Environmental Modelling & Software 22 (2007) 683e692www.elsevier.com/locate/envsoft

Using interactive archives in evolutionary multiobjective optimization: Acase study for long-term groundwater monitoring design

Patrick Reed*, Joshua B. Kollat, V.K. Devireddy

The Pennsylvania State University, Department of Civil and Environmental Engineering, 212 Sackett Building, University Park, PA 16802-1408, USA

Received 25 October 2005; received in revised form 6 November 2005; accepted 15 December 2005

Available online 20 March 2006

Abstract

Monitoring complex environmental systems is extremely challenging because it requires environmental professionals to capture impactedsystems’ governing processes, elucidate human and ecologic risks, limit monitoring costs, and satisfy the interests of multiple stakeholders(e.g., site owners, regulators, and public advocates). Evolutionary multiobjective optimization (EMO) has tremendous potential to help resolvethese issues by providing environmental stakeholders with a direct understanding of their monitoring tradeoffs. This paper demonstrates how3-dominance archiving and automatic parameterization techniques can be used to significantly improve the ease-of-use and efficiency ofEMO algorithms. Results are presented for a four-objective groundwater monitoring design problem in which the archiving and parameterizationtechniques are combined to reduce computational demands by more than 90% relative to prior published results. The methods of this paper canbe easily generalized to other multiobjective applications to minimize computational times as well as trial-and-error parameter analysis.� 2006 Elsevier Ltd. All rights reserved.

Keywords: Groundwater; Long-term monitoring design; Multiobjective optimization; Genetic algorithms; Kriging

1. Introduction

Environmental scientists and engineers address a broad ar-ray of complex pollution problems that range in scale fromglobal emissions of greenhouse gases to micro-scale strategiesfor using bacteria to mitigate water pollution. Although envi-ronmental professionals address a broad array of problems,they share the challenge of developing innovative mitigationand monitoring techniques for polluted environmental systemsthat must successfully protect human and ecologic healthwhile limiting the financial burden placed on society. Balanc-ing these conflicting objectives is extremely challenging be-cause it requires that environmental professionals capture animpacted system’s governing processes, elucidate human andecologic risks, limit design costs, and satisfy the interests ofmultiple stakeholders. In the past decade, researchers have rec-ognized that evolutionary multiobjective optimization (EMO)

* Corresponding author. Tel.: þ1 814 863 2940; fax: þ1 814 863 7304.

E-mail address: preed@engr.psu.edu (P. Reed).

1364-8152/$ - see front matter � 2006 Elsevier Ltd. All rights reserved.

doi:10.1016/j.envsoft.2005.12.021

has significant potential as a decision support tool that canbe used to help resolve these issues by providing stakeholderswith a direct understanding of their design tradeoffs for envi-ronmental systems (Ritzel et al., 1994; Halhal et al., 1997;Loughlin et al., 2000; Reed et al., 2001, 2003; Ericksonet al., 2002; Kapelan et al., 2003).

This paper demonstrates the use of EMO to design ground-water monitoring networks for conflicting objectives. Long-term groundwater monitoring (LTM) can be defined as thesampling of groundwater quality over long time-scales to pro-vide ‘‘sufficient and appropriate information’’ to assess if cur-rent mitigation or contaminant control measures areperforming adequately to be protective of human and ecolog-ical health (Task Committee on Long-Term GroundwaterMonitoring Design, 2003). The LTM problem is ideal for dem-onstrating how EMO can aid environmental decision makingbecause of the tremendous expense and complexity of charac-terizing groundwater contamination sites over long timeperiods.

The four-objective monitoring design problem presented inthis paper is solved using a new version of the Nondominated

684 P. Reed et al. / Environmental Modelling & Software 22 (2007) 683e692

Sorted Genetic Algorithm-II (NSGAII) (Deb et al., 2002),which will be termed the 3-dominance NSGAII in this paperusing the abbreviated notation, 3-NSGAII. The 3-NSGAIIdemonstrates how 3-dominance archiving (Laumanns et al.,2002; Deb et al., 2003) can be combined with a parameteriza-tion strategy for the NSGAII (Reed et al., 2003) to accomplishthe following: (1) ensure the algorithm will maintain diversesolutions, (2) eliminate the need for trial-and-error analysisof parameter settings (i.e., population size, crossover and mu-tation probabilities), and (3) allow users to sufficiently capturetradeoffs using a minimum number of design evaluations. Asufficiently quantified tradeoff can be defined as a subset ofPareto optimal solutions that provide an adequate representa-tion of the Pareto frontier that can be used to inform decisionmaking. A tradeoff solution is termed Pareto optimal when thesolution can only improve its value in one objective by degrad-ing its value for at least one other objective (Pareto, 1896). ThePareto frontier is the full set of Pareto optimal solutions plot-ted in objective space (for example, see the CosteUncertaintytradeoff in Fig. 7).

