Application of Ant Colony Optimization in Water Resource Management

Abstract

The use of optimization techniques in water resource management has received increasing attention in the past few decades. The main objective of optimization in water resource management is to maximize the use of the available resources while minimizing the negative impacts on the environment. Two main approaches are widely used in optimization, including global and local optimization. The main aim of the chapter is to explore the application of the Ant Colony Optimization (ACO) algorithm in water resource management. It is a flexible and powerful tool that can handle various types of hydrological models. The chapter will use different performance measures and case studies to comprehensively compare these techniques in water resource management. The results will be valuable for practitioners in choosing the appropriate optimization technique for their specific application.

1. Introduction

1.1 Background on water resource management

Water is a valuable and limited resource that requires careful management to ensure its sustainable use [1]. Water resource management (WRM) involves modeling and optimization of surface water and groundwater systems to maximize water availability [2], while also minimizing conflicts between users and preserving the ecological balance [3]. The optimization process is critical in WRM because it allows decision-makers to make informed decisions based on accurate predictions and analyzes [3].

Water resource modeling is a complex process that involves the use of mathematical models to simulate the behavior of the water system [4]. These models require the estimation of parameters, such as hydraulic conductivity, recharge rate, and storage capacity [5]. The calibration of these models is essential to ensure the accuracy of the model outputs [6]. The calibration process involves adjusting the model parameters to minimize the differences between the observed and simulated data [7]. The optimization of these parameters is an important step in the model calibration process [8].

2. Optimization in water resource management

2.1 Definition of optimization

Optimization is the process of finding the best solution among a set of feasible solutions to a problem [9]. In the context of water resource management, optimization refers to the process of finding the best allocation of water resources to meet the conflicting demands of different stakeholders while considering the physical and environmental constraints of the system.

2.2 Importance of optimization in water resource management

Optimization is a crucial process in water resource management, and the choice of optimization method should be based on the complexity of the problem and the resources available [10]. Both global and local optimization methods have their advantages and disadvantages, and it is important to carefully consider these factors when choosing an optimization method for water resource management problems.

2.3 Types of optimization techniques

There are two main types of optimization methods: global optimization and local optimization. Global optimization methods aim to find the global optimum solution, which is the best possible solution that can be obtained within the search space [11]. These methods typically involve exploring the entire search space and can be computationally expensive [11, 12]. However, they are well suited for finding the global optimal solution to complex water resource management problems.

On the other hand, local optimization methods focus on finding the best solution in a specific region of the search space [13]. These methods are typically faster than global optimization methods and can be used to refine the solution found by global optimization methods. Local optimization methods are particularly useful when the search space is large, and the objective function is complex.

The choice between global and local optimization methods depends on the nature of the problem and the computational resources available. For instance, when dealing with a large water resource management system with many stakeholders and complex physical and environmental constraints, global optimization methods may be necessary to ensure that the solution found is the best possible one. However, when dealing with a smaller system, local optimization methods may be more appropriate since they can find a good solution more quickly.

3. Global optimization vs. local optimization

Global optimization and local optimization are two distinct approaches to solving optimization problems in water resource management.

3.1 Definition of global optimization

Global optimization is the process of finding the best solution within the entire search space, regardless of the starting point. It involves searching the entire solution space to identify the global optimum, which is the best solution possible. In water resource management, global optimization is used to find the best solution to a problem that may have multiple optimal solutions or where the objective function is complex.

3.2 Definition of local optimization

Local optimization, on the other hand, involves searching for the best solution in the vicinity of a given starting point. It focuses on finding the local optimum, which is the best solution that can be found within a small region of the solution space. Local optimization is used in situations where the objective function is relatively simple and the search space is small.

3.3 Differences between global and local optimization

One of the main differences between global and local optimization is the scope of the search. Global optimization involves searching the entire solution space, while local optimization only searches a small region of the solution space. Another key difference is the starting point. Global optimization does not require a starting point, whereas local optimization requires a starting point from which to begin the search.

3.4 Advantages and disadvantages of each approach

Global optimization has several advantages, including the ability to find the best possible solution, even in complex problems with multiple optimal solutions. It also has the advantage of not being affected by the starting point, making it more robust and reliable. However, global optimization can be computationally expensive and time-consuming, particularly for complex problems.

Local optimization, on the other hand, is faster and more efficient than global optimization, making it suitable for problems with simple objective functions and small search spaces. It also requires less computational resources than global optimization. However, local optimization can be affected by the starting point and may not find the global optimum.

