Chance-constrained multi-objective optimization of groundwater remediation design at DNAPLs-contaminated sites using a multi-algorithm genetically adaptive method

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Highlights

  • A multi-algorithm genetically adaptive multi-objective method is proposed as a multi-objective optimization solver.

  • Multi-gene genetic programming is a promising approach for surrogate modeling.

  • Chance-constrained programming can successfully handle the surrogate-modeling uncertainty.

Abstract

In this paper, a multi-algorithm genetically adaptive multi-objective (AMALGAM) method is proposed as a multi-objective optimization solver. It was implemented in the multi-objective optimization of a groundwater remediation design at sites contaminated by dense non-aqueous phase liquids. In this study, there were two objectives: minimization of the total remediation cost, and minimization of the remediation time. A non-dominated sorting genetic algorithm II (NSGA-II) was adopted to compare with the proposed method. For efficiency, the time-consuming surfactant-enhanced aquifer remediation simulation model was replaced by a surrogate model constructed by a multi-gene genetic programming (MGGP) technique. Similarly, two other surrogate modeling methods—support vector regression (SVR) and Kriging (KRG)—were employed to make comparisons with MGGP. In addition, the surrogate-modeling uncertainty was incorporated in the optimization model by chance-constrained programming (CCP). The results showed that, for the problem considered in this study, (1) the solutions obtained by AMALGAM incurred less remediation cost and required less time than those of NSGA-II, indicating that AMALGAM outperformed NSGA-II. It was additionally shown that (2) the MGGP surrogate model was more accurate than SVR and KRG; and (3) the remediation cost and time increased with the confidence level, which can enable decision makers to make a suitable choice by considering the given budget, remediation time, and reliability.

Introduction

Nowadays, dense nonaqueous-phase liquids (DNAPLs) are frequently detected in groundwater throughout the world on account of the widespread use, improper disposal, accidental spillage, and leakage of petrochemical products (Kueper and Mcworter, 1991). Because of their low solubility, low mobility, and high density in water, DNAPLs may remain in aquifers for a long time. Consequently, they may ultimately become a long-term continuous source of groundwater contamination (Qin et al., 2007). Surfactant-enhanced aquifer remediation (SEAR) is a promising approach to removing DNAPLs from the aquifer (Schaerlaekens et al., 2005). By adding surfactants to the water, the solubility and mobility of DNAPLs in the aquifer can be increased (Delshad et al., 1996), thus making SEAR more efficient than the conventional pump-and-treat technique. However, the cost of the SEAR process is expensive; therefore, the selection of an optimal design that is economical and effective is very valuable. Simulation-optimization techniques have been extensively used for solving such problems (Jiang et al., 2015, Luo et al., 2013).

Previous studies (e.g., Qin et al., 2007, Luo and Lu, 2014, Chu and Lu, 2015, Jiang et al., 2015) aimed to achieve a single objective, that is, minimization of the remediation cost. However, groundwater remediation and management problems typically have multiple conflicting objectives and require the use of a multi-objective optimization technique for appropriate designs (Singh and Chakrabarty, 2011). Instead of a single optimal solution, the multi-objective optimization technique generates a set of Pareto-optimal solutions. A Pareto-optimal solution is one in which an objective cannot be further improved without causing a simultaneous degradation in at least one other objective (Vrugt, 2016). Accordingly, they represent globally optimal solutions to the trade-off problem.

Schaerlaekens et al., 2005, Schaerlaekens et al., 2006 used the shuffled complex evolution (SCE) to minimize the remediation cost and remaining pollutants. Ren and Minsker (2005) applied an elitist non-dominated sorting genetic algorithm II (NSGA-II) to minimize two objectives, specifically, the remediation cost and health risk. Singh and Chakrabarty (2011) adopted NSGA-II to simultaneously minimize the remediation cost and required time. Meenal and Eldho (2014) implemented particle swarm optimization (PSO) to identify optimal groundwater remediation designs under two respective objectives: to simultaneously minimize the cost and time, and to simultaneously minimize the cost and number of wells. According to the above studies, many algorithms exist for solving multi-objective optimization problems. However, it is impossible to choose a single algorithm that is always efficient for a variety of optimization problems.

Recently, Vrugt and Robinson (2007) proposed a multi-algorithm genetically adaptive multi-objective (AMALGAM) method that merges the strengths of the best available individual optimization algorithms, including NSGA-II, PSO, adaptive metropolis search (AMS), and differential evolution (DE) (Zhang et al., 2010). The efficacy of their algorithm was demonstrated by several standard test functions. Zhang et al. (2010) adopted the AMALGAM method to calibrate the SWAT model. They compared it with the strength Pareto evolutionary algorithm 2 (SPEA2) and NSGA-II. The results showed that AMALGAM consistently produced competitive or superior results compared with the other two methods. However, the adoption of AMALGAM in multi-objective optimization problems of groundwater remediation is limited.

