017), MA-ES (Beyer & Sendhoff 2017), Weighted ES (Akimoto et al. 2018), etc. The historical development of various ES variants is also depicted in Figure 4. The different variants use different strategies/adaptation schemes for better evolution and enhanced performance of the ES algorithm while solving a different kind of optimization problems. Apart from these, other EAs and their hybrid variants were proposed and used in solving water resources problems.

The other class of meta-heuristic techniques that are gaining more popularity in recent times for water resources optimization are SI techniques. The SI is based on the claims that intelligent human cognition derives from the interaction of individuals in a social environment. There exist several algorithms that use this socio-cognition, which can be used to solve different optimization tasks (Bonabeau et al. 1999). The individual members of a swarm act without supervision, and each of these members has a stochastic behavior due to their perception in the neighborhood. Swarms use their environment and resources effectively by collective group intelligence. The key characteristic of a swarm system is self-organization, which helps in evolving global level response by means of local-level interactions (Reddy 2009). The SI methods are also called behaviorally inspired algorithms.

The main algorithms that fall under SI algorithms include particle swarm optimization (PSO), artificial bee colony (ABC), ant colony optimization (ACO), honey-bee mating optimization (HBMO), firefly algorithms, etc. Similar to EAs, SI models are population-based iterative procedures. The system is randomly initialized with a population of individuals. These individuals are then manipulated and evolved over many iterations by way of mimicking the social behavior of insects or animals in an effort to find the optima. Unlike EAs, SI algorithms do not use evolutionary operators such as recombination and mutation. Basically, a potential solution flies through the search space by modifying itself according to its relationship with other individuals in the population and the environment (Reddy & Kumar 2012). The two algorithms that attracted the interest of many researchers and received wider applications in water resources are PSO and ACO for solving a variety of problems. PSO is based on the social behavior of fish schooling and bird flocking introduced by Eberhart & Kennedy (1995) and has received wider recognition for numerical optimization with continuous variables whereas ACO is basically inspired by the foraging search behavior of real ants and their ability to find shortest paths and was mainly used for discrete combinatorial optimization (Kennedy et al. 2001). In the following, a brief description of the basic principles and working of these two SI techniques is presented.

Particle swarm optimization

PSO algorithm proposed by Eberhart & Kennedy (1995) is a population-based meta-heuristic search technique that uses co-operative group intelligence concepts. Here the particle denotes individual in a swarm. Each particle in a swarm behaves in a distributed way using its own or cognitive intelligence and the collective or social (group) intelligence of the swarm. As such, if one particle discovers a good path to food, the rest of the swarm will also be able to follow the good path instantly even if their location is far away in the swarm. PSO shares many similarities with GA (Kumar & Reddy 2007). PSOs are initialized with a population of random solutions and searches for optima by updating iterations. However, in comparison to methods like GA, in PSO, no operators inspired by natural evolution are applied to extract a new generation of candidate solutions. Instead, PSO relies on the exchange of information between individuals (particles) of the population (swarm). In effect, each particle adjusts its trajectory towards its own previous best position and towards the best previous position attained by any other member of its neighborhood (usually the entire swarm) (Kennedy et al. 2001).

The PSO algorithm involves the following steps (Kumar & Reddy 2007): initialization of particles with a random position and velocity vectors. Then the fitness of each particle is evaluated by the fitness function. Two ‘best’ values are defined, the global and the personal bests. The global best is the highest fitness value in an entire iteration (best solution so far), and the personal best is the highest fitness value of a specific particle. Each particle is attracted to the location of the ‘best fitness achieved so far’ across the whole swarm. In order to achieve this, a particle stores the previously reached ‘best’ positions in a cognitive memory. The relative ‘pull’ of the global and the personal best is determined by the acceleration constants called social and cognitive parameters. After this update, each particle is then re-evaluated. If any fitness is greater than the global best, then the new position becomes the new global best. If the particle's fitness value is greater than the personal best, then the current value becomes the new personal best. This procedure is repeated until the termination criteria are satisfied. The pseudo-code of the PSO algorithm is given in Figure 5(a). Further to speed up the convergence and to enhance the reliability in optimal solutions, different studies suggested additional mechanisms, like elitist mutation strategy (Reddy & Kumar 2007a), combining PSO with other local search methods and applied for different kinds of problems in water resources. By utilizing the strengths like faster convergence and efficient optimal solutions for single-objective optimization, researchers also developed multi-objective SI algorithms by integrating nondominance principles into single-objective PSO, for example, elitist-mutated multi-objective PSO (EM-MOPSO; Reddy & Kumar 2007b), etc.