In this paper, Section 2 overviews prior 3-dominance ar-chiving and parameterization studies used in the developmentof the 3-NSGAII. Section 3 discusses the four-objectivegroundwater monitoring test case used to demonstrate the3-NSGAII. Sections 4 and 5 provide a more detailed descrip-tion of the 3-NSGAII and its performance for the groundwatermonitoring test case, respectively. Sections 6 and 7 discuss theconclusions and future extensions of this work. This paper as-sumes the reader has prior knowledge of multiobjective evolu-tionary algorithms; readers interested in introductions tomultiobjective optimization and EMO tools should refer tothe texts by Deb (2001) and Coello et al. (2002).

2. Prior work

The 3-NSGAII combines the external archiving techniquesrecommended by Laumanns et al. (2002) and Deb et al. (2003)with automatic parameterization techniques that Reed et al.(2003) developed to eliminate trial-and-error parameter analy-sis for the NSGAII. A primary drawback of using EMOmethods for environmental applications lies in the large com-putational costs associated with assessing performance (i.e.,algorithmic reliability and solution quality). The commonpractice of assessing performance for a distribution of randomseeds employed in the EMO literature is often prohibitivelyexpensive in terms of computational costs and time investedby users because it requires an application to be re-solvedfor each tested random seed. The goal of the automated pa-rameterization approaches developed by Reed et al. (2003)is to eliminate the need to assess algorithmic performancefor a distribution of initial random number seeds and insteadfocus on the NSGAII’s reliability and efficiency for a singlerandom seed. Reliability is addressed in the approach by adap-tively increasing the size of the population. The method usesmultiple runs in which the nondominated solutions are accu-mulated from the results of successively doubled populationsizes. The runs (and successive doubling of population sizes)

continue until either the user-defined maximum runtime isreached or sufficient solution accuracy has been attained.Devireddy and Reed (2004) demonstrated that population dou-bling successfully allows EMO algorithms to attain nearly100% reliability for a broad range of problem types.

The NSGAII parameterization approach enabled Reed andMinsker (2004) to solve the same four-objective monitoringapplication as discussed in this paper. A total of 450,000 func-tion evaluations were required to solve the problem. AlthoughReed and Minsker (2004) demonstrated how EMO can helpenvironmental engineers step beyond two-objective applica-tions, the method employs an inefficient form of archiving,the approach fails to allow users to bias search towards impor-tant objectives, and the method does not take advantage ofearly run results to guide subsequent search. Reed et al.(2003) recommend offline analysis for accumulating nondomi-nated solutions across multiple runs. Offline analysis can beviewed as an unbounded archive (i.e., the number of solutionsstored in memory is not limited) of the Pareto optimal solu-tions found by the NSGAII in every generation of every run.Laumanns et al. (2002) highlight that unbounded archivingleads to memory and nondomination sorting inefficiencies.The 3-NSGAII approach discussed in this paper was specifi-cally developed following Laumanns et al.’s theoretical recom-mendations for bounding archive size and improving solutiondiversity using 3-domination archives. 3-Domination requiresusers to specify the precision with which they wish to quantifyeach objective. User-specified precisions can be used to biassearch towards regions of an application’s objective spacewith the highest precision requirements (see Section 4 formore details). The 3-domination archive was used in this studyto maintain a diverse representation of the Pareto optimal set;moreover the archived solutions found using small populationspre-condition the search with larger populations and minimizethe number of design evaluations required to solve anapplication.

The reader should note that beyond the 3-NSGAII dis-cussed in this paper, Deb et al. (2003) have also proposed anextension of the NSGAII to improve its diversity operatortermed the Clustered NSGAII (C-NSGAII) as well as a steadystate 3-dominance multiobjective evolutionary algorithm(MOEA) that balances convergence speed and diversity. TheC-NSGAII replaces the crowding distance procedure withthe clustering technique that was used in the Strength ParetoEvolutionary Algorithm (Zitzler et al., 2001). Though C-NSGAII performed better than NSGAII, the large computa-tional cost of the clustering algorithm eliminated it from con-sideration since our long-term focus is on high-order Paretooptimization problems. The steady state 3-MOEA (Debet al., 2003) motivated our use of 3-dominance archiving inthis study. 3-MOEA is not used in this study because (1) thealgorithm is limited to real-coded representation and (2) the al-gorithm’s small generation gap (i.e., only 1 population mem-ber is replaced during each iteration) limited its ability totake advantage of small population runs to reduce the overallnumber of function evaluations required to solve anapplication.

