An Elitist Multiobjective Tabu Search for Optimal ... - Wiley Online Library

An Elitist Multiobjective Tabu Search for Optimal Design of Groundwater Remediation Systems by Yun Yang1 , Jianfeng Wu2 , Jinguo Wang3 , and Zhifang Zhou3

Abstract This study presents a new multiobjective evolutionary algorithm (MOEA), the elitist multiobjective tabu search (EMOTS), and incorporates it with MODFLOW/MT3DMS to develop a groundwater simulation-optimization (SO) framework based on modular design for optimal design of groundwater remediation systems using pump-and-treat (PAT) technique. The most notable improvement of EMOTS over the original multiple objective tabu search (MOTS) lies in the elitist strategy, selection strategy, and neighborhood move rule. The elitist strategy is to maintain all nondominated solutions within later search process for better converging to the true Pareto front. The elitism-based selection operator is modified to choose two most remote solutions from current candidate list as seed solutions to increase the diversity of searching space. Moreover, neighborhood solutions are uniformly generated using the Latin hypercube sampling (LHS) in the bounded neighborhood space around each seed solution. To demonstrate the performance of the EMOTS, we consider a synthetic groundwater remediation example. Problem formulations consist of two objective functions with continuous decision variables of pumping rates while meeting water quality requirements. Especially, sensitivity analysis is evaluated through the synthetic case for determination of optimal combination of the heuristic parameters. Furthermore, the EMOTS is successfully applied to evaluate remediation options at the field site of the Massachusetts Military Reservation (MMR) in Cape Cod, Massachusetts. With both the hypothetical and the large-scale field remediation sites, the EMOTS-based SO framework is demonstrated to outperform the original MOTS in achieving the performance metrics of optimality and diversity of nondominated frontiers with desirable stability and robustness.

Introduction Groundwater contamination is an environmental imperative in large parts of the world and lackness of available clean water has brought up new challenge to develop quality cost-effective groundwater remediation techniques (Leeson et al. 2013). During the past few decades, the effective coupled simulation-optimization (SO) models have been commonly used for optimal groundwater remediation design of pump-and-treat (PAT) 1 Corresponding

author: School of Earth Sciences and Engineering, Hohai University, 8 Focheng West Rd., Nanjing 211106, China; +86-15,850,517,513; fax: +86-25-83,787,234; [email protected] 2 Corresponding author: Department of Hydrosciences, School of Earth Sciences and Engineering, Nanjing University, 163 Xianlin Rd., Nanjing 210023, China; [email protected] 3 School of Earth Sciences and Engineering, Hohai University, 8 Focheng West Rd., Nanjing 211106, China Article impact statement: This study aims at enriching the community of optimization techniques and multioptimal design of groundwater remediation systems. Received October 2016, accepted March 2017. © 2017, National Ground Water Association. doi: 10.1111/gwat.12525

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systems which has been focus on complex condition with multiple conflicting objectives (Cai et al. 2003; Wu et al. 2005; Yang et al. 2013; Mategaonkar and Eldho 2014; Majumder and Eldho 2016). As the conventional techniques, in which the multiple objectives were handled to integrate into a weighted sum objective, or one of the objectives was optimized while the rest are constrained, the PAT simulation method was often combined with single objective optimization method, such as linear programming (LP), nonlinear programming (NLP), genetic algorithm (GA), and simulated annealing (SA), to determine the most optimal design based on proper pumping schemes (Gorelick 1982; Gorelick et al. 1984; Dougherty and Marryott 1991; Mckinney and Lin 1994). However, these methods have a variety of restrictions owing to the nonconvexity and discontinuity of problems and are often computationally inefficient in generating Pareto frontiers under circumstance of multiple objectives (Cieniawski et al. 1995; Das and Dennis 1997; Coello Coello 2005). To overcome these difficulties, multiobjective evolutionary algorithms (MOEAs) with high capability of convergence and less computation burden have been proposed to combine with simulation models to determine the optimum

Vol. 55, No. 6–Groundwater–November-December 2017 (pages 811–826)

