In recent years, great progress has been made in optimization technology, and some heuristic solutions with global optimization characteristics have been applied to groundwater management, such as genetic algorithm, simulated annealing algorithm, artificial neural network algorithm, tabu search algorithm and some hybrid intelligent algorithms.
1.2.3. 1 genetic algorithm
Genetic algorithm is a new optimization method founded by Holland and others in the early 1970s and developed by Goldberg [54]. When solving the groundwater management model, genetic algorithm does not require the groundwater system to be linear, so it is more suitable for solving the management problem of complex groundwater system. At present, genetic algorithm has been applied to various fields of groundwater management at home and abroad.
McKinney et al. [55] solved three groundwater management problems with genetic algorithm: the maximum pumping capacity of aquifer, the minimum pumping cost and the minimum cost of aquifer restoration; Katsifarakis et al [56, 57] combined boundary element method with genetic algorithm to find the optimal solutions of three kinds of groundwater flow and solute transport problems, namely, determining hydraulic conductivity, minimizing pumping cost and hydrodynamic control of pollution plume; Morshed et al [58] summarized the application of genetic algorithm in groundwater management and put forward some improved methods. Cai et al. [59] combined genetic algorithm with linear programming to solve large-scale nonlinear water resources management model. Firstly, genetic algorithm is used to identify complex variables, and when these variables are unchanged, the problem tends to be linear, and then linear programming is used to solve the water resources management model in sections. Zheng et al [60] used genetic algorithm to solve the optimal design model of groundwater remediation system established by response matrix method; Ines et al [6 1] combined remote sensing and genetic algorithm to optimize water management in irrigation areas. In recent years, Shao et al. [62], a domestic scholar, took the nonlinear groundwater management model of Yangzhuang Basin in Shandong Province as an example and introduced the concrete steps of applying genetic algorithm to solve this kind of problem. Cui Yali et al. [63] established a groundwater management model with the goal of maximizing the total pumping capacity of three water sources in Yangzhuang Basin, Shandong Province, and solved it with genetic algorithm.
It should be pointed out that genetic algorithm is an approximate algorithm and a global optimization algorithm, and its convergence speed and solution accuracy are controlled by the selection of some parameters of the algorithm. For large-scale and multivariable groundwater management problems, its convergence speed is slow and its calculation time is long, which is the deficiency of genetic algorithm in solving complex groundwater management models.
1.2.3.2 Simulated Annealing Algorithm (SAA)
Simulated annealing algorithm is an extension of local search algorithm. Different from local search algorithm, it selects the state with better objective function value in the neighborhood with a certain probability. Theoretically speaking, it is a global optimization algorithm, which opens up a new way to solve optimization problems by simulating the similarity between the annealing process of metal substances and the solving process of optimization problems [64]. Simulated annealing algorithm has been applied to the field of groundwater management.
Wang et al. [65] solved the groundwater management model with genetic algorithm and simulated annealing algorithm respectively, and compared the results with those of linear programming, nonlinear programming and differential dynamic programming, and evaluated the advantages and disadvantages of the two algorithms. Dougherty et al [66] introduced the application of simulated annealing algorithm in groundwater management. Rizzo et al [67] used simulated annealing algorithm to solve the management problem of multi-stage groundwater remediation, and applied a value function to speed up the search speed of the algorithm. Cunha [68] solved the problem of groundwater management with simulated annealing algorithm, so that the water supply equipment can be selected under the condition of meeting the demand, and the total installation cost and operation cost can be minimized. Kuo et al. [69] proposed an agricultural water resources management model based on field irrigation system and simulated annealing algorithm. Rao et al. [70,765,438+0] established groundwater flow and solute transport model with SEAWAT, and solved groundwater management problem with simulated annealing algorithm.
The experimental performance of simulated annealing algorithm has the advantages of high quality, strong initial robustness, strong universality and easy implementation, but in order to find the optimal solution, simulated annealing algorithm often needs a long optimization time, which is also the biggest shortcoming of this algorithm [72].
artificial neural network
Artificial neural network algorithm is a new subject, which has developed rapidly since the basic concept was put forward in the 1940s. Artificial neural network method belongs to lumped parameter model, which is an intelligent bionic model to simulate the working mode of human brain, and can process information in parallel on a large scale. It has the ability of self-organization, self-adaptation and self-learning, and has the characteristics of nonlinearity and non-locality. Moreover, he is good at association, generalization, analogy and reasoning, and can analyze and extract practical statistical laws from a large number of statistical data [73].
In groundwater management, due to the incompleteness of hydrogeological data caused by the variability of data, the uncertainty of parameters and the spatial variability of aquifer properties, some accurate analysis methods have great limitations in expressing the nonlinear relationship between various parts of groundwater resources system. The introduction of artificial neural network technology has greatly promoted the application research of groundwater management model. 1992, in his doctoral thesis, Rogers first proposed to optimize groundwater management by using artificial neural network technology, and used genetic algorithm for model training and identification. Since then, some scholars have done a lot of research in this field. Ranjithan et al [74] used artificial neural network model to optimize groundwater recharge scheme under the condition of uncertain permeability coefficient; Coppola et al. [75] successfully applied artificial neural network to three kinds of groundwater prediction problems, and solved the complex groundwater management problems; Parida et al [76] used artificial neural network to predict runoff coefficient in water resources management.