685P. Reed et al. / Environmental Modelling & Software 22 (2007) 683e692

3. Monitoring test case problem

3.1. Test case data

The test case developed for this study uses data drawnfrom a 50-million node flow-and-transport simulation per-formed by Maxwell et al. (2000). The concentration dataset used in this study corresponds to the medium testcase from Reed et al. (2004). The simulation provides real-istic historical data for the migration of a hypothetical per-chloroethylene (PCE) plume in a highly heterogeneousalluvial aquifer. The hydrogeology of the test case is basedon an actual site located at the Lawrence Livermore Na-tional Laboratory (LLNL) in Livermore, California. Dataare provided for a total of 58 hypothetical sampling loca-tions within a 29-well multi-level monitoring networkshown in Fig. 1.

The data represent a snapshot in time, 8 years after an un-derground storage tank has continuously released contamina-tion into the aquifer system. The monitoring wells cansample from 1 to 3 locations along their vertical axis andhave a minimum spacing of 10 m between wells in the hor-izontal plane. If the ith monitoring well is selected for sam-pling then PCE is sampled at all possible locations along itsvertical axis. The site is assumed to be undergoing long-termmonitoring, in which groundwater samples are used to assessthe effectiveness of current remediation strategies. Quarterlysampling of the entire network has a potential cost of over$85,000 annually for PCE testing alone, which could trans-late into millions of dollars if the site has a typical lifespan of 20 to 30 years. This paper addresses only spatial re-dundancy analysis, which seeks to identify and remove sam-pling locations that contribute minimally to understandingthe plume’s extent in space, time, or both. This study as-sumes that the spatial sampling plans will be re-evaluatedperiodically as site conditions change. This type of approachhas been applied in several trial-and-error field applications(Johnson et al., 1996; Cameron and Hunter, 2000; Azizet al., 2003).

3.2. Problem formulation

The four-objective monitoring test case used in this papercombines both the spatial redundancy and geostatistical ap-proaches to monitoring design (for a review see Task Commit-tee on Long-Term Groundwater Monitoring Design, 2003).The decision space for this four-objective problem contains229 (over 500-million) possible solutions because sampling de-cisions are modeled using binary variables for each of the 29-monitoring wells. The 3-NSGAII and quantile kriging arecombined to quantify the tradeoffs among the following fourperformance criteria: (1) cost, (2) squared relative estimationerror (SREE), (3) the relative global mass error (MAE), and(4) local uncertainty as measured by kriging estimation vari-ances. Cost is a linear function of the number of PCE samplesthat are used in a given monitoring design. SREE measureshow the interpolated picture of the plume using data onlyfrom wells included in the kth sampling plan compares tothe result attained using data from all available sampling loca-tions. Likewise, the global mass error (MAE) objective mea-sures the relative accuracy of alternative estimates of thetotal mass of PCE in the subsurface. Lastly, local uncertaintyis estimated using the sum of the estimation standard devia-tions (i.e., the square root of estimation variances) from krig-ing (for more details on the problem formulation see Reed andMinsker, 2004).

3.3. Plume interpolation using quantile kriging

Quantile kriging was selected as the plume interpolationmethod used in this study based on the findings of Reedet al. (2004), who present a comprehensive performance anal-ysis of 6 interpolation methods for scatter-point concentrationdata, ranging in complexity from intrinsic kriging based on in-trinsic random function theory to a traditional implementationof inverse-distance weighting. Quantile kriging was shown tobe the most robust and least biased of the interpolationmethods they studied. Additionally, the method’s non-para-metric uncertainty estimates successfully predicted zones of

Fig. 1. Monitoring network sampling a total of 58 locations.

686 P. Reed et al. / Environmental Modelling & Software 22 (2007) 683e692

high estimation error for the same monitoring test case as dis-cussed in this paper.

4. Overview of the 3-NSGAII

Our proposed algorithm aims to reduce user interaction re-quirements and the computational complexity associated withsolving multiobjective optimization problems. EMO algo-rithms require the user to specify various parameters like pop-ulation size, run length, probability of crossover, probability ofmutation, etc. Traditional parameterization techniques usea trial-and-error approach which typically require largeamounts of time and resources. 3-NSGAII enables the userto specify the precision with which they want to quantifythe Pareto optimal set and all other parameters are automati-cally specified within the algorithm. A schematic descriptionof the algorithm is shown in Fig. 2.

The proposed algorithm consists of three steps. The firststep utilizes the NSGAII with a starting population of 5 indi-viduals to initiate EMO search. The initial population size isset arbitrarily small to ensure the algorithm’s initial search isdone using a minimum number of function evaluations. Subse-quent increases in the population size adjust the populationsize commensurate with problem difficulty. In the secondstep, the 3-NSGAII uses a fixed sized archive (which inher-ently results from the user-specified 3 precision) to store thenondominated solutions generated in every generation of theNSGAII runs. The archive is updated using the concept of 3-dominance, which has the benefit of ensuring that the archivemaintains a diverse set of solutions. 3-Dominance allows theuser to define the precision with which they want to evaluateeach objective by specifying an appropriate 3-value for eachobjective.