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design of PAT remediation systems. Some of these techniques mimic certain natural systems, not requiring calculation of a gradient of problems, such as niched Pareto genetic algorithm (NPGA) (Erickson et al. 2002), particle swarm optimization (PSO) (Sepehri et al. 2012), NSGA-II (Singh and Minsker 2008; Singh and Chakrabarty 2010), niched Pareto tabu search (NPTS) (Yang et al. 2013), multiobjective fast harmony search algorithm (MOFHS) (Luo et al. 2014) and Borg MOEA (Piscopo et al. 2015) are common examples of MOEAs. Tabu search (TS) has been deemed to be one of the most representative global metaheuristic optimization algorithms together with GA and SA, which proceeds iteratively from one point (seed solution) to another in the same way as traditional local search or neighborhood move. In order not to be trapped in local optima, tabu list is utilized to record recently selected seed solutions or visited trajectory (the path from one seed solution to another) as tabu, to prevent the repeated search occurring in the previous trajectory based on human memory process (Glover et al. 1993). Previous studies have shown that the improved TS can be designed to solve optimization problems with multiple objectives. Baykasoglu et al. (1999a and 1999b) presented a novel MOEA, the multiple objective tabu search (MOTS), which employed Pareto domination ranking as selection logic of seed solutions. Yang et al. (2013) developed NPTS by introducing the improved selection strategy based on Pareto domination ranking and niche technique, and fitness archiving technology enabling high efficiency for obtaining diverse Pareto optimal solutions. The TS-based MOEAs performed better than the classical GA-based MOEAs (Osyczka and Kundu 1996; Deb et al. 2000) on mechanical design optimization problems. This article is primary aimed at developing a new SO framework in which an elitist multiobjective tabu search (EMOTS) technique combined with finite-difference groundwater flow and transport models, MODFLOW and MT3DMS, is used to identify Pareto frontier to multiobjective PAT groundwater remediation systems. The proposed EMOTS enhances the original MOTS by incorporating the elitist strategy and the neighborhood move rule using Latin hypercube sampling (LHS). Elitism can help to prevent the loss of nondominated solutions once they have been found and recorded in elitist list and candidate list. From that point, we choose the two most remote solutions from the candidate list as seed solutions for next generation, which are Pareto nondominated and widely distributed in the nondominated front. Although MOEAs are commonly used in groundwater remediation design optimization, there is a need to establish a general SO framework by independent modular design. This system follows a modular structure by defining a contract or interface with each other, and thus, minor modifications can be added to original code, or new capabilities can wholly replace the corresponding modules without breaking the system. In terms of the modular design, the SO framework of groundwater remediation includes separate modules of multiple 812

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optimization algorithms, flexible management formulations (e.g., Objective function, multiple constraint type), and groundwater simulators (e.g., MODFLOW/MT3DMS). Zheng and Wang (2003) developed a modular groundwater optimizer (MGO) for optimization of groundwater PAT systems with single objective function. Three evolutionary algorithm, GA, SA, and TS were incorporated into MGO as optimization module; A generally defined cost function accommodating multiple costs of well drilling, well installation, water extraction/injection, or water treatment was included as formulation module; Fully compatible MODFLOW and MT3DMS were modularized as simulator to update the state variables of head and solute concentration. In this study, we inherit the framework of MGO to establish a modular SO framework of groundwater remediation for multiobjective engineering design of groundwater PAT remediation system. The objectives of this study are to develop the EMOTS-based SO framework and apply it to a hypothetical and a large-scale field contaminated site management cases. To study the effectiveness of the proposed algorithm, we firstly performed sensitivity analysis to determine a optimal combination of heuristic parameters of EMOTS including the neighborhood range of each variable (NR), neighborhood population size (N ns ), and tabu list size (LS). We also verified whether different random initial solutions or random neighborhood moves had significant difference in the optimization results to show the robustness of the algorithm. The performances of the EMOTS with different combinations of parameters are assessed and compared by three performance metrics, extent of convergence (dC ), extend width, and uniformity of near Pareto optimal front (d ub + d lb , dU ) (Deb et al., 2000; Yang et al. 2013). Also, in this study, we demonstrated the applicability of EMOTS in the design of a field PAT application for the cleanup of trichloroethylene (TCE) plume in an unconfined sandy aquifer at the Massachusetts Military Reservation (MMR) in Cape Cod, Massachusetts. The optimization results show that the EMOTS is demonstrated to outperform MOTS by generating more convergent and diverse nondominated frontiers with desirable stability and robustness.

SO Framework of Groundwater Remediation The SO framework of groundwater remediation comprises one main program for defining arrays, allocating computer memory, and reading & preparing input data of sink/source information, and three executive subroutines of optimization, objection function evaluation and groundwater flow and contaminant transport simulation, as shown in Figure 1. The optimization package, EMOTS, initiates various optimization individuals and call management formulation module to evaluate objection function value of each individual. MODFLOW/MT3DMS is modified as a subroutine and dynamically linked with optimization package to update state variables of heads NGWA.org

integrated through flow-transport link file of velocity and sink & source terms saved after run by MODFLOW, which are invoked by MT3DMS link package. The three dimensional, transient state partial-differential equations of groundwater flow and solute transport in a heterogeneous and anisotropic aquifer have been detailed described by Harbaugh et al. (2000) and Zheng and Wang (1999b). Multiobjective Management Formulation The SO framework is able to handle two concurrent management objectives that encompass two categories of remediation cost and performance. The remediation cost, RC, is to minimize total cost by accommodating multiple costs of well drilling, well installation, water extraction/injection, or water treatment. The mathematical forms of RC can refer to the previous literatures (Zheng and Wang 2003; Yang et al. 2013). The objective of remediation performance, MR, is to minimize the percentage amount of contaminant mass remained in the aquifer min MR =

massend/mass

0

× 100%

(1)

where mass0 and massend are the initial and ultimate total mass of contaminant plume in the aquifer, respectively. Note that the MR objective function is defined as usersupplied function which is problem dependent. Therefore, it is complied as a Fortran subroutine based on dynamic link library on the Microsoft Windows. This procedure allows the users to write an arbitrary objective function based on their own requirement. A set of commonly used constraints of pumping/injection capacity, solute concentration, hydraulic head, hydraulic gradient are set in order to satisfy the environmental limitation for the designed PAT system (Zheng and Wang 2003; Yang et al. 2013).