When using a simulation-optimization technique, the numerical simulation model is called thousands of times before the optimal design is obtained. This makes the technique computationally cumbersome and time consuming. The common practice is to construct a surrogate model (also known as a meta-model or proxy model), which rapidly runs, to replace the simulation model. Kriging (KRG), artificial neural network (ANN), support vector regression (SVR), and radial basis function (RBF) are common approaches for constructing surrogate models. More recently, multi-gene genetic programming (MGGP) (Hinchliffe et al., 1996, Searson et al., 2007) was designed to produce an input–output relationship of a system. The approach has garnered considerable attention from researchers in a broad range of fields (e.g., Pan et al., 2013, Pandey et al., 2015, Mohammadzadeh et al., 2016). The main advantage of MGGP is its ability to develop a compact and explicit prediction equation in terms of different model variables without assuming the prior form of the existing relationships (Muduli and Das, 2015). However, the use of MGGP to construct a surrogate model for replacing the simulation model in simulation-optimization techniques is limited.

Nevertheless, despite how well the surrogate model fits the simulation model, residuals remain that engender uncertainty in surrogate modeling. This uncertainty, even if minimal, may cause the obtained optimal solutions to deviate from the “true” ones of the problems to a large extent (He et al., 2010). The chance-constrained programming (CCP) technique (Charnes and Cooper, 1963), a type of stochastic programming, is a widely used approach to considering uncertainties in the optimization model. However, although CCP has been used by many researchers for groundwater remediation problems (e.g., He et al., 2010, Singh and Chakrabarty, 2011), its application for handling surrogate-modeling uncertainty in multi-objective optimization remains limited.

To address the above concerns, this study was conducted to identify optimal designs for DNAPLs-contaminated sites using a chance-constrained multi-objective optimization model under the surrogate-modeling uncertainty. The primary contributions of this paper are outlined as follows. 1) An AMALGAM method, which is used to solve the multi-objective optimization model problem of groundwater remediation, is proposed. 2) A recently developed form of artificial intelligence, MGGP, was adopted to construct the surrogate model of the groundwater remediation simulation model in the simulation-optimization technique. 3) The CCP technique was implemented to handle the surrogate-modeling uncertainty in the multi-objective optimization model.

Section snippets

AMALGAM method

The AMALGAM method is combined with two concepts: simultaneous multi-algorithm search, and self-adaptive offspring creation. It thereby ensures a fast, reliable, and computationally efficient solution to multi-objective optimization problems (Vrugt and Robinson, 2007). The method starts with a random initial population P0 of size N by using Latin hypercube sampling (LHS) (Iman, 2008). Then, each individual in P0 is assigned an initial rank by the fast nondominated sorting (FNS) algorithm (Deb

Site overview

The application of the proposed approach was analyzed on a hypothetical perchloroethylene (PCE) contaminated site. UTCHEM (the Center for Petroleum and Geosystems Engineering, 2000) software developed by the University of Texas was used to simulate the SEAR processes. The studied site was a three-dimensional domain with a horizontal area of 100 × 70 m2 and a depth of 20 m. The simulation domain consisted of 20 layers; each layer was discretized into 50 × 30 grid blocks. Each grid block had the

Surrogate model accuracy analysis

Three statistical metrics—R2, root mean square error (RMSE), and relative error (RE)—were used to assess the performance of the three surrogate models, MGGP, SVR and KRG. R2 was used to demonstrate how well the surrogate fit with the simulation model (the higher the value is, the better the fit is). RMSE and RE were used as indicators of the surrogate model's precision (the smaller the value is, the more accurate the surrogate model is). These metrics can be written as:R2=1i=1nyiy^i2i=1nyiy

Conclusions

The AMALGAM method, a multi-objective optimization solver, was implemented in the multi-objective optimization of a groundwater remediation design at a PCE-contaminated site. The most widely used NSGA-II method was adopted to compare with AMALGAM. To save time, the SEAR process simulation model was replaced by the surrogate model constructed by the MGGP technique. Similarly, two other surrogate modeling methods, SVR and KRG, were employed to make comparisons with MGGP. Moreover, the

Acknowledgments

This work was supported by the National Natural Science Foundation of China (NO. 41372237 and 41502221). The authors thank the editor and anonymous reviewers for their insightful comments and suggestions.

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