There were several variants of PSO algorithms, and their hybrid algorithms developed over the years, like Constricted PSO (Shi & Eberhart 1998), Adaptive PSO (Clerc1999), Discrete PSO (Clerc 2004), Elitist-mutated PSO (EMPSO) (Reddy 2006), EM-MOPSO (Reddy 2006), Dynamic niching PSO (Nickabadi et al. 2008), Adaptive PSO (Zhan et al. 2009), Co-evolutionary MOPSO (Goh et al. 2010), Self-adaptive learning PSO (Wang et al. 2011a), Multi-dimensional PSO (Kiranyaz et al. 2011), Hybrid niching PSO (NPSO; Li et al. 2012), Hybrid PSO-Harmony Search (PSO-HS; Li et al. 2012), Hybrid PSO-Firefly Algorithm (PSO-FFA; Bhushan & Pillai 2013), etc. The historical development of the PSO algorithm and its variants are depicted in Figure 6.

Ant colony optimization

The first ACO algorithm was inspired by the foraging behavior exhibited (pheromone trail laying and training behavior) by ant colonies in their search for food (Dorigo et al. 1991). ACO was developed as a population-based, heuristic search technique for the solution of difficult combinatorial and complex problems. The main features of the ACO algorithm are pheromone trail and heuristic information (Kumar & Reddy 2006). The working of the ACO algorithm involves the following phases. First, the system is randomly initialized with a population of individuals. These individuals are then manipulated over many iterations by using some guiding principles in their search, such as a probability function based on the relative weighting of pheromone intensity and heuristic information), in an effort to find the optima. At the end of each iteration, each of the ants adds pheromone to its path (set of selected options). The amount of pheromone added is proportional to the quality of the solution (for example, in the case of minimization problems, lower-cost solutions are better; hence they receive more pheromone). The pseudo-code of the ACO algorithm, depicting the key steps, is given in Figure 5(b).

An important characteristic of ACO one should be aware of is that it is a problem-dependent application. In order to adopt ACO for application to a particular problem, it requires representation of the problem as a graph or a similar structure easily covered by ants and assigning a heuristic preference to generated solutions at each time step. The ACO has many features, which are similar to that of GA (Dorigo & Stutzle 2004; Kumar & Reddy 2006): (a) both are population-based stochastic search techniques; (b) GA works on the principle of survival of the fittest, whereas ACO works on pheromone trail laying behavior of ant colonies; (c) GA uses crossover and mutation as prime operators in its evolution for next generation, whereas ACO uses pheromone trail and heuristic information; (d) in ACO algorithms, trial solutions are constructed incrementally based on the information contained in the environment and the solutions are improved by modifying the environment through a form of indirect communication called stigmergy, whereas in GA, the trial solutions are in the form of strings of genetic materials and new solutions are obtained through modification of the previous solutions.

There were several variants of ACO algorithms and their hybrid algorithms developed over the years, like Ant System (AS; Dorigo et al. 1996), Ant Colony System (ACS; Dorigo & Gambardella 1997), Ant NET (Di Caro & Dorigo1998), Max-Min AS (Stützle & Hoos 2000), Multiple ACS (Gambardella et al. 1999), Multi-Colony Ant Algorithms (Iredi et al. 2001), Population-based ACO for the dynamic environment (Guntsch & Middendorf 2002), ACO for WDNs (Maier et al. 2003), ACO for reservoir system (Reddy 2006), Beam-ACO (Blum 2008), hybrid genetic Simulated Annealing (SA) ACS-PSO (Chen & Chien 2011), Hybrid ACO–PSO (Huang et al. 2013), Parallel ACO (DeléVacq et al. 2013), Hybrid PSO-fuzzy ACO (Elloumi et al. 2014), etc. The evolution of ACO and its variants are also portrayed in Figure 6.

