The work described in the previous sections clearly establishes the importance of accounting for model uncertainty in groundwater management studies and demonstrates the potential benefits of time-varying groundwater management policies. There are, however, other factors that are important and that pose significant obstacles to designing groundwater management strategies. An example is the solution of nonlinear groundwater management problems with discrete decision variables. The techniques described earlier employ gradient-based algorithms that require the objective and constraints to be continuous and differentiable functions of the decision variables. However, certain aspects of groundwater management modeling present difficulties because they are discrete functions of the decision variables. An example is pump-and-treat aquifer remediation design, which is a nonlinear optimization problem because predicted concentrations vary nonlinearly with the pumping rates. In this case, it is very difficult to include discrete decisions, such as well start-up costs, in the management model formulation. It is also difficult to include alternative technologies, such as slurry walls, because the design options are described by a discrete number of configurations. Some recent studies have presented new optimization technologies for solving groundwater management problems with discrete (or discretized) decision variables.
Dougherty and Marryott [1991] present a simulated annealing algorithm to solve groundwater management models formulated as combinatorial optimization problems. Annealing is a process by which a solid's temperature is raised to bring its molecules into a highly mobile state, and then cooled to form a low-energy crystalline structure. The simulated annealing algorithm recasts the groundwater management objective (e.g. minimization of cost) as the energy of the system and the management design (array of discrete decision variables) as the state of the system. The goal is to slowly `cool' the system until the lowest cost management design is achieved. Dougherty and Marryott present a series of examples that demonstrate the simulated annealing algorithm applied to a variety of groundwater management problems---aquifer dewatering, aquifer remediation via pumping/injection, and aquifer remediation via pumping/injection and slurry wall. In each case, the objective is to minimize costs, with the hydraulic and water quality goals incorporated into the objective via penalty terms. The well pumping rates and slurry wall design are defined as discrete decision variables, and the pumping costs, well installation costs, and slurry wall costs are defined as functions of these discrete variables. The penalty `cost' is also a function of these discrete variables and is determined through the aquifer simulation model. The results suggest that the simulated annealing algorithm is a potentially valuable tool for solving groundwater management problems cast in combinatorial form.
McKinney and Lin [1994] and McKinney et al. [1994] present a genetic algorithm approach to solving groundwater management problems with discrete decision variables. The genetic algorithm [ Goldberg, 1989] uses a scheme of directed random search modeled after natural selection. A management strategy is encoded as a string of binary digits, which in the genetic algorithm is analogous to a `creature.' The genetic algorithm models the evolution of a population of `creatures' through successive generations using three probabilistic operators: reproduction, crossover, and mutation. The reproduction process serves to retain those strategies with high fitness (favorable objective), the cross over operator seeks to improve the design by combining the high-fitness strategies, and the mutation operator protects against convergence to a local minimum. The effect of these operators is that designs with high fitness (favorable objective) will persist as the management strategies evolve from one generation to the next. As with the simulated annealing approach, the hydraulic and concentration constraints are incorporated into the objective via penalty functions. McKinney and Lin [1994] demonstrate the methodology for a variety of hypothetical examples and compare the genetic algorithm solution with a linear or nonlinear programming solution. In each case the genetic algorithm solution compares favorably with the solution found using the `traditional' methods.
Rogers and Dowla [1994] combine artificial neural networks with a genetic algorithm to design remediation strategies for a hypothetical aquifer. As mentioned earlier, the artificial neural network (ANN) [ Simpson, 1990] is a `mapping' system that `learns' the relationship between the input signal (e.g. vector of pumping rates) and the output signal (e.g. simulation model predictions). Rogers and Dowla use the ANN to predict the ability of alternative pumping strategies to meet concentration constraints. They then use a genetic algorithm to search through the possible pumping scenarios to identify the optimal strategy. The authors present two examples of pump-and-treat remediation design in which they compare the neural network/genetic algorithm solution with a nonlinear programming solution, and in each case the solutions compare favorably. It should be noted that the neural network approach presented by Rogers and Dowla bears some similarity to the multiple regression approach [see Alley, 1986] in which a regression model is developed to `map' the output (model predictions) as functions of the input (pumping rates). The regression equations are then included as part of an optimization model to determine optimal management strategies.
Karatzas and Pinder [1993] address the problem of groundwater management with fixed charges for well installation. However, rather then solve the problem in its combinatorial form, they incorporate the fixed charges into the objective function using an exponential penalty term. The penalty term takes the value of the fixed charge if pumping is nonzero and takes the value zero otherwise. To solve the problem, the authors introduce the outer approximation method, which is applicable to the global minimization of a concave function---the case that results from incorporating fixed charges in an exponential form. Karatzas and Pinder demonstrate the method for two groundwater management problems---linearly constrained aquifer dewatering with fixed charges and nonlinearly constrained aquifer remediation with fixed charges.
In all of the previously cited studies, candidate pumping well locations are fixed and are assumed to be coincident with nodes of a numerical model. Wang and Ahlfeld [1994] present a groundwater management model that identifies optimal well locations and well rates by defining the spatial coordinates of wells as decision variables to be determined along with the pumping rates. Their method uses a composite objective that minimizes pumping and also constrains the wells to be within a predefined subdomain of the model. Using a hypothetical aquifer system, they study the problem of groundwater remediation design with the goal of identifying the minimum pumping remediation strategy. For their examples, the `moving well' model outperformed the `fixed well' model both in terms of total pumping and number of wells needed.