Figure 1. Overall structure for EMOTS-based SO framework.

and contaminant concentrations under various remedial alternatives, which are the important parameters of objective functions and constraints. Groundwater Flow and Transport Simulations Simulations of groundwater and solute transport during PAT are conducted using MODFLOW (Harbaugh et al. 2000) and solute transport simulator, MT3DMS (Zheng and Wang 1999b). The public domain, finitedifference codes MODFLOW/MT3DMS, are firstly NGWA.org

Elitist Multiobjective Tabu Search The existing MOTS proposed by Baykasoglu and his colleagues is in fact based on the neighborhood search, in which new population or neighborhood set are explored from the single seed solution with a simple neighborhood move. It introduces Pareto domination ranking to selection operator and allows all neighborhood solutions to take part in completion. Candidate seed solutions, which are not dominated by nondominated solutions discovered so far, are identified in the current neighborhood set and added into candidate list. If there exist candidate seed solutions, one of the solutions from the candidate list is then randomly selected as the new seed solution for the coming iteration (Baykasoglu et al. 1999a, 1999b; Baykasoglu 2006). Baykasoglu et al. (1999b) and Baykasoglu (2006) have shown that the MOTS outperforms the classical GA-based MOEAs on mechanical design optimization problems. This article aims at developing EMOTS, a new TS-based metaheuristic algorithm, which inherits the main structure of the existing MOTS and its coupling Y. Yang et al. Groundwater 55, no. 6: 811–826

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Figure 2. Flowchart of the EMOTS algorithm for multiobjective optimal design of groundwater remediation systems. Note that the selection and updating procedure (Stage 2) is described as shown in Figure 3.

with the flow and transport simulators to obtain Pareto optimal solutions for design of groundwater remediation problems. The EMOTS makes improvements on MOTS by adding an elitist strategy, to preserve all nondominated solutions discovered so far as candidate seed solutions for enhancing the convergence of nondominated solutions to the true Pareto front. Also, the EMOTS incorporates the LHS (McKay et al. 1979) to generate uniform neighborhood set in the bounded neighborhood space. The elitist strategy was often used in GA-based MOEAs. The main idea of elitist strategy is to preserve an overall nondominated set of solutions to form the parent population of the next iteration. The parent population is competitive selected based on a systematic comparison of previous nondominated solutions and current population, or comparison of previous parent population and offspring population. With this strategy, MOEAs was proved that had ability to direct the search to true Pareto optimal front (Zitzler and Thiele 1998; Knowles and Corne 1999). Figure 2 is a flowchart representing the improved procedure of EMOTS based on original MOTS. The framework of EMOTS is mainly composed of three following components labeled as Stages 1-3. 814


Stage 1: Random Initialization and Generation of Neighborhood Solutions An initial solution vector consisting of all continuous decision variables is randomly generated in feasible solution region. Neighbor solution sets with N ns solutions are explored from the initial solution for first generation and created based on seed solutions for the following generations using LHS. Each neighborhood solution consists of nd continuous decision variables. The neighborhood range, NR, determines the value range of each decision variable after neighborhood move. For all neighbor solutions, the flow and transport subroutines must be run repeatedly to calculate fitness values of multiple objectives functions and to evaluate whether all prescribed constraints are satisfied. Stage 2: Selection and Updating Procedure Selection and updating procedures are to select seed solutions and update elitist list and candidate list. Selection strategy is based on Pareto optimality logic (Pareto domination ranking) (Yang et al. 2013) and updating procedure incorporates the elitist strategy. To implement the selection and updating NGWA.org

Figure 3. Sketch map of Stage 2 in Figure 2 showing the selection procedure of seed solution from iteration t to iteration t + 1, as well as updating strategies of both elitist list and candidate list from iteration t−1 to iteration t. The crossed blocks ( ) represent the dominated solution set removed from neighborhood solution set, elitist list, and candidate list. Si for i = 1, 2, and 3, are the nondominated solutions in the neighborhood solution set, elitist list, and candidate list, respectively.

strategy, the algorithm needs to cope with the following operations: Elitist Strategy As the optimization proceeds, we maintain an external pool (elitist list) recording all nondominated solutions (SE t , as shown in Figure 3) discovered so far beginning from the initial solution. If the neighborhood solution after each iteration is dominated by any of the solutions in elitist list, the solution is discarded. On the other hand, if the neighborhood solution is not dominated by any solutions in external pool, the neighborhood solution is accepted as the candidate seed solution, and is put into candidate list and elitist list. Stage 2 of the EMOTS (see Figure 2) includes three parts as shown in Figure 3: selection of seed solution, update of elitist list, and update of candidate list. The elements of selection and updating procedure are defined as follows: Selection of Seed Solution Part 2 shown in Figure 3 is the selection strategy, where the two new seed solutions for iteration t + 1 (SS t+1 ) are selected from the updated candidate list for current iteration t (S 3 ) after Pareto domination sorting. Selection procedure used in EMOTS is an elitist one that is modified from MOTS in which one of the candidate solutions is randomly selected as the new seed solution. Each individual in neighborhood set (St NS ) is compared with the solutions in the elitist list for previous iteration (SE t−1 ) based on Pareto domination ranking to justify its degree of Pareto domination. All the nondominated solutions from St NS are considered as the candidate seed solutions (S 1 ) and put into candidate list for previous iteration, SC t−1 , then the nondominated solutions, S 3 , is selected from the combined set with S 1 and SC t−1 . Solutions in S 3 are sorted based on the ascending order of the first objective function value. Selection of SS t+1 is illustrated in Figure 3, the preference being given to the two solutions at both ends of nondominated front (b1 NGWA.org

