_{1}

^{*}

This study proposes a groundwater management model in which the solution is performed through a combined simulation-optimization model. In the proposed model, a modular three-dimensional finite difference groundwater flow model, MODFLOW is used as simulation model. This model is then integrated with an optimization model, in which a modified Pareto dominance based Real-Coded Genetic Algorithm (mPRCGA) is adopted. The performance of the proposed mPRCGA based management model is tested on a hypothetical numerical example. The results indicate that the proposed mPRCGA based management model is an effective way to obtain good optimum management strategy and may be used to solve other type of groundwater simulation-optimization problems.

Groundwater is a vital resource throughout the world. Nowadays, with increasing population and living standards, there is a growing need for the utilization of groundwater resources. Unfortunately, the quantity and quality of groundwater resources continues to decrease due to population growth, unplanned urbanization, industrialization, and agricultural activities. Therefore, sustainable management strategies need to be developed for the optimal management of groundwater resources [

Groundwater management models are widely used to determine the optimum management strategy by integrating optimization models with simulation models, which predict the groundwater system response [

Many researchers have adopted non-heuristic optimization approaches in conjunction with groundwater simulation models to solve groundwater management problems [

Groundwater management problems are commonly nonlinear and non-convex mathematical programming problems [

Many studies deal with groundwater management problems using genetic algorithms. Mckinney and Lin (1994) integrated GA based optimization model with a groundwater simulation model programming to solve three management problems (maximum pumping problem, minimum cost pumping problem, and pump-and- treat design problem) [

But similar to other heuristic optimization approaches, GAs are also unconstrained search technology and lack a clear mechanism for constraint handling [

In trying to solve COPs using GA or other optimization methods, penalty function methods have been the most popular approach [

Thus, many researchers have developed sophisticated penalty functions or proposed other various constraint handling techniques over the past decade. Relevant methods proposed for constraint handling for heuristic optimization approaches can be categorized into: 1) penalty function methods; 2) methods based on preserving feasibility of solutions; 3) methods which make a clear distinction between feasible and infeasible solutions; and 4) hybrid methods [

Among these constraint handling techniques, methods based on multi-objective concepts have attracted increasing attention. Deb (2000) introduced a constraint handling method that requires no penalty parameters, this method used the following criteria: 1) any feasible solution is preferred to any infeasible solution; 2) between two feasible solutions, the one with better objective function value is preferred; and 3) between two infeasible solutions, the one with smaller degree of constraint violation is preferred [

However, it is worth noting that the newly-defined multi-objective problem (MOP), which is transformed from single objective COP, is in nature different from the customary MOP. That is, the philosophy of customary MOP is to obtain a final population with a diversity of non-dominated individuals, whereas the newly-defined MOP would retrogress to a single objective optimization problem within the feasible region [

In this study, methods based on multi-objective concepts are utilized to handle the constraints in groundwater management models. We firstly adopt multi-objective concept to transform single objective COPs to bi-objec- tive optimization problems. Next, Pareto dominance is introduced for comparison of vectors and then individual’s Pareto intensity number is used to substitute for fitness value in GA. Furthermore, generalized generation gap model and a modified SPX operator are utilized to increase the performance of real-coded genetic algorithm (RCGA).

The remaining of this paper is organized as follows: firstly, the formulation of groundwater management model (simulation model and optimization model) is described; secondly, a modified Pareto based Real-Coded Genetic Algorithm (mPRCGA) with generalized generation gap model and a modified SPX operator is proposed; thirdly, performance of the proposed mPRCGA based management model is tested on a hypothetical example.

The main purpose of groundwater management model is to determine an optimal management strategy that maximizes the hydraulic, economic, or environmental benefits. Two sets of variables (decision variables and state variables) are involved, and the management strategies are usually constrained by some physical factors including well capacities, hydraulic heads, or water demand requirements. A groundwater management model is coupled with two main parts: simulation model and optimization model.

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 ^{–1}], h is the hydraulic head [L], ^{–1}], t is time [T], W is the volumetric flux per unit volume (positive for inflow and negative for outflow) [T^{–1}], and

In this study, the computer model of MODFLOW [

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 a_{1} is the fixed capital cost per well in terms of dollars or other currency units [$], a_{2} is the installation and drilling cost per unit depth of well bore [$/L], a_{3} is the pumping costs per unit volume of flow [$/L^{3}], 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], ^{3}·T^{–1}], ^{3}·T^{–1}],

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

where _{1}) and equality constraints (g_{2}) 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:

where

where

In this step the parameter sets of mPRCGA should also be defined: n_{pop} (population size), Iter_{max} (maximum generation),

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 [

where SI(i) is the Pareto intensity number of ith individual in the population, P_{t} is the population in generation t,

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 [

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

where

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

where

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)

4b: Modified SPX procedure

4b.1: From the chosen

4b.2: Construct a simplex spanned by the chosen

4b.3: Select a point

4b.4: Perform Gaussian mutation at point

4c: Choose two parents randomly from population P_{t};

4d: Combine the randomly selected two parents ( 4c ) and

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 performance of the mPRCGA based management model is tested on a hypothetical example considering multiple management periods.

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).

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

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