It should be emphasized that ANN model is not a true description of nonlinear process and can not reflect the real structure of the system, so it can not completely replace the mechanism model of the system. When establishing various groundwater nonlinear system management models, we must first consider the essence of artificial neural network model. At present, the application and research of artificial neural network technology in the research of groundwater resources management in China are still relatively few, especially the comprehensive application of artificial neural network technology in groundwater resources management, which is far behind foreign countries.
1.2.3.4 tabu search algorithm (TSA)
The idea of gradually optimizing tabu search algorithm was first put forward by Glover [77]. It is an extension of local neighborhood search, a global algorithm and a simulation of human intelligent process. Tabu search algorithm avoids circuitous search by introducing flexible storage structure and corresponding taboo criteria, and forgives some taboo good states by flouting criteria, thus ensuring diversified and effective exploration and finally realizing global optimization.
Zheng et al. [78] combined tabu search algorithm and linear programming method to solve the design problem of groundwater pollution restoration, mainly applying the advantages of tabu search (more effective in optimizing discrete well locations) and linear programming (more effective in optimizing continuous pumping capacity); Zheng et al [79] used tabu search algorithm and simulated annealing algorithm to determine the optimal parameter structure, and evaluated and compared the effectiveness and flexibility of the two methods. Lee et al [80] made an empirical comparison of eight heuristic algorithms for solving nonlinear integer programming problems. The application results in monitoring network design show that simulated annealing algorithm and tabu search algorithm are outstanding; Yang Yun and Wu Jianfeng [8 1] applied tabu search algorithm and genetic algorithm to solve groundwater management model respectively, and the results showed that tabu search was more effective than genetic algorithm.
Tabu search algorithm has a strong dependence on the initial solution. A good initial solution can make tabu search find a good solution in the solution space, while a poor initial solution will slow down the convergence speed of tabu search. Whether tabu search can be well applied to practical problems should fully consider the influence of initial solution on optimization results, which needs further study. In addition, the iterative search process is serial, only moving in a single state, rather than parallel search, which makes the optimization time of the algorithm often longer. In order to improve the efficiency of optimization, the current trend is to combine tabu search with other heuristic methods, such as combining tabu search algorithm with genetic algorithm [82, 83].
1.2.3.5 hybrid intelligent algorithm
Simulated annealing genetic algorithm is an optimization algorithm which combines genetic algorithm and simulated annealing algorithm. Sidiropoulos et al. [84] used simulated annealing algorithm and genetic algorithm to study the groundwater management problem with the goal of minimum pumping cost, and finally put forward a more effective solution method of groundwater management model-simulated annealing genetic algorithm; Shieh et al [85] applied simulated annealing genetic algorithm to optimize the design of in-situ bioremediation system; Han et al [86] used simulated annealing genetic algorithm to study the optimal allocation of water resources in Shiyang River Basin. Pan Lin [87] and others applied simulated annealing genetic algorithm to optimize the allocation of irrigation water in irrigation areas; Wu Jianfeng et al. [88] solved the groundwater management model by using genetic algorithm and simulated annealing penalty function method, and successfully applied this method to the evaluation and management model of groundwater resources in Xuzhou, with satisfactory results. Simulated annealing genetic algorithm not only overcomes the shortcomings of gradient-based optimization algorithm, but also ensures that the optimal solution (or near optimal solution) of the problem can be obtained effectively through simulated annealing process. However, how to reduce the population size and effectively improve the optimization speed of genetic algorithm in solving the problem of large-scale and multi-decision groundwater optimal management needs further study.
There are also many studies on the combination of artificial neural network algorithm and genetic algorithm to solve groundwater management model. Rogers et al. [89] used artificial neural network algorithm and genetic algorithm to design the optimization scheme of groundwater remediation, and used artificial neural network to predict the simulation results of water flow and solute; Aly et al. [90] put forward the optimal design methods of aquifer purification system under uncertain conditions-artificial neural network algorithm and genetic algorithm; Brian [9 1] and others combined genetic algorithm with artificial neural network algorithm to solve the water quality management problem of aquifer system with linear objective function, and compared this method with the traditional algorithm based on gradient function.
In addition, some other hybrid algorithms are often applied to groundwater management. Tung et al [92] studied the management of groundwater exploitation by combining pattern classification with tabu search algorithm. Hsiao et al. [93] applied the hybrid algorithm of genetic algorithm and constrained differential dynamic programming to solve the optimization problem of phreatic aquifer restoration. Mantawy et al [94] combined genetic algorithm, tabu search algorithm and simulated annealing algorithm to solve the unit transportation problem. The core of the algorithm is genetic algorithm, which generates new population through tabu search and accelerates the convergence speed through simulated annealing.
There are many methods to solve the groundwater management model. In addition to the optimization algorithm mentioned in this paper, intelligent methods developed rapidly in recent years, such as chaotic optimization algorithm and ant colony algorithm, provide new ideas for solving this problem. The groundwater resource system itself is a highly complex nonlinear system, and its functions and functions are various and multi-level. The input of the model is deterministic and random. Therefore, in order to realize more scientific and effective management of groundwater, the solution method of groundwater management model must be more accurate and practical.