The final step checks a user-specified performance and ter-mination criteria to determine if the Pareto optimal set hasbeen sufficiently quantified. If the criteria are not satisfied,the population size is doubled and the search is continued.When increasing the population, the initial population of thenew run has solutions injected from the archive at the end ofthe previous run. The algorithm terminates if either a maxi-mum user time is reached or if doubling the population size

Fig. 2. Schematic overview of the Epsilon NSGAII.

fails to significantly increase the number of nondominated so-lutions found across two runs. The following sections discussthe 3-NSGAII in greater detail.

4.1. Searching with the NSGAII

Development of the 3-NSGAII was motivated by theauthors’ goal of minimizing the total number of function eval-uations required to solve computationally intensive environ-mental applications, eliminate trial-and-error analysis forsetting the NSGAII’s parameters, and eliminate the need forrandom seed analysis. The dynamic population sizing and in-jection approach applied in the 3-NSGAII exploit computa-tionally inexpensive small populations to expedite searchwhile increasing population size commensurate with problemdifficulty to ensure the Pareto optimal set can be reliablyapproximated.

The initial population size, N0, is set to some arbitrarilysmall value (e.g, 5), as it is expected that the adaptive popula-tion sizing scheme will adjust for an undersized population. Arandomly selected subset of the solutions obtained using thesmall population sizes are injected into subsequent larger pop-ulations, aiding faster convergence to the Pareto front. Thiscan be viewed as using a series of ‘‘connected’’ NSGAIIruns that share results so that the Pareto optimal set can be re-liably approximated. Computational savings should be viewedin two contexts: (1) the use of minimal population sizes and(2) elimination of random seed analysis. Note that the numberof times the population size needs to be doubled varies withdifferent random seeds, though exploiting search with smallpopulations will on average dramatically reduce computa-tional times. Moreover, our approach eliminates the need to re-peatedly solve an application for a distribution of randomseeds.

The NSGAII’s remaining parameters are set automaticallybased on whether an application is being solved using a realor binary coding. The four-objective problem solved in this pa-per is solved using binary coded variables, uniform crossoverwith a probability of 0.5, and a probability of mutation equalto 1/population size (for more details see Reed et al., 2003).Additional parameter settings include the recombination scal-ing factor, nc, which was set to 15 and the polynomial muta-tion operator scaling factor, nm, which was set to 20 for thisstudy.

4.2. Archive update

Recent studies in the EMO literature have highlighted theimportance of balancing algorithmic convergence speed andsolution diversity (Laumanns et al., 2002; Deb et al., 2003).These studies maintain that the NSGAII remains one of thefastest converging methods available, but its crowding opera-tor fails to promote diversity for challenging EMO problems.The 3-NSGAII overcomes this shortcoming by using 3-domi-nance archiving (Laumanns et al., 2002). The 3-dominance ar-chiving approach is particularly attractive for environmentalapplications because it allows the user to define the precision

687P. Reed et al. / Environmental Modelling & Software 22 (2007) 683e692

with which they want to quantify their tradeoffs while bound-ing the size of the archive and maintaining a diverse set of so-lutions. Fig. 3 adapted from Deb et al. (2003) illustrates the3-dominance approach.

The concept of 3-dominance requires the user to define theprecision they want to use to evaluate each objective. Theuser-specified precision or tolerance vector 3 defines a gridfor a problem’s objective space (see Fig. 3), which biasessearch towards the portions of a problem’s decision spacethat have the highest precision requirements. Fig. 3 illustrateshow 3-domination allows decision makers to extend a solu-tion’s zone of domination based on their required precisionfor each objective (i.e., 31 and 32). Under traditional nondomi-nation sorting solution P dominates region PECF, whereas us-ing 3-domination, the solution dominates the larger regionABCD. The 3-dominance archive improves the NSGAII’s abil-ity to maintain a diverse set of nondominated solutions by onlyallowing 1 archive member per grid cell. In the case whenmultiple nondominated points reside in a single grid cell,only the point closest to the lower left corner of the cell (as-suming minimization) will be added to the on-line archivethereby ensuring convergence to the true Pareto optimal set(Deb et al., 2002; Laumanns et al., 2002). For example, solu-tion 1 in Fig. 3 would be stored in the archive because it iscloser to Point G than solution 2.