and b2) of S 3 . Compared with the one randomly selected seed solution by the existing MOTS, the two most remote seed solutions chosen by the proposed EMOTS increase the diversity of the searching space. Updating Strategy Part 1 in Figure 3 deals with the procedure of updating elitist list (SE ). The neighborhood solutions for current iteration (St NS ) are added into the elitist list for iteration t−1 (SE t−1 ), and the dominated individuals among SE t−1 + St NS are discarded from the combined solution set, then the nondominated solutions are survived and composed of the updated elitist list for iteration t, SE t . Similarly, candidate list is updated from iteration t−1, SC t−1 to S 3 as shown in Part 3 of Figure 3. Moreover, the selected seed solutions, b1 and b2, are removed from the S 3 , and candidate list is updated for iteration t, SC t = S 3 −(b1 + b2). Stage 3: Tabu Strategy and Termination To avoid repeated search and endless cycle in local solutions space, tabu strategy is employed in which the last seed solutions of Tabu list size, LS, are recorded in the tabu list. The two candidate seed solutions outcompeting from Stage 2 of selection are firstly identified whether they are stored in tabu list. If one of them is revisited, the closest individual to the tabu seed solution is choosed as the alternative seed solution from candidate list, or if there is no solution in candidate list, randomly generate new seed solution for next iteration. After the two seed solutions are determined, the selected seed solutions are stored and tabu list is updated for iteration t. As shown in Stage 3 of Figure 2, if the maximum number of iterations is reached, the algorithm terminates and outputs the discovered Pareto optimal solutions. Performance Metrics In order to identify the algorithmic performances in terms of its computational efficiency and solution quality, we define performance metrics of convergence, Y. Yang et al. Groundwater 55, no. 6: 811–826

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diversity, and uniformity. It is well-known that it is difficult or even impossible to achieve the true Pareto optimal solutions to nonlinear multiobjective optimization problems. For this purpose, we combine the near Pareto optimal solutions found by different single EMOTS runs with each other, and define the nondominated solutions among the aggregated solutions as the “true” Pareto optimal set. The metrics used in this study to assess the quality of Pareto optimal set are defined as below. N OFE — —the number of objective function evaluations associated with groundwater simulation model for a separate optimization run. N NPS —the number of near Pareto optimal solutions for a separate run. N TPS —the number of “true” Pareto optimal solutions for a separate run. N OFE —the average N OFE to obtain each “true” Pareto optimal solution for a separate run. dC —the average value of the minimum scaled Euclidean distance between the j th near Pareto optimal solution to the “true” Pareto optimal solutions, j = 1, 2, . . . , N NPS . d ub + d lb —the sum of two scaled Euclidean distance between the boundary solutions of “true” Pareto optimal frontier and the extreme solutions of the obtained near Pareto optimal set. dj —the scaled Euclidean distance between the adjacent two solutions, j and j + 1. d, σ d —the mean and variance of dj The computational efficiency can be depicted by N OFE , because repetitive objection function evaluation associated with groundwater simulation model accounts for more than 90% of total computational time. The performance metric dC indicates the extent of convergence to the “true” Pareto optimal set. d ub + d lb expresses the extend of the obtained near Pareto optimal solutions. Furthermore, we define the uniform degree among the near Pareto optimal solutions, dU , which can be given by dj − d

NNPS

dU =

j =1

(NNPS − 1) d

(2)

It is noteworthy that the solution quality can be denoted by the above three metrics of convergence, diversity, and uniformity, in which the smaller the each value of dC , d ub + d lb , and dU , the closer, wider, and more uniformly the near Pareto optimal solutions spread along the Pareto frontier.

Groundwater Remediation Design We describe the application of EMOTS-based SO framework to typical PAT groundwater remediation applications. Case A is the hypothetical groundwater remediation system, intended to illustrate the performance of EMOTS in terms of parameter sensitivity and superiority over original MOTS. Case B is a real-world application on 816


the Chemical-Spill 10 (CS-10) site, located in the southeast corner of MMR, with the focus of discussion on the universal applicability of groundwater quality management. Case A: A Hypothetical Groundwater Remediation System The hypothetical groundwater remediation systems applied in this study to evaluate the performance of the proposed EMOTS was the same as the case studied by Yang et al. (2013). Figure 4 is the hypothetical field area with the two-dimensional, homogeneous, isotropic, and confined aquifer, contaminated with an existing plume. All the relevant flow and transport parameters and boundary conditions for this application are also shown in Figure 4. The problem formulation contains two objectives that encompass minimization of remediation cost through PAT technology and the contaminant mass remaining in the aquifer and two base constraints of pumping capacity and the allowable contaminant concentration. Figure 4 shows the locations of the four potential extraction wells (W1∼W4) with pumping capacity constraint, and also shows the area of compliance where the cleanup level is imposed at the end of remediation period. The RC objective functions are formulated as follows.