Other swarm-based meta-heuristics

The other meta-heuristic algorithms that were proposed and applied in various fields for optimization include Harmony Search (HS; Geem et al. 2001), Shuffled Frog Leaping Algorithm (SFLA; Eusuff & Lansey 2003), Honey-Bee Mating Optimization (HBMO; Abbass 2001), Glowworm Swarm Optimization (GSO; Krishnanand & Ghose 2006), Firefly algorithm (FFA, Yang 2007), ABC (Karaboga 2005), Cuckoo Search (CS, Yang & Deb 2010), Bat Algorithm (Yang 2010), Multi-colony Bacteria Foraging Optimization (MC-BFO; Chen et al. 2010), etc. Table 1 gives brief details of these SI-based meta-heuristic algorithms and their working principles. The evolution of these meta-heuristics over the years and their hybrid variants are also showed in Figure 6.

The performance of meta-heuristic search methods is generally influenced by the parameter of the algorithm. Similar to EAs, these SI algorithms are also quite sensitive to set-up parameters (Reddy 2009). So it is important to fine-tune the parameters for a particular problem of interest before actually applying the same to the problem (Reddy & Kumar 2012).

The EA and SI methods have emerged as a powerful tool for optimization and management of water resources problems. There are numerous applications of EAs for water-related problems, namely, reservoir operation, water distribution systems design, groundwater remediation, parameter estimation in hydrological modeling, watershed management, and fluvial systems, etc. Since there exist several thousands of papers on applications of these algorithms, here, some of the important applications in water resources are reviewed.

Applications in water distribution systems

WDS comprises a system of interconnected nodes, via pipes, supply sources, such as reservoirs, tanks, and a set of hydraulic control elements, such as pumps, valves, regulators, etc. The network of interconnected nodes, pipes, and other hydraulic control elements is collectively termed as a water distribution network (WDN). A typical WDN design is formulated as an optimization problem requiring minimization of cost, satisfying the minimum pressure and flow requirements at different nodes. A variety of EA were applied for design and rehabilitation of WDNs, like GA (Simpson et al. 1994; Mackle et al. 1995; Savic & Walters 1997; Dandy & Engelhardt 2001; Munavalli & Kumar 2003; van Zyl et al. 2004; Keedwell & Khu 2005; Vairavamoorthy & Ali 2005; Wu & Walski 2005; Rao & Salomons 2007; Kadu et al. 2008); Ant Colony optimization (ACO; Maier et al. 2003), Differential Evolution (DE) (Suribabu 2010; Vasan & Simonovic 2010; Sirsant & Reddy 2018), SFLA (Eusuff & Lansey 2003); Harmony search (HS) algorithm (Geem 2006), Tabu search (TS) algorithm (Cunha & Ribeiro 2004), Honey-bee mating optimization (HBMO; Mohan & Babu 2010) Cross entropy (CE) optimization (Shibu & Reddy 2014); Gravitational search algorithm (GSA, Fallah et al. 2019), etc. More details of these applications are given in Table 2.