The choice between global and local optimization in water resource management depends on the complexity of the problem, the size of the search space, and the resources available. Global optimization is suitable for complex problems with multiple optimal solutions, while local optimization is suitable for simpler problems with smaller search spaces. Both approaches have their advantages and disadvantages, and the choice between them should be based on the specific requirements of the problem at hand.

4. Applications of global optimization in water resource management

4.1 Global optimization techniques

Global optimization techniques are widely used in water resource management to find the optimal parameter set for models that involve a high degree of nonlinearity. These techniques seek to minimize the objective function by searching for a global minimum in the parameter space.

4.1.1 Ant colony optimization (ACO)

Ant colony optimization (ACO) is a metaheuristic optimization technique based on the foraging behavior of ants in nature [14]. In recent years, ACO has gained increasing popularity in solving optimization problems in various fields, including hydrology, particularly in groundwater flow and solute transport [15]. ACO has been used to solve inverse problems, which have wide applications in groundwater management, contaminant remediation, and environmental monitoring [16].

The main idea behind ACO is to simulate the behavior of ants that deposit pheromone trails as they search for food [17]. In ACO, the optimization problem is modeled as a graph, where the nodes represent the possible solutions, and the edges represent the quality of the solutions [18]. The ants follow a probabilistic decision-making process based on the pheromone trail [19] and the quality of the solutions represented by the edges [20]. The more frequently a solution is visited, the higher the pheromone concentration on the corresponding edge, and the higher the probability of choosing that solution [21].

4.1.2 Ant colony optimization in inverse modeling

The ACO algorithm is used to find the global minimum of the objective function in inverse modeling problems [22]. In the context of groundwater flow and solute transport, the objective function is typically a measure of the difference between observed and simulated data [23]. ACO is well suited to these problems because it can efficiently search a large parameter space and avoid being trapped in local optima [24, 25].

4.1.3 Application of ACO to water resource management problems

It was noted that the ACO approach has only been utilized since 2001 in the field of water resources [26]. ACO algorithms have been shown to be particularly effective for a range of optimization problems, including complex nonlinear systems [27]. Some studies have utilized ACO algorithms in the field of groundwater management and remediation [28]. For example, Li et al. [29] applied an ACO algorithm for the optimization of spatial sampling in long-term groundwater monitoring networks by reducing the number of monitoring wells while maintaining an adequate representation of groundwater conditions. Wang et al. [30] used ACO and numerical modeling to optimize remediation strategies and determine optimal locations for groundwater remedial systems.

Water distribution systems and reservoir optimization are the primary areas where ACO has been predominantly utilized. For example, Jalali et al. [31] and Kumar and Reddy [32] used the ACO algorithm to optimize reservoir operation. Jalali et al. [31] proposed an improved ACO algorithm that uses a new heuristic function and a new pheromone updating rule to improve the convergence speed and the quality of the solution. The proposed improved ACO algorithm is tested on a case study of the Hirakud reservoir in India, and the results show that it outperforms the traditional ACO algorithm and other optimization algorithms in terms of solution quality and computational efficiency. Kumar and Reddy [32] ACO technique to derive operating policies for a multi-purpose reservoir system.

Many studies utilized ACO in Water Distribution Systems. Simpson et al. [33] proposed methodology can be used to improve the performance of ACO in the optimal design of water distribution systems by selecting the parameters of ACO, including the number of ants, the heuristic information, and the pheromone evaporation rate. The proposed method tested on a case study of a water distribution system in Singapore. The results showed that the selected parameters affected the performance of the ACO algorithm. After that, Maier et al. [16] proposed modified ACO algorithm can be used to improve the performance of ACO in the design of water distribution systems. The modified ACO algorithm uses a new pheromone, a new heuristic function and updating rule to enhance the quality of the solution and the convergence speed. The proposed algorithm tested on the same case study of a water distribution system in Singapore [33]. The results proved that ACO is a promising approach for the design of water distribution systems and can be used to generate optimal designs that meet the required performance criteria. El-Ghandour and Elbeltagi [34] compare the performance of Five algorithms (GA, particle swarm optimization (PSO), DE, Artificial Bee Colony (ABC), and ACO) in terms of solution quality and computational efficiency using a case study of a water distribution network in Iran. The results showed that DE and ACO are promising approaches for the optimization of water distribution networks and can be used to generate optimal designs that meet the required performance criteria.