min J1 = α1 Nw + α2

Nt Nt Nw Nw Qi, t tt + α3 Mi, t i=1 t=1

i=1 t=1

(3) where J 1 is the total remediation cost (RC), Qi,t denotes the extraction rate associated with well i during the tth remediation period, Nw is the number of extraction wells (in this case Nw = 4), Nt is the total number of remediation periods, and tt is the management duration of the tth period, Mi,t is the amount of solute mass removed by well i during the tth period, and α 1 , α 2 , and α 3 are the costcoefficients of well drilling, extraction, and contaminant treatment. The MR objective function is defined by Equation 1. The constraints are specified as Cj,t ≤ C ∗ ,

j = 1, 2, · · · , mC , t = 1, 2, · · · , Nt (4)

Qli,t ≤ Qi,t ≤ Qui,t ,

i = 1, 2, 3, 4

(5)

where the contaminant concentration, Cj,t , at node j at the end of tth remediation period must not be greater than the maximum concentration level C * . mC is the total number of the nodes contained in the area of the compliance; Ql i,t and Qu i,t are the lower and upper bounds of Qi,t with well i during the tth period. Owing to the simplicity of the application, we suppose that the pumping rate during each management period is constant. The model parameters associated with objective NGWA.org

Figure 4. A hypothetical field area showing the aquifer properties, initial plume, location of the potential remediation wells, and the area of compliance where a concentration limit is imposed.

functions and constraints used in this implementation are assigned as Nw = 4; Nt = 1; C * = 3 ppb; Ql i,t = 0 m3 /d; Qu i,t = 10,000 m3 /d; α 1 = $10,000 per well; α 2 = $0.4/m3 . Of course, groundwater simulation models, MODFLOW and MT3DMS, are applied to update the head and concentration variables under different PAT strategies. Sensitivity Analysis of EMOTS Parameters The parameter settings of EMOTS, including neighborhood range, NR, neighborhood population size, N ns , and tabu list size, LS, control the quality of the obtained near Pareto optimal sets. Thus, sensitivity analysis is made for these parameters based on the performance metrics described above to evaluate the applicability of optimization results. Considering that the intervals of neighborhood width of each decision variable around the seed solution rely on NR and N ns , and can be calculated by NR/N ns , we take parameters, NR and N ns , as a parameter combination of (NR, N ns ). To characterize the results influenced by (NR, N ns ), we design nine schemes with different NR (NR = 200, 400, or 600) and N ns (N ns = 5, 10, or 15). The “true” Pareto optimal solutions can be assumed as the nondominated solutions among the aggregated near Pareto optimal solutions obtained from nine single EMOTS runs. Table 1 shows how well the number of “true” Pareto optimal solutions, N TPS , depends on the neighborhood range. For the same neighborhood range, NR = 600, NGWA.org

400, or 200, the N TPS value increases with N OFE which is dependent on the value of N ns , that is, with NR = 600, the EMOTS evaluate 3136, 7991, and 11,986 objective function evaluations and produces 52, 263, and 309 “true” Pareto optimal solutions for N ns = 5, 10, and 15, respectively. On the other hand, for the same neighborhood size, N ns = 5, 10, or 15, the smaller the value of NR is, the more “true” Pareto optimal solutions is obtained, that is, with N ns = 5, the EMOTS obtains 52, 70, and 180 near Pareto optimal solutions with the smaller NR value from 600, 400, to 200. Therefore, for the same N ns scenarios, less individuals are evaluated to obtain one “true” Pareto optimal solution with the NR decreases. Obviously, the smaller the NR is, the more “true” Pareto optimal solutions the EMOTS generates, and the efficiency of generating “true” Pareto optimal solutions is steadily raised. Figure 5 shows nine final trade-off curves obtained from nine EMOTS runs for different parameter combinations of (NR, N ns ), and constant LS (LS = 10). Note that 100 uniformly spread near Pareto optimal solutions are plotted in Figure 5 for each scheme (similarly hereinafter). We can find that the final Pareto offline solutions shown in Figure 5d, 5g, and 5h spread over relatively narrow span of trade-off curves. For the same neighborhood range, NR = 600, 400, or 200, the EMOTS generates near Pareto optimal solutions spreading wider span of tradeoff curve as the value of N ns increases from 5, 10, to Y. Yang et al. Groundwater 55, no. 6: 811–826

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Table 1 Comparison of the Performance Results for Quantity of EMOTS-Based “True” Pareto Optimal Solutions for Nine Parameter Combinations of (NR, N ns ) Parameter Combination of (NR, N ns ) Item NOFE N TPS N OFE dC d ub + d lb dU

(600, 5)

(600,10)

(600,15)

(400, 5)

(400,10)

(400,15)

(200,5)

(200,10)

(200,15)

3136 52 61 0.00142 0.09935 0.695

7991 263 30 0.000893 0.0157 0.787

11,986 309 39 0.000646 0.00122 0.822

3266 70 47 0.00129 0.275 0.687

7991 248 33 0.000792 0.0372 0.637

11,986 424 29 0.000502 0 0.754

3996 180 23 0.000970 0.369 0.688

7991 362 23 0.000686 0.308 0.843

11,986 472 26 0.000405 0.0561 0.673

Figure 5. Comparison of the final Pareto optimal solutions based on the nine different parameter combinations of neighborhood range and neighborhood size. The plus symbol (+) denotes the near Pareto optimal solutions by EMOTS.