Further, there were several studies that have used multi-objective EA for multi-objective optimization of WDNs. In order to ensure satisfactory performance of WDNs at different failure conditions, the objectives such as reliability, minimum surplus head, etc., are incorporated into the model in addition to the minimization of cost of the network. The reliability expressed as the performance of the network in terms of demand satisfaction considering these failure conditions. Failures can be hydraulic or mechanical; here, hydraulic failure occurs due to uncertainty in input parameters like nodal demands, and pipe roughness coefficients, whereas mechanical failure occurs due to failure of one or more components such as pipes, pumps, valves, etc. (Sirsant & Reddy 2018). Different reliability indicators are employed in different studies as the objective function to be maximized along with minimization of cost. Different variants of GA techniques were used for the multi-objective design of WDNs such as Multi-objective GA (MOGA; Halhal et al. 1997; Savic et al. 1997; Vamvakeridou-Lyroudia et al. 2005; Dandy & Engelhardt 2006), Strength Pareto Evolutionary Algorithm (SPEA; Cheung et al. 2003), Nondominated Sorting GA-II (NSGA-II; Prasad et al. 2004; Prasad & Park 2004; Farmani et al. 2005; Kapelan et al. 2005; Ostfeld & Salomons 2006; Jayaram & Srinivasan 2008; Prasad & Tanyimboh 2008; Creaco et al. 2014; Sirsant & Reddy 2020); Multi-objective PSO (Patil et al. 2020), etc. More details and discussion of these applications is given in Table 3.

Applications in urban drainage and sewer systems

Urban drainage and sewer systems need to be designed such that the required flow capacity is met at minimum cost. The consideration of networks where both stormwater and sewage are transported through the same channel makes the problem a little more complex. Various studies used different EAs for the design of urban drainage and sewer systems like GA (Walters & Lohbeck 1993; Walters & Smith 1995; Liang et al. 2004; Afshar et al. 2006; Guo et al. 2006) SA (Karovic & Mays 2014), TS (Liang et al. 2004), DE method (Yazdi 2018), etc. More details of these applications are explained in Table 4. In addition to carrying the required flow during normal conditions, the high and extreme flow conditions such as flooding overflow should be considered to make the system more robust to such situations. This calls for the need to perform the multi-objective design of these systems considering minimization of the flooding overflow volume or flood damage cost, in addition to the minimization of cost. Different MOEAs were engaged for solving these problems, such as NSGA-II (Barreto et al. 2006; Muleta & Boulos 2007; Barreto et al. 2010; Penn et al. 2013; Vojinovic et al. 2014; Yazdi et al. 2017a; Wang et al. 2018), Nondomination Sorting Differential Evolution (NSDE; Yazdi et al. 2017b), etc. The specific details of these applications are also given in Table 4.

Applications in reservoir operation and irrigation systems

Reservoirs and irrigation systems need to be operated in a cost-effective manner such that the deficits are minimum as well the benefits achieved in terms of minimum cost or maximum energy production. Thus, the optimization problem is formulated as determination of the optimal release or operating policies such that the deficits are minimum and the benefits are maximum. Several meta-heuristic techniques were applied to solve different problems (Rani & Moreira 2010). To solve single-objective problems, the techniques used include GA (Oliveira & Loucks 1997; Wardlaw & Sharif 1999; Nixon et al. 2001; Merabtene et al. 2002; Dessalegne et al. 2004; Raju & Kumar 2004; Kerachian & Karamouz 2006, 2007; Kumar et al. 2006; Kuo et al. 2006; Zahraie et al. 2008), Shuffled complex evolution (SCE) algorithm (Lerma et al. 2015), PSO (Kumar & Reddy 2007; Reddy & Kumar 2007a; Ghimire & Reddy 2013); for solving multi-objective problems, the MOGA (Reddy & Kumar 2006), elitist-mutated MOPSO (Reddy & Kumar 2007b, 2009), multi-objective DE (MODE, Reddy & Kumar 2007c, 2008; Raju et al. 2012), etc. More details of these applications are described in Table 5.

Applications in water supply and wastewater system

The water supply and wastewater systems are subjected to many dynamic loadings, such as rain, the release of stormwater from storage tanks, etc. These loadings need to be regulated efficiently such that the required flow quality and quantity levels are maintained at minimum costs. There were several studies that have used EAs for solving these problems, such as GA (Rauch & Harremoes 1999; Tsai & Chang 2001; Wang & Jamieson 2002; Lavric et al. 2005; Murthy & Vengal 2006; Montaseri et al. 2015; Swan et al. 2017; Raseman et al. 2020), SFLA (Chung & Lansey 2009), etc. Also, for considering different conflicting issues, such as minimizing the total system cost and satisfactory performance of the systems under different dynamic loadings or contaminant additions, multi-objective optimization models developed and engaged MOEAs to solve the problems like MOGA (Chen et al. 2003), NSGA-II (Guria et al. 2005; Yandamuri et al. 2006; Fu et al. 2008; Muschalla 2008), etc. Table 6 gives more details of the applications and findings of the studies.