The archive is updated in every generation of the3-NSGAII runs with a diverse set of ‘‘3-nondominated’’ solu-tions. The values specified for 3 also directly impact the algo-rithm’s convergence speed. A high-precision representation ofthe Pareto optimal set can be captured by specifying the pre-cision tolerances 3 to be very small. Small precision toler-ances will increase the number of Pareto optimal solutionsthat are 3-nondominated, increase the archive size, and in-crease population sizing requirements (see p. 74 Khan,2003). The 3-NSGAII has the advantage of allowing usersto dramatically reduce computation times, by acceptinga lower resolution (i.e., specifying larger values of 3) repre-sentation of the Pareto frontier. Note that lower resolution ap-proximate representations of the Pareto frontier can behelpful by reducing the number of designs decision makersmust consider.

ε

ε

B

A D

P F

G

21 f2

f1

E C

1

2

Fig. 3. Illustration of the 3-dominance concept (adapted from Deb et al., 2003).

4.3. Injection and termination

The 3-NSGAII also seeks to speed convergence by pre-con-ditioning search with larger population runs with the priorsearch results attained using small populations. In prior efforts(Reed and Minsker, 2004) attempts to inject solutions foundusing a small population into subsequent runs made theNSGAII prematurely converge to poor representations of thePareto optimal set, especially for problems with greater thantwo objectives. The 3-domination archive’s ability to preservediversity plays a crucial role in overcoming this limitation.The 3-NSGAII begins search with an initial population offive individuals from which the 3-nondominated solutionsare identified and stored in the archive. Eq. (1) presents the in-tra-run termination criterion d used in the 3-NSGAII to assesssearch progress for a single run (i.e., a single population size):

if d<

�jAðiÞ �Aði�wÞj

Aði�wÞ

�100% then double N

else continue search: ð1Þ

Search progress is rated in terms of a user-defined crite-rion that specifies the minimum percentage (d) change in thenumber of 3-nondominated individuals that must be foundwithin a moving window of w generations to justify continuingthe run with the current population size. In Eq. (1), d is com-puted based on the absolute difference in the number of ar-chived solutions A(i) present in the current generation i fromthe number of solutions that were in the archive w generationsbefore the current generation, A(i� w). d is set equal to 10%based on the empirical analysis of Devireddy and Reed (2004).

The search window, w, used in this application was setequal to 12 generations to give 3-NSGAII sufficient time tostabilize at each population size. Assessing search progress us-ing a moving window guarantees each population will be ableto search for a minimum of w generations. The maximum runduration for each population size is set based on the conver-gence theory of Thierens et al. (1998) to be equal to 2l wherel is equal the number of binary decision variables. For theLTM application solved in this study l equals 29, which corre-sponds to the total number of available monitoring wells. Thesearch w used in the 3-NSGAII corresponds approximately to20% of maximum run length (i.e., 0.2� 58 generations). Eq.(1) simply defines the two cases when the current populationN will be doubled: (1) if search within w generations fails toyield a 10% increase in archived solutions or (2) the maximumrun duration has been reached.

The archive at the end of each run contains 3-nondominatedsolutions that can be used to guide search in future runs andspeed convergence to the Pareto front. This is achieved by in-jecting 3-nondominated solutions from the archive at the endof the run with population size N into the initial populationof the next run with a population size 2N. Fig. 4 illustratesthe two scenarios that arise when the 3-NSGAII injects solu-tions from the archive generated with a population size Ninto the initial generation of a run with a population size2N. In scenario 1 shown in Fig. 4a, the archive size A issmaller than the subsequent population size 2N. In this case,

688 P. Reed et al. / Environmental Modelling & Software 22 (2007) 683e692

100% of the 3-nondominated archive solutions are injectedinto the first generation of the subsequent run with 2N individ-uals. We have found that the number of injected solutionsshould be maximized to aid rapid convergence. The 3-domi-nance archive in combination with successive doubling of pop-ulation size guarantees the 3-NSGAII will maintain sufficientsolution diversity to find solutions along the full extent ofthe LTM application’s tradeoffs. Fig. 4b shows the second in-jection scenario, which occurs when the archive size A isgreater than the next population size 2N. In this case, 2N 3-nondominated archive solutions are selected randomly and in-jected into the first generation of the next run, again maximiz-ing the impact of injected solutions.

The termination of search across all runs (i.e., inter-run ter-mination) compares how the archive size changes at the end oftwo successive runs of the 3-NSGAII. For example, a run thatuses a population of N sampling designs to evolve an 3-nondo-minated set composed of A individuals will be compared toa second run that used a population of 2N designs to evolvean 3-nondominated set of K individuals. The results of theseruns are used in Eq. (2), to define which of the two followingcourses of action will be taken: (1) population size is againdoubled, resulting in 4N individuals to be used in an additionalrun of the 3-NSGAII or (2) the algorithm stops to allow theuser to assess if the 3-nondominated set has been quantifiedto sufficient accuracy.D was set to 10% for this study as rec-ommended by Reed et al. (2003).

if D<

�jK�Aj

A

�100% then double N and continue search

else stop search:

ð2Þ

The solutions obtained in the archive at the end of the fi-nal run represent a sufficient approximation of the true Paretofront based on user-defined accuracy goals. Section 5 demon-strates the efficiency of the 3-NSGAII in solving high-orderPareto optimization environmental problems (i.e., problemswith more than two objectives).