818


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Table 2 Comparison of the Performance Results for Quantity of EMOTS-Based “True” Pareto Optimal Solutions for Different Tabu List Sizes, LS Tabu List Size Item N OFE N TPS N OFE dC d ub + d lb dU

LS = 5

LS = 10

LS = 15

LS = 20

7991 355 23 0.000914 0.0292 0.717

7991 395 21 0.000838 0 0.835

7991 365 22 0.000792 0.00623 0.745

7991 394 21 0.000730 0.00273 0.757

15, respectively. For the case of NR = 600, in Figure 5a5c, the obtained near Pareto optimal solutions are always spreading wide along the trade-off curves whatever the N ns is. However, for the case of NR = 400, N ns value of 10 and 15 as in Figure 5e and 5f result in wider range of the trade-off curves than the N ns of 5 as in Figure 5d, where there are no solutions with the value of RC > $12M (or 12 million US dollars, similarly hereinafter). Similarly, in the case of NR = 200, the EMOTS produces solutions covering wider range of trade-off curve with the N ns value of 15 as in Figure 5i than those with the N ns values of 5 and 10 as in Figure 5g and 5h, where there are no designs with the value of RC > $10.2M and RC > $11.1M, respectively. This shows that moderately large NR value could help EMOTS overstep local optimum and expand to global optimum covering whole span of Pareto frontiers, whereas too small NR value may trap the algorithm into local optima, although large neighborhood size can help realize global search. Figure 5 qualitatively analyses the diversity in near Pareto optimal solutions corresponding to different parameter combinations of (NR, N ns ). Table 1 shows the quantitative analysis on the convergence and diversity of near Pareto optimal set compared to the “true” Pareto set. As given in Table 1, the results of d ub + d lb verifies well with the results shown in Figure 5 that the value of d ub + d lb for (NR, N ns ) of (400, 5), (200, 5), and (200, 10) are a lot larger than others, which indicate that the boundary solutions of the three near Pareto optimal set are distant from the extreme solutions of the “true” Pareto optimal set. With the analysis results of N TPS in Table 1, the results of performance metrics of dC are similar,

that more “true” Pareto optimal solutions are obtained, smaller the dC indicating that the solutions converging nearer to the true Pareto front. However, the results of dC are all small indicating that the convergence of obtained near Pareto optimal solutions are closely similar and also good for all nine scenarios. Similarly, the results of dU show that the uniform degree of the near Pareto optimal solutions are resemble to each other, where more than 80% solutions as in Figure 5 crowd the range of the trade-off curves with the value of RC between $3.0M and $10.0M. To sum up, with larger NR value, although EMOTS evaluates a bit more feasible solutions to obtain one “true” Pareto optimal solution, the obtained near Pareto optimal solution maintains a wider span and similar degree of convergence and uniformity compared to the solutions generated by EMOTS with smaller NR value. However, with smaller NR value, the diversity of near Pareto optimal set is very sensitive to the value of N ns , EMOTS being hard to jump out local optima and tend to unexplored feasible solution space departing from local area for the case of small N ns value. Given in Table 2 is the sensitivity analysis of the EMOTS with NR = 600, and N ns = 10 associated with the different values of LS. It is shown that the values of N OFE , N TPS , dC , dU , and d ub + d lb are quite similar for each LS scenario. This indicates that the performance metrics of EMOTS are not sensitive to the value of LS. Randomization Strategies For some search techniques, starting with a good initial solution could enhance the convergence efficiency, but it is not straightforward to select an appropriate initial solution. The degree to which the quality of solutions by EMOTS depends on the generation of initial solutions will affects its applicability in practice. Table 3 shows the variance of convergence (σ dC ) and diversity (σ (d ub + d lb ) and σ dU ) metrics by nine single runs based on nine initial solutions generated randomly are quite small. So if the values of the EMOTS parameters are appropriate, for example, NR = 600, N ns = 10, and LS = 10, the results by EMOTS are insensitive to the randomly initial solutions. Another randomization strategy is the neighborhood move, which generate the neighborhood solution set using LHS for each iteration. To verify whether the proposed rule affects the optimal results, nine calculations are made with the fixed initial pumping rates from four extraction wells, 1225, 4030, 3057, and 1770 m3 /d, respectively. The results illustrated as in Table 3 show that the mean

Table 3 Mean and Variance of the Convergence and Diversity Metrics for EMOTS-Based Near Pareto Optimal Solutions with Two Random Strategies Random Strategy Random initial solutions Random neighborhood moves