Applications in watershed management and fluvial systems

Watershed management and planning require modeling the hydrologic and fluvial characteristics properly and efficiently, such that the required water management conditions can be achieved in a cost-effective manner. Thus, the various watershed management techniques, such as the design of detention systems, flood management practices, and other best management practices (BMPs), need to be designed considering cost minimization and system reliability and efficiency maximization. Several studies used EAs for solving the relevant optimization models like GA (Harrell & Ranjithan 2003; Muleta & Nicklow 2005; Artita et al. 2013; Aminjavaheri & Nazif 2018), DE (Hang & Chikamori 2017), PSO (Shamsudin et al. 2014), MOGA (Yeh & Labadie 1997; Perez-Pedini et al. 2005), NSGA-II (Lee et al. 2012; Karamouz & Nazif 2013; Aminjavaheri & Nazif 2018), etc. More details of the applications and their findings are given in Table 7.

Applications in parameter estimation of hydrological models

For simulating the various hydrological processes, there exist several hydrological models (namely, lumped, semi-distributed, distributed models). The model may consist of a large number of components with several parameters, so one has to choose the appropriate model depending on the purpose and accuracy of the model variables of interest. To represent and simulate various hydrological processes accurately, it may require proper calibration and validation of the model parameters using historical data. A typical calibration process thus requires minimization of the error between the model simulated and actual values of the hydrologic variables (e.g., runoff). Sometimes, the model responses vary and may be suitable for a particular event or application. Also, the different performance indices used for the calibration process may give different results. For improving the performance of the simulation models under different scenarios, multi-variate calibration may be needed, which can be done using multi-objective calibration. For calibration of the hydrological models, several studies used different EAs like GA (Wang 1991; Franchini & Galeati 1997; Liong et al. 2001; Zou & Lung 2004), SCE algorithm (Zhang et al. 2009; Tigkas et al. 2016; Krishnan et al. 2018; Shin & Choi 2018), MOPSO (Mostafaie et al. 2018), NSGA-II (Khu & Madsen 2005; Alamdari et al. 2017; Tian et al. 2019), AMALGAM (Koppa et al. 2019), etc. More details of EAs and MOEA applications for single and multi-objective parameter estimation of hydrological models are presented in Table 8.

Applications in groundwater remediation, groundwater systems monitoring network design

The purpose of the GW monitoring networks is to capture the information about the contamination of GW and its source/location. In order to design a cost-effective system, the number of monitoring wells should be minimum and should not be redundant while being able to capture the desired information about the contamination. Different types of EAs were used for solving the groundwater systems and monitoring network design models formulated with different complexities. The popular techniques used include GA (McKinney & Lin 1994; Ritzel et al. 1994; Chadalavada & Datta 2008), SA (Dougherty & Marryott 1991), DE (Alizadeh et al. 2018), Probabilistic Pareto GA (PPGA; Luo et al. 2016), NSGA-II (Tang et al. 2007), ε-multi-objective noisy memetic algorithm (ε-MONMA; Song et al. 2019), etc. More details of the applications of EAs for groundwater systems are given in Table 9.

Groundwater systems are redundantly subjected to various pollutants at different times and locations, which are dynamic in nature. In order to ensure that the required GW quality levels are maintained, observation, as well as pumping wells, need to be designed. The problem can thus be formulated as determining the location and number of wells as well as the required pumping rates at these wells such that the cost is minimum and the desired quality levels are maintained. Several studies used EAs for solving the groundwater remediation problems via the simulation–optimization framework such as GA (Huang & Mayer 1997; Wang & Zheng 1997; Sun & Zheng 1999; Smalley et al. 2000; Yoon & Shoemaker 2001; Zheng & Wang 2002; Babbar & Minsker 2006; Wu et al. 2006; Park et al. 2007; Bayer et al. 2008; Seyedpour et al. 2019), SA (Kobayashi et al. 2008); MOGA (Erickson et al. 2002; Mantoglou & Kourakos 2007; Singh et al. 2008), NSGA-II (Singh & Chakrabarty 2011; Ouyang et al. 2017), etc. More details of these EAs applications are also elaborated in Table 9.

