The simulation model is the principal part of groundwater management model, since its solution is necessary in predicting the hydraulic response of aquifer system for different management strategies. The three-dimensional groundwater flow equation may be given as:

where is the hydraulic conductivity tensor [L·T^–1], h is the hydraulic head [L], is the specific storage [L^–1], t is time [T], W is the volumetric flux per unit volume (positive for inflow and negative for outflow) [T^–1], and are the Cartesian coordinates [L].

In this study, the computer model of MODFLOW [27] is used to simulate the groundwater flow process.

The optimization model is also absolutely necessarily for groundwater management models. In a groundwater optimization problem, the often-used objective is to maximize the total pumping or to minimize the total cost of capital, well drilling/installing and operating at a fixed demand. In this study, we use the minimization of total pumping cost as the objective of optimization model.

The objective function consists of capital cost, cost of well drilling/installing, and operating costs. Decision variables are pumping rates of candidate wells. The constraint set include some physical factors such as well capacities, hydraulic heads, or water demand requirements. The optimization model can be given as follows:

subject to,

where a₁ is the fixed capital cost per well in terms of dollars or other currency units [$], a₂ is the installation and drilling cost per unit depth of well bore [$/L], a₃ is the pumping costs per unit volume of flow [$/L³], y_i is a binary variable equal to either 1 if ith well is active or zero if ith well is inactive, d_i is the depth of well bore of ith

well [L], is the minimum hydraulic head value at ith well at time j [L], and are the ranges of allowable pumping rates for ith well at time j [L³·T^–1], is the water demand at time j [L³·T^–1], is the land surface elevation at ith well.

In this section, a modified Pareto dominance based real-coded genetic algorithm (mPRCGA) is proposed. The main features of mPRCGA are as: 1) vector combination of objective function and the total degree of constraint violation is preferred to weight combination; 2) Pareto intensity number is substituted for individual’s fitness; 3) real-coded representation is used in GA; 4) generalized generation gap model (G3 model) is adopted as the population-alternation model; 5) modified SPX operator is used as recombination operator. The details of mPRCGA are described and explained below.

Step 1: Problem initialization and setting mPRCGA parameters

Let be an objective function to be minimized, N be the number of decision variables, be the ith decision variable to be determined, be the vector, and T is the transpose operator. Based on these definitions, the mathematical optimization problems can be stated as follows:

subject to (4)

where and are lower and upper bounds of the decision variables. In addition, there are M constraints including inequality constraints (g₁) and equality constraints (g₂) in the constrained optimization problem:

where q is the number of inequality constraints and M-q is the number of equality constraints.

To solve this optimization problem using mPRCGA, the constraints in Equation (5) should be converted into objective function. Vector combination of objective function and the total degree of constraint violation is used as follows:

, subject to (6)

where is the vector composed of and. is the total degree of constraint violation, and can be obtained according to Equation (7) and Equation (8).

where is weighing of jth constraint, is the degree of jth constraint violation.

In this step the parameter sets of mPRCGA should also be defined: n_pop (population size), Iter_max (maximum generation), (parameters for G3 model), (expanding rate in SPX operator), c (parameter for Gaussian mutation in modified SPX operator).

Step 2: Generation of initial population

Make n_pop real-number vectors randomly and let them be an initial population P_t (t = 0).

Step 3: Individual ranking in population

As shown in Equation (6), the objective function is not a scalar but a vector. Thus, Pareto dominance is used to compare the vector [28] . On the basis of the vector comparison, Pareto intensity number is adopted to rank the individual in the population. The Pareto intensity number can be obtained as follows:

where SI(i) is the Pareto intensity number of ith individual in the population, P_t is the population in generation t, means Pareto dominate, # is cardinality of the set.

Step 4: Population improvement and updating

Population-alteration models and recombination operators are of great significance to real-coded GAs’ performance. Generalized Generation Gap model (G3 model) is modified from MGG model and it is more computationally faster by replacing the roulette-wheel selection with a block selection of the best two solutions [29] [30] .

UNDX and SPX are the most commonly used recombination operators. The UNDX operator uses multiple parents and Gaussian mutation to create offspring solutions around the center of mass of these parents. A small probability is assigned to solutions away from the center of mass. On the other hand, the SPX operator assigns a uniform probability distribution for creating offspring in a restricted search space around the region marked by the parents.

A modified SPX operator below is the combination of UNDX and SPX and can overcome some of their shortcomings. For simplicity, considering a 3-parent SPX in a two dimensional searching space as shown in Figure 1, where, and are parent vectors, is the center of the three parents. The inner triangle is formed by the three parent vectors firstly, then to be expanded to form the outer triangle. The vertex (of a triangle) is calculated as follows:

where is the expanding rate. Thus a simplex is accomplished.

Then, the Gaussian mutation borrowed from UNDX operator is performed as follows:

where is the unit coordinate vectors; r is the mean value of distances between each parent vector and center; is zero-mean normally distributed variables with variance. Zhou et al. (2003) suggested and observed that c = 1 to 1.3 performed well.

In Step 4, G3 model is employed as the main process, and the modified SPX is embedded and used as a sub process. Detailed process is as follows:

4a : Select μ(= n + 1) parents (best parent and μ-1 other parents randomly) from population P_t; Repeat (4b) times.

4b: Modified SPX procedure

4b.1: From the chosen parents to compute their center;

4b.2: Construct a simplex spanned by the chosen parents and its center;

4b.3: Select a point randomly in the spanned simplex;

4b.4: Perform Gaussian mutation at point to create an offspring.

4c: Choose two parents randomly from population P_t;

4d: Combine the randomly selected two parents ( 4c ) and created offspring (4b) to form a population S;

4e: Rank individual of population S, choose the best two individuals;

4f : Replace the chosen two parents ( 4c ) with these two individuals to update P_t.

Step 5: Repeat the above procedure from Step 3 to Step 4 until a certain stop criteria is satisfied.

The example is to deal with the minimization of pumping cost from an unconfined aquifer system and it is assumed that the numbers and locations of the candidate wells are known. This example was previously solved using DDP (Differential Dynamic Programming) by Jones et al. (1987), GA and SA by Wang and Zheng (1998), and HS (Harmony Search algorithm) by Ayvaz (2009). Figure 2 shows the plan view of the aquifer system under consideration.

Groundwater is pumped from an unconfined aquifer with a hydraulic conductivity of 86.4 m /day and specific yield of 0.1. As can be seen from Figure 2, boundary conditions of the aquifer include the Dirichlet boundary at the north and no-flow at the other sides. The distance between land surface and aquifer bottom is 150 m . The flow model is transient; it is assumed that initial hydraulic head value is 100 m everywhere.

The total management period is one year, which is divided into four stress periods of 91.25 days each. There are eight candidate pumping wells, and the water demands for each period are 130,000, 145,000, 150,000, and 130,000 m ³ /day, respectively. The hydraulic head must above zero (bottom) anywhere in the aquifer, and each pumping rates must be in the range of 0 to 30,000 m ³ /day. The objective function to be minimized is in the form of Equation (2) with T = 4. Note the first two terms in Equation (2) is neglected and Equation (2) is reduced to the last term.

Using the parameter sets given in Table 1, the optimum pumping rates and total cost has been solved through the proposed mPRCGA based management model. Table 2 summarizes the results of mPRCGA as well as other studies.