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dC

σdC

dub + dlb

σ (d ub + d lb )

dU

σdU

0.000951 0.00102

0 0

0.00792 0.0130

0.000266 0.000151

0.765 0.760

0.00104 0.00294


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and variance of the convergence and diversity metrics for EMOTS-based near Pareto optimal solutions are quite small. Thus, the results obtained from EMOTS, on the premise of appropriate EMOTS parameters to be the same as in Table 3, are insensitive to the random neighborhood moves with NR = 600 and N ns = 10 for this application. The final Pareto offline solutions by EMOTS based on the above two randomization strategies are plotted in Figures S1 and S2 in Supporting Information to this paper. It can be seen in Figure S1 and Figure S2 that the distribution characteristic of Pareto optimal solutions are quite similar in both shape and range of their trade-off curves. To summarize, there is virtually no difference between Pareto optimal sets generated by both randomization strategies, random initial solutions, and neighborhood moves, verifying the robustness of the proposed EMOTS. Comparison of Different Algorithms To investigate the algorithm performance, we compare the proposed EMOTS with MOTS and single objective tabu search (SOTS) in terms of solution quality for this groundwater remediation implementation. Procedural details on MOTS and SOTS could be found in many literatures (Baykasoglu et al. 1999a, 1999b; Zheng and Wang 1999a; Baykasoglu 2006). For the sake of comparison, we enhance the MOTS and SOTS with real number coding by using the modified neighborhood strategy with LHS. Meanwhile, the controlling parameters of SOTS and MOTS, such as LS, N ns , and NR, are set identically to those of EMOTS. The “true” Pareto optimal solutions are the nondominated solutions among the aggregated solutions from the EMOTS- and MOTS-based near Pareto optimal solutions. We carry out SOTS by transforming the RC objective given by Equation 3 into a total remediation cost constraint, RC = RC* . Three single SOTS runs with RC* = $3.1M, $5.5M, $9.2M call 160, 210, and 200 individual evaluations and converge to three single optimized solutions with MR = 9.21%, 2.9%, and 0.81%, respectively (see Figure 6). By the same way, the trade-off curve as in Figure 6 is constructed by executing a series of SOTS runs with continuous RC* values, approximately implementing 190 evaluations to generate one optimal solution. However, EMOTS searches for 263 near Pareto optimal solutions concurrently by implementing 7991 evaluations in a single optimization run, 30 evaluations converging to a single solution on average (see also Figure 6 and Table 4). Compared with the SOTS, the EMOTS has remarkable advantage in computational efficiency increasing by about 6 times. Furthermore, the Pareto optimal solutions with RC = $3.1M, $5.5M, $9.2M among the near Pareto optimal set converge to MR = 9.175%, 2.776%, and 0.766%, respectively. Thus, compared with SOTS-based PAT strategies, EMOTSbased pumping schemes lead to more contaminant mass removal after PAT. The preselected optimal solutions obtained by EMOTS (Figure 7) show that the pumping rates are centralized at well locations of W2 and W3. The 820


Figure 6. Comparison of the final Pareto optimal solutions by EMOTS and three single optimal solutions by SOTS. Note that the three single SOTS runs with constrained RC of $3.1M, $5.5M, and $9.2M, respectively.

Table 4 Comparison of the Performance Results for Quantity and Quality of EMOTS- and MOTS-Based “True” Pareto Optimal Solutions MOEA Item

EMOTS

MOTS

N OFE N TPS N OFE dC d ub + d lb dU

7991 101 80 0.001362 0 0.29452

3994 26 154 0.003134 0.76215 0.62361

direction of groundwater flow is from west to east, so the upstream pollutant in the aquifer will move east and could be mainly captured by wells W2 and W3 even though W1 well locates at the center of plume. Figure 8 shows the comparison of near and “true” Pareto optimal solutions found by EMOTS and MOTS. Both near Pareto optimal solutions spread along the trade-off curves which reflect the potential relationship between RC and MR objectives as shown in Figure 8a. However, the range of the EMOTS-based trade-off curve is much wider than the MOTS-based results, where there are no solutions with the value of RC > $13.4M for the MOTS-based near Pareto front. Furthermore, the EMOTSbased near Pareto optimal solutions with RC = $3.3M, $6.2M, $10.9M converge to MR = 7.984%, 2.047%, and 0.549%, respectively. With the same RC values, EMOTS-based near Pareto optimal solutions converge to MR = 8.274%, 2.102%, and 0.574%, respectively. This NGWA.org

Figure 7. Optimal pumping rates at different pumping well locations for three preselected Pareto optimal solutions with RC = $3.1M, $5.5M, and $9.2M, respectively, obtained by EMOTS.