Meta-heuristics for decision making?

The wide range of the above applications shows that meta-heuristics is an emerging research area for solving a variety of water resources problems. There is also a growing interest for integrated water resources management frameworks, where the conventional disciplinary boundaries in water resources need to be reconsidered, and future management frameworks have to address different complexities that may arise due to high nonlinearities, a wider range of uncertainties, integration of large system components, etc. These issues pose significant challenges and motivate the need for EAs applications to advance adaptive decision making under uncertainty. Also, it is important to identify the problem properties across the different water resources field domains (such as watersheds, surface water, groundwater, reservoirs, water supply, etc.) that are posing computational barriers for large-scale water resources systems. There is a greater need to engage meta-heuristics-based optimization frameworks for improved decision making in water resources management.

Advantages of EAs as compared with conventional methods of optimization

While applying for practical problems, the EAs offer several benefits that include conceptual simplicity, flexibility, and robust response to changing environments and their ability to self-adapt the search for optimum solutions on the evolution. EAs have broad applicability, since, with the same procedure, they can be applied to any problem that can be expressed as a function optimization task (e.g., discrete combinatorial problems, continuous-valued parameter optimization problems, mixed-integer problems, and others). In contrast, the conventional techniques might be applicable only to continuous values or other restrictions on constrained sets. Many times, the objective or response surfaces modeled in practical problems are often multi-modal, and the conventional gradient-based approaches rapidly converge to local optima, which may return unsatisfactory performance. For simpler problems, where the response surface is strongly convex, the EAs do not perform as well as traditional optimization methods (e.g., gradient-based methods are designed to take advantage of the convex property of surface). EAs offer significant advantages for real-world problems with multi-modal functions. Also, in case of applying LP method to problems with nonlinear objectives and/or constraints, which offers an almost certainly incorrect solution because of the simplifications or assumptions required for the technique. In contrast, EAs can directly incorporate any kind of arbitrary objectives and constraints. Also, the conventional methods of optimization are not robust to dynamic changes in the environment and often require a complete restart to provide a solution (such as DP technique). In contrast, EAs can be used to adjust solutions to changing environments; and the available population of evolved solutions offers a basis for further improvement, and in most cases, it is not needed to reinitialize the population at random. Thus, EAs proved to be effective, especially for problems that are intractable by classic methods of optimization, and for cases where heuristic solutions are not accessible or generally lead to unsatisfactory results.

Which evolutionary algorithm to use?

There were several variants of EAs. Each of the meta-heuristic algorithms has its own advantages and disadvantages. Among the nature-inspired EAs, GA is one of the oldest and popular techniques that has several applications in the water resources field. The main advantages include its ability to handle nonlinear, nonconvex, nondifferentiable functions, multi-modal solutions and can provide optimal or near-optimal solutions to a given problem. Real-coded GA has advantages over binary-coded GA for real-valued decision variable problems. However, GA requires the selection of appropriate genetic operators among several versions available and proper tuning of the parameters such as probabilities of crossover and mutation, population size, etc. Also, the optimization process may take higher computational time for complex water resources problems such as groundwater systems monitoring design and remediation, WDN problems, etc., as it may involve a time-consuming simulation–optimization process. The other popular algorithm, differential evolution, also has advantages similar to that of GA; apart from that, it is proved that DE has faster convergence and reliable, optimal solutions for numerical optimization. DE also has limitations in selecting an appropriate version of DE and algorithm parameters. Similarly, other EAs (like SCE, ES) are found to have similar capabilities and difficulties while solving water resources optimization problems. A brief summary of the comparison of basic characteristics and quality performance features of different EA (GA, DE, ES) are given in Table 10. However, the recent developments of self-adaptive EAs (e.g., SADE) are found to be overcoming this issue and helping the tuning of the algorithm parameters as the search progresses.