Fig. 4. Schematic representation of injection when (a) the archive size A is

smaller than the next population size 2N and (b) the archive size A is larger

than the next population size 2N.

5. Results

5.1. Enumerated test case

In order to accurately compare the performance betweenthe NSGAII and the 3-NSGAII algorithms, a 25 well subsetof the previously described 29 well LTM test case was enu-merated. Enumeration provided the true Pareto optimal refer-ence set from which each algorithm could be assessedrigorously using performance metrics. To generate this smallertest case, the four least important wells were eliminated basedon the findings of Reed and Minsker (2004) in order to allowfor a timely enumeration of the problem. This 25 well casecontains over 33-million possible sampling schemes, butonly 2439 solutions are Pareto optimal. It should be notedthat enumeration took nearly 1 week of continuous computingon a Dell Pentium IV 3.0 GHz Processor running WindowsXP, whereas the 3-NSGAII took an average of 2 h per trial(i.e., 2 h per random seed). To assess algorithm performance,three metrics are used. The first metric proposed by Deb andJain (2002) measures the average distance of the approxima-tion set generated by the algorithm to a known reference setand will hereafter be referred to as convergence. This metricis normalized to range from zero to one, with zero indicatingan average distance of zero to the reference set. The secondmetric, again proposed by Deb and Jain (2002), attempts tomeasure diversity by taking into account the distribution of so-lutions with respect to a reference set. The values of this met-ric range from zero to one with one indicating perfect diversityaccording to the definition of the metric. The third metric, 3-performance, proposed by Kollat and Reed (2005) measuresthe proportion of solutions that fall within a user-specified 3

distance of a reference set. This metric inherently providesa measure of both convergence and diversity because it ac-counts for the distance of the solutions from the referenceset while allowing only one solution to be matched withineach 3 region surrounding a reference point, thus preventingclustered solutions from having additional weight in determi-nation of the metric. This metric again ranges from zero to onewhere a value of one would indicate 100% convergence towithin 3 of the reference set.

The running performance of the 3-NSGAII using an 3 res-olution setting equivalent to setting 1 shown in Table 1 is com-pared to that of the NSGAII using a population size of 100 (asrecommended in the literature Deb and Jain, 2002; Deb et al.,2002, 2003). Each run was conducted using 50 random seedsand scatter plots were produced for each performance metric

Table 1

Resolution settings used to solve the long-term monitoring problem

Setting no. 3Cost 3SREE 3Uncertainty 3MAE

1 1 10�5 0.01 10�6

2 1 0.01 0.01 10�4

3 1 0.01 1 10�4

4 1 0.1 1 10�4

5 1 0.1 1 0.01

689P. Reed et al. / Environmental Modelling & Software 22 (2007) 683e692

reflecting the performance versus the total function evaluationsfor all random seeds tested. The 3-NSGAII was permitted toself-terminate in its usual manner using a lag window, w, equalto 10 generations and both termination criteria d and D set at5%, while the NSGAII was terminated after 200,000 functionevaluations to ensure that its runtime would exceed that of the3-NSGAII. The 3-NSGAII used an initial population size offive individuals and was permitted 50 generations withineach run. Fig. 5 displays the scatter plot comparing perfor-mance between the NSGAII using a population size of 100and the 3-NSGAII using 3 resolution setting 1. At resolutionsetting 1, 2411 epsilon nondominated solutions exist, whichis only a 28 solution decrease from the full Pareto optimalset. In addition, the reader should note that the 3 resolution set-ting used in evaluating the 3-performance metric followed thatof setting 5 shown in Table 1 so as to provide a bit more le-niency in comparing the approximation set to the reference

set, hence giving a more easily displayed measure for that par-ticular metric.

Fig. 5 shows that the 3-NSGAII out performed the NSGAIIin all metrics. In fact, while all of 3-NSGAII’s random seedswere able to attain an 3-performance of 0.4 in under 100,000function evaluations, the NSGAII only achieved less thanhalf of that 3-performance value (about 0.15) after 100,000function evaluations. A majority of the 3-NSGAII’s seedswhere able to attain 3-performance values of 0.30 using asfew as 25,000 function evaluations, demonstrating both im-proved algorithmic search and reliability. The scatter plotsfor the convergence and diversity metrics also show that the3-NSGAII is able to maintain diversity while rapidly converg-ing to a much better approximation of the enumerated Paretoset for this reduced size test case. Overall Fig. 5 verifies ourassumption that on average adaptive population sizing and ep-silon dominance archiving serve to improve the efficiency and

Fig. 5. Running performance metric results for both the NSGAII using a population size of 100 and the 3-NSGAII using 3 resolution setting 1. Plots are perfor-

mance metric versus total design evaluations.