Ant colony optimization (ACO) has been applied in various hydrological problems, including groundwater flow and solute transport. For example, ACO is used to estimate groundwater model parameters, resulting in more accurate and reliable groundwater flow and transport predictions. Li et al. [35] proposed a hybrid ACO system that combines the traditional ACO algorithm with a local search algorithm. The hybrid system employs the ACO algorithm to generate a set of candidate solutions and the Simulated Annealing (SA) algorithm to refine the solutions by accepting some of the worst solutions with a certain probability. The proposed hybrid ACO system is tested on a case study of a groundwater system in China. The proposed hybrid ACO system is a promising approach for parameter estimation in groundwater hydrology and can lead to more accurate and reliable predictions of groundwater flow and transport. Irani and Nasimi [36] proposed ACO algorithm used to improve the performance of neural networks in the estimation of the permeability of a reservoir. The proposed new method is called ACA-BP, which combines the ACO algorithm with the Backpropagation (BP) algorithm. The method was tested on a case study of a reservoir in Iran. The proposed ACA-BP method is a promising approach for the estimation of the permeability of a reservoir and can be used to generate accurate and reliable predictions of the performance of neural networks in the estimation of the permeability of a reservoir. Dobre and Drobot [37] proposed ACO-Artificial Neural Network (ANN) method can be used to improve the accuracy and efficiency of the estimation of soil hydraulic parameters, which are important for the prediction of soil water movement and plant growth. A new proposed method called ACO-ANN, which combines the ACO algorithm with the Artificial Neural Network (ANN) to improve the accuracy and efficiency of the estimation process. The proposed method is tested on a case study of soil in Romania.

Ghorbani et al. [38] used ACO algorithm to solve the fuzzy optimization problem for river water quality management. The proposed model handled multiple objectives and multiple pollutants in a river system. The proposed model is tested on a case study of Sefid-Roud River in Iran and uses a fuzzy optimization model to determine the optimal waste load allocation for the pollutants in the river. The proposed model could effectively allocate pollutant loads, leading to improved river water quality. Guo et al. [39] applied ACO algorithm to synthetic and field 3D resistivity imaging examples. In both synthetic and field, the a priori constrained ACO method obtained better models with higher resolution and accuracy. The a priori constraints were based on geological knowledge of the geological feasibility, survey area, and enforce physical on the resistivity models. The constraints applied on thickness of geological layers, resistivity range, and layer sequence. The results showed that incorporating prior geological constraints into the ACO algorithm can significantly improve imaging quality and feasibility for 3D electrical resistivity imaging applications.

The ACO method developed by Jamei et al. [40] was utilized as one of the techniques to assess the salinity of multi-aquifers in Bangladesh’s coastal regions. The ACO model used a new hybrid neuro-computing approach that combined the ACO algorithm with the Adaptive Neuro-Fuzzy Inference System (ANFIS) and the Slime Mould Algorithm (SMA) to estimate the spread of salinity in Bangladesh’s coastal aquifers. The new hybrid neuro-computing approach demonstrates a promising approach for assessing groundwater salinity distribution and can be utilized to provide accurate and reliable projections of the salinity trend in Bangladesh’s coastal groundwater. Ahmed et al. [41] developed two novel deep learning (DDDL) algorithms for forecasting river water levels: DDDL-ACO and DDDL-PSO. These approaches optimize the structure of the deep neural network using ant colony optimization (ACO) and particle swarm optimization (PSO), respectively. The ACO feature selection technique was utilized to determine the most important input variables from climatic indices, satellite data, and ground-based data. The fuzzy multi-objective model introduced by Ghorbani et al. [42] in conjunction with the ACO algorithm is an excellent decision-support tool for reducing numerous contaminants in rivers. The ACO algorithm iteratively seeks optimal pollution reduction solutions that meet the model’s objectives. The study discovers that the ACO algorithm may generate a collection of Pareto optimal solutions that represent the trade-offs between the objectives. The ACO algorithm outperforms other evolutionary algorithms in terms of convergence speed, solution variety, and computational efficiency. It is capable of dealing with the uncertainties and complexities of real-world river pollution concerns. For the bivariate simulation of river flows in the subbasins of Lake Urmia, Eslamitabar et al. [43] suggest the hybrid intelligence models ANFIS-ACO and ANFIS-PSO. These correspondingly combine ant colony optimization (ACO) and particle swarm optimization (PSO) with an adaptive neuro-fuzzy inference system (ANFIS). The ANFIS model’s parameters are optimized using the ACO and PSO algorithms. They look for the ideal parameter setting that reduces the discrepancy between the simulated and actual flow data. According to the study, the ANFIS-ACO model outperforms the ANFIS-PSO model in terms of accuracy and simulation error for both low and high river flows. The ANFIS-ACO model performs better than the study’s individual and hybrid models as well. This shows that PSO and ACO are both capable of optimizing the parameters of the ANFIS model.