Which SI algorithm to use?

Several variants of the SI algorithms (like PSO, ACO, ABC, HBMO, FFA, BA, etc.) were proposed in different studies. The SI methods are basically inspired by co-operative group intelligence principles of the swarm and proved to be other classes of alternative meta-heuristic techniques for solving different kinds of optimization problems in water resources. Similar to EAs, SI algorithms are population-based random search techniques guided with some heuristics, which can also handle different types of problem complexities (like nonlinear, nonconvex, nondifferentiable functions, multi-modal solutions, etc.). But their applicability and convergence characteristics may vary from problem to problem. For example, PSO is a technique applicable for real-valued decision variables, has advantages of easiness in coding the algorithm, provides fast convergence to simple numerical problems, and requires low computational time. However, fine-tuning of algorithm parameters (e.g., inertia weight, social, and cognitive parameters) is required for getting optimal solutions and consistently good performance from the method. But, the basic PSO may face difficulties like premature convergence to locally optimal solutions for higher dimensional, large-scale water resources problems. Several studies proposed different strategies for the improvement of the PSO performance (like EMPSO proposed by Kumar & Reddy 2007). The ACO method is also a random search method guided by probabilistic rules and some heuristics (i.e., pheromone laying behavior of ants) and can produce global-optimal solutions. But ACO is applicable effectively only for a set of problems that involve decision variables with discrete values (for example, the WDNs problem can be represented in graphical form/shortest path that can be easily covered by the ants, and it can be modeled with pheromone trail laying behavior). There are several variants of ACO algorithms. It has similar advantages to other EAs in handling different kinds of complexities. However, the performance of ACO is sensitive to its algorithmic parameters, so it may require tuning of the parameters such as relative pheromone trail, heuristic information, evaporation rate, etc. while solving large-scale problems. Other SI algorithms (like ABC, HBMO, CS, FFA, BA algorithms) also have similar capabilities and limitations like that of PSO. A brief summary of basic characteristics and quality performance features a comparison of different EAs (PSO, ACO, and ABC) that are presented in Table 10.

It should also be noted that different studies have presented different algorithms for solving water problems and noted that some algorithms are relatively better than other methods. But, the inferences made may be applicable for those problems, which may not be generalized to one or not applicable for all types of problems, as each problem may have varying complexity depending on the dimension of the problem and existing relationships for the problem in hand. So, proper selection and usage of the algorithm for a given problem should be based on the problem type and its characteristics.

Hybrid meta-heuristic methods?

Research studies also reported that hybrid meta-heuristic methods that combine global-search with local search algorithms found to be providing improved performance for large-scale water resources optimization problems. Also, several strategies like problem-specific heuristics, chaotic concepts into the initialization of population, elitism concepts, self-adaptive schemes, etc., are incorporated into SI algorithms to improve the performance of the existing methods and to get consistent satisfactory results. Also, to handle the issues related to multi-objective optimization, different studies have suggested different schemes for getting an array of uniform widespread and true Pareto-optimal solutions for different kinds of practical problems.

There are several issues which require more research efforts to improve the solutions to the practical problems in the water resources engineering area. Some of these issues and future research directions are listed below.

Improvement in algorithm and solution methodology

There is a great deal of research on-going on the development of the new algorithms, a better understanding of the algorithm's performance and search behavior, analyzing the suitability in handling the different complexities of the problems, etc. As the meta-heuristic methods are population-based search algorithms and involve several parameters (like population size, maximum generations/iterations, other algorithm-specific parameters) to start the optimization procedure, and also the model performance is sensitive to the selected parameters. This warrants for proper fine-tuning of parameters before applying to a given problem. To address this issue of how to overcome the effect of the sensitivity of the parameters on the performance of the algorithm, recently, several studies explored self-adaptive schemes for overcoming this issue. For improving computational efficiency, studies also explored guided initialization of population, parallelization schemes (parallel processing that uses two more CPUs for computations saves time, which is well suited for population-based EA/SI methods that use simulation–optimization framework for solving complex real-world problems), the multi-algorithm search that combines two or more (global and/or local) search algorithms, etc. But still, there is a scope for improvements and more research is needed in this area. Moreover, it is important to develop knowledge related to the suitability of various optimization methods for a particular type of water resource problem and the incorporation of domain knowledge in the search, which can also help in improved solution methodology.