690 P. Reed et al. / Environmental Modelling & Software 22 (2007) 683e692

reliability of multiobjective search relative to the originalNSGAII.

5.2. Large test case

To further demonstrate the 3-NSGAII’s performance rela-tive to the NSGAII, we compared the algorithms’ perfor-mances for the full 29 well four-objective groundwatermonitoring test case described in Section 3. The full 29 welltest case has an overall decision space of 500-million solutionsand the best known approximation to its Pareto optimal setwas provided by Reed and Minsker (2004) using the NSGAII.They used four runs with varying population sizes rangingfrom 500 to 4000 individuals resulting in a total of 450,000 de-sign evaluations. In this study, the same problem was solvedusing the suite of 3 precision settings described in Table 1.The resolution of the solutions obtained decreases from setting1 to 5, with setting 1 having the highest resolution require-ments; and hence the smallest 3 tolerance values. Setting 1would be ideal for a user who requires a highly precise repre-sentation of the entire tradeoff surface and can tolerate com-promising the computational time required to solve theapplication. The four additional settings demonstrate howthe 3-NSGAII allows users to directly control the computa-tional time required to solve applications by decreasing reso-lution requirements. In setting 2, the values of 3SREE and3MAE are set to 0.01 and 10�4, respectively, based on the rangeof objective values obtained from setting 1. Setting 3 is ob-tained by increasing the value of 3Uncertainty by a factor of100 while keeping the other resolutions’ constant. Similarlysettings 4 and 5 are obtained by increasing 3SREE and 3MAE.3Cost is not varied in the experiments because it is a discreteinteger function of the number of sampling points used.

The effects of varying the resolution on the number of so-lutions found by the 3-NSGAII as well as the number of func-tion evaluations required are demonstrated in Fig. 6. Thenumber of nondominated solutions found by the 3-NSGAIIdoes not significantly decrease when the resolution is reducedfrom the values used in setting 1 to those in setting 2. Even

Fig. 6. The change in the number of solutions found and the number of func-

tion evaluations required for different resolution settings.

though setting 1 had the highest resolution requirements, the3-NSGAII required only 82,740 design evaluations, which rep-resents a computational savings of more than 80% relative toprior published results (Reed and Minsker, 2004). This is a sub-stantial result because the original solution for the 29 wellLTM test case generated by Reed and Minsker (2004) usingthe NSGAII required an entire day of computational time.Fig. 6 demonstrates that as user-specified resolution require-ments decrease, the number of function evaluations requiredby the algorithm in turn decreases to a minimum value of37,260 for setting 5, representing an order of magnitude de-crease relative to the 450,000 evaluations utilized by Reedand Minsker (2004).

In the LTM application a single constituent is being moni-tored at 29 monitoring wells, which results in a decision spaceof more than 500 million possible sampling designs (i.e., 229

sampling designs). Although the four-objective Pareto surfacecannot be visualized, the surface can inform decision making.Reed and Minsker (2004) demonstrated the quality of theirevolved tradeoffs using two-objective tradeoffs that can beeasily interpreted by stakeholders. These two-objective trade-offs are subsets of the overall four-objective Pareto front andare nondominated in terms of two specified objectives, inde-pendent of the remaining objectives’ values.

Fig. 6 clearly shows that the 3-NSGAII is capable of dra-matically reducing computational costs, but it is very impor-tant to also show that these computational savings do notrequire a substantial loss of solution quality. Figs. 7e9 demon-strate the 3-NSGAII’s performance for the CosteUncertainty,CosteSREE, and UncertaintyeMAE tradeoffs. These figurescompare the solution quality attained by Reed and Minsker(2004) with results of settings 1 and 5 (i.e., the most and leastprecise runs of the 3-NSGAII). We are only showing a sampleof the total number of two-objective tradeoffs within the over-all four-objective problem. The results shown in Figs. 7e9 arereflective of the 3-NSGAII’s overall performance.

Fig. 7 shows the CosteUncertainty tradeoff generated byReed and Minsker (2004) designated as the ‘‘prior method’’as well as the results attained by the 3-NSGAII using 3 settings1 and 5. The 3-NSGAII closely approximates the results of the

Fig. 7. Results for the CosteUncertainty tradeoff.