Another application of ACO in Optimal Crop and Irrigation Water Allocation, Nguyen et al. [44], offered an enhanced ant ACO formulation for the allocation of crops and water to different irrigation areas as another use of ACO. The improved ACO formulation is compared with that of other ACO algorithm variants (without and with domain knowledge) for two case studies, including one from the literature and one introduced in the paper for different water availability scenarios within an irrigation district located in Loxton, South Australia, near the River Murray. The revised ACO formulation with domain knowledge is a viable technique for optimal crop and irrigation water allocation problems, particularly those requiring more complex crop models than those utilized in the case studies.

4.1.4 Advantages of ACO

Ant colony optimization (ACO) has several advantages over other optimization methods. One of the main advantages is that it can efficiently explore a large parameter space without getting trapped in local optima. This is particularly useful in inverse modeling problems where there are often multiple local optima. Another advantage is that ACO can handle discrete and continuous parameters, which makes it useful for problems with mixed variables. Additionally, ACO has been found to be computationally efficient and scalable.

4.1.5 Limitations of ACO

Despite its advantages, ACO also has some limitations. One of the main limitations is that it requires the definition of a pheromone update rule. The pheromone update rule is critical for the algorithm’s performance, but it can be difficult to determine the optimal rule for a particular problem. Additionally, ACO is prone to premature convergence, which can limit its ability to explore the parameter space.

4.1.6 Conclusion

Ant colony optimization (ACO) is a promising and effective optimization tool for inverse modeling of groundwater flow and solute transport. It has been found to be effective in finding the global minimum of the objective function in complex problems. Its ability to efficiently search large solution spaces and its adaptability to different problem types make it a valuable tool for hydrogeologists and other researchers in related fields. However, careful consideration must be given to the selection and tuning of its parameters, and researchers should be aware of its limitations and explore hybrid and modified versions of the algorithm to improve its performance in specific applications. Overall, ACO offers a powerful and efficient alternative to traditional optimization methods for solving complex inverse modeling problems. Its ability to search the parameter space efficiently and effectively makes it a suitable option for optimizing complex and computationally expensive hydrological models. However, further research is needed to fully understand its capabilities and limitations and to explore its potential applications in other areas of hydrological modeling. In addition, further research is required to address the limitations of the algorithm and optimize its performance to fully understand its capabilities and limitations in different contexts. Moreover, the effectiveness of ACO can be affected by various factors, such as the choice of parameters and the complexity of the model.

4.2 ACO algorithms’ fundamental mathematical equations

These mathematical expressions aid in the understanding of the fundamental optimization principles behind techniques such as ACO algorithm. Specific formulations then differ depending on the algorithms and applications.

The basic mathematical equations for ACO algorithm is provided here:

4.2.1 ACO

Global optimization technique inspired on ant foraging behavior [45].

$$P_{i,j}=\frac{\left({\tau }_{i,j}^{\alpha }\right)\left({\eta }_{i,j}^{\beta }\right)}{\sum \left({\tau }_{i,j}^{\alpha }\right)\left({\eta }_{i,j}^{\beta }\right)}$$E1

where

$P_{i,j}$ is the probability of choosing path j from i.

${\tau }_{i,j}^{\alpha }$ is the pheromone trail concentration.

${\eta }_{i,j}^{\beta }$ is the heuristic value (inverse of distance).

α and β are parameters to control influences for ${\tau }_{i,j}^{\alpha }$ and ${\eta }_{i,j}^{\beta .}$

4.2.2 Local pheromone update

After an ant traverses an edge (i, j), the pheromone level on that edge is locally updated as follows:

$${\tau }_{i,j}=\left(1-\rho \right){\tau }_{i,j}+\rho ∆{\tau }_{i,j}$$E2

where $\rho$ is the pheromone evaporation rate, and $∆{\tau }_{i,j}$represents the amount of pheromone deposited by the ant.

4.2.3 Global pheromone update

After all ants complete their tours, the pheromone level on the edges is globally updated based on the quality of the solutions found. The update is performed as follows [46]:

$${\tau }_{i,j}=\left(1-\rho \right){\tau }_{i,j}+\sum \left(∆{\tau }_{i,j}^k\right)$$E3

where $∆{\tau }_{i,j}^k$ represents the amount of pheromone deposited by ant k on edge (i, j).