Search space reduction and improving computational efficiency

The complexity of the real-world problems (especially for problems in the areas of water distribution systems, groundwater systems management, calibration of distributed hydrological models) is increasing day-by-day, and it is expected to continue critical and promising research areas in the future also. Recently several studies are focusing on the issues of how to reduce the search space and guide the algorithms to speed up the convergence and/or improve the reliability in achieving global-optimal solutions. Schemes like fitness approximation, parallelization, and multi-algorithm search frameworks have been explored. But, still much more research is needed to come up with better strategies.

Model inputs and uncertainty

Since the model inputs may involve different kinds of uncertainty, there is a necessity to explore how to best represent various types of uncertainties (such as aleatoric and epistemic uncertainty) in the optimization model and incorporate them into decision making. Also, future uncertainties due to climate change, urbanization, etc., might affect the planning of water resources systems, and exploring new frameworks and evolving flexible design is one of the promising areas for future research.

Solution post-processing and decision making

The solutions obtained by solving an optimization model are sensitive to a given input variables condition. This poses doubts of how reliable are the solutions, considering the changing conditions or input variable values. In the case of multi-objective optimization (with more than two objectives), the MOEAs generate a large number of Pareto-optimal solutions. Then the challenge is how to use them for decision making, as it is difficult to visualize and choose the solutions. So, more efforts are needed to come up with effective decision-making procedures, which can help and equip the users in envisaging the trade-offs and assist in arriving at few practical alternatives and/or facilitate effective decision making under several conflicting goals.

Real-life water resource planning and management problems may involve several complexities and may pose several challenges for decision-makers. The use of evolutionary algorithms for solving numerical and practical optimization problems has become very attractive and extensive in the last three decades. Several new meta-heuristic algorithms including EA (such as GA, DE, SCE, etc.) and SI algorithms (such as PSO, ACO, ABC, SFL, HBMO, HS, FFA, CS, BA, etc.) are being proposed, which showed improved performance for solving a variety of water resources problems. The main advantage of EAs is the usage of a population of potential solutions that explore the search space simultaneously, exchanging information among them, and uses only objective function values. Also, EAs are stochastic search algorithms, can move to any complicated search apace and locate near-optimal (or optimal) solutions in reasonable computational time. They can provide solutions to any complex optimization problem that is difficult to be solved with the conventional NLP methods due to their nature that may imply discontinuities of the search space, nondifferentiable objective functions, imprecise arguments and function values.

Some of the remarks on the current state-of-the-art in EAs:

  • There is no general algorithm that can be applied efficiently to all problems, as the efficiency varies as a function of problem size and complexity. However, the incorporation of problem-specific knowledge and heuristics may help to achieve faster and efficient solutions to a real-world problem.

  • EAs may require calibration of the search parameters to ensure efficient convergence. The self-adaptive EA as part of improving the solution methodology can help to arrive quickly at near-optimal (or optimal) solutions, thereby helping water resources engineers in a better decision-making process.

  • For complex problems, EAs may require a large number of simulations to find optimal solutions. In such cases, the use of meta-modeling, fitness approximation, parallelization schemes may help to speed up the simulation–optimization process, thereby helping in improving computational efficiency.

  • The different types of problem complexities and their associated uncertainties motivate a great need for advances in multi-objective search, interactive optimization, and multi-algorithm search frameworks.

Therefore, EAs continue growing interest among the research community, and there is a tremendous potential for solving many practical water resources problems.

Authors acknowledge Ms Swati Sirsant (Civil Engineering, IIT Bombay) for assistance in compiling some of the literature used in the study.

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