the trade-off curve, whereas the MOTS-based solutions unevenly distributed along Pareto front with the value of RC < $10.3M as shown in Figure 8b. The results of the algorithmic performance of convergence and diversity are listed in Table 4. The value of dC and d ub + d lb associated with convergence and diversity of the EMOTS-based near Pareto optimal solutions are smaller than the MOTSbased solutions, indicating that EMOTS-based near Pareto optimal solutions approach nearer to the “true” Pareto optimal front. Therefore, the proposed EMOTS, inheriting the advantages of elitist strategy and selection of two most distant seed solutions, outperforms the MOTS in finding near Pareto optimal solutions which covers a full span of trade-off curve and converges closer to the “true” Pareto optimal front to synthetic PAT remediation system. Case B: A Field-Scale Groundwater Remediation Site

indicates that compared with MOTS-based PAT strategies, EMOTS-based pumping schemes lead to more amount of contaminant mass removal after PAT remediation. Given in Table 4 is the comparison of quantity and computation effort of EMOTS and MOTS in searching for “true” Pareto optimal solutions on the PAT remediation problem. As shown in Table 4, Although EMOTS requires more computational time to implement 7991 evaluations, the solution quality is greatly enhanced as EMOTSbased “true” Pareto optimal solutions make up 79.5% (101 solutions) of the aggregated “true” Pareto optimal solutions. Therefore, compared with the MOTS, the EMOTS improves the search efficiency which can be approximately characterized by N OFE value. It indicates that the efficiency of EMOTS (N OFE = 80) can be twice as high as that with MOTS (N OFE = 154) for obtaining “true” Pareto optimal solutions. In addition, the EMOTS-based “true” Pareto optimal solutions spread a whole span along

Site Description and Management Formulation The Chemical-Spill (CS-10) site, located in the southeast corner of MMR at Cape Cod, Massachusetts, USA has been chosen as a large-scale test case (Figure 9). This 57 km2 (22 square miles) military reservation was established in 1911 and the most intensive periods of activity occurred from 1940 to 1946 and 1955 to 1970. TCE is the primary contaminant at the CS-10 site. The TCE plume is approximately 5 km (16,400 ft) long, 2km (6500 ft) wide, and about 43m (140 ft) thick (Zheng and Wang 2002). To remediate this contamination, the PAT remedy consisting of six pumping wells and two infiltration trench are operating at the site. Three in-plume wells (numbered 2, 4, and 5) have been constructed in the center of pollution zone, and three perimeter wells (numbered 1, 3, and 6) have been created to contain the contaminant plume within the MMR base boundary.

Figure 8. Comparison of the final near Pareto optimal solutions and “true” Pareto optimal solutions by EMOTS and MOTS.

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Figure 9. The MMR field area and the initial plume representing the peak value of all vertical layers at each horizontal location. The solid dots (•) represent the optimal pumping well locations (source Zheng and Wang 2003).

The extracted water from Wells 1 ∼ 6, after treatment, is to be reinjected into the infiltration trenches. The main hydrogeological parameters and the configuration of the MMR PAT model domain can be found in previous studies of Yang et al. (2013). This study aims at seeking optimal PAT strategies by the EMOTS-Based SO Framework which can achieve the cleanup standards of the allowable contaminant concentration beyond the MMR base boundary and within the modeled domain at the end of the remediation horizon, with simultaneous minimization of total pumping rates over the entire project horizon and the contaminant mass remaining in the aquifer. Consequently, the first objective of the problem can be expressed as: min J1 = TPR =

6

|Qi |

(6)

i=1

The second objective is defined by Equation 1. The constraints can be stated as ∗ , Cj ≤ CTCE

0≤

6

j = 1,2, . . . , mnode

|Qi | ≤ TPR∗ ,

j = 1,2, . . . , 6

(7)

(8)

i=1

where C * TCE is the maximum cleanup level, 5ppb; and m node is the number of points at which concentration is required to be less than or equal to C * TCE ; The 822


constraint imposed by Equation 8 denotes that the total pumping rate cannot exceed the total capacity of the existing on-site treatment plant, TPR* . For this application, TPR* = 14,720 m3 /d, or about 2700 GPM (gallons per minute) (Zheng and Wang 2002). All other constraints, including the physical pumping capacity (2400-4100 m3 /d, or 440-750 GPM) of each individual well, the water balance between pumping and injection through the infiltration basins, were kept the same as those used by Yang et al. (2013). Optimization Results For this field application, as described as above, the physical pumping capacity of each individual well is nonuniform, that is, Qk max =3800 m3 /d, 2400 m3 /d, 4100 m3 /d, 3700 m3 /d, 3400 m3 /d, 3800m3 /d, for k = 1, 2, . . . , 6 (Zheng and Wang 2002). In this study, the maximum number of iterations for the EMOTS runs was set to 200. The main EMOTS parameters used for this application were set as follows: neighborhood range of each individual well is NRk = 300 m3 /d, 190 m3 /d, 325 m3 /d, 295 m3 /d, 275 m3 /d, 300 m3 /d, for k = 1, 2, . . . , 6, neighborhood size, N ns = 10; tabu list size, LS = 10. Plotted in Figure 10 are the all searched solutions and the final Pareto optimal solutions among them between the two conflicting objectives: TPR and MR, for this application. The distribution of feasible solutions is mainly strip-shaped reflecting the course of converging to optimal trade-off curve. Among these solutions as in Figure 10a, it can be seen that the solutions with the value NGWA.org

Figure 10. The feasible solutions and the final near Pareto optimal solutions to the MMR problem obtained by the EMOTS. The plus signs (+) indicate those solutions that satisfy the concentration constraints, while the open triangles () represent those solutions that violate the constraints, and the circle dots () are the near Pareto-optimal solutions among the feasible solutions.

Figure 11. Optimal pumping rates at different pumping well locations for Pareto optimal solutions with 8300 < TPR < 9000 m3 /d, 10,000 < TPR