4.3 Recommendations

The optimization approach used should be determined by the characteristics of the water resource problem, such as model type, data, objective function, restrictions, and uncertainty. The problem complexity, initial parameter estimations, objective function, computing demand, desired optimality, trade-off analysis, and hybrid approaches should all be taken into account while selecting the ACO algorithm for managing water resources. If these parameters are carefully taken into account, the ACO algorithm can offer practical answers to challenging optimization issues in the management of water resources. Here are some suggestions based on various viewpoints:

4.3.1 Problem complexity

Ant colony optimization (ACO) may not be appropriate for issues with a very wide or dynamic search space, or for situations that demand accurate answers. ACO, on the other hand, can handle complicated problems with discrete or continuous variables, nonlinearities, numerous objectives, and restrictions.

4.3.2 Initial parameter estimates

If initial parameter estimations are far from ideal, it is more likely to find the genuine optimum. ACO can explore the search space and adapt to the problem’s features, therefore, it does not need much prior information or initial parameter estimates. Nevertheless, ACO would need some tuning of its parameters, such as the number of ants and the relative importance of pheromones and heuristic data.

4.3.3 Objective function

Ant colony optimization (ACO) may collaborate with any objective function that assesses the quality of a solution. ACO could work better with objective functions that are smooth, continuous, and differentiable, though, since they can provide the ants more information to help them with their search. In order to capture various parts of the issue, ACO may potentially profit from the use of several objective functions.

4.3.4 Computational demand

Ant colony optimization (ACO) is a population-based algorithm. ACO could also be computationally costly because it takes several iterations and evaluations to converge to a satisfactory solution. ACO is a relatively computationally demanding algorithm. The computational demand can be reduced by using parallelization techniques and distributed computing. If the pheromone trails become too dominant or too weak, ACO may potentially experience stagnation or premature convergence.

4.3.5 Desired optimality

Ant colony optimization (ACO) is a metaheuristic algorithm that can discover approximations of solutions that are sufficient for use in real-world situations. ACO does not promise to identify the global or even a local optimum. ACO could also have trouble identifying other options or escape from local optima. It is not a given that ACO will identify a problem’s ideal resolution. ACO, however, frequently finds effective solutions to issues that other optimization methods find challenging. ACO is not always the best algorithm. However, it frequently comes up with effective answers to issues with water resource management.

4.3.6 Trade-off analysis

Think about the trade-offs that exist between solution quality and computing efficiency. ACO is appropriate for multi-objective optimization or issues with complicated trade-offs because it strikes a solid balance between exploration and exploitation. To select the best method, weigh the trade-offs between solution quality, computational efficiency, and resource availability. When managing water resources, ACO may be used to undertake trade-off analyzes between several goals. ACO, for instance, may be used to identify a solution that maximizes water supply while minimizing the cost of water treatment. ACO can be used to balance computing demand with solution optimality. This makes it a suitable alternative for cases where finding a decent solution fast is critical. ACO can be used to trade-off multiple goals, such as lowering costs while enhancing water quality. As a result, ACO is a useful technique for water resource management problems with various objectives.

4.3.7 Hybrid methods

Ant colony optimization (ACO) can be hybridized or combined with other methods to enhance its efficacy and flexibility. To improve the solutions discovered by the ants, for instance, ACO can employ local search or other heuristics. It can also use additional metaheuristics or evolutionary algorithms to diversify the search or improve convergence. For particular water resource management issues, hybrid techniques can improve performance by combining the advantages of several algorithms. These frequently offer greater performance than relying just on ACO. As a result, it is a flexible tool that may be used to a variety of water resource management issues.

Ant colony optimization (ACO) has various disadvantages, such as convergence to the best solution can be slow, especially for big and complicated problems. Furthermore, performance is dependent on parameter adjustment, which might be problem-specific. ACO might be a fantastic starting point for finding high-quality solutions in a timely manner. Global optimization techniques can be used to refine solutions and strive for the accurate global optimum if more accurate optimization is required.

Finally, in water resource management, the decision between ACO and global optimization is determined by the specific issue being handled and the computational resources available. ACO can be a strong and adaptable strategy to tackling complicated water resource management problems, although global optimization approaches may be better suited to situations requiring the global minimum of a complex objective function. The issue complexity, computing demand, intended optimality, and trade-off analysis should all be considered while selecting an algorithm such as ACO as global optimization approaches is well suited for solving complex and nonlinear water resource problems with many objectives, constraints, and uncertainties.