691P. Reed et al. / Environmental Modelling & Software 22 (2007) 683e692

prior method using up to 90% fewer function evaluations.Fig. 8 shows that the 3-NSGAII again closely matches the re-sults of Reed and Minsker (2004) for the CosteSREE tradeofffor both epsilon settings. Note setting 5 has fewer solutionsthat spread throughout the full extent of the CosteSREE trade-off. This occurs because course epsilon settings yield a courseepsilon dominance grid (see Section 4) where the number ofsolutions that are archived reduces because there are fewergrid cells. Also, the distance between the solutions increasesso that even though a smaller set of solutions is being archived,they cover the entire tradeoff. This has the advantage of allow-ing stakeholders to review fewer solutions while still accu-rately demonstrating diminishing returns for objectivetradeoffs.

The CosteSREE tradeoff found by the 3-NSGAII is slightlyless accurate than the original result by Reed and Minsker(2004). This actually demonstrates an advantage of the 3-NSGAII’s automatic termination criteria (d equal to 10%, D

equal to 10%), which allow the algorithm to dramatically re-duce computational time by avoiding the large computationalcosts associated with finding unnecessarily precise results. Inthis case, if the user wanted higher accuracy for the CosteSREE tradeoff at the expense of significantly more

Fig. 8. Results for the CosteSREE tradeoff.

Fig. 9. Results for the UncertaintyeMAE tradeoff.

computational time, the values 3-NSGAII’s termination crite-ria could be reduced and the run continued. Interestingly, insome cases 3-NSGAII’s injection and diversity preservationenabled the algorithm to actually find better solutions thanthose found originally by Reed and Minsker (2004) as demon-strated by the UncertaintyeMAE tradeoff shown in Fig. 9.Figs. 7e9 highlight that a tremendous amount of the compu-tation time originally used to solve this application was spentseeking high-precision results. Nondomination sorting at 6-digits of precision and beyond is computationally expensiveand does not significantly improve the representation of thetradeoffs used for decision making.

6. Discussion

The 3-NSGAII gives users more direct control over balancingtheir accuracy needs and the computational demands associatedwith evolving the Pareto frontiers for their applications. A keyresult presented in this paper is that the number of function eval-uations required to solve the four-objective LTM problem de-creased nearly linearly with decreasing user-specifiedresolution requirements. The 3-NSGAII was able to approxi-mate the LTM application’s tradeoffs general shape by identify-ing a diverse set of solutions along the entire extent of the Paretofront. The approximate representation could be used by environ-mental decision makers to make reasonable assessments of theirdiminishing returns. For example, the CosteUncertainty trade-off in Fig. 7 demonstrates that the 34-sample solution would re-duce sampling costs by over 40% while resulting in only a 10%increase in system uncertainty.

The computational efficiency of the 3-NSGAII will aid futureefforts in exploring the use of EMO to solve other high-orderPareto optimization problems such as non-point source pollu-tion management and integrated model calibration. Reed andMinsker (2004) introduced the value of considering more thantwo objectives for water resources and environmental design ap-plications. High-order Pareto frontiers allow decision makers tobetter understand interactions between their objectives. As anexample, Reed and Minsker’s four-objective monitoring appli-cation highlighted previously unknown objective conflicts thatsignificantly impact the design of LTM systems.

The 3-NSGAII has significant potential for dramatically re-ducing the computational costs of evolving high-order Paretofronts. The algorithms 3-dominance archive enhances deci-sion-making abilities by bounding the size of the Pareto opti-mal set that stakeholders must consider. For manyenvironmental applications, high-order Pareto optimizationcan be augmented with spatiotemporal visualization. Stake-holders can visualize members of the reduced set of Pareto op-timal solutions evolved by the 3-NSGAII to better understandhow their objectives impact designs and to exploit low costimprovements in their design objectives.

7. Conclusions

The 3-NSGA2 demonstrates how 3-dominance archivingcan be combined with a parameterization strategy for the

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NSGAII to accomplish the following goals: (1) ensure the al-gorithm will maintain diverse solutions, (2) eliminate the needfor trial-and-error analysis for parameter settings (i.e., popula-tion size, crossover and mutation probabilities), and (3) allowusers to sufficiently capture tradeoffs using a minimum numberof design evaluations. A sufficiently quantified tradeoff can bedefined as a subset of nondominated solutions that provide anadequate representation of the Pareto frontier that can be usedto inform decision making. Results are presented for a four-objective groundwater monitoring case study in which thearchiving and parameterization techniques for the NSGAIIcombined to reduce computational demands by greater than90% relative to prior published results. The four-objective Par-eto surface that was obtained was explored using two-objectivetradeoffs between selected pairs of objectives. The methods ofthis paper can be easily generalized to other multiobjectiveapplications to minimize computational times as well astrail-and-error parameter analysis.

Acknowledgements

This work was partially supported by the National ScienceFoundation under Grant No. EAR 0310122. Any opinions,findings and conclusions or recommendations expressed inthis paper are those of the writers and do not necessarily reflectthe views of the National Science Foundation.

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